Elementos de Armonía - Aristóxeno de Tarento [m-a-000274-f3] ()

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THE ELEMENTS OF HARMONY

the sum is a concord. But this is not the case with thetwo smallest concords. For the doubling of a Fourth orFifth does not produce a concord; nor does the additionto either one of them of the concord compounded of theoctave and that one; but the sum of such concords willalways be a discord.

A tone is the excess of the Fifth over the Fourth; the 46Fourth consists of two tones and a half. The followingfractions of a tone occur in melody: the half, called a semi- [tone j the third, called the smallest Chromatic diesis; thequarter, called the smallest Enharmonic diesis. No smallerinterval than the last exists in melody. Here we have twocautions for our hearers; firstly, many have misunderstoodus to say that melody admits the division of the tone intothree or four equal parts. This misunderstanding is due totheir not observing that to employ the third part of a toneis a very different thing from dividing a tone into threeparts and singing all three. Secondly, from an abstractpoint of view, no doubt, we regard no interval as the small-est possible.

The differences of the genera are found in such a tetra-chord as that from Mese to Hypate, where the extremes arefixed, while one or both of the means vary. As the variablenote must move in a certain locus, we must ascertain thelimits of the locus of each of these intermediate notes. Thehighest Lichanus is that which is a tone removed from theMese. It constitutes the genus Diatonic. The lowest is .that which is two tones below the Mese; this is Enharmonic.The locus of the Lichanus is thus seen to be a tone.The interval between the Parhypate and Hypate cannot,plainly, be less than an enharmonic diesis, for this latter is 47the minimum melodic distance. It is to be observed alsothat it can only be extended to twice that distance; for

the Lichanus in its descent, and the Parhypate in its199

ARISTOXENUS

ascent reach the same pitch, the locus of each note finds itslimit. Thus it is seen that the locus of the Parhypate is notgreater than the smallest diesis.

This proposition has afforded some students great per-plexity. 'If,' they ask in surprise, 'the interval betweenthe Mese and the Lichanus (assuming it to be any one ofthe above-mentioned intervals) be increased or diminished,how can the note bounding the new interval be a Lichanus ?There is admittedly but one interval between the Mese andParamese, and again between the Mese and Hypate, and infact between any pair of the permanent notes. Why thenshould we admit a plurality of intervals between the Meseand the Lichanus ? Surely it would be better to change thenames of the notes; and restricting the term Lichanus toany one of them, the two-tone or any other, to employ otherdesignations for the rest. For notes that bound unequalmagnitudes must be different notes. And one might addthat the converse is equally valid, namely, that the bound-aries of equal magnitudes must have the same designations.'To these objections the following reply was given. In thefirst place, to postulate that a difference in notes necessarilyimplies a difference in the magnitudes bounded by them isa startling innovation. We see that the Nete and Mesediffer in function from the Paranete and Lichanus, and theParanete and Lichanus again from the Trite and Parhypate,and these latter again from the Paramese and Hypate; and

48 for this reason each pair has names of its own, though thecontained interval is in every case a Fifth. Thus it is seenthat a difference in the contained intervals is not necessarilyimplied by a difference of notes.

That the converse implication is equally inadmissible willappear from the following remarks. In the first place, ifwe seek particular designations to suit every increase anddecrease in the intervals of the Pycnum, we shall evidently

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need an infinite vocabulary, since the locus of the Lichanusis infinitely divisible. For as a matter of fact, to which of 49- 7the disputants as to the shades of the genera should we giveour adherence ? Every one is not guided by the same divi-sions in harmonizing the chromatic or enharmonic scale.Why then should the term Lichanus be applied to the two-tone Lichanus rather than to one slightly higher ? Which-ever division be employed, the ear equally recognizes anenharmonic genus; yet it is plain that the magnitudes ofthe intervals are different in the two divisions. In the 48. 15second place, if we have eyes exclusively for equality andinequality we shall miss the distinction between the likeand unlike. Thus we shall have to restrict the term Pycnumto one particular magnitude; as likewise evidently theterms Enharmonic and Chromatic; for they too are deter-mined not to a point but to a locus. But it is evident thatsuch a restriction is not in accordance with the mode inwhich sense forms its representations. It is by consideringthe common qualities found in some one class, not themagnitude of some one interval, that sense employs suchterms as Pycnum, Chromatic, Enharmonic. That is to say,it constitutes a class Pycnum to embrace every case inwhich the two intervals occupy a smaller space than theone ; for in all Pycna, though they are unequal in size, thereis evident to the ear the sound of a certain compression.Likewise it constitutes a class Chromatic to embrace allcases in which the Chromatic character is apparent. Forthe ear detects a motion peculiar to each of the genera,though each genus employs not one but many divisions of 49the tetrachord. Thus it is clear that, while the magnitudeschange, the genus may remain unaltered, for up to a certainpoint changes in the magnitudes do not involve a change ofgenus. And if the genus remains the same, it is reasonableto suppose that the functions of the notes may be permanent

ARISTOXENUS

also. For the species of the tetrachord is the same, and forthis reason we must hold that the boundaries of the intervalsare the same notes. In general, as long as the names ofthe extreme notes remain the same, the higher being calledMese, and the lower Hypate, so long will the names of theintermediate notes also remain the same, the higher beingcalled Lichanus, and the lower Parhypate. For the notesbetween the Mese and Hypate are always stamped by theear as Lichanus and Parhypate. To demand that all notesbounding equal intervals should have the same names, orthat all notes bounding unequal intervals should havedifferent names, is to join battle with the evidence of thesenses. For in melody we make the interval between theHypate and Parhypate sometimes equal and sometimes

50 unequal to that between the Parhypate and Lichanus.Now in the case of two equal consecutive intervals it isimpossible that the notes bounding each of them should bedesignated by the same terms, unless the middle note is tohave two names. The absurdity is also evident when theabove-mentioned intervals are unequal. For it is impossiblethat one of any pair of such names should change while theother remains the same; since the names have meaningonly in their relation to one another. So much for thisobjection.

The term Pycnum we shall employ in all cases when, ina tetrachord whose extremes form a Fourth, the sum oftwo of the intervals occupies a lesser space than the third.There are certain divisions of the tetrachord which standout from the rest as familiar, because the magnitudes of theintervals in them are familiar. Of these divisions, one isEnharmonic, in which the Pycnum is a semitone, and itscomplement two tones; three are Chromatic, namely, theSofi, the Hemiolic, and the Tonic Chromatic. The divisionof the Soft Chromatic is that in which the Pycnum consists

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of two of the smallest Chromatic dieses, while its com-plement is expressed in terms of two quanta, namely, a•semitone taken thrice, and a Chromatic diesis taken once,so that the sum of it amounts to three semitones and thethird of a tone. This is the smallest of the ChromaticPycna and its Lichanus is the lowest in this genus. Thedivision of the Hemiolic Chromatic is that in which the 51Pycnum is one and a half times the Enharmonic Pycnum,and each Diesis one and a half times an Enharmonicdiesis. It is manifest that the Hemiolic Pycnum is greaterthan the Soft, since the former is less than a tone by anEnharmonic diesis, the latter by a Chromatic diesis. Thedivision of the Tonic Chromatic is that in which the Pyc- .num consists of two semitones, and its complement of a toneand a half. Up to this point both the inner notes vary;but now the Parhypate, having traversed the whole of itslocus, remains at rest, while the Lichanus moves an enhar-'monic diesis. Thus the interval between the Lichanusand Hypate becomes equal to that between the Lichanusand Mese, so that the Pycnum does not occur in thisdivision as in the preceding. The disappearance of thePycnum in the division of the tetrachord is coincident withthe first appearance of the Diatonic genus. There are twodivisions of the Diatonic genus, the Soft and the SharpDiatonic. The division of the Soft Diatonic is that inwhich the interval between the Hypate and Parhypate isa semitone, that between the Parhypate and Lichanus three,". ;.Enharmonic dieses, that between the Lichanus and Mesefive dieses. The division of the Sharp Diatonic is that inwhich the interval between the Hypate and Parhypate isa semitone, while each of the remaining intervals is a tone.Thus, while we have six Lichani, as there are six divisions 52of the tetrachord, one enharmonic, three chromatic, andtwo diatonic, we have but four Parhypatae, that is, two

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less than the divisions of the tetrachord. For the semitoneParhypate is employed for both diatonic divisions, and forthe Tonic Chromatic. Thus, of the four Parhypatae, oneis peculiar to the Enharmonic genus, while the Diatonicand Chromatic between them employ three. Of the in-tervals in the tetrachord, that between the Hypate andParhypate may be equal to that between the Parhypateand Lichanus, or less than it, but never greater. That itmay be equal is evident from the Enharmonic and Chro-matic division of the tetrachord; that it may be less isevident from the Diatonic scales, and also may be ascer-tained in the Chromatic by taking a Parhypate of the Soft,and a Lich&nus of the Tonic Chromatic; for such divisionsof the Pycnum sound melodious. But to adopt the oppositeorder produces an unmelodious result; for instance, to takethe semitone Parhypate, and the Lichanus of the HemiolicChromatic, or the Parhypate of the Hemiolic, and theLichanus of the Soft Chromatic. Such divisions producean inharmonious effect. On the other hand, the intervalbetween the Parhypate and Lichanus may be equal to,greater than, or less than that between the Lichanus andMese. It is equal in the Sharp Diatonic, less in all theother shades, and greater when we employ as Lichanusthe highest of the Diatonic Lichani, and as Parhypate anyone lower than that of the semitone.

We shall next proceed to explain, beginning with a general53 indication, the method by which we should expect to deter-

mine the nature of continuity. To put it generally, ininvestigating continuity the laws of melody must be ourguide, nor must we imitate those who shape their accountof continuity with a view to the massing of small inter-vals. Such theorists plainly disregard the natural sequenceof melody, as appears from the number of dieses thatthey place in succession; for the voice's power of con-

204.


necting dieses stops short of three. Thus it appears thatcontinuity must not be sought in the smallest intervals, norin equal nor in unequal intervals; we must rather followthe guidance of natural laws. Now, though it were noeasy matter at present to offer an accurate exposition ofcontinuity before we have explained the collocation of inter-vals, yet the veriest novice can see from the followingreasoning that there is such a thing as continuity. It willbe admitted that there is no interval which can be dividedad infinitum in melody, and that the natural laws of melodyassign a maximum number of fractions to every interval.Assuming that this will be, or rather must be, admitted, we .necessarily infer that the notes containing fractions of the •said number are consecutive. To this class belong thenotes which, as a matter of fact, have been in use fromthe earliest times, as for instance the Nete, the Paranete,and those that follow them.

Our next duty will be to determine the first and mostindispensable condition of the melodious collocation ofintervals. Whatever be the genus, from whatever note one 54starts, if the melody moves in continuous progression eitherupwards or downwards, the fourth note in order from anynote must form with it the concord of the Fourth, or thefifth note in order from it the concord of the Fifth. Anynote that answers neither of these tests must be regarded'as out of tune in relation to those notes with which itfails to form the above-mentioned concords. It must beobserved, however, that the above rule is not all-sufficientfor the melodious construction of scales from intervals.It is quite possible that the notes of a scale might formthe above-mentioned concords with one another, and yet •that the scale might be unmelodiously constructed. Butif this condition be not fulfilled, all else is useless. Letus assume this then as a fundamental principle, the vie-

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ARISTOXENUS

i lation of which is destructive of harmony. A law, in somerespects similar, holds with regard to the relative position

• of tetrachords. If any two tetrachords are to belong toj ' the same scale, one or other of the following conditions'•; must be fulfilled; either they must be in concord withJ • each other, the notes of one forming some concord or;< other with the corresponding notes of the other, or they;:; must both be in concord with a third tetrachord, with. i which they are alike continuous but in opposite directions.'M . This, in itself, is not sufficient to constitute tetrachords ofj | the same scale: certain other conditions must be satisfied,I ( 55 of which we shall speak hereafter. But t h e absence of t h e1; . condi t ion renders t h e rest useless.I ' W h e n we consider the magni tudes of intervals, we find'/. tha t while the concords either have no locus of variation,/ and are definitely determined to one magnitude, or have': an inappreciable locus, this definiteness is to be found in

\j a much lesser degree in discords. For this reason, the(/, ear is much more assured of the magnitudes of the con-jf. cords than of the discords. It follows that the mostj j accurate method of ascertaining a discord is by the principlej ; of concordance. If then a certain note be given, and itI j be required to find a certain discord below it, such as thej ' ditone (or any other that can be ascertained by the method

of concordance), one should take the Fourth above thegiven note, then descend a Fifth, then ascend a Fourthagain, and finally descend another Fifth. Thus, the intervalof two tones below the given note will have been ascer-tained. If it be required to ascertain the discord in theother direction, the concords must be taken in the otherdirection. Also, if a discord be subtracted from a concordby the method of concordance, the remaining discord isthereby ascertained on the same principle. For, subtract,the ditone from the Fourth on the principle of concordance,.

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and it is evident that the notes bounding the excess of thelatter over the former will have been found on the sameprinciple. For the bounding notes of the Fourth are con- 56cords to begin with; and from the higher of these a concordis taken, namely, the Fourth above; from the note thusfound another, namely, the Fifth below; from this againa Fourth above, and finally from this a Fifth below; andthe last concord alights on the higher of the notes bound-ing the excess of the Fourth over the Ditone. Thus itappears that if a discord be subtracted from a concord bythe method of concordance the complement also will havebeen thereby ascertained on the same principle.

The surest method of verifying our original assumptionthat the Fourth consists of two and a half tones is thefollowing. Let us take such an interval, and let us findthe discord of two tones above its lower note, and the same .discord below its higher note. Evidently the complementswill be equal, since they are remainders obtained by sub-tracting equals from equals. Next let us take the Fourthabove the lower note of the higher ditone, and the Fourthbelow the higher note of the lower ditone. It will be seen ,that adjacent to each of the extreme notes of the scalethus obtained there will be two complements in juxta-position, which must be equal for the reasons already given.This construction completed, we must refer the extremenotes thus determined to the judgement of the ear. If theyprove discordant, plainly the Fourth will not be composed 57of two and a half tones; and just as plainly it will be socomposed, if they form a Fifth. For the lowest of theassumed notes is, by construction, a Fourth of the higherboundary of the lower ditone; and it has now turned outthat the highest of the assumed notes forms with the lowestof them the concord of the Fifth. Now as the excess ofthe latter interval over the former is a tone, and as it is

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< ARISTOXENUS

,| here divided into two equal parts; and as each of theseJ equal parts which is thus proved to be a semitone is ati the same time the excess of the Fourth over a ditone,

• it follows that the Fourth is composed of five semitones.• It will be readily seen that the extremes of our scale

cannot form any concord except a Fifth. They cannotform a Fourth; for there is here, besides the original Fourth,an additional complement at each extremity. They cannotform an octave; for the sum of the complements is lessthan two tones, since the excess of the Fourth over theditone is less than a tone (for it is universally admittedthat the Fourth is greater than two tones and less thanthree); consequently, the whole of what is here added tothe Fourth is less than a Fifth; plainly then their sumcannot be an octave. But if the concord formed by the

58 extreme notes of our construction is greater than a Fourth,and less than an octave, it must be a Fifth; for this isthe only concordant magnitude between the Fourth andOctave.

BOOK III

Successive Tetrachords are either Conjunct or Disjunct.W E shall employ the term conjunction when two succes- 58. 15

sive tetrachords, similar in figure, have a common note; theterm disjunction, when two successive tetrachords similar infigure are separated by the interval of a tone. That successivetetrachords must be related in either of these ways, is evidentfrom our axioms. For a series, in which each note formsa Fourth with the fourth note in order from it, will constituteconjunct tetrachords; while disjunct tetrachords result, when 59each note forms a Fifth with the fifth from it. Now as allsuccessions of notes must fulfil one or other of these con-ditions, so all successive similar tetrachords must be eitherconjunct or disjunct.

Difficulties have been raised by some of rriy hearers onthe question of succession. It has been asked, Firstly,what is succession in general ? Secondly, does it appear inone form only, or in several? Thirdly, are conjunct anddisjunct tetrachords equally successive ? To these questionsthe following answers have been given. In general, scalesare continuous, whose boundaries either are successive orcoincide. There are two forms of succession in scales ; inthe one, the upper boundary of the lower scale coincideswith the lower boundary of the upper scale; in the other,the lower boundary of the higher scale is in the line ofsuccession with the higher boundary of the lower scale. Inthe first of these forms, the scales of the successive tetra-chords have a certain space in common, and are necessarily

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ARISTOXENUS

similar in figure. In the other form, they are separatedfrom one another, and the species of the tetrachords may

I [ be similar, only on condition, however, that the separating\ j interval is one tone. Thus we are led to conclude that two^ | similar tetrachords are successive, if they are either separated• ! by a tone, or if their boundaries coincide. Consequently

i similar successive tetrachords are either conjunct or dis-{ junct .

! i W e also assert that two successive tetrachords either6 0 must b e separated by no tetrachord whatsoever, or must not

b e separated by a tetrachord dissimilar to themselves.Tet rachords similar in species cannot b e separated by adissimilar tetrachord, and dissimilar bu t successive tetra-chords cannot be separated by any tetrachord whatsoever.H e n c e we see that tetrachords similar in species can bearranged in succession in the two forms above ment ioned.

The interval contained by successive notes is simple.For if the containing notes are successive, no note is

wanting; if none is wanting, none will intrude; if noneintrudes, none will divide the interval. But that whichexcludes division excludes composition. For every com-posite is composed of certain parts into which it is divisible.

The above proposition is often the object of perplexityon account of the ambiguous character of the intervallicmagnitudes. ' How,' it is asked in surprise,' can the ditonepossibly be simple, seeing that it can be divided into tones ?Or, how again is it possible for the tone to be simple seeingthat it can be divided into two semitones ?' And the samepoint is raised about the semitone.

This perplexity arises from the failure to observe thatsome intervallic magnitudes are common to simple andcompound intervals. For this reason the simplicity of aninterval is determined not by its magnitude, but the relationsof the notes that bound it. The ditone is simple when


bounded by the Mese and Lichanus; when bounded by theMese and Parhypate, it is compound. This is why we 6lassert that simplicity does not depend on the sizes of theintervals, but on the containing notes. "' '

In variations of genus, it is only the parts of the Fourth thatundergo change.

All harmonious scales consisting of more than one tetra-chord were divided into conjunct and disjunct. But conjunctscales are composed of the simple parts of the Fourth alone,so that here at least it will be the parts of the Fourth alonethat will undergo change. Again, disjunct scales comprisebesides these parts of the Fourth a tone peculiar to disjunc-tion. If then it be proved that this particular tone does"not alter with variation of genus, evidently the change canaffect only the parts of the Fourth. Now the lower of the'notes containing the tone is the higher of the notes con-taining the lower of the disjunct tetrachords; as such wehave seen that it is immovable in the changes of the genera/Again, the higher of the notes bounding the tone is the . /

lower of the notes bounding the higher of the disjunct \tetrachords; it likewise, as we have seen, remains constantthrough change of genus. Since therefore, it appears thatthe notes containing the tone do not vary with a change ofgenus, the necessary conclusion is that it is only the partsof the Fourth that participate in that change.

Every Genus comprises at most as many simple intervals (5aas are contained in the Fifth.

The scale of every genus, as we have already stated, takesthe form of conjunction or disjunction. Now it has beenshown that the conjunct scale consists merely of the partsof the Fourth, while the disjunct scale adds a single intervalpeculiar to itself, namely the tone. But the addition ofthis tone to the parts of the Fourth completes the intervalof the Fifth. Since therefore it appears that no scale of any'

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genus taken in the one shading is composed of more simpleintervals than those in the Fifth, it follows that every genuscomprises at most as many simple intervals as are containedin the Fifth.

In this proposition the addition of the words 'at the most'sometimes proves a stumbling-block. ' Why not,' it is asked,' show without qualification that each genus is composed ofas many simple intervals as are contained in the Fifth?'The answer to this is that in certain circumstances each ofthe genera will comprise fewer intervals than exist in theFifth, but never will comprise more. This is the reasonthat we prove first that no genus can be constituted of moresimple intervals than there are in the Fifth; that everygenus will sometimes be composed of fewer, is shown inthe sequel. .

63 A Pycniim ca?mot be followed by a Pycnum or by part ofa Pycnum.

For the result of such a succession will be that neitherthe fourth notes in order from one another will form Fourths,nor the fifth notes in order from one another Fifths. Butwe have already seen that such an order of notes is un-melodious.

The lower of the notes containing the ditone is the highestnote of a Pycnum, and the higher of the notes containing theditone is the lowest note of a Pycnum.

For as the Pycna in conjunct tetrachords form Fourthswith one another, the ditone must lie between them;similarly since the ditones form Fourths with one another,the Pycnum must lie between them. It follows that thePycnum and the ditone must succeed one another altern-ately. Therefore it is evident that of the notes containingthe ditone, the lower will be the highest note of the Pycnumbelow, and the higher will be the lowest note of thePycnum above.

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The notes containing the tone are both the lowest notes ofa Pycnum.

For in disjunction the tone is placed between tetrachordsthe boundaries of which are the lowest notes of a Pycnum ;and it is by these notes that the tone is contained. For the.lower of the notes containing the tone is the higher of those;containing the lower tetrachord; and the higher of thosecontaining the tone is the lower of those containing the-higher tetrachord. Therefore it is evident that the notescontaining the tone will be the lowest notes of a Pycnum,

A succession of two Ditones is forbidden, 64Suppose such a succession; then the higher ditone will

be followed by a Pycnum below, and the lower ditone wilLbe followed by a Pycnum above, for we saw that the notethat forms the upper boundary of the ditone is the lowestnote of a Pycnum. The result will be a succession of twoPycna; and as this has been proved unmelodious, the sue- •cession of two ditones must be equally so.

In Enharmonic and Chromatic scales a succession of two.tones is not allowed. Suppose such a succession, first illthe ascending scale; now if the note that forms the upperboundary of the added tone is musically correct, it mustform either a Fourth with the fourth note in order from it,or a Fifth with the fifth in order; if neither of these con-ditions is satisfied, it must be unmelodious. But thatneither of them will be satisfied, is clear. For if it beEnharmonic, the Lichanus, which is the fourth note inorder from the added note, will be four tones removedfrom it. If it be Chromatic, whether of the Soft or Hemi-olic colour, the Lichanus will be further removed than-a Fifth; and if it be of the Tonic Chromatic, the Lichanuswill form a Fifth with the added note. But this does notsatisfy our law which demands that either the fourth noteshould form a Fourth, or the fifth a Fifth. Neither condition

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ARISTOXENUS

is here fulfilled. It follows that the note constituting theupper boundary of the added tone will be unmelodious.

Again, if the second tone be added below it will render65 the genus Diatonic. Therefore it is evident that in the

Enharmonic and Chromatic genera a succession of twotones is impossible.

In the Diatonic genus three consecutive tones are permitted;but no more. For let the contrary be supposed; then thenote bounding the fourth tone will not form a Fourth withthe fourth note from it, nor a Fifth with the fifth.

In the same genus a succession of two semitones is notallowed. For first suppose the second semitone to beadded below the semitone already present. The result isthat the note bounding the added semitone neither makesa Fourth with the fourth note from it, nor a Fifth with thefifth. The introduction, then, of the semitone here will beunmelodious. But if it be added above the semitonealready present, the genus will be Chromatic. Thus itis clear that in a Diatonic scale the succession of twosemitones is impossible.

It has now been shown which of the simple intervals canbe repeated in immediate succession, and how often theycan be repeated; and which of them on the contrary it isabsolutely impossible to repeat at all. We shall now speakof the collocation of unequal intervals.

A ditone may be succeeded either above or below by aPycnum. For it has been proved that in conjunct tetra-chords these intervals follow alternately. Therefore eachcan succeed the other either in an ascending or descendingorder.

A ditone can be followed by a tone in the ascending scaleonly. For suppose such a succession in the descending

66 order. The result will be that the highest and the lowestnote of a Pycnum will fall on the same pitch. For we saw

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that the note that forms the lower boundary of the ditonewas the highest note of a Pycnum, and that the note thatforms the upper boundary of the tone was the lowest noteof a Pycnum. But if these notes fall on the one pitch,it follows that there is a succession of two Pycna. As thislatter succession is unmelodious, a tone immediately belowa ditone must be equally so.

A tone can be followed by a Pycnum in the descending orderonly. For suppose such a succession in the opposite order;the same impossibility will be found to result again. Thehighest and lowest note of a Pycnum will fall on the samepitch, and consequently there will be a succession of twoPycna. This latter being unmelodious, the position of thetone above the Pycnum must be equally so.

In the Diatonic genus, a tone cannot be both preceded andsucceeded by a semitone. For the consequence would bethat neither the fourth notes in order from one anotherwould form a Fourth, nor the fifth a Fifth.

A pair of tones, or a group of three tones may be bothpreceded and succeeded by a semitone; for either the fourthnotes from one another will form a Fourth, or the fiftha Fifth.

From the ditone there are two possible progressions upwards,one only downwards. For it has been proved that theditone can be followed in the ascending scale by eithera Pycnum or a tone. But more progressions upwards from'the said interval there cannot be. For the only othersimple interval left is the ditone, and two consecutiveditones are forbidden. In the descending order there is 67but one progression from the ditone. For it has been

> proved that a ditone cannot lie next a ditone, and thata tone cannot succeed a ditone in the descending order.Consequently the progression to the Pycnum alone remains.It is clear then that from the ditone there are two possible

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progressions upwards, one to the tone, and one to thePycnum; and one possible progression downwards, to thePycnum.

From the Pycnum, on the contrary, there are two possibleprogressions downwards, and one upwards. For it has beenproved that in the descending scale a Pycnum can befollowed by a ditone, or a tone. A third progression therecannot be. For the only remaining simple interval is thePycnum, and a succession of two Pycna is forbidden. Itfollows that there are only two possible progressions froma Pycnum downwards. Upwards there is but one, to theditone. For a Pycnum cannot adjoin a Pycnum, nor cana tone succeed the Pycnum in the ascending scale; there-fore the ditone alone remains. It is evident then thatfrom the Pycnum there are two possible progressions down-wards, one to the tone, and one to the ditone; and onepossible progression upwards, to the ditone.

From the tone there is but one progression in either direction :downwards to the ditone, upwards to the Pycnum. It hasbeen shown that in the descending scale the tone cannotbe followed by a tone or by a Pycnum. Therefore theditone alone remains. And it has been shown that inthe ascending scale the tone cannot be followed by a toneor a ditone. Therefore the Pycnum alone remains. Itfollows that from the tone there is but one possible pro-

68 gression in either direction, downwards to the ditone, andupwards to the Pycnum.

The same law can be applied to the Chromatic scales,except of course that one must substitute for the ditone theinterval between the Mese and Lichanus, which varies,according to the particular shade, with the size of thePycnum.

The same law will also hold good of the Diatonic scales.From the tone common to the genera there is one possible

216


progression in either direction; downwards to the intervalbetween the Mese and Lichanus, whatever it may happento be in any particular shade of the Diatonic scales;upwards to the interval between the Paramese and Trite.

Some persons have been much perplexed by this pro-position. They are surprised that we do not arrive atquite a contrary conclusion; for they think that the pro-gressions in either direction from the tone are innumerable,since there are innumerable possible magnitudes of theinterval between the Mese and Lichanus, and of thePycnum as well. To this objection we offered the followinganswer. To begin with, the same observation might bemade equally well in the other cases we have considered. -Evidently one of the two descending progressions from thePycnum admits of innumerable possible magnitudes; like-wise one of the two" ascending progressions from the ditone.For such an interval as that between the Mese and Lichanusadmits of innumerable magnitudes, and the same may besaid of such an interval as the Pycnum. Nevertheless thereare but two progressions from the Pycnum downwards, andtwo from the ditone upwards; and similarly one from thetone in either direction. For the progressions must be 69ascertained in accordance with one individual shade in oneparticular genus. In making any musical phenomenon theobject of scientific knowledge, its definite side should beinsisted on, its indefinite features left in the background.Now in respect of the sizes of intervals and the pitch ofnotes, the phenomena of melody are indefinite, while inrespect of functions, common qualities, and orders of'arrangement, they are definite and determined. To takethe first example that occurs, the progressions downwardsfrom the Pycnum are in function and character determinedas two in number. The first proceeds by the tone andbrings the scale into the disjunct class; the second pro-

217

ARISTOXENUS

ceeding by the other interval (whatever its size may be)brings the scale into the conjunct class. Hence we see also

ij that there is but one possible progression in either directionfrom the tone, and that both these progressions alike

;'. produce but one class of scale—the disjunct. But it isj • • quite plain from these observations, and from the naturei of the facts, that if one seek to discover the possible pro-

gressions by considering not one shade of one genus atI a time, but all shades and all genera together, one willI come upon an infinity of them.( In the Chromatic and Enharmonic scales every note partici-| ' pates in the Pycnum. For every note in the said genera isi the boundary either of a part of the Pycnum, or of the tone,

or of an interval such as that between the Mese and Licha-t 70 nus. The case of notes that bound the parts of the Pycnum

requires no proof j it is immediately evident that they partici-| pate in the Pycnum. And we proved already that the notes: containing the tone are both the lowest notes of a Pycnum ;j we showed also that the lower of the notes containing the1 remaining interval was the highest of a Pycnum, and the•j higher of them the lowest of a Pycnum. Now as these are1 the only simple intervals, and each of them is contained by

notes both of which participate in the Pycnum, it followsi that every note in the Chromatic and Enharmonic genus•j participates in the Pycnum.• One will readily see that the positions of the notes situated

in the Pycnum are three in number, since, as we know, a! Pycnum cannot be followed by another Pycnum or part ofI one. For it is evident in consequence of this latter law,I that the number of the said notes is so limited.

It is required to prove that from the lowest only of the notesin a Pycnum there are two possible progressions in eitherdirection, while from the others there is but one. It hasalready been proved that from the Pycnum there are two

218


progressions downwards, one to the tone, and one to theditone. But to prove that there are two progressionsdownwards from the Pycnum is the same as proving thatthere are two progressions downwards from the lowest ofthe notes situated in the Pycnum; for this note marks thelimit of the Pycnum. Again, it was proved that from theditone there are two progressions upwards. But to saythat there are two progressions upwards from the ditone isthe same as saying that there are two progressions upwardsfrom the higher of the notes bounding the ditone. Forthis note marks the upper boundary of the ditone. But it 71is clear that the same note which forms the upper boundaryof the ditone also forms the lower boundary of the Pycnum;being the lowest note of a Pycnum (for this too was proved).Hence it is evident that from this note there are two

' possible progressions in either direction.

It is required to prove that from the highest note of aPycnum there is but one progression in either direction. Itwas proved that from a Pycnum there is but one pro-gression upwards. But to say that there is one progression ,upwards from the Pycnum is (for the reason given in- theformer proposition) the same as saying that there is butone from the note limiting it.

Again, it was proved that from the ditone there is butone progression downwards : but to say that there is but oneprogression downwards from the ditone is (for the reasongiven) the same as saying that there is but one from thenote bounding it. But it is evident that the note whichbounds the ditone below is at the same time the upperboundary of the Pycnum; being the highest note of aPycnum. It is plain, then, that from the given note there /is but one possible progression in either direction.

/ / is required to prove, that from the middle note of aPycnum there is but one progression in either direction. Now

219

ARISTOXENUS

since the given note must be adjoined by some one orother of the three simple intervals, and there lies already

>; a diesis on each side of it, plainly it cannot be adjoined on~~ ; either side by either a ditone or a tone. For suppose

• a ditone to adjoin i t; then either the lowest or the highestj • note of a Pycnum will fall on the same pitch as the given'; note, which is the middle note of a Pycnum; consequently\ I there will be a succession of three dieses, no matter on/ i 72 which side the ditone be located. Again, suppose a tone'/j :"• to adjoin the given note; we shall have the same result.' '; The lowest note of a Pycnum will fall on the same pitch asy., the middle note of a Pycnum, so that we shall again have' j three dieses in succession1. But this succession is unmelo-V; dious; therefore it follows that there is but one possibleI i progression from the given note in either direction.

It has now been shown that from the lowest of the notesj' - of a Pycnum there are two possible progressions in eitheri direction ; while from the others in either direction there is;• but one.• ; It is required to prove that two notes that occupy dissimilar1 positions in the Pycnum cannot fall on the same pitch without

J' * violating the nature of melody. Suppose, firstly, that thet highest and lowest note of a Pycnum fall on the same pitch.j: The result will be two consecutive Pycna, and as this isj unmelodious, it must be equally unmelodious that notes dis->[ similar in the Pycnum in the manner of the assumed notesf should fall upon the same pitch.

Again, it is evident that the notes also that are dissimilarin the other possible manner cannot have a common pitch.For if the highest or lowest note of a Pycnum coincide inpitch with a middle note, there necessarily results a succes-sion of three dieses.

It is required to prove that the Diatonic genus is composedof two or of three or of four simple quanta. It has been


already shown that each genus comprises at most as manysimple intervals as there are in the Fifth. These are four 73in number. If then three of those four become equal,leaving but one odd,—as happens in the Sharp Diatonic— -there will be only two different quanta in the Diatonicscale. Again, if two become equal and two remain unequal,which will result from the lowering of the Parhypate, therewill be three quanta constituting the Diatonic scale, namely,an interval less than a semitone, a tone, and an intervalgreater than a tone. Again, if all the parts of the Fifthbecome unequal, there will be four quanta comprised in thegenus in question.

It is clear then that the Diatonic genus is composed oftwo or of three or of four simple quanta.

It is required to prove that the Chromatic and Enharmonicgenera are composed of three or four simple quanta. Thesimple intervals of the Fifth being four in number, if theparts of the Pycnum are equal, the genera in question willcomprise those quanta, namely, the half of the Pycnum,whatever its size may be, the tone, and an interval such asthat between the Mese and Lichanus. If on the otherhand the parts of the Pycnum are unequal, the said •genera will be composed of four quanta, the least, aninterval such as that between the Hypate and Parhypate,the next smallest one such as that between the Parhy-pate and Lichanus, the third smallest a tone, and thelargest an interval such as that between-the Mese andLichanus.

On this point the difficulty has been raised, How is itthat all the genera cannot be composed of two simple 74quanta, as is the case with the Diatonic? We can nowsee the complete and obvious explanation of the difference.Three equal simple intervals cannot occur in successionin the Enharmonic and Chromatic genera; in the Diatonic

221

ARISTOXENUS

they can. That is the reason that the last-named genus issometimes composed of only two simple quanta.

Passing from this subject we shall proceed to considerthe meaning and nature of difference of species. We shalluse. the-terms 'species' and 'figure' indifferently, applyingboth to the same phenomenon. Such a difference ariseswhen the order of the simple parts of a certain whole isaltered, while both the number and magnitude of thoseparts remain the same. Proceeding from this definition wehave to show that there are three species of the Fourth.Firstly, there is that in which the Pycnum lies at thebottom; secondly, that in which a diesis lies on each sideof the ditone; thirdly, that in which the Pycnum is abovethe ditone. It will be readily seen that there are no otherpossible relative positions of the parts of the Fourth.

223

NOTES

[The references in these notes are to the pages and lines ofthe present edition.]

Page 95, line 3. The term fiiXos signifies a song, and as suchincludes the words, the melody proper, i. e. the alternation ofhigher and lower pitch, and the rhythm. But as the secondof these factors is evidently that which is characteristic of song,it came to appropriate to itself the term peXos. Then reXeiovfiiXos was used in the wider sense. Cp. Anonymus, § 29, TiXetov8e /ieXoy etrri TO (TVyKei/jLevov ?K re Ae£fa>s KCU fiekovs Kai pufyiou.

See also Aristides Quintilianus (ed. Meibom, p. 6, line 18). /icXosthen in the narrower sense signifies in Aristoxenus that momentof music which consists in the employment of higher and lowernotes, always with the implication that the complete series ofcompossible higher and lower notes is determined by a naturallaw. This quality of fifkos by which it is obedient to a law, orrather the embodiment of a law, is called TO fjpfioo-fievov: andconsequently all true melody is an r\p\iao-p.ivov fUXos. Thus forthe Greeks Harmony is the law of Melody, q fiovo-iKt] on theother hand is a term of very wide signification. Aristides Quin-tilianus (ed. Meibom, pp. 7, 8) gives the following analysis of it—

xpij<m/ca

II j

pvdfiiKovirolrjcris

pyd/ioirouaviroKpi.TiK.6i>

uftlKOVopyaviKov

Now in which sense is the term f»eXouj used in the passage223

ARISTOXENUS

before us? Marquard supposes in the general sense of theobject-matter of fiova-iKf). (In support of this view he mighthave quoted Anonymus, § 29, MOUO-IKIJ iariv ima-Tripi) 6ea>pr)TiK.riKal TrpaKTiKr) fxiKovs rekuov re K.a\ opyaviKov.) But this is not inaccordance with Aristoxenus' use, and probably Westphal isright in interpreting it in its close and strict meaning. If so,

v what are the other sciences of it besides apfwviKfj ? Westphalreplies, fu\onoua, opyaviKt], cidiicr] (i.e. the sciences of composition,of instrumental music, of singing).

1. 4. jxiav Tiva avrav \mo\afieiv del K.T.\. — T h e construction ofthis sentence is Set {mo\at3elv rqv apfioviKtjV KaXovnevrjv irpayfiaTeiaveivai jxiav riva avrSiv (i. e. T£C l&ewv), rr} re raêt irpwrrfv ovtrav, K.T.X.

Marquard and Westphal construe 8« in-oXaêlv piav rivaavratVj Trjv dpfLOfiKrjv KaKovfievrjv, aval Trpayfictreiav Tfj re r aê t irptBTTjv

ova-av, K.r.'k., and translate ' we must regard one of them, namelyHarmonic, as primary.' But the Greek for ' to be a good man 'is not tlvai avrjp aya66s &v.

rfjv dpfioviKTjv. The English word ' Harmony' in no wisecorresponds to the Greek apfiovia. This latter properly signi-fies an adjustment or fitting together of parts. Hence, bybeing transferred from the method to the concrete object whichembodies it, it is used to connote (a) a scale or system asa whole whose parts have been adjusted in their proper rela-tions, (6) the enharmonic scale, because in that genus three notesof the Tetrachord are fitted most closely to one another, that is,placed at the smallest possible intervals. The term cipjwviKT]signifies then the science of scales, that is the science by whichwe constitute a system of related and compossible notes.Harmony in the modern sense of the word was in its infancyamong the ancient Greeks.

1. 6. Tvyxavu yap oucra TO>V Trparav dfaprjTiKr)' ravra 8 i(TTivoa-a. The MSS reading is here plainly ungrammatical. If weretain irpaiTT] TS>V deuprynKav, we must change ravra to TaurTjj, ' tothis science belong,' &c. [cp. 1. 12, OVKCTI Tavrrjs earif]. ButI prefer to read as above with Westphal, in which case of courseTavra refers to T<J npcora. Cp. Anonymus (a mere echo ofAristoxenus), § 31, nparevou de fiepos TTJS HOV<TIKTJS r) dp/xoviKi eort"ra yap iv fiovmKrj npara avrrj deapei. Also § 19, rav 8e rijs

224

NOTES

IxovtTiKtjt iiepav Kvpiitrarov earl Kal irpurov TO ipy.ovi.K6v' ra>v yap

npmTcov ixov<n.Krjs Tre(j)VKe 6ea>prjriKT]. Cp. a lso 1. 14 of th is p a g e , 8'1

5>v Trdvra detopeirai ra Kara fiov<nicf)v.

For the relation between Harmonic and Music, cp. Plutarchde Musica, 1142 F, <pavepbv 8' hv ye'coiro, ei TO CKaonjv e£erd£oixorav iiruiTr]fJMv, TIVQS iaTi 8ea>pr)TiKT]' SfjXov yap o n fj /lev apjioviKTj

yevav re rav TOV fippoo-fievov Kal Siaorrj/idrav Ka\ <rv(XTt]fia.Tav Kai

<p86yya>v Kal rovav Kal fierajioX&v uviTTr]p.aTiKa>v cirri yvaxrriKrj' irop-

paiTcpa 8' ovKeri Tairr] irpoekdeiv 0*61/ re, atrr ov8e fijT«ri< napa

ravrtjc TO SiayvUvai bivaaBai, Trorepov oiKelcos ttKrjfyev 0 noirjTrjS . . .

rbv 'Y7roSc!)piov rovov em r<]v apyj\v r) rbv Mi£oAu8iov re Kat Aapiov

em rf)V eKâtriv rj TOV 'Yirtxppvyiov re Ka\ $>pvyiov eiti rf/v jiearfv.

1. 16. The point of the passage lies in the possible ambiguityof the term ippoviKos, which properly signifying 'concernedwith scales' [cp. dpfioviKrj — science of scales] might also mean' concerned with the enharmonic scale.' Cp. note on 1. 4.

P . 96, 1. 2. Kai roi ra biaypififiard y1 aircov. See end of noteon p. 101, 1. 1.

1. 4. itep\ fie ran aWav fuyeBav re Kal (rxtl^drav. I havechanged the MSS reading yevav to iieytBatv for three reasons:(1) quite sufficient stress has been laid on the early theorists'omission of the Chromatic and Diatonic genera, and further refer-ence to it is not required; (2) a reference to their omission of' other magnitudes ' is required in view of what follows (cp. 1. 7 ) ;(3) the close connexion of yevS>v and o-xruMirav by re Kai wouldmake it necessary to supply the qualification ev alra re r<3 yeVtrovra Kal rols Xoin-ois with both, which is obviously impossible.

crxopu, which we shall translate by 'Figure,' signifies thearrangement or order of the parts of a whole, and two thingsdiffer in <rxwa if they have the same parts, but these parts arearranged in a different order. Thus the scale from C to c andthe scale from B to b on the white notes of the piano arecomposed of the same intervals, five tones and two semitones,but they differ in crxrj/xa or the arrangement of those intervals.

1. 6, anoTefivofxevoi. . . . TO hta jraa-av. By the phrase rb rpirovliepos rijs S\rjs pe\<p8Las is meant the Enharmonic genus, just asa few lines above rfjv irao-av rrjs fieXabias raîv means the Enhar-monic, Chromatic, and Diatonic Genera.

MACEAN Q 225

ARISTOXENUS

Hence the MSS reading lv n -yevoy peyeBos Be is untenable.What is the rpirov fiepos of /zeXcoSm from which the Harmonistscan be said to have selected one genus ? According to Mar-quard &p/j.ovia (in the sense of 'melodic element in music').But even granting that /ueXwfii'a here means music in general,and that music in general may be divided into dpfiovia, pvdfias,and \6yos, could this division have been so universally familiarthat Aristoxenus would presuppose it, and employ the phraserp'iTov fnipos without explanation ?

I omit yhos and 8e. The former might easily be inserted byan ignorant scribe, who not understanding TOV rpirov pipovsmissed the necessary reference to the enharmonic genus. Theintrusion of yivos naturally entailed the addition of de.

1. II . An unknown polemic.1. 18. (pavrjs. The term <pa>vq in Aristoxenus comprehends

the human voice, and the sounds of instruments. See Ari-s to t l e , de Anima, 4 2 0 b , fj 8e (j>a>vfj \j/6<pos rls etrriv ipAJ/vôv' TS>V

ynp a\jrv)(<0i> oidev <f>mvei, dXKa Kad' ofioioTrjTa Xtycrni (pcovc'iv, olov

av\6s Kai Xvpa Kai 8<ra aWa TS>P a^rv\v anoTCuriv ? x « (tat fiikos Kal

diaXeKTOv.

P . 97, 1. 2. I read eVt/ieXfj for <?jrtjueX5r of the MSS which(1) gives a weak construction to ycyivrjrat, and (2) requires, asMarquard saw, the biopurBivros of I. 4 to be supplemented byan adverb.

1. 6. Ado-oy. Lasus of Hermione, the well-known dithy-rambic poet, and teacher of Pindar. Suidas credits him withthe authorship of the earliest work on the theory of Music.See Suidas s. v.; Athenaeus x, 455 c and xiv, 624 c; Herodotusvii. 6 ; Plutarch, de Musica, 1141 B-C.

'Emyovelcov. Disciples of Epigonus of Ambracia, a famousmusical performer. See Athenaeus iv, 183 d and xiv, 637 f.

1. 7. TTMTOS. The spatial image, under which Aristoxenusrepresents the pitch relations of notes, is that of an indefiniteline x-y , ,

o n w h i c h t h e s e v e r a l n o t e s a p p e a r a s p o i n t s a b e d [ c p . N i c o -

m a c h u s (ed . M e i b o m , p . 2 4 , 1 . 2 1 ) , <f)d6yyog «<m (j>a>vq STOHOS, olov

226

NOTES

novas KaT dKof]v], and the intervals as the one-dimension spacesbetween them. The obvious objection to this conception isthat it attributes quantity and so reality to the spaces betweenthe notes, while it denies it to the notes themselves, whereasour senses tell us that the notes are the realities, and theintervals only their relations. This objection lies at the basisof the contending theory, here quoted by Aristoxenus, whichassigns to notes a certain quantity or'breadth.'

1. 16. h irfi fiiv irrj S1 oC. For Aristoxenus' answer to thequestion see p. 107,11. 13-19.

1. 17. I conjecture \CKT£OV for SUaiov of the MSS. Cf. noteon p. 143,1. 13.

1.19. Probably Marquard's SieAtfoWa is correct. SieXoVra isnot objectionable in itself (cp. p. 98,1. 5, p. 108,1. 18, &c.); butif we retain it, the passage lacks any reference to the generaltreatment of the scale.

1. 22. TrXeiour eltA (pitrcis fiekovs. See p. 110.P. 98, 1. 9. The meaningless mVijr of the MSS may have

been interpolated to produce a show of connexion between thisparagraph and the preceding.

1. iy. o*s a/ia . . . (runfialvei.

The distance between e and a, regarded as a whole, is aninterval; regarded as a series of smaller distances, between e andf,f and g, g and a, it is a scale.

1. 21. Of Eratodes nothing is known beyond what we learnfrom Aristoxenus himself.

1. 22. on OTTO . . . fic'Xos. That is, one has a choice betweenconjunction and disjunction.

Conjunction.Conjunction.

Disjunction.

At the point EBE

Disjunction.

the ascending melodic progression

Q 2 ;"7

ARISTOXENUS

diverges into

Similarly at

branches into

the descending melodic progression

1. 23- « anb TTCLVTOS . . . yiyverai. Evidently the law onlyholds of those Fourths of which the boundaries are fixed notes.

If we take the Fourth F?§—J there is but one

method of completing the melodic progression in each direction ;thus—

P. 99,1. 12. For the Perfect System or Scale see IntroductionA §29.

1. 14. Kara crvvBetrtv,' in respect of the method of their com-position,' according as that may be by conjunction, disjunction,or a combination of both these methods. See Introduction Apassim.

1. 15. Kara o^fta. Cp. note on p. 96, 1. 4.H probably supplies the true reading here. Marquard inserts

(cat Kara Biaiv on account of fujre deais in 1. 17. But the latterwords (which do not appear in H) are probably a dittograph topyre orivBeais. Though 8e<ns does not occur as a technical termin Aristoxenus, it might conceivably mean ' key' on the analogyof riBeadai (see e.g. p. 128, 1. 7) ; but key-distinctions belong toa later part of the subject (p. 100, 11. 14-20) and are out ofplace here, Aristoxenus being well aware that such distinctionsare not essentially scale-distinctions (see p. 100, 1. 16).

I. 25. dvimoSeiKTa? . .. yiyveaBat deUvvrai. Eratocles, accord-ing to the criticism of Aristoxenus, would seem to have presup-posed the constitution of the octave scale

mand to have arrived at the enumeration of its Figures by showing

228

NOTES

that after proceeding through the various arrangements to beobtained by beginning successively with e, f, g, a, 6, c, d, one isbrought back again to the first Figure with which one started.Against this superficial empiricism Aristoxenus very justlyurges that the Figures of the Fourth and Fifth and the laws oftheir collocation must be demonstrated prior to the enumerationof the Figures of the Octave. Otherwise we are not justified inlimiting these Figures to seven. Why, for example, should wenot admit the Figure

Here we have a scale that is illegitimate though it consists offive tones and two semitones, because it violates the law of theFigures of the Fourth and Fifth and their collocation.

P . 100, 1. io. Several words must have been lost herethe substance of which I have supplied. Aristoxenus is evi-dently insisting that the enumeration of the scales cannot becomplete unless account be taken of the scales of mixedgenus: therefore after the number of possible scales in eachgenus has been ascertained, we must, he tells us, mix generaand repeat the process of enumeration. But what is the senseof giving as a reason for the necessity of this process the factthat ' they,' whoever ' they' may be, ' had not even perceivedwhat mixture is ' ?

1. 17. Marquard inserts TOO TOTTOV before alrov and translates1 though the space is in itself homogeneous.' Westphal rightlyreads with the MSS and understands avrov as equal, to TOV

I. 22. The question here raised is one of great importance.Are there any affinities between scales and keys ? By scales wemean so many series of notes in which abstraction is made ofpitch and regard is had solely to the order of intervals. Bykeys we mean so many series of notes, in which the intervalsand their order are identical, while each series is situated ata different pitch from every other.

See Introduction A, § 22.P . 101,1. 1. Aristoxenus here contrasts two principles by

229

ARISTOXENUS

which one might be guided in determining the relative positionsof the keys proper to the several scales. One is the falseprinciple of KaTairvKvacris, or ' close-packing' of intervals ; theother the true principle of the possibility of intermodulation.To understand the difference between these principles let ustake the seven modes or scales of Table 20 in Introduction A,in the Enharmonic forms as follows :

TonicMlXOLYDIAN

LYDIAN

PHRYGIAN

DORIAN

HYPOLYDIAN

HYPOPHRYGIAN

HYPODORIAN

«J=„ [ rJ-

TonicI I

Tonic

Tonic

T otTonic

•j J j

Tonic J_

and let us place all the notes supplied by these scales betweenn

V&>—J— and in one series as follows:

^

Now.we see that in this series there is no

no no

. no

, that is,

230

NOTES

there are several intervals of a semitone which are notdivided into their apparently possible quarter-tones. At thesame time it is evident that the tonics of these keys are sorelated to one another that it will be possible to pass directly orindirectly from any one to any other. (See note on p. 129,1.4.)

Once more let us again take the same seven enharmonicmodes, but changing the keys let us arrange them as follows :

Tonic

Tonic

w=3z3n& *S* "f

Tonic

J—j^—««l—g»L-

MlXOLYDIAN

LYDIAN

PHRYGIAN

DORIAN

HYPOLYDIAN

HYPOPHRYGIAN

HYPODORIAN

Writing in one series all the notes of these keys between

H& J ~ = and TOT — we obtain the following result:

IW) J~sJ—jJ-g]fJ=—^" «J~^- )iilJ * "* Jt*^t* * ***=

Here we have an unbroken series of the absolutely smallestintervals (i. e. quarter-tones); but the keys are so related to oneanother, their tonics being spaced by the interval of three

231.

ARISTOXENUS

quarter-tones, that a modulation from one to another of them isimpossible. (See note on p. 129,1. 4.)

The first of the above sets of scales is arranged on theprinciple of possible intermodulation ; the second on the prin-ciple of KaranvKvains, or arrangement at the closest possibleintervals. It is obvious that the former is the true principle ofmusic. The unbroken series of small intervals may satisfythe eye, but to use the words of Aristoxenus [p. 129, 1. 1] it isfKfieXrjs Kai irdura rpartov &xpô-ros, that is, at variance with thenature of melody which forbids a succession of more than twoquarter-tones; and of no practical value, because the onlyobject in a relative determination of keys is to render inter-modulation possible.

We can now understand the statement of Aristoxenus [p. 96,1. 2] that the tables of the early harmonists, though only con-structed with a view to the Enharmonic Genus, exhibited thewhole melodic system. In such a series as that last given allthe chromatic and diatonic scales are implicitly presented. [Itis however possible that ihrjkov in this passage may signify' professed to exhibit.']

1. 2. I read rivwv for MSS TZV*1. 3. jrepi TOVTOV . . . rovff fjfxiv. I have corrected the read-

ings of the MSS by inserting on before eVi Ppaxv. Then oniviois o~vp{3el3T]icev Trepl TOVTOV TOV fiipovs uprjKivai, ov8(v\ 8e o-vp,-j3e(3r]Kev KadoXov elprjKevai is t h e sub jec t of (pavepov yeyivr]Tai.

1. 7. 7reiriyr)Tai of Me. for ireiroiijTai is an interesting exampleof a mistake arising from dictation. Such mistakes are frequentin the MSS of Aristoxenus. Compare p. 144, 1. 12 fj TOVTOISo-vvex*ls t<H ol TOVTOIS crvvexeis, p . 139, 1. 18 Beiitvvcriv ( in R ) forSq Kivt]o-Lv, p. 139, 1. 13 ua\v as (in R) for cb iv6s, p . 137,1. 15VTrapvirarri (in B) for ij Trapvna.Tr) ; also such spellings as dweTOvv,aKiacnv, mKvd, axpio-Ta, clpeio-6u>, for cmaiTouv, aKkoia>o-iv, irvxva,"XPV0"rai elpwdco, and the constant confusion of subjunctive andindicative forms.

P . 102,1. 8. ironpov... io-ri o-nfyi-ns. See Introduction B § 2.Aristoxenus is not concerned with the truth or falsity of thephysical theory of sound.

1. I I . TO 8c Kivrjo-at Tovrav iKarepop. T h e true reading here232

NOTES

is hard to conjecture. Marquard's first idea was to omit8e and understand nivrjo-ai in the sense of' to raise or moot aquestion'; but he afterwards abandoned this view on the groundthat Ktveh occurring so often in the same passage in the technicalsense of 'motion' could not in this one case bear a differentmeaning. [On this point Mr. Goligher aptly cites Berkeley'sP r i n c i p l e s of H u m a n K n o w l e d g e , § 7 7 : ' I f w h a t y o u m e a n b ythe word matter be only the unknown support of unknownqualities, it is no matter whether there is such a thing or no,since it in no way concerns us.'] His final conjecture is 81a-Kplvai for 8e Kivr)<rm, and he gives as the meaning of the passage' for the purposes of the present argument it is not necessaryto decide this question.' But this is, I think, quite untenable.Even if we grant t ha t ' it is not necessary to discriminate eachof these things' is a possible expression of the meaning ' it isnot necessary to decide for either of these alternatives,' yet itis clear from 1. 7 that inaTepov TOVTCOV must here mean ' each ofthese phenomena,' namely, the two kinds of voice-motion. Oncewe admit this, we must reject TO Sicucplvai; for it is obviouslyfalse to say tha t ' the, discrimination of these phenomena fromone another is unnecessary for our argument.'

I believe the true reading to be TOV SievKpivijo-ai (or some suchword) TOVTIOV (Karepov, where TOV 8ievtcpivrjo-ai is the genitive ofthe material after riji' ivea-Taxrav 7rpayfiareiav: and the meaningto be ' the question of the objective possibility of rest and motionof the voice belongs to a different sphere of speculation, andis irrelevant to our present purpose, which is to discriminateeach of these two phenomena from the other.1

1. 26. 8ta iraBos. As in the case of impassioned recitation.Cp. Aristides Quintilianus (ed. Meibom, p. 7,1. 23), q pkv ovv o-vv-€Xhs (.Kim/tris) idTiv, y SiaXeyd/ie(9a' fiecrij 8e, y TCIS TO>V iroujfiarmv

avayvwcreis rroiovfi^Ba' 8iaoT7j/«mK^ be r\ Kara \ikaov T£>V &ir\a>v

<pa>v£>v 7ro(ra Troiovpivr) SiaorijfiaTa Kai p.ovds, rj Tts Kai fiekaSticri

KaXelrm.

P . 103,11. 1-6. As the monotone of declamation is a licenseof speech, so is the tremolo a license of music; and the use ofeither, if not justified by the presence of an exceptional emotion,is a sin against nature.

233

ARISTOXENUS

1. 3. Probably oera> yap &v . .. noirja-afuv, the reading of B andR, is right.

I..16. eniraa-is and <1ve<ris signify the processes, not the states,of tension and relaxation. Though properly applying only tostrings, they are used metaphorically of the human voice andthe sounds of wind-instruments.

P . 104,1. 14. eVl TOV ivavriov TOTTOV, the reading of B, is un-doubtedly right. Cp. p. 145, 1. 9 ; also the phrases eVi ro <5£u,ori TO fiapv.

1. 20. rp'iTov. Westphal's conjecture of liiimrov is, I think,unnecessary, in spite of p. 106, 1. 9. For the purposes of theargument cmraa-is and avems may be regarded as subdivisionsof one conception, and similarly dgvTrjs and fiap(iTr)s.

1. 23. fiij TapaTTiraa-av K.T.X. Aristoxenus very rightly in-sists that the validity of his distinction is not injured by thefact that it is verbally incompatible with the theory of thePhysicists. When he speaks of motion and rest of the voice,he refers to certain phenomena which the ear distinguishes asmotion and rest, though this distinction may directly contradictthe ultimate nature of these phenomena as apprehended by theintellect. Thus, when the Physicist presses upon him thetheory that all sound is vibration or motion, and urges thatmotion at rest is a contradiction, he replies : ' According to theevidence of the ear (which, for my purposes, is the final testof truth) the voice is at rest in cases where, according to yourtheory of objective facts, the rate of its vibration is constant;consequently, to distinguish the phenomena before us, we mayemploy the language of the ear just as well as the language of.physics.'

P . 105 , 1. 15. The MSS read here o ff r)fiels Xeyo/xev Kiviyjiv reK(il T^pijiiav (pavrjs Kai 6 cKe'iaoi Kivr/cnv which is translated ' it isfairly evident what we mean by rest and motion of the voice,and what they mean by motion.' But this is unsatisfactory, notonly on account, of the weakness of the conclusion thus drawn,but also because 5 6'... Kivqvw being a relative sentence andnot an indirect-question, the correct translation would be ' thething to which we give the name of rest and motion of thevoice is a fairly patent thing, as is also the thing to which

234

NOTES

they give the name of motion,' which does not give the requiredmeaning.

P . 107, 1. 3. 8i£<r«a)s rrjs eXnîori;?. That is a quarter-tone.Aristoxenus uses bUa-is for any interval less than a semitone.

1. 5. WITTS Ka\ l-vvievai K.T.\. Aristoxenus does not mean thatwe cannot hear any interval smaller than a quarter-tone, butthat though we may be conscious of such a smaller interval,we can have no perception of it as a musical entity, since wecannot estimate its magnitude in reference to other musicalintervals.

P . 1 0 8 , 1 . 21. Kaff f)v TO. o-ifjKpava TS>V 8uxpo>va>v. The only con-cords recognized by Greek theorists are the Fourth; the Fifth ;the Octave ; the sum of two or more Octaves : the sum of oneor more Octaves and a Fourth ; the sum of one or more Octavesand a Fifth.

In his note on this passage Marquard has collected severaldefinitions of concords and discords.

According to Gaudentius [ed. Meibom, p, 11,1. 17] o-vufyavmSe Z>v ci/Aa KpovofjUvutv jj av\ov[i€va>v det TO /zeXoff TOV fiapvTtpov 7rpbs

TO 6£v Kal TOV 6£vTepov np6s TO (iapv TO airo fj... bia<f><ovoi 8e av &fui

K.povo)iiva>v Tf avKovfiivav ovbiv n (palveTai TOV /xeXour elvai TOV

Hapvrepov irpos TO b^v rj TOO o^vripov irpos TO fiapv TO avro.

' The nature of concordant sounds is that when they are struckor blown simultaneously, the melodic relation of the lower noteto the higher is identity, as likewise the relation of the higherto the lower; but when discordant sounds are struck or blowntogether, there seems to be nothing of identity in the relationof the lower note to the higher, or of the higher to the lower.'[Practically the same definition is given by Aristides Quintilianus(ed. Meibom, p. 12,1. 21), and Bacchius (ed. Meibom, p. 2,1. 28).]

Marquard professes himself unable to find any meaning inthis definition. The language is certainly not happy; but Ithink the sense is clear enough. If two sounds are discordant,when they are sounded together, the particular character ofeach will stand out unreconciled against the other; that is,the relation of the higher to the lower or of the lower to thehigher will not be one of identity in which differences are sunk.On the other hand, when concordant sounds are heard together,

235

ARISTOXENUS

the resulting impression is that of the reconciliation of differences,the merging of particular natures in an identical whole. Thisis well illustrated by the concord called the Octave, where therelation of identity is so predominant that we regard the notesof it as the one note repeated at different heights of pitch.

According to the Isagoge (ed. Meibom, p. 8,1. 24) eari Se o-v/i-(pavia jiiv xpao-is bio cpdoyyav d£vrepov Kai fiapvTfpov' dioKpavia de

rovvavriov bio (pBoyyav a/u£ta more ixr] KpaBrjpat, aXKa rpaxvvBrjvai

TTJV aKotjv. ' Concord is the blending of two notes, a higher anda lower ; discord, on the contrary, is the refusal of two notes tocombine, with the result that they do, not blend but grate onthe ear.' The same conception is more clearly expressed in thedefinition quoted, by Porphyrius:—<rvfj.<pavia 8' eorl bvolv <p86y-y<ov df-vrrjTi Kai fiapvTT)n buKpepovrcov Kara TO avro ITTSHTIS Kai

Kpao-is' bei yap rois (pdoyyovs o-vyKpovo-Oivras ev Tt erepov elbos

(f>86yyov arroTehelv Trap' eiceivovs ££ <Sv (pdoyyav fj avjAtpavLa yeyovev.

' Concord is the coincidence and blending of two notes of differ-ent pitch, for the notes, when struck together must result ina single species of sound distinct from the notes which havegiven birth to the concord.'

The following definition of Adrastus is quoted by Theo.Smyrn., p. 80, and Porphyrius, p. 270, <rvp<pavovo-i 8e (j>66yyoijrpos aX\rj\ovs &v Oarcpov Kpovadevros iirl TWOS opydvov TO>V ivrarfov

Ka\ 6 XOHTOS Kara Ttva olxeiirrjTa Kai ovfmadeiav avvrjîj' Kara TO

airo Se afia afUporepav Kpovo-devrav Xfia Kai irpoarjvt]S e/c TTJS Kpd-

o-eas ft-aKoierai <pa>vq. ' Notes are in concord with one anotherwhen upon the one being struck upon a stringed instrument,the other sounds along with it by affinity and sympathy; andwhen the two being struck simultaneously one hears, in con-sequence of the blending, a smooth and sweet sound.'

Most philosophic of all is Aristotle's definition in Problemsxix, 38, cvfKpavia be \alpofuv on Kpao~is io~Ti \6yov exovravevavTiav jrpos aWtjka. 6 /icv ovv Xoyos rdf-is, o TJV (pvarei fjbv. ' T h e

reason that we take pleasure in concord is that it is a blendingof opposites that have a relation to one another. Now rela-tion is order and we saw that order naturally gave pleasure.'C p . a l s o Ar i s t o t l e Trepl alo-6f]O-€as Kai alo-drjTav C. 3 , p . 439 *,

TO. ptv yap iv api.Bjj.ois ev\oyio~Tois xP^>tulTa> ^djrep CKEI Tas

236

NOTES

crvficpavias, ra rj8i<TTa rav xpa>yL°â>v «"<"

agreeable colours, like concords, depend upon the easily calcu-lable relations of their ingredients.'

Later theorists introduced jrapd<pui/o? as an intermediate termbetween o-iprpcovos and bidrpnvos. According to Gaudentius[ed. Meibom, p . I I , 1. 30], rrapa(f>a>voi be oi fitVoi jxev avfMpavov Kai8ia(j>&ivov' iv 8e 777 upoiaei (fraivo/ievoi crv/Jiâvoi, fitnrep eni rpiavTovav (paiverai, dno itap\ma.Tr)s fie'crap ejrl jrapaj«Vi)»', Kai. eVl 8uoTOVCOV, <wro piaatv 8tar6vov in\ irapafii<rr)v, ' Paraphone sounds standmidway between concords and discords; when struck' [thisprobably means ' when not prolonged by voice or wind instru-ment, but sounded momentarily on strings'] ' they give theimpression of concord; such an impression we receive in thecase of the interval of three tones between the Parhypate Mesonand the Paramese; and in the case of the interval of two tonesbetween the Lichanus' [the term ' Diatonus' is sometimes usedfor Lichanus] ' Meson and the Paramese.'

The term 6fi6<pa>voi is applied to notes which differ in function,but coincide in pitch. Thus the Dominant of the key of D andthe Subdominant of the key of E fall alike on A. See AristidesQuintilianus, ed. Meibom, p. 12,1. 25.

1. 22. TO. (rvvdira TS>V atrvvQeToiv. Aristoxenus means by asimple interval one that is contained by two notes between whichnone can be inserted in the particular scale to which they belong.

Thus in the enharmonic scale,

interval between_/"and a is simple, because in this scale no note can

occur between them; but in the diatonic

scale the interval between f and a is compound, because in thisscale g occurs between them. Thus the same piytdos or mag~nitudef-a, which as a neyeOoc is of course composite [the simplemagnitude of music being a quarter-tone], may sometimes beoccupied by a simple, sometimes by a composite interval.

1. 23. Kad' rjv &taxf)epu ra ptjra TOIV aXdyo)!'. This 8ia0opa is notwithout difficulty. The terms fara and akoya naturally apply toquanta in relation to one another. 4 is aXoyo>> in relation to 7,tfie area of a square in relation to that of a circle. But where

237

m

ARISTOXENUS

in the case of an interval are the two quanta the relation betweenwhich constitutes it rational or irrational ? Not inside theinterval, for Aristoxenus, as we have already seen, has nothingto do with the Pythagorean view of intervals as numericalrelations. An interval then must be rational or irrational invirtue of the relation it bears to some quantum outside itself.Marquard supposes this quantum to be the twelfth of a tonebecause that is the smallest measure used by Aristoxenus incalculating the comparative sizes of intervals. (See p. 117,11. 1-19.)' But this supposition, as we shall presently see, isdirectly forbidden by Aristoxenus himself. The true explanationis supplied by the following interesting passage from theElements of Rhythm (Aristoxenus, ed. Marquard, p. 413, 29) :—

"Qpiarai Se TCOV iroSiov eKaarros rjrot \6ya> nvl rj akoyiq roiavrr],

i]Ti» fiuo \6ya>v yvtopip.a>v rrj alcrdTjcrei dva /j.eo'ov coral. TevoiTO

8' av TO elpr/pevov a>8e KaTa(paves' el \r)tf>8eii)0~av 8io irodes, 6 pAv Itrov

TO avto T(3 Kara câv Kal St'oTj/iOj/ eKarepov, 6 tie TO /lev Kara dlmj/tov,

TO 8e ava ijfiiav, TpiTOS 8e TIS Xijipdeir/ TTOVS irapa TOVTOVS, rf/v /lev

|3acrtv icnjv hv TOIS ap.(pOTtpois ixav> Thv ^ apcriv (ie(rov fiiyedos

cXowav TZV apareav. 'O yap TOIOVTOS nobs SKoyou piv e£ei TO mica

7Tpos TO Kara' f orm 8' r) akoyia /leTaî) dvo \6ytov yvapi/uov Trj

al(r8t]o~£i, TOU re io"ou Kal TOV 8wrAa<riou. . . .

A « de fir/S' ivravda SiafiapTfic, dyvor)6evros rov re pijTov Kal TOV

aKoyov, Tiva rponov iv TOIS nepl TOW pvdfiovs Xap-fidvcrai. "Qcnrep

ovv iv rots 8iaoT7j(iaTiKoiy orotêioif TO p.ev Kara p.e\os pijrov £\r)(p8r),

0 npa>Tov fiiv IOTI p.ekabsovfj.cvov, (ireira yvapiftov Kara peyedos, TJTOI

li>s TO. T£ o-vp.(pa>va Ka\ 6 TOVOS, r\ &>s TO. TOVTOIS avuixtrpa, TO he Kara

TOVS TB>V dpidfiatf fiSvov \6yovs prprov, 0) o~vvefiaivev dficKaSrjTa elvai'

OVTIO Kal iv TOIS pvd/j.ols viro\rjTrreov exeiv T6 Te ptjTov Kal TO akoyov.

T6 ptev yap Kara TT)V TOV pv6p.ov tyvaiv \apJSdverai pryrov, TO hi

Kara TOVS TO>V dpiBfiav povov \6yovs. To p.ev ovv iv pvBpxo Xajuj3avd-

fievov prjrov xpovov p.eyedos irpatTOV pev tSei TS>V Ttnttovrav els TTJV

pvBuoiToilav elvai, eirena TOC 7ro8os iv <S reraKTai fiepos elvai pr\TOV

TO 8e KOTCI TOVS TSIV dpidfioiv \6yovs \ap.âv6jievov prjTov TOLOVTSV TI

del voeiv oioi> iv TO"IS SuumntaTUCois TO fSa&eKaTriiiopiov TOV TOVOV Kal

ie? TI TOIOVTOV aXXo iv rats TSIV BtaoTrnxdrav irapaWaya7s \ap.jBdveTat.

$avepbv Se 81a TS>V elprjjievav, on f/ \Li<rr\ Xr](p6iio'a TOIV apcreav oitt

carat avfifierpos Trj fiacrei' ovhev yap avrav fierpov carl Koivbv

238

NOTES

' Every foot is determined either by a ratio (betweenits accented and unaccented parts) or by an irrational relationsuch as lies midway between two ratios familiar to sense.This statement may be illustrated as follows: take twofeet, one of which has the accented and unaccented partsequal, each of them consisting of two minims of time, whilethe other has its accented part equal to two minims, but itsunaccented only half that length.' [Assuming the minimto be, what it once was, the sign of the shortest possiblemusical time, the first of these feet would be of the form| Q Q | > ihe second of the form | Q ej | •] ' Now takea third foot besides, having its accented part equal to the ac-cented part of either of the first two, but its unaccented, a meanin size between their unaccented parts.' [Its form will be

| Q cJ • |•] 'In s u c n a foot the relation between the ac-cented and unaccented parts will be irrational, and will liebetween two ratios familiar to sense, the equal,' [ Q : Q ] 'andthe double' [ Q : cj ] • • • ' Nor must we be led astray here byignorance of the principle on which the conceptions " rational"and "irrational" are determined in matters of rhythm. In theElements of Intervals we assumed on the one hand a " rationalin respect of melody" which is firstly something that can besung, and secondly, something whose size is well known, either[directly] as the concords and the tone, or else [indirectly] asthe intervals commensurate with these; and on the other hand,a "rational in respect of numerical ratios," which, as a fact,was something that could not be sung. A similar view mustbe taken in the case of rhythm, and we must distinguish therational in respect of the natural laws of rhythm from therational in respect of numerical ratios only. According tothe first reference, a rational time-length is one which, firstly,can be introduced into rhythmical composition, and secondly,is a rational fraction of the foot in which it is placed. Accord-ing to the second reference, it must be conceived as somethingin the sphere of rhythm corresponding to the twelfth of a tonein the sphere of melody, or to any other similar quantumassumed in the comparative measurement of intervals. It is

239

ARISTOXENUS

clear from these remarks that the mean between the two un-accented parts will not be commensurate with the accentedpart; for they have no common measure with a rhythmicalexistence.'

We see here that the reason why the foot | Q cJ • | isirrational is, that though cJ . is a possible rhythmical element,and though the relation of ^ . to Q is known as that of 3 to 4,yet the length ^ . , while mathematically commensurate witha, is rhythmically incommensurate. For their common measure,being half the minimum time length, has no existence in thepractice of rhythm.

The case is similar with regard to Melody. If any intervalcan be sung; if its length be readily cognisable, either imme-diately as a concord or tone, or because it is commensuratewith one of these, the common measure being an actual melodicinterval, then it is prjrov. If these conditions be not fulfilled, itis aXoyov. Thus a twelfth of a tone is not a rational intervalin respect of melody, because it cannot be sung ; neither is theinterval of three sevenths of a tone rational; because though itcan be sung, and though its length can be mathematicallyexpressed in relation to a tone, yet the common measure ofit and of a tone is one seventh of the latter; which is not anactual melodic interval.

1. 24. Tar 8e XotTras K.T.X. Cp. Aristides Quintilianus [Mei-bom, p . 14) 1. l o ] , ?TI 8' CLVTSIV a \iiv Itniv aprta, a Be nepiTTa.apria fiev ra els "ura 8i.aipovp.eva, as TJ/UTOPIOV Ka\ rovos' irepiTTa.8e TO. els avura' as al y diecrfis icai 7reiTe Ka\ f, and [Meibom, p. 14,1. 20], eri TO>V bicurrqiiaTan/ a \ikv iariv apaia & 8e nvKvd' nVKva fi£vTa iXaxurra as al Sieaeis, apaia de ra /leyurra its TO 81a Teaadpav.

P . 1 0 9 , 1 . y . TOVTOV ye TOV rpoirov K.T.X. Aristoxenus impliesby this reservation the possibility of dividing scales into thosewhich are composed of other scales (as for instance an octave,which is a compound of a Fourth and a Fifth), and those which

are not so composed, as for instance IM) J~~j£ ^ • Buteven this last scale, though it cannot be analysed into otherscales, is composed of certain parts, namely intervals, and so canhardly be called simple.

249

NOTES

1. 16. an6 nvos fieyidovs. The meaning i s , ' Every scale froma certain magnitude upward.' Evidently a scale of a Fourthor any smaller scale need not exhibit either conjunction ordisjunction.

I.18. TOCTO. ' This phenomenon of the blending of conjunc-tion and disjunction.'

iv ivlois, i. e. a-va-Trjfiaariv. See Introduction A, § 20.1. 19. The term vnepfiarov signifies that the scale skips certain

notes which would naturally belong to it by the laws of continuityor sequence. See Introduction A, § 26.

1. 20. airkovv ieai SwrXoBv K.T.\. Cp. Aristides Quintilianus[ed. Meibom, p. 16,1. 2] , ml ra fiev &ir\a a <ad' tva rpoirov eKKeirai,TO 8e ov\ dVXS a Kara TTXCIOUCOV rponav TTKOKTIV yivfrat. ' Singlescales are those that are composed in one • mode; manifoldscales those that are based on a complex of several modes.'

Cp. also Isagoge [ed. Meibom, p. 18,1. 20], TJ 8e TOV afieraâXovKai E/x/iera/3dXou SioiVei Ka6' f/v Siarpepct ra cnrXa <marr\}uxTa TO>V fir]airkSiv" d?rXa fxev oiv iari ra npbs p-iav ixicnjp rjpfioo'iJLeva, dmXa 8c rairpos Svo, TprnXa 8e ra rrpos rpeis, iroXkairkaaia di ra npos 7rX«'ovas.'The difference between the modulating and non-modulatingscale will be the difference between single scales and those thatare not single. Single scales are those that are tuned to oneMese, double those that are tuned to two, triple those that aretuned to three, multiple those that are tuned to several.'

The distinctions here referred to we have already consideredin our comparison of the three ancient Harmonies [IntroductionA, § 14]. The Mixolydian scale on the old reading of it [Intro-duction A, § 20] was a o-ioTrjfia SLTTXOVV.

Cp. p. 131, 11. 9-10 where Aristoxenus contrasts dtrKovv and

P. 110,1. 5. \oySsbis TI fifkos. For the relation between Greekspeech and Greek song, see Mr. Monro's Modes of AncientGreek Music, § 37:

1. 14. I read Kadokov for Kal TTOV. Some such word is calledfor by the following tfiidri/ra.

1. 21. on noWas . .. Zv re xai rairov K.r.X. Aristoxenus. meansthat in spite of the great variety of forms that consecutionadopts, there underlies this variety one immutable law, which

MACRAN R 241

ARISTOXENUS

decides in any case whether any given sounds may or may notsucceed one another.

P. I l l , 1. 7. T&V els ravro xipjioafUvav is my suggestion for theimpossible TO>V a? TO ^pfioo-pivov of the MSS. Aristoxenus isobliged to add this qualifying phrase to show that his divisionof the nekos is not inconsistent with mixture of genus. Thusthe meaning is ' every melody that observes one genus through-out falls into one of the three classes of diatonic, chromatic, andenharmonic.1

1. 8. rjroi biarovov eirriv fj xPa>PLaTiK0V K.r.X. A r i s t i d e s Q u i n -

tilianus (ed. Meibom, p. 18, 1. 19), gives the following deri-vations of these names : Enharmonic, OTTO TOC o-vvt)pp.6o-8ai, i. e.from the close fitting of intervals exhibited in its Pycnum; Dia-t o n i c , ineidlj <r<po8pdrepoj' 17 (jxovr) /car' avTo dtareweTai (Hiarovos is

t o 8iareiva> a s o~vvrovos t o CTVVTUVO>) ; C h r o m a t i c , a>s yap TO fiera^v

XevKov Kai fieXavos xp£>p.a /caXeirm* OIITO) xal TO 81a \xi(Tii>v a/i(poiv

6eapovp.evov ^pa/ia 7rpo(reiprjTat.

Cp. Nicomachus (ed. Meibom, p. 25, 1. 32), xai i< TOVTOV yeSiaroviKov KaXctrat, e< TOV irpo^topdv 81a TS>V TOVCOV OVTO povaTarov

TO>V aXKav. (p . 26 , 1. 27) , <UOT' avriKeiaBai TO ivappoviov T& Siarova'

ixeirov 8' airav vnapxsiv TO •^puyp.aTiicov. fuKpbv yap Traperpetyev, ev

J]fiiToviov OTTO TOC btaroviKOv' evBev 8c Kai ^p5fia ex€LV Xeyo/MV

cvrptirrovs avBpanrovs.

Cp. also the interpolated passage in Aristides Quintilianus( M e i b o m , p . I l l , 1. 8), xPa>liaTlKOV &* KaXeirai Trapa TO xpd>£eiv avTOTO. Xowra tSiacTTtjiiaTa, fir) hkio~8ai 8c TWOS eKfivcov. [ A c c o r d i n g t o

Bellermann [Anonymi Scrifitio, p. 59) xp°>&lv Ta Xora-a SiaorijfiaTa= attingere cetera genera; the ptj &uo-8ai he TWOS imlvav is unin-t e l l ig ib le ] . . . TO 8' ivapnoviov 81a TO iv rrj TOV iir\pfJLO(rfi.ivov TeXeia

8tatrrao"et \anJ3dpeo~8ac' ov yap SITOVOV ifKiov, aire dieaetos ifXaTTOV

ivdexCTai. ( M S S e'8e'êTo) Kara alaBrfaiv \aficiv Ta 8iao"TijâTa i . e .

the Enharmonic genus derives its name from the fact that ituses to the full the liberty of variation permitted by the laws ofHarmony. It uses quarter-tones, than which there is no smaller,and ditones, than which there is no greater (simple) interval.

1. I I . If avaTorov be correct, it means ' highest' in the processof development and so furthest from the state of nature. But

v, the reading of H, is very tempting.243

NOTES

1. 24. TO fitv IKCLXUTTOV. The Greeks did not recognize theGreater or Lesser Thirds as concords.

P. 112,1. 11. TO yap rpls K.T.X. Marquard reads iiexpi 7"P T0^'

I prefer to read TO yap with VbBRS, and am quite willing to con-strue it either as a direct accusative after hiaTdvopsv (just as wecan say 'to stretch an interval' as well as ' to stretch the voice')>or as an accusative of length with bi.aTeivop.sv used in a neutersense.

1. 13. al\£>v. For a full description of the ai\6s the readeris referred to the exhaustive article of Mr. A. A. Howard,in Vol. IV of the Harvard Studies in Classical Philology.A few general remarks will suffice here.

The term av\6s commonly denotes a reed instrument ofcylindrical bore; whether the reed was double-tongued as inthe oboe, or single as in the clarinet, or whether both theseforms of mouthpiece were employed, there is no conclusiveevidence to prove. The musician generally performed on a pairof these instruments simultaneously, playing the melody on one,and an accompaniment (which in Greek music was higherthan the melody), on the other. These double pipes weredivided according to their pitch into five classes, trapBivwi,jraifiiKoi, KiOaptcTTripioi, TeXetoi, and iirepTeXeioi, correspondingclosely to the soprano, alto, tenor, baritone, and bass rangesof the voice.

1. 15. KaTaoTrao-$fioT)f ye T!JS trvpiyyos. According to the in-genious theory of Mr. Howard (see last note), the term o-vpiy£,which commonly signifies a pan's-pipe, was used to denotea hole near the mouthpiece of the ai\6s, like the ' speaker' ofthe clarinet, the opening of which facilitated the production ofthe harmonies by the performer. The passages which he quoteson the matter are the following :—

(i) Aristotle (de andib. p. 804 a), $10 Kai TO>V avbpmv eio-i iraxi-Tepat. Kai T&V rcKeiav av\S>v, Kai ftSXXov OTOV irXrjpaxrr) rig alrovs rod

Trvevp-aros' (pavepov 8' e'oriV Kai yap av mecrr) TLS TO. £evyt] ( i . e . ' i f

one squeezes the reed between the lips or teeth') paKkov ogvrepar) (pavi) yiyverai Kai XenroTepa, Kav KaTaairaoi] TIS Tat avptyyas, Kav

be iirCKafirj, irap/irXeiav 6 oyKOs yiyverai Trjs (pavijs bia TO TT\TJ6OS TOU

os mxQcmep <a\ curb T&V va^yTepav ôpSSK.

R 2 243

ARISTOXENUS

From this passage, as from the passage of Aristoxenus beforeus, it is evident that the effect of the operation Karairirav ri]va-vptyya was to raise the pitch of the instrument.

(2) Plutarch inon posse suaviter, p. 1096 a),8ia ri T5>V "wav av\5>v6 (TT£ita>T€pos (6£vTfpov, 6 ff evpvreposy fiapvrepov <])$£yyeTaL' KOL dta

ri Trjs tripiyyos avaoirtoiiivr)s iraaiv oîverai roil 4>66yyois, KKivojxivqs

de iraktv ftapvveL (read fiapvveTaC) Kai avvaxdels irpbs rbv irepov

(ffapirepov), diaxdits 8e o&repov ^xe ' ! From this passage welearn that the effect of the operation avamrav rfjv a-vpiyya wasto raise all the tones of the instrument.

(3) Anecdota Graeca Oxoniensia, Vol. II, p. 409, (crOpiy£)trrjfialvfi rf/v OITTJV r£>v /J.OV(TIK£>V avKwv.

(4) Plutarch (de Musica, p. 1138 a), Klrka Tr]\e(pdvris 6McyapiKbs ovrcos eiro\e/jL!]<Tf rats <rvpiyt-iv, &are TOVS avXoTroioiis oi&'

£m8eivai jramore eiacrev e'irl Toiiy avKovs, aXXa (cal TOV HVBIKOV ayavos

fid\i(TTa 81a TOVT anetjTT).

[Mr. Howard gathers from this passage that Telephanes asa virtuoso objected to mechanical shifts such as the arvpiyt-which brought elaborate execution within the reach of poorperformers. I am rather disposed to think from the contextthat this musician was a lover of the simplicity and reserve ofancient art, and resisted innovations in the direction of com-plexity.]

The only difficulty offered by these passages is in the appar-ently indifferent use of avcunrav and Karacnrav to signify the sameoperation (or operations with the same effect). Mr. Howardthinks that the <rvpiy£ might have been covered when not in useby a sliding band, which in some instruments was pushed up toopen the hole, and in other cases pulled down for the samepurpose. I might suggest that possibly avcunrav and Karao-nuvin these passages are not direct opposites; that Karaa-nav maybe used in its primary sense of ' to draw down,' and dvaornav inits secondary sense of ' to open' (being answered inKkivew, ' to shut').

Von Jan supposes {Phil. XXXVIII, p. 382), that thewas a joint at the lower end of the ai\6s which could bedetached from it. But this view, as Mr. Howard points out,does violence to the passage of Aristoxenus before us, as may

=44

NOTES

be seen from his own explanation of it. ' Der Theil also, aufwelchem man nach Abnahme der Syrinx weiter blasen kann,heisst selbst Syrinx, und das Blasen darauf a-vphreiv.'

P. 113,1. 5. (kr&> is the excellent emendation of Westphalfor <FK TWV of the MSS. The eight concordant intervals are, TheFourth : The Fifth : The Octave : The Fourth and an Octave :The Fifth and an Octave : The interval of Two Octaves : TheFourth and Two Octaves : The Fifth and Two Octaves.

11. 7-12. For Aristoxenus the Concords are the elements ofintervals, and from them are derived directly or indirectly,by processes of addition and subtraction, all the discordantintervals. Even the quarter-tone must be thus ascertained:From a Fifth subtract a Fourth, and divide the result intofour equal parts. The latter part of this construction is un-satisfactory, for how is the ear to assure itself of the equalityof those parts? It could apparently do so only by such anijnmediate recognition of the interval in question as wouldrender any tnethod of ascertaining it nugatory.

1. 8. The contrast between the Pythagorean and Aristo-xenian views of musical science comes out strongly in thedefinitions of a tone. For the Pythagoreans a tone is thedifference between two sounds whose rates of vibration stand inthe relation 8:9; for the school of Aristoxenus, the differencebetween a Fourth and a Fifth. The latter explain the pheno-mena of music by reducing these to more immediately knownmusical phenomena, the former by reducing them to theirmathematical antecedents.

Ttov TTpwTcov av/j-rpavav. That is, the Fourth and Fifth.1. 18. For KaXovfuvov rd re irXfla-ra of the MSS I read Karcx°-

fifvov rd ye nXua-ra. If KaKoififvov be retained it necessitates theinsertion of the phrase 81a retrcrdpav, to give it a meaning;similarly, vno reacrdpav (pdoyyav, being left without any con-struction, calls for some such word as <nrexoiievov.

rd ye TrXeiora. Usually, not always; see note on p. 115,1. !•1. 20. Tira 8ij ra£iv . . . Kivovvrai. This is undoubtedly, as

Westphal has pointed out, a marginal scholium that has creptinto the text and displaced the conclusion of the precedingsentence. Observe the use of elvi instead of eon..

245

ARISTOXENUS

1. 21. For the meaning of the terms 'variable' and 'fixed'notes* see Introduction A, § 8.

P. 114, 1. 14. TovToav fie ro /ieu ZXarrov K.T.X. According toMarquard's explanation (accepted by Westphal) of this difficultsentence, TO eAarrov and ro êlfov are used by brachylogy for TO' OVK eXaTTov d<£i'orarai,' and TO ' OU iieiôv afpiorarat,' and thusrepeat the eXarrov and jueifov of the preceding sentence. Againstthis it may be urged that the brachylogy is a very violent one;and also that on this interpretation the latter clause of the sen-tence implies that the existence of a Lichanus further than twotones from the Mese was a matter of dispute. But of such aLichanus we have no evidence. Mr. Monro would avoid thelatter difficulty by supposing TO fieTfav to be used illogically inthe sense of' the question of the greater limit.'

I consider that the misinterpretation of this passage is due tothe natural but false assumption that TO eXarrov refers to the eXar-TOV of the preceding sentence. On my view TOVTWV — TOVTWV TS>VbuurrrjuaTav = row Toviaiov Siaorjj/xaroy Kal TOV SITOVOV : the geni-tive is a partitive o n e ; TO tXarrov TOVTOW (TO>V biaaTrjfiaTwv) andTO fi*i£ov TOVTCOV mean respectively the tone interval and theditone interval. The general object of the sentence beginningat TOVTWV is to justify not the smallness but the largeness of thelocalization of the Lichanus. In fact Aristoxenus would say,'The interval between the Lichanus and Mese cannot be lessthan one tone or greater than two tones. The lesser of thesedistances (which I have assigned as the minimum limit of thespace between the Lichanus and Mese), is found in the Diatonicgenus, and is consequently of unquestionable legitimacy; thegreater of these distances (which I have assigned as the maxi-mum limit of the space between the Lichanus and Mese) isadmissible, though often disputed in the present day, and wasthe distinguishing feature of the Ancient Enharmonic music'

1. 15. ovx is plainly wrong, as is seen from the followingovyxcopoiT* av.

1. 16. iiraxOevTtov. iirayeiv means to lead one on to therecognition of a general principle through the consideration ofparticular cases. Hence iirayayrj — induction.

P . 116,1 .1 . T&V apxaUav rponcov Tots Teirparois Kal rois devrepois.246

NOTES

Besides the enharmonic scale of the form

there was another enharmonic scale (commonly called after its

inventor Olympus), of the form Efe J j "-' '• which in-

troduced but one note of division into the tetrachord. It ispossible, as Marquard thinks, that these two scales are herereferred to as the earlier and later of the ancient modes ; butthe phrase is a strange one.

1. 3. ol jj£v yap K.T.\. Aristoxenus here records the fact,familiar to us from other sources, of the gradual extinction ofthe old enharmonic music. The intervals it employed were sofine and required such delicacy of ear and voice, that it cannever have been popular. But, as we saw in the Introduction A,§ 6, the cause which not only accounts for but justifies itsabandonment is the necessarily imperfect determination of itsintervals. Aristoxenus himself was quite aware of this deficiency,though not alive to the seriousness of it. In a passage quotedby Plutarch (de Musica, cap. 38, 1145 B), after assigning as onecause of the disuse of the enharmonic music the difficulty ofhearing such a small interval as a quarter-tone, he proceeds tosuggest another explanation, etra Ka\ TO /xq SivavBai Xij flijvai 8taavfi(j>a»iias TO jieyeOos naddwep TO re ljpiToviov KOI TOV TOVOV Kal Ta

'Konra Be rav Toioxnav Snumjuarav. ' Besides, there is the fact thatthe magnitude of this interval (i.e. the quarter-tone) cannot bedetermined by concord, as can the semitone, the tone, and thelike.' For this important principle of the determination ofdiscordant intervals by concord, see pp. 145, 146.

1. 6. ykvKatveiv. Anonymus (§ 26) contrasts the Diatonicgenus as ' avdpiKarepov . . . ml avo-rrjpoTepov' with the Chromatica s ' jj&iaTov re Kai yoepwraTOv.'

1. 20. The subdivisions of the genus are called xPâL o r

' shades.' See note on p. 116,1. 4.P. 118,1. 1. For convenience, the word Pycnum will be

retained in the translation to denote the sum of the two smallintervals of the tetrachord, when that sum is less than theremainder of the Fourth. For the meaning of the term seep. 139,11. 29-30.

247

ARISTOXENUS

Pycnnm

In the Enharmonic tetrachord lm J ,J J ~'3—= thesum of the intervals between e and xe, and between e and/ is a Pycnum, because it is less than the interval between /and a.

Plenum

For the same reasons in the Chromatic

tetrachord the sum of the intervals between e and / , and /and $ / is a Pycnum.

\ J( i i i

But in the Diatonic tetrachord \-m J ~~j J -- — thereis no Pycnum, for the sum of the intervals between e and/, and/and g is greater than that between g and a.

1. 4. Tourav 8' OVTCOS K.T.\. We have already seen that theGreeks recognize three genera, differentiated by the magnitudesof the intervals into which they divide the tetrachord; and wehave given as the plan of the Enharmonic, quarter-tone, quarter-tone, ditone; of the Chromatic, semitone, semitone, tone anda-half; of the Diatonic, semitone, tone, tone. But it willimmediately be asked, ' Are not other divisions intermediatebetween these equally permissible ? Why not for instancedivide your tetrachord into third of a tone, third of a tone,eleven-sixths of a tone ? Or into five-twelfths of a tone, semitone,nineteen-twelfths of a tone ?' Certainly, Aristoxenus replies,the possible divisions of the tetrachord, the possible locationsof the Parhypate and Lichanus, are as infinite as the points ofspace. But the ear ignoring the mathematical differencesattends to the common features in the impressions which thesedivisions make upon it, and constitutes accordingly three genera,the Enharmonic, Chromatic, and Diatonic, subdividing thelatter two again into xP°ah t n a t ' s colours or shades of distinc-tion ; the Chromatic into the Soft, the Hemiolic and the Tonic ;the Diatonic into the lower or Flat, and the Sharp or higher.It is evident then that each of these subclasses covers manydifferences of numerical division; but one division is taken byAristoxenus as typical of each.

The exact proportions of these typical divisions are exhibited248

NOTES

in the following table in which the tetrachord is in each caserepresented by a line divided into thirty equal parts, each partconsequently being the twelfth of a tone. The places of theParhypate are definitely marked as they are given in pp. 141,142;in this present passage their positions are less accurately stated.

TABLE OF THE GENERA AND SHADES.

_ 1 r = one-twelfth of a tone.

. * . 2 . 3 . = a quarter-tone, or the least Enharmonic diesis.1 2 3 4 = a third of a tone, or the least Chromatic diesis.

. i . 2 , 3 , 4 . 5 . 6 . = a semitone.1 2 3 4 5 6 7 8 9 10 11 12 _ a tone.

ENHARMONIC

Parhypate Lichanus

1 2 3 I4 ] il T g 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

CHROMATIC (SOFT)

Parhypate Lichanus

1 2 3 4 I 5 6 7 8 I 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

CHROMATIC (HEMIOUC)

Parhypate Lichanus

i a 3 4 J 5 6 7 8 9 110 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

CHROMATIC (TONIC)

Parhypate Lichanus

1 2 3 4 5 6 I 7 8 9 10 11 12113 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 80

DIATONIC (FLAT)

Parhypate Lichanus

1 2 3 4 5 6 J 7 8 9 10 11 12 13 14 15 I 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

DIATONIC (SHARP)

Parhypate Lichanus

1 2 3 4 5 el 7 8 9 10 11 12 13 14 15 16 17 18119 20 31 22 23 24 25 26 27 28 29 SO

1. 19. TO xPs>lxa> ' the particular species of chromatic' ij/u-6\wv, ' in the ratio of three to two ' ; because this was the

249

ARISTOXENUS

relation between the Pycnum of the Hemiolic Chromatic andthe Pycnum of the Enharmonic scale (9 and 6 respectively inthe above table).

3?. 117,1. 4. del yap K.T.X. These words are followed in someof the MSS by a detailed proof of the fact that the third ofany quantity exceeds the fourth of the same quantity by atwelfth. It runs as follows : inei8rj7rep 6 TOVOS iv p.ev XP<*>H-aTl e '?Tpia diaipeirai, TO be TpiTtjjj.6pi.ov KakeiTai xpafiaTiKr) Sieois' iv ap-fiovia 8e els 8 (TC(TCFapa M) Statpetrat, TO 8e TETapTijfiopiov (8 fiopLov M)KaKelrai dpfioviKr) 81'ecrij, TO OVV Tpirrjuopiov (y popiov M) TOU avTovKal evbs TOV TerapTrjuopiov (8 fiopiov ? M) TOV avTov 8a>8eKara> inrep-€Xe ')- °*ov ®s e7r' To" 'l^- *'" S ' ^ * 0 TOV «|8 els y . 8. Kai naKivTOV avTov ij3 els 8.8 (8. y . restituit Marquard), ev nev TJJ els y . 8.Siaipecrei yivovrai revcrapes rpiades, iv 8e TJj els 8. 8. (8. y . restituitMarquard) Tpels TerpdSes. lirepexet ovv rj 8 rrjs y . 8. (7 restituitMarquard) T6 rpirqixopiov TOV Teraprrifiopiov /iovd8i, oirep corl TOVSKov 8a>8iKaTov. Marquard very properly relegated this glossto the Critical Commentary.

P. 118, 1. 3. aweipovs TOV apidp.6v. Aristoxenus means ofcourse not that there can be more than one Lichanus in anyone scale, but that, given any note and its Fourth above asboundaries, one can constitute an infinite number of scalesdifferentiated by the positions of their variable notes, that isof their Lichani and Parhypatae.

1. 15. Marquard, followed by Westphal, changes the orderof the sentences here and reads xoivavel yap TO 8VO yevr\ TS>VnapvTtaTaiv—6 8' erepos 1810s TTJS ipuovlas, on the ground that theformer sentence gives the explanation of 6 pJv KOIVOS TOV TE Siarovov

t Kal TOV xpiip-aros and so must immediately follow it. But the MSS\ order is correct. Kowavel yap K.T.X. explains not the phrase 6 p.ev

KOIVOS K.T.X., but the principal sentence irapviraT-qs 8e 8vo ela\ TOTTOI,and 6 p.h KOIVOS . . . TTJS apfiovias is a parenthesis. The senseis, 'The loci of the Parhypate are not three, like those of theLichanus, but two (one common to two genera, and one par-ticular) ; for the Chromatic and Diatonic have their Parhypataein common.'

For TO 8io ykvr\ compare p . 126, 1. 8, ov yap inpayiuxTevovTo ittplTS>V dvo yevZv, dWa irep\ avrrjs rrjs apjxovlas.

250

NOTES

I. 17. xPa>lxaTLI<'l &* K.T.X. There are two loci of the Parhy-pate; the line 4 in the above table, which is peculiar to the

• Enharmonic genus, and the line consisting of 5 and 6 whichis common to the Chromatic and Diatonic. The meaning ofthis last assertion is that the Diatonic and Chromatic generaborrow one another's Parhypatae, so that you may melodiouslycombine in a tetrachord any Parhypate in 5 and 6 with anyLichanus in the lines from 8 to 18 inclusive with this importantexception however that the lowest interval of the tetrachord mustnever be greater than the one above it. See Introduction A, § 7.

II. 18-21. Of this most important law Aristoxenus offers noproof beyond an appeal to the ear—yiyvirai yap ififieXk rerpa-

I. 21. avtvov aficporepias, ' unequal in both ways' that is ' greaterand less.'

II. 23, 24. The substitution of irapvirdrr]! re xptoV-af-Kys T y /3apu-Tarijr for the jrapvirdrrfs re \pa>)xaTiKrjs irapvnaTqs of the MSScompletely restores the sense. Aristoxenus proves his state-ments that the Chromatic and Diatonic genera borrow eachother's Parhypatae by appealing to the extreme case. A melo-dious tetrachord is obtained from the combination of the lowestChromatic Parhypate, and the highest Diatonic Lichanus.

P . 119, 1. 2. I retain o-vvredds the reading of M V B R S .Aristoxenus means that he has exhibited the extent of the locusof the Parhypate, both as divided into the loci peculiar tocertain genera and colours, and as a whole embracing all thosedivisions. In p. 115,1.19, he says that having determined the locias wholes {rav S\av roirav) he must proceed to determine theirdivisions according to genus and colour. Here he sums up hisaccount of the locus of the Parhypate by stating that he hasdealt with it from both these points of view.

Marquard, followed by Westphal, reads evredeU, and trans-lates, 'The locus of the Parhypate is clear (from the aboveremarks) as to its division and its place of insertion.' But thistranslation conveniently ignores the words oa-os itrriv, whichshow that the size of the locus is what is here considered ; andthe space of a locus is not affected by its place.

1. 15. Aristoxenus here returns to his criticism of the method251

ARISTOXENUS

of KaTairvKvaa-is (cp. note on p. 101, 1. i), and shows that itsupplies a false conception of musical continuity or sequence;in other words, that it gives a false answer to the question,' Starting from a given note, how are we to determine whatis the next note to it above or below ? • For it ignores theSwam? of the given note, that is, its function in the systemof which it is a member; and regarding it merely as a pointof pitch, it declares that the next note to it is that point ofpitch which is separated from it by the smallest possible interval.But Aristoxenus sees that though there may be a certain truthin this answer from the point of view of Physics, it is musicallyabsurd. Let us take the notey, and ask what is the next noteabove it. But for the purposes of music / is nothing exceptas a member of a system or scale, and the question of the nextnote to it is meaningless until its function in a scale is deter-mined. Let us then restate our question thus : ' what is the nextnote above an f which is the second passing note in an enhar-monic scale ascending from e ?' Now the answer to this cannotbe xf, as the theory of KarawiKvacns would lead us to believe;for that would imply the possibility of singing three quarter-tones one after the other; whereas it is a law of the voice, andconsequently a law of music, that only two dieses can occur insuccession. In fact, the theory of naTawvKvaia-is in its completeapplication would imply the possibility of singing in successionas many quarter-tones as are contained in the whole compassof the scale.

I. 19. <n>x on like oix OTTOS is an elliptical phrase signifying'not to speak of,' and is used for oh y.6vov oh. Cp. p. 130,1. 7,ov yap on irepas -rijs &pp.oviKr]s. The corruption of the M S Sreading here might be traced through the following stages; theinsertion of ov after on by a scribe who, ignorant of the ellipse,felt the want of a negative; the misreading of on oh as rod;the consequent change of bwarov to bivaoBai to supply aninfinitive for the article, the addition of prj to supply the placeof the lost oh ; the change of luXaSrjo-ai to ne\adelo-6ai to explainTTJ <pa>vrj, the true construction of which had been hidden bythe corruption of dvvar6v.

SUa-eig OKTU) Kai CIKOO-IV. Why twenty-eight quarter-tones352

NOTES

and not rather twenty-four, seeing that there are six tonesin an octave ? Because some scales, such as the Dorian, con-sisted of seven tones. See Introduction A, § 20.

1. 24. fj fiiKpa K.r.X. This seems to be a somewhat contemptuousreference of Aristoxenus to the fact that in strict mathematicalaccuracy a Fourth is not quite two tones and a half. As wehave often seen already, Aristoxenus is concerned with musicalphenomena with a view to their artistic use, not their physicalinvestigation.

P. 120,1. 2. oi Si) 7rpoa-eKTeov «. Marquard retains the readingof the MSS and translates ' Nicht also ist fur die Aufeinanderfolgedarauf zu sehen, wann sie aus gleichen, wann aber aus unglei-chen entsteht.' But ore is relative usually, demonstrative some-times ; but never interrogative.

The general meaning of the passage is clear. The natureof melodic consecution, Aristoxenus would say, cannot be ex-pressed by any law enjoining a succession of so many equalor so many unequal intervals. Thus, we cannot say, 'Twoequal intervals must be followed by two unequal,' for while thisrule is fulfilled by the Enharmonic scale, it is violated by theDiatonic, which has three tones in succession. Nor can wesay 'three equal intervals may follow one another'; for whilethis is possible in the Diatonic genus, it is impossible in theEnharmonic. [Cp. p. 143,11. 21-23.] Translate,' We must notfix our attention on the fact that in certain cases,' &c.

1. 13. I read perd for fiiv of the MSS. /iev is out of place,as there is no antithesis between this assumption and thefollowing ; and some preposition is required to give a con-struction to T6 TTVKVOV . . . (rvcriTjfui.

1. 16. imoKeio-da 8<= tai T5>V i£fjs K.T.\. Here Aristoxenus statesfor the first time his fundamental law of continuity; that if aseries of notes be continuous, any note in that series will formeither a Fourth with the fourth note in order from it above orbelow, or a Fifth with the fifth note in order from it aboveor below, or will fulfil both these conditions.

Thus I

253

ARISTOXENUS

is a legitimately continuous scale. A, though it does not forma Fourth with c, forms a Fifth with e; B, though it does notform a Fifth with xe, forms a Fourth with e; xB does not forma Fifth with /, but forms a Fourth with xe; c does not form aFifth with a, but forms a Fourth with/"; e forms a Fourth witha and a Fifth with b; and so on.

On the other hand,

is not a legitimate scale; for b forms neither a Fourth with ttenor a Fifth withy1!

1. 22. i>s enl TO no\v i. e. in the Enharmonic and Chromaticscales, but not in the Diatonic.

1. 25. ivavr'ws rl8f<r8ai K.T.X., ra 8io lira are the two equalintervals of the Pycnum : TO dvo av«ra are (1) the complementof the Fourth and (2) the disjunctive tone. Now in the scaledescending from the Pycnum

the disjunctive tone lies next the Pycnum, and the complementof the Fourth second from it; while in the scale ascending fromthe Pycnum

1 ss Ii §1 H£ i£ -In

d° 3

we find the complement of the Fourth next the Pycnum, and, the disjunctive tone second from it.

P . 121,1. 5. Every compound interval can be analysed intosimple intervals but not into simple magnitudes. Thus, a Fourth

254

NOTES

in the Enharmonic scale is analysed into quarter-tone, quarter-tone, ditone. Now quarter-tones are simple intervals and simplemagnitudes at the same time; for quite apart from any con-sideration of systems or scales, no smaller musical magnitudethan a quarter-tone exists for ear or voice. But the ditonethough a simple interval in this scale, since the voice in thisscale cannot divide.it, is not by any means a simple magnitude.For if we abstract rom consideration of systems and scales, aditone as a space is obviously reducible to two tones, andeven farther.

1. 7. This passage is quite corrupt in the MSS. I readaxpav for apxiov, ev for iv, and e<ra6ev for 'i£a>dev ; insert &i> after(pdoyyav, and omit it after aitpav, and insert eraoTou before

It must be remembered that ol egrjs (fidoyyoi are not neces-sarily consecutive or immediately successive notes; the phraseapplies equally to notes that are in the same line of successioneven if at a distance from one another. Thus, in our majorscale of C, the notes D, A, B, are e£>jr, because members ofthe same legitimate scale. Now an ayayt) is a sequence ofconsecutive or immediately successive notes, and this couldnot be expressed by saying merely that it proceeds 8m TS>Ve£rjs (pduyyav. The further necessary qualification is given bythe following words: the successive notes must be separatedfrom one another by simple intervals; must, in other words,be the nearest possible notes to one another in their scale.

Direct sequence is a species of sequence in general. Thus

is a sequence, but not direct;

is a direct sequence.tcnaBev T&V axpatv means 'within the extremes,' that is 'between

the first and last notes.' The first note of a sequence is notpreceded, the last note not succeeded, by a simple interval.[Mr. Monro would retain egaBev in the sense of' except.']

255

. . ARISTOXENUS

P. 122, 1. 10. TouTtav. For OVTOS in the sense of iste, cp.p. 132,1. 24.

1. 13. There may be an allusion here to such a doctrineas we find in the Philebus, or possibly ro mpas may be an ac-cusative in apposition to the following sentence, and mean ' asthe sum or final conclusion of the matter.' In the latter caseI should prefer to read rayadov.

1. 20. Marquard quite unnecessarily reads €tX?j/i/ieV>/ forflprinevrj, and gives the following reason for the change ; ' Kannman denn eine prior opinio griechisch einfach eine elpmitv-qvrrok-q-^ns nennen, wenn vorher von einem Aussprechen garkeine Rede gewesen ist ?' f) tlptifievti mroAi?\ i? refers back to

' vTroKaitfidvovTa of 1. 9.P . 123,1. 1. i>s £$77. The MSS read i>s etpnv which Marquard

retains, translating ' aus den genannten Grunden.' But a>s ecprjvis not the same as as clnov, and must refer, not to airas ravrasTas alrias, but to 81* aiiras ravras ras atria? nyjoeXfye 'Apiororekrjs,and Aristoxenus has not said that.

1.11. Marquard ruins the sense of this passage by his insertionof Kal between 5™ and naff 00-ov, and his mistranslation of oltfaKovcravres oXcor—' das aber, dass die Musik und in wie weit sieniitzen kann, verstehn sie gar nicht.' The sentence TO 8' STL . . .<2><£eXf iv is elliptical. The complete statement which Aristoxenush a d m a d e w a s Sri fj fiiv TOKUJTJJ JXOV(TIKTJ jSXairrei. fj Se ToiavTr/

<J</>cX«, <ad' otrov fiovaiKr) SWOTOI dxfreXelv. The careless listenersjust caught the first part of the statement on 17 fitv . . . roi-avrr) &>0eXeI : the concluding qualification on [17 fiev . . . roiavrr)aXpeXet] Ka8' oaov novaiKr] dvvarai w(j)e\etv escaped their earsaltogether. In such a sentence as this on serves the samepurpose as inverted commas in English.

Westphal rewrites the whole sentence and destroys itsmeaning.

1. 13. I read t^meipoi for aireipoi. If liireipoi be retained wemust suppose a deficiency in the MSS. Marquard supplies itby inserting ayvoelv irpoo-eunv after iurlv. As he translates' kommen aber herzu,' it would seem that he has confused theforms of dfii and el/u.

1. 15. i>: vvv ex€l °f t n e MSS is meaningless. The present256

NOTES

condition of the science has nothing to do with the argu-ment.

1. 18. fmapxei KaQcmtp aii \iytrai. Marquard retaining the17 of the MSS translates 'many other things are indispensableto the musician than those that are constantly said to be s o ' ;but both the grammar and sense of this sentence are doubtful.Is there any evidence or any likelihood that there was ^perpetualmisunderstanding of the qualification of a musician ? Wouldnot iraWa erepa rj mean ' many things different from' rather than' many things in addition to ' ? And why not erepa j) a ratherthan crepa tj Kadairep. Kadanep del \fyerai, if we omit the rj, means'as we consistently assert* [see, for example, p. 95,11. 13-15].For a similar use of the present passive of Xeyco, cp.-p. 130, 1.16,oTi 8'dXrjdrj Ta Xeyofxva, 'that our assertion is t rue ' ; also p. 153,1. 6. Westphal secures the right sense by the clumsy insertion_of TOVTO after 17.

P . 124, 1. 2. In this paragraph Aristoxenus defines hisposition in relation to the question What is the foundation ofmusical science ? On the one hand, he rejects the intellectualor mathematical theory of the Pythagoreans on the ground thatthe principles, from which they seek to deduce the facts ofmusic, lie outside the sphere of music altogether, and fail toaccount for those facts. On the other hand, he rejects equallythe blind empiricism which takes the single facts and registersthem without any attempt to ensure completeness, or ascertainthe general law. See Introduction B, § 2.

1. 17. Let us suppose that as we are listening to a passageof music in the diatonic scale

1

the voice passes from FTO—^—— to E£BEEifc=; to apprehend

this musical phenomenon, what faculties must we employ?In the first place we obviously require our sense of hearingto tell us that a semitone has been sung; but that is notenough. We require our intellect also to form a conceptionof the system in which the e and / occur, and to identify their

MACRAN S 257

ARISTOXENUS

functions in it; so that the phenomenon before us may be for usQ

something quite distinct from the passage from \-C\) ~^j^^

in the enharmonic scale

1. 18. T&V <j)86yya>v. Tovrav of the MSS is wrong. Thehiaorrj/juiTa Aristoxenus always regards as mere distances ;functions he attributes only to the notes. Cp. p. 127,1. 3, OUKavrapKTj ra fiiaoTJj/iaTfi K.T.X.

Swdfiets. Bivaixts signifies the function which a note dis-charges in relation to the other notes of a scale. Thus inmodern music the Siva/iis of b is that of a leading note inthe key of c, that of a dominant in the key of e, that of a tonicin the key of b.

P. 124,1. 22-P. 125,1. 2. Marquard and Westphal have com-pletely missed the meaning of this passage. r& povaiKa is notthe musician in the sense of the musical artist; nor is Aristo-xenus labouring at the obvious fact that keenness of senseis a sine qua non of artists in general as distinguished fromstudents of science. T<» /ioueruoa is the student of musicalscience; and the point to which Aristoxenus would draw ourattention is that Music presents us with a science for whichaccuracy of sense is indispensable. In this respect musical andgeometrical science differ from one another. The propositionsof Geometry are deduced from principles which, though possiblyin the last resort principles of sight in the sense that withoutsight we never could have conceived them, are yet so abstractand fundamental that their acceptance accompanies the lowestuse of that faculty. But the principles of musical science rest,not on the presuppositions of hearing in general, but on theevidence of the developed and cultivated ear. That a straightline is the shortest distance between two points may be a prin-ciple of sight in the sense that ' straight,' ' distance,' ' two,' &c.are phenomena of sight; but it does not require sharp eyes toapprehend it. On the other hand Aristoxenus' proof of themagnitude of the Fourth [pp. 146-147] depends on an appeal

258

NOTES

to the ear, by no means universal, that can distinguish a concordfrom a discord.

P. 125,1. 6. From consideration of the faculties Aristoxenusturns to the object matter which those faculties are to appre-hend. Of this object matter he finds the all-pervadingcharacteristic to be identity under difference, the co-existenceof a permanent and a changeable element; and cites in supportof his statement several cases which may be made clearer bythe following illustrations:

(i) 1. 7. [eldea? yap K.r.X.]. •DIATONIC r Q . I I ENHARMONIC ft

JHere we have as permanent element the relation between thefixed notes ; as changeable the position of the intermediate notes.

(2) 1. 8. [waAiy orav fiivovros K.T.X.].

Compare the interval between E and A, and the intervalbetween b and e. Here we have as permanent the magnitudeof the intervals (a Fourth); as variable the tivvaius of the notescontaining the interval.

(3) 1. II . [KCH irakiv orav TOV mrrov fieyidovc K.T.X.].

Here we have the same magnitude, a Fifth, appearing in twodifferent figures, that is with its intervals arranged in differentorders.

(4) 1. 13. [acravTas 8e KOI orav K.T.X.].

In the two scales

and

s 2 259

ARISTOXENUS

compare the tetrachord between b and e in the former, withthat between a and d in the latter. Here we have aspermanent the size and figure of the interval; as variable thefunction of the tetrachord which in one case is modulating, inthe other, not modulating.

(5) 1. 16. [fern yap fifvovros TOV \6yov K.T.X.].Compare the three following feet or bars :

iv oKTaarjfua fieycdet

(taking the crotchet as the unit).

iv «£a(7ij/i<j> ptytQti.

A JJ

yivos TO

In these three we have as permanent the Dactylic characterwith its ratio of equality between the arsis and thesis ; while thelengths of the feet differ, their difference being due to the differentrate of movement.

(6) 1. 18. [KV /leyedmv fievovrav K.T.X.].

Compare the two following bars or feet:

T 6 BOKTVKIKOV yivos TO iv roi

lira

TO lafifltKov yivos TO iv TU

A- J J .iv f£a<Tt]fi&i fuyidti.

Here we have the iieyedos permanent, six crotchets; but thegenus varies, the first being ' dactylic' with the arsis equalto the thesis, the second being ' iambic' with the arsis doublethe thesis.

(7) 1. 19- [tol TO avro peyeBos noba K.T.\,].Compare (a) and (&).

(*)A A A J J J

Here the same-quantity, eight crotchets, appears in (a) as a singlefoot, in (6) as a pair of feet.

(8) 1. 2O. [ai 8ia<f>opai . . . SiaipfVetoi'].The same magnitude, say | IWI | may be divided into two

260

NOTES

semibreves, or four minims, or one semibreve and two minims,or eight crotchets, or one semibreve, one minim, and twocrotchets, &c.

(9) 1- 20. [al 8ia(popa\ . . 1 (Txrip.cnav\.Let us suppose a certain magnitude, say of three crotchets

divided into a minim and a crotchet, these parts may be arranged* • ' ' * ,

in the order J A or in the order d J .(10) 1. 21. [KatfoXou 8' tmtiv K.T.\.].

In general, rhythmical science reduces the infinite variety andmultiplicity of verse to combinations of a few primary elements,namely feet.

1. 10. The omission of ydp, suggested to me by Mr. Bury,restores the construction of this sentence.

P. 126, 1. 20. I have changed the MSS yevea-i to /ne'Xeo-i.The corruption might easily be explained both e rei matetiaand also through the proximity of yiyvopivais. For the plural of/x=Xos used of the concrete, cp. p. 130,1. 2.

yevea-i is plainly wrong. ' That we must distinguish thegenera if we are to follow the distinctions that occur in the genera'is an absurd tautology. A comparison with p. 126, 1. 25, ou8el 8' ayvoelv K.T.X. makes clear the meaning of Aristoxenus' warn-ing :—' if we neglect the scientific determination of any differ-ence, we shall fail to detect the concrete cases of that differencewhich meet us in any musical composition.'

[Since writing this note I have discovered, in collating theSelden MS, the letters pik crossed out before yheo-i.]

P. 127, 1. 3. eWi 8' iariv OUK K.T.X. For example, part of theconnotation of the terms Mese and Hypate is that they are theupper and lower boundaries of a Fourth ; but more is requiredto determine the conception of these notes ; for the same mightbe predicated of the Nete and Paramese.

1. 8. See Introduction B, § 2.1. 14. ovtSerepov . . . ra>v rpovav. One method is to exhaust the

acts by a faithful enumeration ; the other is to deduce the factsfrom the principle on which they depend.

1. 24. Pythagoras of Zacynthus was the inventor of a stringedinstrument called the rp'movs. See Athenaeus, xiv, 637.

261

ARISTOXENUS

1. 25. Agenor of Mitylene is quite unknown. See Porphyry,p. 189.

P . 128,1. 6. tsifmrov &' eWl K.T.X. On the whole paragraphcp. Introduction A, §§22-26, where I have explained also theuncertainty as to the key of the Mixolydian mode.

1. 19. rpiai hiia-ecnv. The separation of keys by intervals ofthree quarter-tones would be an application of the principleof KaxairvKvuMTis. Cp. note on p. 101,1. 1.

P. 129,1. 4. fieTajiokijs. The modulation with which Aristo-xenus is here primarily concerned is the fieTaj3o\ri a-uorqfiaTiKijwhich is thus defined by Bacchius [ed. Meibom, p. 14,1.1], orav «TOC ifiroKtip&vov av&TTifuiTOS els (Tepov aiarr]fia avaxtoprjO'r] f/ /icXo>8ta

hipav fie<rt]v KaTcurKevdfavo-a, ' t h e transition which a melodymakes from one scale into another by providing for itself a dif-ferent Mese.' But a different Mese can mean nothing else thana tonic of different pitch, so this transition means simply modu-lation into a different key. The conditions of its possibilityare given in the following passage of the Isagoge [ed. Meibom,p. 20,1. 33] : -

Tlvovrai Se at pcTafioXal cmb TTJS q/uroi/uuar ap£afievai pcXP1 T°vfita TTauav, &v ai fxtv Kara trvjKpwva yivovrai SiacrTy'jfxaTa, ai 8e Kara

&ia<j>a>va. TovTtov &' ai fiev (HfiiKels TJTTOV rj eV^fXfif, ai Se fiaXkov.

iv oa-ais fiev oSv airav jrXeiav ff Koivavia, eixfiekeo-Tcpai' iv ocais 8«

iXarrcov, fKjLieXeorcpai* fireiStj dvayKcuov Tracrp fieTajSoX^ KOIVOV TI

\mapX*w, T (pdoyyov, fj bia<TTi)fia, i] (ruoT^/ia. Xa^(3d«Tai 8e i) KOI-

vavla Kaff 6fxoiOTT)Ta <j>66yya>v. orav yap in' aXXjjXous iv rais jitra-

/3oXa(? TTfcraxriv ofioioi <\>66yyoi Kara TTJV TOU TTVKVOV iieroxqv, ip.fu\r)s

yiverai q fieTafiokr), OTQV 8« dvofiowi, «KfifX y. ' Modulations begin

with modulation by the semitone, and proceed to the octave.Some of these are by concords and others by discords. Someof them are more melodious than otherwise; others less so.The greater or less the community of elements, the more or lessmelodious the modulation. For every modulation demandssome common element, whether note, interval, or scale. Butthis community is ascertained by the similarity of notes; fora modulation is melodious or unmelodious, according as thenotes that coincide in pitch are similar or dissimilar as regardstheir participation in the Pycnum.'

262 1

NOTES

The last phrase of this passage requires some explanation.The Greeks considered that every note of every scale wasactually or potentially the lowest, the middle, or the highestnote of a Pycnum. Thus in the Enharmonic scale

D . i i I - I1 / , . i—H ;gF=~J J- -rJ —

E is actually the lowest, xE actually the middle and F actuallythe highest note of the Pycnum E-xE-F. Similarly b, xb and care respectively the lowest, middle, and highest notes of thePycnum b-xb-c. Similarly e is the lowest note of the Pycnumof the conjunct tetrachord by which we might extend the scaleupwards. Finally A, though not actually participating in anyPycnum in the above scale, does so potentially as the lowestnote of the Pycnum A-xa-Vb, in the possible conjunct tetra-chord

Representing the lowest, middle, and highest notes of aPycnum by the signs LP, MP, and HP, we find these notesthus distributed in the Enharmonic scale :

LP MP HP LP LP MP HP LP

The same terms naturally apply to the Chromatic Genus;and may be applied analogically to the notes of the DiatonicScale: thus—

LP MP HP LP LP- MP HP LPi f , 1-1 J J J -

This distinction in notes is a deep and essential one, in whichthe Siva/us of the note is conceived in relation to the tetrachordin general, abstraction being made of the difference betweenthe individual tetrachords.

If then it be asked whether two scales admit of melodiousintermodulation, the answer is 'Yes, if they have a commonelement; and the more common elements they possess, themore melodious will be the modulation.' But when we speak

263

ARISTOXENUS

of a common element, we mean not only certain points of pilchcommon to both scales, but certain coincident points of pitchoccupied in both scales alike by lowest, by middle, or by highestnotes of a Pycnum. In other words there must be a coincidencein pitch of notes of the same Siva/us in relation to the tetra-chord.

Let us consider then in particular the possibilities of inter-modulation between- the keys of the seven modes.

MeseLP MP HP LP MP HP LP LP

MIXOLYDIAN r j ? " . ' "i . J J — ~J ••

MeseMP HP LP MP HP LP" LP MP

LYDIAN

PHRYGIAN

DORIAN

HYPOLYDIAN

MeseHP LP MP HP LP LP MP HP

i I 1 «> 1 J * •

P=3=Mese

LP MP HP LP LP MP HP LP

m =g= =Mese

MP HP LP LP MP HP LP MP

MeseHP LP LP MP HP LP MP HP

HYPOPHRYGIAN I P ~ — i -i - ! - j J

J **—"L

MeseLP LP MP HP LP MP HP LP

HYPODORIAN i I J J J

A semitone separates the tonics of the Mixolydian and Lydian264

NOTES

keys. Similarly related are the Dorian and Hypolydian. Takingthe first pair as typical we find that although there are severalcoincident points of pitch in the two scales such as E and A,there is no common element, because these points are occupiedin the two scales by notes of different Swajur in relation to thePycnum, A for instance being LP in the Mixolydian key, butMP in the Lydian. Hence between scales separated by a semi-tone there is no direct modulation.

A tone separates the Lydian and Phrygian; the Phrygianand Dorian; the Hypolydian and Hypophrygian, the Hypo-phrygian and Hypodorian. Taking the first pair as typical wefind that of the coincident points of pitch E, $F, A, b, %c, e,one alone, Jc, is occupied in the two scales by notes of thesame Sivapis, namely the lowest notes of a Pycnum. Hence amelodious modulation is possible between scales separated bya tone, though the common element is the smallest possible.

A tone and a half separates the Mixolydian and Phrygian;the Phrygian and Hypolydian ; the Dorian and Hypophrygian.In such pairs we find no common element; and hence they donot admit of direct intermodulation. Two tones separate theLydian and Dorian; and the Hypolydian and Hypodorian.Here again we find no common element, and no direct modu-lation.

Two tones and a half, or the Concord of the Fourth, separatethe Mixolydian and Dorian; the Lydian and Hypolydian; thePhrygian and Hypophrygian; the Dorian and Hypodorian.In the first pair we find several common elements E, F, G, A, e.In general, any two scales separated by a Fourth have manycommon elements, and modulation between them is highlymelodious.

Three tones separate the Mixolydian and Hypophrygian keys.Here we find no common elements.

Three tones and a half, or the Concord of the Fifth, separatethe Lydian and Hypophrygian; and the Phrygian and Hypo-dorian. In the first pair we find as common elements #G, A,b, c. Hence in general one may modulate most melodiouslyietween scales separated by a Fifth.Four tones separate the Mixolydian and Hypophrygian. Here

265

ARISTOXENUS

there are no common elements. Four tones and a half separatethe Lydian and Hypodorian. Here again there are no commonelements.

Five tones separate the Mixolydian and Hypodorian. Herewe have E and e as common elements, and direct modulationis possible.

The general result we arrive at is that when two scales areseparated by a Fourth or Fifth, modulation between them ismelodious in the highest degree; when they are separated bya tone or five tones, modulation between them is again melo-dious though in an inferior degree; but when they are separatedby other intervals then these, melodious modulation cannot beeffected between them directly, but only by the intervention ofother keys. It follows that the limits of indirect modulation arestrictly defined. Since direct modulation exists only betweenkeys whose tonics are spaced by a tone, by a Fourth, by a Fifth,or by five tones, indirect modulation can only connect keys thespace between whose tonics can be arrived at by addition andsubtraction of these four intervals. But the only intervals thatcan result from the addition and subtraction of a tone, two tonesand a half, three tones and a half, and five tones are the semi-tone and its multiples. Hence, if two keys have their tonicsseparated by any other intervals than these, modulation betweenthem, direct or indirect, is impossible. See note on p. 101,1. I.

Beside the ptrafioKri orvorij/jaTiK^ Bacchius (ed. Meibom, p. 13,1. 26) mentions three other /ueraj3o\ai affecting melody: yiviKfj, ' ofgenus' ; KOTO rpteov, ' of mode' ; Kara TJ8OS, ' of emotional char-acter.1

1. 6. I read TI'VOJ for MSS TWOS. ~kiya> de introduces an alter-native statement, and the alternative statement of a question isa question.

1. 7. Kara 7rd<ra SiaaTq/xaTa. The answer to this question asappears from the last note is 'four,' Kara ra crvn<j>cDva fitaorij/naTa,KO.\ Kara rbv TSVOV Kal Kara TOVS irivrt rovovs.

1. 10. fifXimouas. The other parts of Harmonic science havesupplied the material of melody, notes, intervals, and scales;it remains for the composer to make a judicious use of it. Thescience of the use of musical material is the science of eXo-

266

NOTES

7roun. One of the functions of this science will be to determinewhich class of melody is adapted to any particular subject;whether the energetic style suits the chorus of a drama, or theHypodorian tragedy, or the Enharmonic lamentation. Butthis function manifestly lies beyond the limits of appjoviKt\. Tothis latter science, however, belongs the classification of theseveral melodic figures by which a composition takes its shape.

In the Isagoge (ed. Meibom, p. 22,1. 3), we find the followingaccount of this subject: Me\o7roua eWi xpij(ris T3>V irpoeipr]-fiivav fiep&v rrjs apfioviKtjs Kal \moKeip.evu>v bivap.iv ixivrav' &*$>v be p.e\oiroita orweXeiTai recraapd ianv' <rya>yij 7rXo/cq ireTTeiarovtj. ayatyfj [cp. above, p . 121, 1. 7] p-ev ovv eariv TJ bta ravfffis (pdoyyav 686c TOV p.e\ovs, rrkoKT] be r) «Va\Xa£ TS>V re 8tatrr>)-fidTuiv deals TrapdXXijXor, nerrfla Sc r] e(j> evus rovou 7roXXa/«y yiyvo-\ievx] n\rj£is, rovi] be t) en\ ir\eiova xpo"ov fiovr] Kara jiiav yivop-ivrj7rpo(popav T?is<p<ovrjs.

' Melopoeia is the employment of the above mentioned partsof Harmonic science which serve as a material to it. Thefigures through which Melopoeia takes final shape are four;the sequence, the zigzag, the repetition, and the prolongation.

The Sequence is the progression of the melody throughconsecutive notes; the Zigzag, the irregular progression withalternate location of the intervals [i.e. every second intervalis ascending, every second descending]; the Repetition, theconstant iteration of one note; the Prolongation, the dwellingfor a length of time on one utterance of the voice.'

'Ayayi) again is divided into three species (see Aristides Quint-ilianus, ed. Meibom, p. 29,1. 11), tbdeta, or r\ 81a rav e'£ijr <f>86yycovTT)V iiriTaoriv nowvfievi) (ascending by consecutive no tes ) ; di/axa/i-TTToutra or fj biii TO>V eirofuvaiv cmorekovcra rr\v j3apurijra (descending*by consecutive notes) ; irtpifpfpfis or r/ <ara (rwijp.fievcw pev eV:-Teivovara, Kara Siefevyp-evai' be avieltra' fj evavriws (ascending byconjunction and descending by disjunction, or vice versa). Amore general definition of n-XoKij is supplied by Aristides Quint-ilianus (ed. Meibom, p. 19,1. 20), n-Xoxij be, ore 81a T&V <ad' vnep-f3a<riv \afif3avofihav (iroti>p.e6a rrjv p.e\<pbiav), ' t h e zigzag occurswhen our melody proceeds by notes that have been taken with

' a skip between them.'267

ARISTOXENUS

If we accept this more general definition of v\oKrj, and regardthe more particular definition given in the Isagoge as descriptiveof one special case of the class, it is easy to see that everymelody is capable of being analysed into these four figures asfinal elements. I subjoin a few examples of such analysis.

070171)

(4)

=££ J£L

(s) iyaiyij thOua iyayi) dSeia

268

NOTES

yyi ayayfjei0tta h&

1.17. In this sentence I insert e'ori after 8c, read 7rapaKo\ov8eivfor irapaKokovBei and insert SijXov.

Either this paragraph is defective in the MSS, or its brevityamounts to obscurity. Yet it is not wholly unintelligible asit stands. In the first sentence Aristoxenus asserts that tounderstand a musical composition means to follow the processof its melody with ear and intellect. We have already learnedfrom Aristoxenus what parts these two faculties play. The eardetects the magnitudes of the intervals as they follow oneanother, and the intellect contemplates the functions of thenotes in the system to which they belong. But the phrase'process of the melody' turns the speculation of Aristoxenusinto another channel. It reminds him of the difference thatexists between music and such an art as architecture, the pro-ducts of which present themselves to our senses complete atone moment. Melody, on the contrary, like everything inmusic, is a process of becoming, in which one passes, andanother comes to be ; and we require here memory as well assense, to retain the past as well as to apprehend the present.

But although this is undoubtedly the general sense of thepassage, the logical connexion of the sentences is by no meansobvious. 'Ec yevecrei yap K.T.X. justifies the previous use of TOKyiyvojxcvois, but how is the sentence e< Svo yap TOVTIOV K.T.\.

related to what goes before ? The fact that the understandingof music requires memory as well as perception is a consequencerather than an explanation of the fact that melody is a process ;and roirav implies that a'lcrOqaris and fivrjiirj, if not alreadymentioned, have at least been indicated.

Of course the contrast between aKorj and Sidvoia [cp. p. 124,1. 17] must not be confused with the contrast between ato-drio-ttand iivrjiii).

P. 130,1. I. & Si rivet irotovvrai re'Xij K.T.X. This paragraphcontains a polemic against (a) the absurd theory that one who

269

ARISTOXENUS

/ can notate a melody has reached the pinnacle of musical know-i! , ledge ; and (b) the equally absurd theory, which, basing the law;•; of harmony on the construction of clarinets, reduces musicalV,! science to the knowledge of instruments and their construc-

ii I tion"I. 6. oXou TWOS is governed by dirjimprTjKOTos, ' of one who hasI: missed some whole ' = ' missed something completely.' But

perhaps we should read S\ov, the accusative neuter used as(> : an adverb in the same sense as the cognate accusative SAoe

;

j ; &/idprriij.a, and construe nvos in agreement with SujfiapTqicoTos.

! ! 1. 7. Marquard, followed by Westphal, inserts an ov between,j j on and irfpas, being ignorant apparently of the use of ohx, on =I' / OV flOVOV OV.

I! V 1. 13. Marquard is wrong in bracketing oh yap avaynaiov eWi/( j . . . io-Ti TO (ppiyiov iieXos as a gloss. He does so on the sup-jj j position that its presence in the text involves apetitioprincipii;\ i because, he would say, Aristoxenus proves his statement' thatji ; the capacity to notate a melody does not necessarily imply the\\ ; : understanding of it' by an appeal to a parallel case in metricalj| ' science; and then proceeds to justify his analogy by assumingji ! the truth of the statement.5! j But Marquard has missed the course of the reasoning, which

1 is as follows: You admit that to mark a metre is not thej . end-all of metrical science. On what grounds then ? Because! it is a fact that a man may mark a metre, and yet not under-' stand its nature. Very well then. The same fact holds good

{• with regard to melodic science (as I shall prove hereafter); it isj namely (yap) a fact that a man may notate a melody without1 understanding its nature. Therefore you are logically bound; to admit that to notate a melody is not the end-all of melodic! science.I 1.17. This argument is based on two premises ; (1) Notation

ij takes account of nothing beyond the bare magnitudes of intervals.(2) Perception of the bare magnitude of intervals is no part of

i musical knowledge.In support of the first premiss he appeals to the following

facts :(a) The notation makes no distinction of genus. Thus [see

NOTES

table 22 in Introduction A] the notes _ ^ stand for the

progression Igj; i whether in the diatonic scale

J * J j - J

or in the chromatic scale

though the interval in the first case is compound and diatonic,in the second case simple and chromatic.

(b) The notation makes no distinction of Figure. Thus the-

notes jt. p mark the interval of the sixth

both in the diatonic scale

where its schema is tone, semitone, tone, tone, tone ; and in thediatonic scale

where its schema is tone, tone, tone, semitone, tone.(c) The notation makes no distinction of the higher and lower

R C

tetrachords of the scale. Thus the notes . _ apply to the

interval 1 ^ T - ' whether in the scaleHypat6n Mes6n

or in the scaleMesfln

ARISTOXENUS

j ; yet in the first case the interval belongs to the tetrachor^Mes6n, in the second to the tetrachord Hypat6n.

The second premiss is evident from the undeniable fact thatthe perception of the distance between two sounds leaves all the

' vital distinctions of music untouched.1. 25. To the reading adopted in the text Marquard would

object (1) that Aristoxenus never refers to the tetrachords Hyper-;; bolae6n and Hypaton ; (2) that we know of no signs that were1/; employed to denote tetrachords. But (1) in p. 99,1. 12 we have

a reference to the Complete System of which the said tetrachords,'•/ ;. were p a r t s ; (2) when Aristoxenus speaks of the notation of a

tetrachord, he means of course the notation of the notes of thetetrachord. The singular T& aira o-Jineia is used because thesense is ' the same sign is used to represent a note of the tetra-chord Hypaton and a note of the tetrachord Meson,' &c.

Marquard's reading (given in the corrections at the beginningof his volume) TO yap vfjr^s Kal iiitrrjs KOI TO irapafiio-r^s Kai iman)shas the fatal defect that these intervals are Fifths, not Fourths.Sense might be obtained by reading with Westphal TO yap vijri/yKai irapaiiitTTjs KO.1 TO fieo-tjs Kal {mart)!, but this is rather far fromthe MSS.

P . 131, 1. 6. oSre yap . . . yvapipov. An anacolouthon.1. 10. TOIIJ T<OV p.eXoiroiiav rpoirovs. See Aristides Quintilianus

(ed. Meibom p. 29, 1. 34), Tponoi fie fieXovroitas •yeVft p.cv rpus' 8i0v-pafij3iic6s} vofUKos, rpayiKos. 6 fttv ovv VO/IIKOS Tpoiros eori vr)Toeibf)s(i. e. its prevailing character is that of the tetrachord Neton),6 fie ftiBvpapfiiK.os, iMe<Toei8!)s (i. e. its prevailing character is thatof the tetrachord Meson), 6 Se TpayiKos vnaToeidtjs (with thecharacter of the tetrachord Hypaton). «8ci 8« evpio-Kovrcu TrXeiovs,ovs SVVOTOV 81 6p,oioTTjTa rois yeviKols iiro(3d\\eiv. cpanicai T« yapKa\ovvrai TIVCS, &>V 18101 ciri8a\afuoi, Kai Kai/iiKoi, Kal eyKaftiacmKol.Tponot 8e \eyovTai 81a TO avvcfKpuipuv was TO r/Bos Kara ra pcXi] Trjsdiavoias.

1. 21. Marquard, followed by Westphal, has made sad havocof the following passage by changing the order of the sentences.In fact, the reading of the MSS calls for very little emend-ation, nepas must be inserted in 1. 22 ; and I have omittedi) before rds in P . 132,1. 3, and inserted fie after it; and omitted

272

NOTES

fj in 1. 4, after Trvev/ia. No other changes are necessary, exceptin punctuation. The course of the argument is sufficiently clearfrom the translation.

P. 132,1.12. )xtyi<TTov fih ovv. fth ovv signifies a correction orstrengthening of the preceding statement, ' No less absurd, nayrather most absurd of all.' I have followed Marquard in readingaronov though I am not at all sure that the addition is necessary.KaOo'Kov fidXia-Ta rav afiaprrjiMTiov might mean ' the most complete

mistake possible.' Cp. note on p. 130,1. 6.1.17. KoiXias. The plural is very strange, if the word means,

as it seems to mean, the main bore of the instrument.Mr. Howard (Harvard Studies in Class. Phil. Vol. IV, p. 12)

quotes in support of this rendering Porphyrius ad Ptol. p. 217,ed. Wallis : naKiv 8e iav Xd/Sj/s Sio av\ois, TOIS fiiv jajKeariv i<rovs,

rais §€ fipvTrjcri TO>V Koikiav tSiadjepovras' icaddirep eôuo-iv 01 $piyioi

irpbs TOVS 'EXXtivucovf' evprjtrns •napairKrjo-'uits TO evpvKoiKiov oj-irepov

npotefievoy <p66yyov TOV orej/o/cotXiou' deapovfiev y€ TOI TOVS &pvyiovs

(TTeuovs Tais KoCKims ovras enl JTOXXCB fiapvrtpovs rjxovs npofiaKXovTas

TS>V 'EXXTJVIKO))/. Also Nicomachus (ed. Meibom, p. 8, ]. 33), avairdkiv

Si TS>V ifiTrveviTTaiv ai iiei£oves KoiXidxrcif Ka\ ra jiei^pva /ITJKTJ, vcodpuv

Ka\ CKXVTOV. He cites too the parallel use of the Latin cavernaeby Servius ad Aen. ix, 615.

If it were not for the strength of these passages, one mightsuppose KoiKias here to refer to the sidetubes with which someai\oi were furnished, and which served, when in use, to lowerthe pitch of the instrument (see Mr. Howard's article, p. 8).

1.18. Marquard inserts 6 avXijrrjs unnecessarily. He assumesthat ols in 1. 19 must be an instrumental dative, and that n£<pvicemust be used personally, in which case the construction will be6 av\6s TT€<f>v<ev emTciveiv Ka\ avievat, and imTeivew and aviivai will

be used intransitively. But there is no reason why ols may notbe a dative after irtrpvicev >= [those other parts] to which it isnatural [to raise and lower tone].

1. 24. raCro. Cp. p. 122, 1. 10.1.25. KOI yap a<paipavvrts. For the violent ellipse by which yap

is left without a finite verb, cp. p. 145,1. 6,7} yap avfiâveiv.Should we read itapaipovvres for a<paipoivres ? For this

expedient of bringing the two pipes together, and drawingMACRAN X 273

IIIARISTOXENUS

If'< them apart, and for its effect on the pitch, see the last clause ofthe sentence from Plutarch (non posse suaviter 1096 a) quoted inthe note on p. 112, 1. 15.

P . 133, 1. 2. ovhiv Siacpepei Ae'yetv K.r.A. Here Marquard'stranslation is distinctly amusing, 'daher macht es offenbarkeinen Unterschied, ob man sagt " gut die Floten" oder"schlecht.'" Westphal is equally ridiculous: ' sodass es meistens

r:' j eigentlich dasselbe besagen will, wenn das Publikum beim/,'/ j Aulosspiel " gut" oder " schlecht" ruft.' The meaning simply

is that the goodness or badness of the music does not dependupon the instrument.

1. 21. BavfuxTTov 8' (I K.T.\. One more argument. Clarinetsare changeable instruments, and their music must alter with thealteration in themselves.

P . 134,1. 5. The MSS TO dprifihov Spyavov cannot be right.The argument plainly is (1) instruments in general will notserve as bases for the laws of harmony; and (2) least of all willthat very defective instrument, the clarinet, do so. For Spyavovused alone cp. p. 133,1. 4.

1. 14. 7rpS>Tov /iev airaiv K.T.\. It is required of us firstly toascertain the phenomena correctly, secondly, to distinguishtruly in these phenomena what is primary and what is derived,thirdly to grasp aright the result and conclusion. In otherwords we must first observe accurately, then analyse our factsand find the essentials, then sum the results of our observationand analysis in a generalization. The generalizations, whichwe shall thus obtain, will be the dpxai, or fundamental principlesof our science, from which its other propositions will be deduced.It is indispensable that such fundamental principles should be(a) indisputably true ; (b) recognizable by our sense perceptionas primary truths of music.

The science of Harmonic then as conceived by Aristoxenusstarts from the observation of individual facts, and proceeds byinduction to general principles, which serve in turn as foundationsfor a train of deductive reasoning.

1.17. TOV av/xftaivovTos . . . o-vvo(f)6evTos. Th is passage is mis-translated by Marquard 'die methodische Beobachtung desZufalligen und Uebereinstimmenden/ that is ' the methodical

374

NOTES

observation of the contingent and constant'; by Westphal' somuss der Sache gemass erkannt werden was sich (erst) alsSchlussfolge ergiebt, und was in die Kategorie des allgemein An-genommenen gehort,' that is, 'we must distinguish in accordancewith the facts what is only arrived at as a conclusion, and whatbelongs to the category of the universally admitted.' But (i) TOovpfialvov and TO ofioXoyoi/ievov are technical terms for the resultand conclusion; (2) a-vvopav means' to see the connexion of things'not to 'seethe difference1 between them ; (3) if TO o-vy-fiaXvov andTO SfMoXoyovfievov are distinct and contrasted classes, we shouldrequire TOV o-v/xfialvovTos KCU rou ofiokoyovfiivov.

1. 25. KadoKov 8' iv TO) K.T.X. W e must neither trace back ourmusical phenomena to physical and non-musical principles;nor be content till we have resolved them into the ultimate lawsof music.

1. 27. For fj of the MSS I read y in the sense of qua ' re-garded as.'

P . 135,1.1. ndfiirTovTes ivTos. A metaphor from the race-course.1. 7. fj fitKTov . . . fj KOIVOV. See Isagoge [ed. Meibom, p. 9,

1. 34]) ôivov Se TO i< TS>V ecrTtoTwv avyKelfievov. fuxrhv Se TO iv(S 860 $i rpels xa9aKTVPes yeviicol ifxpaivovTai. A melody is commonwhen it employs only the fixed notes, which, of course, arecommon to all three genera ; it is mixed, when it employs notesof different genus.

1. 12. 7re/>ie'xeTai 8' rj vo-repa . . . irpoTepq. That is the differ-ence between concords and discords in one special case of thedifference between larger and smaller intervals. The conno-tation of the b\a<popd between concords and discords containsthe connotation of the 8ia<popd of size, but the denotation of thehia<popd of size contains the denotation of the 8ia<j)opd betweenconcords and discords.

1. 18. The MSS are corrupt here. It is absurd to say thatthe Fourth is determined as the smallest interval by its ownnature. It is so determined by the nature of melody or song,inasmuch as all the smaller intervals which the latter producesare discords. The correction is due to Westphal.

F . 136, 1. I. Tavra fitv ovv \iyofttv & wapa TS>V f/iirpoadep irapti-\r](pa[iev. Marquard rejects this sentence on the ground that

T 2 275

1 • ' i '

,,. ARISTOXENUS

jj i the sense required is no t ' we say what we have learned,' but'what we say, we have learned.' But, just as ravra \iyop.(voKqdq means 'in saying this we are speaking the truth' (thepredicative force lying in the aXjjflfj), so here the meaning is' in the above statements we are repeating what we have learnedfrom our predecessors.'

1. 6. irddos. Cp. the use of irao-xa> in p. 145, 1. 17 ; p. 156,1. S ; p. 159,1. 8.

1. 10. ovre TO ef eKarcpov K.T.X. Meibom, Marquard andWestphal alike find this sentence unintelligible. Is it nota fact, they ask, that the sum of a Fourth or Fifth and anoctave is a concord? Accordingly they correct the readingby inserting Sis redivros after eKarepov airrav. But the MSS areperfectly right, and the commentators construed wrongly.Written in full with the ellipse supplied, the whole sentencer u n s , OVTC yap TO "urov (Karipto avroav avvTeBiv TO OXOV o~vp.(pa>vov

TroieF, OVTC TO t£ cKarepov avTav KCU TOV dta iraamv avyKii/ievov

€KaT(p(f avrSiv ovvn6iv TO oXov <rifi<pa>voti Troici, a n d t h e m e a n i n g

is ' Add to a Fourth or a Fifth an interval equal to itself; theresult is a discord. Add to a Fourth or Fifth respectivelythe sum of an Octave and a Fourth or Fifth ; again the resultis a discord.'

According to the absurd misconstruction of Meibom, Marquardand Westphal, the second part of the sentence in its complete-ness is as follows ." ovre TO (£• eKarepov avTtov Bis redivTOs Kat TOV

81a irturSiv' o-vyKeifievov TO SKOV crviKpavov TTOKI. N O W it is quite

correct to say ' 4 added to 6 causes the whole to be 10' or' the addition of 4 to 6 causes the whole to be 10,' but surelynot to say ' the sum of 6 and 4 causes the whole to be 10.'

1. 18. Aristoxenus introduces two warnings. When he saysthat it is possible to sing the third or fourth part of a tone, hemust not be misunderstood as saying that one can in singingdivide a tone into three or four parts. For that would implythe possibility of singing three thirds of tones or four quarter-tones in succession which is against one of the fundamentallaws of melody [see p. 119,1. 20].' Again, he has mentioned no smaller division of the tone than

the quarter-tone, because the voice can sing and the ear dis-•3.76

NOTES

criminate none smaller. But it must not be forgotten that inthe abstract there cannot be a minimum interval any more thana minimum space or time.

P. 137,1. 4. ore 8e darepov K.T.\. Between the Diatonic andChromatic scales there is only variation of the Lichanus, asthese genera have their Parhypatae in common.

P. 137,1. 18-P. 138,1. 6. Marquard is greatly disconcertedby the abrupt transitions which he finds in this passage from theindicative to the accusative and infinitive construction. Besidescorrecting rightly Set to Sttv in p. 138, 1. 3, he omits ecrn inp. 137, 1. 20 to remove the incongruity. As a fact, with theexception of the blunder Sei for SeiV, the reading of the MSS isquite unexceptionable, and the construction normal. The quotedquestions are in the indicative, the quoted statements in theaccusative and infinitive. The dvat. that follows 8erfov in p. 137,1. 23 is grammatically dependent on it, and not the infinitive oforatio obliqua, as Marquard supposes.

1. 18. The objection cited in this paragraph, and the answerof Aristoxenus to it, raise again the conflict between the super-ficial view of notes as points of pitch, separated by certain spaces,and the deeper view of Aristoxenus according to which notesare essentially members of a system with special functions. Theobjection is stated in 1. 18-p. 138,1. 5 and here again Marquardhas quite wantonly perverted the order of the sentences. Theargument of the objection may be stated thus: 'We object toapplying one term, say the term Lichanus, to several points .ofpitch at different distances from the Mese. The term Hypatesignifies one certain point at one certain distance from theMese; why not similarly restrict the term Lichanus to someone point, say the point two tones below the Mese, yourEnharmonic Lichanus; and use other names for what youcall the Chromatic and Diatonic Lichani ? For we hold thatnotes which bound unequal magnitudes must be different notes;or, to put it more plainly, that a difference in the size of thecontained interval necessarily implies a difference in the con-taining notes. We hold equally, by simple conversion of thisproposition, that different notes must bound different intervals,or that a difference in the containing notes necessarily implies

277

ARISTOXENUS

a difference in the size of the contained intervals. Consequentlya proper nomenclature will always employ the same terms todenote the points bounding the same magnitudes of intervals ;and will always employ different terms when the boundedintervals are unequal.'

1. 19. Marquard reads redivros for KivTjdevros on the groundthat it is when one posits, not when one changes, one of thepossible intervals between the Lichanus and Mese that aLichanus results. But the sense is rather this: The objectorsurge that between any two notes there must be but one interval;if this interval be changed, then there must, say they, be a changeof notes also.

P. 138,1. 2. The addition of Xi ayds is perhaps unnecessary;K\t]6jj might stand by itself for ' receives the name.'

1. 3. Probably S is right in omitting TO.1. 5- The sentence ra yap icra TS>V fityedav rots avrols 6vop.acn

nepi\r]iTTiov ehai is the simple converse in sense, though notin form, of 8uv yap irepovs eivai (pdoyyov: TOVS ro erepov peyeBosoplôvras. For the former sentence = ' equal intervals should bebounded by identically-named notes ' = ' no notes should havedifferent names unless they bound unequal intervals'='no notesare really different unless they bound unequal intervals'='alldifferent notes bound unequal intervals,' which is the simpleconverse of' all notes that bound unequal intervals are differentnotes.'

1. 9. Before dealing with the original proposition of the ob-jectors Aristoxenus disposes of its converse by insisting thatthe essential feature of a note is its bivapis, and that nomen-clature cannot overlook the distinction between the notes a ande in the scale

when they are Mese and Nete, and the notes a and e in thescale

when they are Lichanus and Paranete.278

NOTES

1. 14. I read tv, TO for iv ro>of the MSS.1. 16. on 8' oufie Tovvavriov K.T.X. Having disposed of the

converse Aristoxenus turns to the original proposition, whichrequires a special refutation; for the two propositions arerelated to one another as a Universal Affirmative and its simpleconverse; and the falsity of one does not prove the falsity ofthe other. Aristoxenus has to prove not only that inequalityin the contained intervals is not the sole ground for distinguishingnotes by name, but also that it is no sufficient ground for doingso at all. His arguments are two:

' In the first place, if you insist on having different nameswherever there is a difference of interval, you will require aninfinite vocabulary. The voice, for example, may make its secondresting place in the passage of the tetrachord at any point betweena semitone above the Hypate and a tone below the Mese. Thenumber of such points is infinite. We call them all Lichanus,but you who insist that a difference of interval demands a differ-ence of name will require an infinity of names. Perhaps youwill think that this is the quibble of a casuist; that as a matterof fact three terms would do, one for the Enharmonic Lichanus,one for the Chromatic, and one for the Diatonic. But it is noquibble. For consider seriously (i>s aKrjdas): different schoolsor theorists assign different positions to the Lichani of thedifferent genera; and there is no earthly reason for giving one'sadherence to one of these schools rather than another. Takea special case. Some theorists locate the Enharmonic Lichanusat two tones below the Mese; some place it a little higher.Supposing, then, that we even went so far with you as to restrictthe term Lichanus to the Enharmonic Lichanus, we should havejust the same difficulty again. For here are two upper passingnotes, one two tones below the Mese, and one a little higher;both of them to the ear give an Enharmonic scale, so that bothhave equal claims to the name of Lichanus: yet they boundunequal intervals from the Mese, therefore, on your theory, theone name will not apply to both.'

'In the second place, your demand ignores the fundamentalcharacter of sense perception which, abstracting from the pettydistinctions of quantity, looks to the similarity of things through

279

ARISTOXENUS

their possession of common qualities. Thus the juxtapositionof two small intervals produces on the ear an impression ofa certain sort, which remains the same whatever the exact sizeof the intervals may be; and one uses the general term Pycnumfor this juxtaposition. But on your principle, one has no rightto employ this term, since Pycna are of different sizes. Similarly,one has no right to speak of Enharmonic, or Chromatic, orDiatonic, for all these classes imply the ignoring of mathematicaldifferences. If, on the other hand, we do admit a class Pycnum,a class Enharmonic, why not also a class Parhypate and a classLichanus ? For just as in the case of Pycna you have a generalfeature, namely, a certain compression, and as in each genusyou have a certain character common to the particular cases ofit, so here you have as common features the species or figureof the tetrachord, that is, a plan of four notes, the two outerfixed at an interval of a Fourth with the upper as tonic, and twopassing notes between them.'

I.17. For aKo\ovdt]Teov of the MS S I read a.Ko\ov6eiv dereov. T h epreceding sentence asserts that A is not a necessary result of B ;nor, continues Aristoxenus, must we allow that B is a necessaryresult of A. But aicdkovdeiv cannot mean ' to assert a necessarydependence.'

Tovvavriov aKoKovduv = ' the opposite order of dependence.'1. 21. £>s aKrjdas . . . iv eKarepq TO>V dimpecreav, I have

transposed this passage from its unintelligible position afterSiafieveiv in p. 140,1. 1. In its proper place it is most serviceablein answering the certain objection that to talk of an infinity ofLichani is mere casuistry.

F. 139,1. 2. It is quite unnecessary to insert with Marquardand Westphal ov ndpv p'adiou crvvibeiv. axne may very wellintroduce a conclusion pressed against an adversary in the formof a question.

1. 13. \iya> 8e is parenthetical, and nOelo-a agrees with eW«7and stands in apposition to els o/notdr^ra . . . /3X«rouo-a.

1. 14. I read «or for i>s in 1. 14, and fie el&os e<as av for hi 17SieVf cor av in 1. 17. For Zcos in the sense of' to cover all cases inwhich' cp. p. 141,1. 1.

1. 16. nvKvov rtvis (pavrj. If the reading is correct, TTVKVOV280

NOTES

nvos must be construed as a genitive of the material: ' a voice-utterance consisting in a compression,' i.e. in a succession ofclose-lying notes.

1. 21. I insert ixiveiv after o-vfifiaiva.

P. 140,1. 9. Finally, Aristoxenus shows a palpable absurditythat would result from the acceptance of this principle—theabsurdity of one note bearing more names than one in the samescale. In the first place let us take two equal intervals insuccession; for instance, the interval between e and / , and

between/and J / in the Chromatic scaleX Y

If we insist on using the terms X and Y universally for thelower and higher notes of an interval of this size, the / of theabove scale will be both X and Y.

In the second place, let us take two unequal intervals, theinterval between e and/and that between/and £• in the Diatonic

n M N

scale FTO _) J «J -'J '- On the principle under exami-X Y

nation, inasmuch as the names signify no function or intrin-sic qualities of notes, but merely a space relation betweentwo points whose only quality is that they are so far from oneanother, every such name of a point must connote its relation toanother point at some certain distance; and cannot be employedoutside this relation. Thus every change in the size of aninterval will demand a new pair of note-names. Hence in thepresent case the intervals between e and / and between / andg will bear two distinct pairs of names, say XY and MN; and

'/will bear two names, Kand M.P. 141,1. 1. In this paragraph we have another exposition of

the genera and their 'shades.' See pp. 116-118.P. 142,1. 23. The missing words have been well supplied by

Westphal.P. 143,1.13. I have little doubt that we should read \cicreov for

fieiKre'ov. Cp. p. 147,1.25, where all the MSS read XeKreav insteadof the plainly necessary SeiKreov.

281

ARISTOXENUS

1. 18. dymyrjs: cp. p. 121, 1. y. The term is here used, not ofa particular melodic figure, but of the general consecution ofmelody.

1. 19. I omit the words ov yap Sici Totrovrtov 8vvt]deir) TIS av asa gloss which has crept into the text. They are meaninglessby themselves, and require the addition of fteXaSeiv, or thelike; even when thus emended they present a singularly weak,and at the same time wholly unnecessary statement. The glosswas occasioned by the ambiguity of the following /«xpi.

1. 20. jiexp1 n e r e = ' UP t0> but excluding.' It more often means'up to and including* (see p. 131,1. 3). The same ambiguityattaches to ems. Cp. p. 144, 1. 1, and p. 140, 1. 4. Perhaps,however, we should read ddwarel here.

1. 21. TO egrj? OVT iv K.T.X. The nature of melody brings it topass that (a) sometimes the next note to a given note is separatedfrom it by the smallest possible interval, as in the Enharmonic

scale Nfy J ; J J "' the next note above xe is / .

(b) Sometimes the next note to a given note is separated fromit by an interval of considerable size, as for instance in the samescale the next note above / i s a. (c) Sometimes a consecutiveprogression moves by equal intervals as from f to b in the

f 283

Diatonic scale Fm~~^~=^='J ^ ^ = id) Sometimes

I I Ia consecutive progression moves by unequal intervals as from

/ to b in the Chromatic scale h fi 1 —I—H—^—&--

g§ 2 2" " -3

Consequently, the true conception of continuity is not derivedfrom the notions of the minimum, the equality, or the inequalityof intervals.

P. 144,11. 8-9. After much hesitation I have accepted Mar-quard's reading, though I believe his interpretation of it to bequite erroneous. The difficulty lies in the genitive rou •Kpoeipt)-

dptdpov: the general argument is clear. If we admit that

NOTES

the maximum number by which the distance AB can be dividedis four

A

it is evident that the points A, x, y, z, B are consecutive, andadmit of no intermediate points of section. Aristoxenus refersto these points A, x,y, z, B as ' the notes that bound fractionsof the said number.' Marquard identifies the number with thedistance AB, and regards TOV Trpoei.prjfi.hov apid/iov as a partitivegenitive. But, to take the above illustration, dpifyiou evidentlyrefers not to the distance AB but to the number four by whichit has been divided. For it would not be true to say that thepoints which bound parts of the said interval are consecutive;A, y, B for example bound parts of it, and are not consecutive.

We must therefore understand the partitive genitive TOVtSiao-TTjuaTos with \i*pr\, and interpret TOV irpoetprifievov apidfiov as'having the said number as denominator.' To recur again toour illustration, the whole phrase TOV npoeiprjiiivov apidpov fidpr)TOV Siao-ny/MiTos would mean ' fractions-of-four' (or 'fourths') 'ofthe distance AB.'

1. 18. I read Aa/ aveVo) for XapPdveTu of the MSS, as themiddle voice is out of place. \anJ3aptra is parallel to eKfiekfjsforo) that immediately follows.

Meibom wished to read /iijficVepoi/ for fujSerepa. But Marquardpoints out that each alternative here referred to comprehendstwo relations, those of any given note to a certain note aboveit and to a certain note below it.

1. 20. ov Set 8* dyvoeh K.T.X. For instance, the scale

obeys the above law; yet it is illegitimate, because it violatesthe law of the tetrachord that the interval between the lowerfixed note and the first passing note must never be greater thanthat between the two passing notes.

P . 145,1. 5. tle'i yap rots K.T.\. Th& law of the sequence oftetrachords is as follows: two tetrachords belong to the onescale either if the notes of one form some one concord with the

283

i

ARISTOXENUS

corresponding notes of the other, or if the notes of both forma concord with the corresponding notes of a third tetrachordof which they are both alike continuations, but in opposite direc-tions, one upwards, one downwards.

Thus, in the Greater Complete System (see Introduction A,

§29)

^J^J-

the notes of any one tetrachord form some one concord (Fourthor Fifth or Octave) with the corresponding notes of anyother.

Again, in the Lesser Complete System (see Introduction A,§ 29)

MesSn Synemmen8nHypat8n ^ " j j

the corresponding notes of the Hypaton and Meson tetrachordsform Fourths with one another; as do also the correspondingnotes of the Meson and Synemmenon tetrachords. But whatabout the Hypat6n and Synemmen6n tetrachords? Theyevidently belong to the one scale, and yet the notes of onedo not form a concord with the corresponding notes of theother. Here the second clause of the law applies. TheHypat6n and Synemmen6n tetrachords are both continuouswith the Meson, but in different directions (^7 eVi rw alrovTOTTOV), one lying below it and one above, and the notes of theHypaton and Synemmen6n form concords with the correspond-ing notes of the Mes6n.

1. 9. Marquard, followed by Westphal, wrongly altered rbvavrbv TOTTOV to T£ airm TOITIU, and supposing it to refer to thecoincidence of the extremities of conjunct tetrachords proposedto omit the (ifj of I. 8.

1. II . It is uncertain what are the other conditions of thelegitimate synthesis of tetrachords, to which Aristoxenus here

284

NOTES

alludes. One may perhaps have been a certain order in theemployment of conjunction and disjunction. Thus the scale

might be regarded as illegitimate, because the conjunction anddisjunction do not occur alternately.

1. 15. The MSS here read dXX' iv p.*ye8ei apia-rat, which I havecorrected to dXX' iv\ fueyiQei a>pitrdm. topiadai is the infinitive afterSoKfi, and with iravrekas aicapiaiov nva one repeats ?Xet|/ Soxet TOITOV.-Marquard reads OVK *x.(lv ôicel TOITOV dXX' J) el fieyedet upurrat, i)navTe\ws aKapiaiov nva and translates absurdly' seem only to takeplace when they are determined in magnitude, or at any rateonly in a highly limited degree.' Of course Z\€lv T°ôv means' to have a locus of variation.' The same misconception under-lies Westphal's reading OVK tx(lv Soxel ^ n-aireXcoj aKapiaiov nvaTonov dXX' ^ el ret juyiBr] apurrai.

1. 19. aKpi^ta-TOTT] K.T.X. Note Aristoxenus' recognition ofthe truth that the determination of all intervals must in thelast resort fall back upon the elementary relations of theconcords.

8', deleted by Marquard, may be an example of the 8e airobo-

1. 22. TS>V dwaraiv. Intervals smaller than semitones cannot bedetermined by concords. For the Fourth consists of two anda half tones, the Fifth of three and a half tones, and the Octaveof six tones; and no repetition, addition, or subtraction of thesenumbers will lead to any fraction smaller than a half.

1. 23. eVi TO al-v K.T.\. If it be required to ascertain by concordsthe note that lies two tones below G, the following will be theprocess:

ARISTOXENUS

i

I

The note that lies two tones above G is ascertained thus :

1-

P. 146,1. 5. yiyverai 81 Kal K.TX This is evident. If in the

Fourth M) -J li—*3— we determine the ditone between

a and / by concords we have in so doing also determined byconcords the semitone between e and/ For e is given in concordwith a, and / has now been determined by concord with a;and e and/are the bounding notes of the semitone.

1. 20. irorepov 8' 6p65>s K.T.\. The following is Aristoxenus'demonstration that a Fourth consists of two tones and a half(a tone being the excess of the Fifth over the Fourth). Take

4th 4th

a Fourth e-a, and determine by concords the note/two tonesbelow a, and the note $g two tones above e. It follows that theremainder £-/=the remainder #g-a because each of them=thewhole Fourth, e-a, less by two tones. Now take the Fourthabove/namely #<z, and the Fourth below $g namely $d. Therewill now lie side by side at each extremity of the scale tworemainders, which must be equal for the reason already given ;that is, §d-e, e-f, %g-a and a-#a are all equal, because each ofthem equals a Fourth less by two tones.

Now if $d and Ja, the lowest and highest notes of the scale,be sounded, our ears will assure us that they form a concord.This concord, as greater than a Fourth by construction andobviously less than an octave, must be a Fifth. But since%d-$a is thus found to be a Fifth, and §d-$g by constructionis a Fourth, %g-$a must be the difference between a Fourth

286

NOTES

and a Fifth; in other words, a tone. But we have already seenthat >jj,g-a=a-%a :. I^g-a—z. semitone. But by the construction<?-#£•= two tones; therefore e-a being the sum of e-$g and %g~amust be equal to two tones and a semitone.

P. 147, 1. 4. The MSS read 8vo. o-we^cis tso-ovrai nai fir) tv aiwepoâi which Marquard and Westphal following Meibom correctby changing Hv to pia. But (1) how did the grammaticallyobvious nia come to be corrupted to Zi> ? (2) what is the sense ofinsisting that the remainders are ' not one' ? (3) the article beforeIncpoxai is objectionable, as the meaning is ' there will be tworemainders. ' I read Keifievai for xal 17 ev ai. nei/ievai <rvvexeis=' lying side by side,' ' in juxtaposition.'

1. 9. The absurd rerrapa in this line and in 1.15 arose of coursefrom the scribe mistaking the 8 of 877X01/ and the 8' before oîrarovfor numerals.

P . 148, 1. I. The MSS read birovov' crvyxapeiTai irapa wavravK.T.\. Marquard followed by Westphal inserts aXka beforeo-uyx&jpemu ; but I prefer o-uûparai yap, because (1) the sentencesupplies a reason, (2) yap might easily have been lost before napd.

P. 149,1. 12. Before we consider Aristoxenus' exposition ofthe continuity of tetrachords, there are two points to be noticed.Firstly, whereas in his former sketch of the matter [p. 145,11.3-13]he considered the relation of similar tetrachords only, here histreatment takes into account the differences of Figure. Secondlythere is an ambiguity in the terms o-uvexijr and i^s, which some-times signify merely 'in the same line of succession,' at othertimes ' next in the line of succession.'

In general, Aristoxenus asserts, tetrachords are in the sameline of succession if their boundaries are in the same line ofsuccession or coincide. In this general definition are explicitlygiven the two species of succession of which tetrachords arecapable. We have a case of the one species when the lowerboundary of the higher of two tetrachords coincides with theupper boundary of the lower; a case of the other species, whenthe lower boundary of the higher of two tetrachords is in the oneline of succession with the upper boundary of the lower.

Now we must not confuse this distinction with the distinctionbetween conjunct and disjunct tetrachords. The latter distinction

287

ARISTOXENUS

I

divides successive tetrachords into (a) those whose extremitiescoincide; and (b) those whose extremities are divided by onetone. The former distinction divides successive tetrachordsinto (x) those whose extremities coincide; and (y) those whoseextremities are in the same line of succession. Now the class(«)=the class (x), but (6) is only one subdivision of the class (y).Thus in the legitimate scale

a i I i. ! J

the tetrachords E-F-G-A and c-d-be-f fall into the class {y),since A and c are in the same line of succession, but not into theclass (b), since they are separated not by one tone but by a toneand a half.

Now if two tetrachords belong to the class (a) (and con-sequently to (x) also) they must be similar in figure. Otherwiseas in the pair

we shall find a violation of the fundamental law of continuity[p. 120,1. 16].

On the other hand, if tetrachords belong to the class (y)they will sometimes be similar, sometimes dissimilar in figure :similar, when they belong to the class (6), that is when theirextremities are divided by a tone (and also, of course, if they areseparated by a full concord); dissimilar, if they are separated byany other interval. -

Thus in the scales

and

I I J

288

NOTES

E-F-G-A and \?B-C-D-\?e, E-F-G-A and be-f-g-ba in thefirst, and E-F-G-A and C-D-e-f, E-F-G-A and f-g-a-b,E-F-G-A and B-C-D-e in the second are all examples ofclass (y); but only the last pair are examples of class \b) andonly the last are similar in figure.

Since then we have seen that all successive tetrachords maybe divided into (x) and (y), and since all (a) are (x) and aresimilar in figure and only those (y) are similar which are also(6), it follows that all similar tetrachords in the same line ofsuccession are either (a) or (d). As Aristoxenus says, TO cgrjsreTpdxopSa Sfiota ovra fj (rvvijitfieva avayxaiov etvai fj Sieêvyfieva.

P. 140, 1. 14. In general, tetrachords in the same line ofsuccession cannot be separated by a tetrachord dissimilar tothemselves; for

1. Similar tetrachords in the same line of succession cannotbe separated by a tetrachord dissimilar to themselves.

For if it be possible, between the similar tetrachords E-F-G-Aand d-be-f-g\e£ the dissimilar tetrachord A-B-$C-d'be inter-posed.

The resulting scale is illegitimate, because f neither forms aFourth with the fourth note below it, nor a Fifth with the fifth.

2. Dissimilar tetrachords in the same line of succession cannotbe separated by a tetrachord of any figure.

For if it be possible, let the two dissimilar tetrachordsE-xE-F-A and d-§f-x #f-gbe in the same line of successionand separated by a tetrachord of any of the three figures.

(«)

ARISTOXENUS

Any one of the resulting scales is illegitimate. In (a) for examplexA neither forms a Fifth with the fifth note above it nor a Fourthwith the fourth; and the other scales suffer from the samedefect.

P. 151,1. 4. For OP io-Ti I read 3 y iarrl for two reasons. Firstly,the sentence is thus made exactly parallel to the next; andAristoxenus is fond of such parallelism. Secondly, if we readov, the meaning is ' People take the ditone as simple and thenwonder how it can be divided'; but we require rather ' Peopleknow that the ditone can be divided, and then wonder how itcan be simple'; and this sense is secured by reading 0 7' e'ort.The difficulty which Aristoxenus here resolves arose from thecommon misconception by which one decides an interval tobe simple or compound by its dimension, without taking intoaccount the scale to which it belongs, and the functions of itscontaining notes.

1. 17. I omit TO 8' ISiov rrjs 8iafev£eo>j OKIVT^TOV CCTTIV. T h e factthat the disjunctive interval (the tone) does not vary is usedto prove the theorem, and therefore cannot be part of thestatement of it.

1. 22. The disjunctive interval is constant because the notesthat contain it are fixed notes.

P. 152, 1. 14. For MSS da-ivdera TrXetora I read aa-ivSera rair\ei(TTa, Cp. p. 153, 1. I.

1. 18. For the MSS eprpoo-dei' rede'ura Marquard and West-phal read npocrride'icra, supposing the efiTrpoa-dev to have creptin from 1. 16. I read tv Trpoo-riBela-a ; ev helps to account for thecorruption, and strengthens the expression of the argument.

P . 153, 1. II . Sri 8c KCU c'£ eXaTTovap K.T.\. Defective ortransilient scales [see Introduction A, § 26] contain fewerintervals than the simple parts of the Fourth. Also in theEnharmonic scale of Olympus [see note on p. 115, 1. 2] theFourth was only divided into two intervals.

1. 13. nvKvbv tie Trpbs irvKvq K.T.~K. The next eleven pagesare occupied by a series of special rules as to the successionof notes and intervals, all of which rules derive themselvesimmediately from two fundamental laws. One of these laws,that by which the order of intervals of the original tetrachord is

290

NOTES

determined, is always presupposed by Aristoxenus; the otherwhich demands a Fourth between fourth notes or a Fifth betweenfifth notes [see p. 120,1.16] isexplicitly quoted. To understand thenall these special rules, it is only necessary to keep before one'smind (a) the form of the original tetrachord, and the functions of

LP MP HP LP

its notes as regards the Pycnum M^J—^^*7J J r^ • [ see

note on p. 129,1. 4] and (5) the possibility of choosing betweenconjunction and disjunction both in the ascending scale

LP MP HP LPLP MP HP LP

0 1 1

1 '— 1 ~ —t

LP MP HP LPand in the descending scale

LP HP MP LP HP MP LP

mLP HP MP LP

P . 156, 1. 5. I read with M TOVVOVTIOV iriirovBev &n\Zs oi 8vva-fifva. The other M SS have bwdpeda for Swdpcva which Meibomretains, inserting a before &TT\S>S. Marquard, rightly urgingthat the explanation of the general phrase roivavriov nenovdcvwould not be given in a relative sentence, reads rolvavriovireirovde KM air\S>s, and is followed by Westphal. But thereading of M is quite unexceptionable. Marquard's objection tothe two participles bwafuva and "a-a Svra, which are not co-ordinated in sense, is groundless. In the active one mighthave oi Svvafi(6a Tavra riGivai laa Svra t^f/s, which wouldbecome in the passive oi bivarai. ravra Tidftrdai lira ivra it-rjs,and if used participially ov Swa/ieva Ti8e<rdai itra ovra e£ijr.Another objection to the readings of Meibom and Marquardis that they would require nShat, not riQeaBai.

P . 157, 1. 6. Before anb 8e rov birovau, the MSS have ani> t5/u-TOVI'OU fuv iirl TO d£v 8io 680I Kal eVi TO j3api 8io. This sentencecannot be retained; for in the first place it makes a false

u 2 291

ARISTOXENUS

assertion, there being but one progression upwards from thesemitone or first interval of the Diatonic tetrachord (that is,of course, in the scale' of any one shade, see p. 159, 1. 12);and in the second place, referring as it must, along with thepreceding paragraph, to the Diatonic genus only, it couldnot stand in such close connexion with the following propo-sition, which as it concerns the ditone can only apply to theEnharmonic Genus.

1. 10. eiri 8e TO fiapv irvKvbv p.6vov which in some of the MSSfollows inl TO oi-i is a most silly interpolation. The sentencein 1. 11, Xeiwerai fih yap K.T.A., introduces the proof of theassertion irKtiovs 8e TOVTIOV OVK ecrovrai in 1. 9. The considerationof the descent from the ditone does not begin till 1. 13, em. 8e TO(iapii fxia' 8£8eiKTat yap K.T.X.

P . 168, 1. 15. I read Kara with R. The other MSS haverat. But whichever we read, TO TOV WVKVOV piyedos is accusative(whether governed by xa0' or Kara) and not nominative, asMarquard and Westphal suppose. Evidently the chromaticinterval that corresponds to the enharmonic ditone (which willdiffer in size as we pass from one shade to another) will varyinversely as the size of the Pycnum. TO ye fiea-rjs of the MSS,earlier in the sentence, is quite correct.

P. 159,1.15. I have corrected el to g. Cp. p. 101,1.13, whereWestphal has corrected e'Lirep to f/irep. The MSS of Aristoxenusexhibit perpetual confusion of 1, e, i;, v, «, 01. Cp. note on p.101,1. 7.

1. 18. 8vvdpeis . . . etSi; . . . Bta-tis are used in a general nota technical sense here.

P . 161, 1. 24. The absurd eVi which appears in the MSS isreally the iitd of p. 162,1.1. This is proved by the Selden MS, thewriter of which after the pia 686s i(p' iKarepa e<rrai of 11. 23-24missed a line, and proceeded to write the SeiKreov eVi (for end)of p. 162, 1. 1. Then discovering his mistake he drew his penthrough these latter words.

P . 162,1. 4. Whether we retain KOT' oibirepov T5>V rpoirav ofthe MSS or read as I prefer /tar' oidfrepov TS>V TOWCOV the senseis ' neither above nor below.'.

1. 8. The MSS read Snore pas &v reBji TO blrovov' ry ToVng) rdVou292

NOTES

TtOeipevov. Marquard followed by Westphal reads inorepas hvTtBrj TO dirovov' eir\ Be TC5 nira TOTTO) rovov Te6eip.evov K.r.X., takingonorepas in the sense of' whether above or below' on the analogyof KOT' oiSerepov T&v rponmv (1. 4) ; and «rl T£> aira T<$JT. But this last is very hardto accept; the phrase would much more naturally mean ' inthe same direction of pitch' i.e. either ascent or descent.I prefer, having read xar' ovSerepov TS>V ronav in 1. 4 = ' inneither of the directions,' to read here (moripas hv TeOrj TOhirovov TS>V To7r<oy='in whichever manner the ditone be placedin regard of the directions.' The two Ton-ot are 6 im TO 6£V and0 eVi TO /3ezpu.

1. 21. The MSS reading is obviously defective. The words1 have introduced restore the sense simply. Marquard's in-sertion of the article before (pdoyyovs is quite inadequate.Westphal reads cjrl Tt)v airfiv raaiv TOUS elpr^fievovs iv TTUKVIO(pdoyyovs.

P. 163, 1. 4. on fit TO SidroKov aiyKeirai TJTOI K.T.X. The pro-position of this paragraph seems at first sight inconsistentwith Aristoxenus' exposition of the shades (see p. 142,11. 9-14);according to which exposition there are only two shades of theDiatonic genus, (a) the soft Diatonic, the tetrachord of whichis thus divided

x 6 aa

1 1 1

(in which 1 = ^ of a tone)(i) the sharp Diatonic with the tetrachord

&

If we complete the Fifth by adding to each of these tetra-chords the disjunctive tone = 12, we shall have in the sharpDiatonic 12 and 6 as the only dimensions of intervals. Inthe flat Diatonic, on the other hand, we shall have four

293

NOTES

dimensions, 6, 9, 12, 15. But how can there be a Diatonic withthree dimensions? In this way, that it is allowable for theDiatonic scale to borrow the Chromatic Parhypatae. Thus, bya combination of the Sharp Diatonic Lichanus and the softChromatic Parhypate we obtain a Fifth of the form

I 4 II

12

which may be called Diatonic from its prevailing character.In it there are three dimensions, 4, 12, 14.

P. 164, 1. 13. elSos here = schema = the ' figure' or order ofdisposal of the given parts of a whole.

?94

I N D E X

ayaB6s 122. 8, 10, 13.'Ayqvwp 127. 25.ayvoia 125. 3 ; 126. 26; 127. 2 ;

136. 19; 144.21.aTôialSl. 12, 14; 151. 7.070) 104. 5, 6, 9, 24; 128. 11;

144. 16; 159. 22.ayorfi) = ' rate of movement' 105.

14; 125. 17.= 'sequence'121. 7; 143.18.•= ' keeping,' ' observance'

128. 10.aSia<popos 129. 11.dSwareoi 106. 25.aSoi 102. 25.a-qp 134. 27.d9eti7»jTos 126. 25; 127. I .'A6ijvaTos 128. 11.aXaOavoiiai 98. 20; 125. I, 2, 7;

129. 22.aiaetjais 99. 21; 101. 22, 24; 102.

8; 103. 5; 104. 5 i 111. 13;124. 4, 23, 27; 129. 22; 139.3, 10, 19 ; 140. 9 ; 145. 18.

ciaBrfrds 99. 6.a*Tto 98. 24; 114. 18; 123. I ;

124. £, 9 ; 126. 18;. 133. 1;134. 2, 3 ; 138. 12; 145. 17;151. 10; 153. 9; 159. 26; 160.21; 161. 17, 20; 164. 7, 10.

atno<i 114. 9; 115. 6; 118. 22;122. 16.

d/atpiaibs 145. 16.&KIVI)TOS 113. 23; 151. 17; 152.

6 , 9 , 1 1 .d/coij 102. i s ; 106. 24; 107 ; 124.

16, 17 ; 129. 17.d«o\ou0e'iu 126. 19; 138. 16, 17;

143. 23; 154. 14, 16.aicovai 108. 8 ; 122. 8, 19 ; 123. 6,

12; 149. 12.

iueplBaa 125. I.anptfip 97. 8; 103. 6; 108. 9;

124.5; 144. 1; H5. 19; 146.23-

<xKpi$o\oyeancu 126. 15.d«pi/33s 119. 3 ; 124. 19.attpoaopiat 123. 2.aicp6a.<m 122. 8.awpos 121. 8; 137. 2 ; 141. 2;

147- 8, 20; 148. 7.dXAoiowris 130. 24; 164. 17.&AAOT/>IOAO7C<U 124. 4.&\\6Tpios 124. 8.0X070* 108. 23 ; 109. 12.&p&prt]fia 132. 13.i/napTia 128. 5.ilitK^SrjTos 113.12 ; 117.8 ; 120.1.an<pta0t]Teaj 138. 22.&/i(ptc(tfiTT]ms 118. 7-dcâ) 124. 15 ; 132. 14; 133. 5,

II.avaycayfi 133. I ; 184. 2.avatpioi 110. 24, 25; 145. 3.dva/id/JTijTos 133. 6.dra7T<iSci«Tos 99. 2, 17,19; 129. 9.ivip/ioaros 110. II, 17; 143. 6.avacpopi. 106. 23.dvem\T]VTOs 108. 14.area-:? (see note on 103. 16) 97.

10; 103; 104; 105. 19; 106.11 ; 114. 9.

av-fip 112. 18.avSpaiirtKos 106. 17.&vii)p.i 103. 12 ; 104. 3, 5; 110. 1;

115.17 ; 123. 24 -r 132.19; 183.1 ; 137. 15.

dv6iwios 125.18; 139. 7; 150. id,17; 162. 16, 23.

aVTierpltpa 138. 4.avuTtpov 95. 11.avartpa 101. 12.

ARISTOXENUS

dvcOroros 111. II.d£i6\oyos 98. 14.d£i6ai 95. 9; 138. 7 ; 140. 9 ; 143.

14.do/wo-TOs 99. I.dTraiT&u 134. 25.dnaW&TTa 104. 10; 124. 2 2.fiiraf 141. 12, 13.direipia 160. 3.air«pos97.15; 104. 4 ; 106; 107;

112. 2; 138. 20, 31; 144. 3, 5;158. 22, 24; 159.5, 7, 15.

Airex* 137. 7; 155. 6, 8.dn-AoSs 109. 20, 22; 110. 25; 125.

24; 129.3; 131.?-mrkCbs = (1) 'in plain speech, not

in accordance with strict philorsophical truth' 102. 14.

= (2) 'roughly speaking,overlooking particular excep-tions'125.6; 126.2,25; 127.5.

= (3) ' m general terms, sum-ming up particulars' 131. 8;143. 15.

= (4) 'absolutely, withoutexception' 127. 20; 150. 15;1 5 3 . 4 ; 156.5-

= (5) 'in the abstract' 136.24.

airoP&\\a> 139. 6.a,Tro0\iiTa 104. I.avoî-iviiUKai 122. 18.ivaSelxwiu 99.9,11; 118. 4 ; 124.

13 ; 153. 9.airoSeiKriKos 99. 2 ; 129. 9.dir<55e<£ir 124. 2, 10; 134. 25.dnoSiSw/u 98. 9, 10; 99. 13 ; 103.

15; 108. 8; 113.3; 119. 16;128.9; 131.17; 143.17; 144.1, 2.

dir6So<ris 128. II.diroSeffirifa 124. 9.i-ttoXiimiva 135. 2.anopta 149. 1 2 ; 164. 5.dirorifivai 96. 6.dnotpaiva 105. 6; 119. 17.&im» 95. 17; 96. 10; 98. 16; 99.

24; 114. 6, 21.amixvos 120. 13.dpiffKai 122. 19.'ApiaTori\T)s 122. 7, 30.

296

dppovia (see note on 95. 5) 95.10; 115. 9; 116. 9; 118. 15;126. 9, 11; 127. 23; 135. 5;139. I, 3, 9, 12; 142. 19; 154.22; 155. 15; 160. 5, 16; 163.19; 164.8.

dp'noi>tic6s 123.19 ; 180. 7 ; 134. 24.appovticSs, 6 95. 18; 96. 12 ; 98.

• 19; 101. 1 ; 119. 15; 128. 10,13; 131.13-

dpnovucij, fj 95. 5; 101. 11; 126.3 ; 130. 1; 134. 10.

appovucd, T& 123. 7.dp^6TT<o 104. 2 ; 107. 23; 110. 8;

133.17, 18,19; 139.1; 147.13.dpxdiKos 115. 2.

h 98. 13; 108. 14; 119. 4;123. 4; 124. 11, 27; 131. 5;184. 19, 20; 145. 2; 146.21;147. 21.

dpxo€iSjjr 134. 22, 25.apxoiuu 101. 13; 109. 16; 126.

11; 134. 26; 135. 1; 142. 7.darpaPfc 133. 6.aarpoXofia 122. 13.davpfieTpos 115. 23.dovn<po>vos 120. 21; 144. 20.davvBeros (see note on 108. 22)

passim,drafta 99. 4.drotrla 131. 19.OTOJTOS 131. 14, 21; 182. 12, 13;

140. 17.av\icu 112. 16; 180. 5 ; 134. 2, 3.ouA.oiroifa 134. 8.av\6s (see note on 112. 13) 112.

13; 128. 17, 18; ISO. 4; 132.12,16; 133.3,5,7; 134.1,4,7.

ai£6.V(a 106. 20; 112. 2; 137. 14.avâts 97. 16; 107.17 ; 119. 11;

138. 18.avrdpicris 100. 13; 123. 16; 127.

3 ; 144. 21; 145. 10.bpaipea 132. 25; 146. 7, 8, 19;

147. I.dânjs 103. 10.bpiriiu 100. 7; 108. 25.dipiKvioftat 105. 24; 115. 9, 16;

137. 15-&plari;iit 114. 13; 115. 14; 128.

32; 133. 22.

INDEX

d<popi'f<D 98. 3 ; 108. 5 ; 109. 24 ;111. 2, 20, 22 ; 118. 8,15 ; 118.13, 18; 143.14; 144.I3J 146.24; 151. 11 ; 164. 17.

a<poptoii6s 111. 3.axpr)0Tos 129. I ; 145. 13.&f>vXos 182. 5.

foi 122. 4./3opur (' low' in pitch) passim.tfa/wTTjr 97. 11; 108; 104; 105;

106./3cj8aio'iu 133. 7 ; 134. 4.0€A.TIW 122. 3 ; 123. 3, 7.

xiis 101. 4.

yiveais 129. 19.7«'os (see Intr. A, § 6) passim.7€<U/t6Tp7JS 124. 22.yeaifieTpla 122. 13.y\vxalvai 115. 6.yvaiplfa 127. 7.yviiptfios 105.4; 106.13 ; 107. 20;

113. 7; 131. 12; 135. 10, 16.ypi/i/M 119. 6, 8 ; 128. 2, 3.7pa/if»iJ 124. 21.7P<£<p 96. 2.Siaftaivw 102. I, 16.Sicfyvwffts 100. 14; 127.4; 139. 7.Siaypafifia 95. 21 ; 96. 2 ; 101. 6 ;

119. 16; 124. 20.8ia(evyvviu 109. 17; 149. 1, 15;

150. 14.Sidfeufis ('disjunction'; see Intr.

A, § 12) passim.Staipeais and Siaiplu passim.Sicuae&vonai 107. 5 ; 126. lo, 13 ;

130. 18 ; 131. 4.Siaictvos 118. 5.SiaxpiPSa 108. 14.$ia\lyoiuu 96. 21; 102. 20, 25;

110. 7 ; 119. 7 ; 128. 3.SiafutpT&vio 99. 21 ; 110. 18 ; 180.

7 ; 182. 9, 23 ; 136. 19.

iva 122. 19; 133. 14; 189.22 ; 140. 1, 6.

Siavoico 182. 5.Siivota 124. 16, 18; 126. I ; 129.

18.Stcmopia 140. 21.Siaaa<piw 107. 4.Siaoicoiria 111. 16.

1 Siaaraais 97. 15 ; 106. 14, 21;107. I, 9,12,15; 112.12 ; 128.24.

SiiaTrjfia (' interval') passim.StaaTTinaTiicSs passim.8iao<£i(a> 133. 15.Sioreiva 112. 11; 125. 6.

co) 102. 19.(see Intr. A, § 6) passim,

pw 115. 8 ; 126. 10.Sta(pv\a.TTOi 119. 13.Sia<poiv4<u 136. 12.8ia<p(uvia 111. 16.Sicupavos (see note on 108. 21)

passim.8U[eini 96. 14; 101. 21; 103. 13.

103.12 ; 106.18; 184.

px 97. 19; 101. 16; 106.13; 126.3; 142.2.

Sieais (any interval smaller thana semitone) passim.

Si X1" 119- 18.Snjyio/mt 122. 7.Swpata 127. 10.fitopifa passim.Sin\aows 117. 7; 120.8; 187. 14.$iir\ovs 109. 20, 22.hirovot passim.5i'Xa 98. 22.8<5fa 96. 12 ; 104. 24.So(<L£a 108. 25.Svva/ns 95. 6 ; 96. 14; 113. 4 ;

124. 18, 23; 125. 11; 127. 5,9 ; 180. 24; 181. 2, 7; 188.10; 140. 1; 159.18, 20.

$ai$eKa.Ti]iiApiov 117.dwpws 128.

iia 159. 15.kyyiyvoiuu 98. 8.477OT 115. 9, 13 ; 120. 7.iOifa 124. 20, 23.

297

ARISTOXENUS

eTSos 97. I ; 189; 150. 9, 16, 19 ;159; 164.

liearepweev 121. 8.cicKfinai 95. 12 ; 135. 6, 16.iKK\ivw 124. 4.i P 108. 7.

vm 124. 4 ; 150. 24, 25./ ^ ('violating the laws ofmelody') passim.

iKTt]/t6piov 117. 7.Jd 97. 16; 188. 18.

w 108. 12.(' musically legitimate *)

passim.l/iireipos 123.13 ; 124.12 ; 126. 14.hinr'nrrtti 134. 27 ; 150. 24; 160. 3.Ipupaivoftai 139. 15, 17.! i f 102. 18 ; 154. 1 ; 156. 9.

/js 108. 14.i (see Intr. A, § 6)

passim.Ivei/u 96. 23.evepyiw 133. I.evlarrjfU 102. 10.cwoia 95. 20; 98. 12.evreiva 133. 13.4rT<5r 185. I.iytrnipx<" 180. 20.((aSwaria 107. 3 ; 119. 23.e(alperos 141. 4.k}api6nloi 99.19, 24; 100. 8 ; 124.

10; 127. 19, 21, 24.ifcr&fa 99. 22; 115. 21; 146. 22.?fir98. 7; 123. 19.Ifopifo) 115. 5.!mi7a< 114. 16; 115. 1.ivaywyq 97. 23; 144. 4.iTroXXaTTo; 115. 14; 150. I, 12.kiravA.ya> 147. 9.knatpaojiai 98. 2.lir<0f$» 124. 19.em0\iirio 126. 12; 159. 3.'EmySvetoi 97. 6.

96. 2.105. 17.

!m7roXijs 164. 7.Mtnceif/is 111. 18; 127. 17, 23.hm<TKOttio> 96. 11; 98. 25; 108.

22; 107. 19; 114. 7; 130. 18;160. 2.

ioj 132. 3 ; 133. 10.

298

7ifir) 95; 101.11; 180. 8,15 ;131. 16, 20 ; 134. 18 ; 159. 15.

Mraais (see note on 103. 16) 97.10; 103; 104; 106. 11; 114. 9.

eirnelvw 103. 13 ; 104. 3, 4 ; 110.6 ; 115. 16; 123.23; 132. 19;133. 1; 137. 16.

'EpaToicKTJs 98. 21; 99. 18.epyov 181. 18 ; 132. 1, 2.epianuds 122. 17.ep/irjvda 108. 14.evBcu/iovia 122. II.cvSr;\os 114. 21.eietas 125. 6 ; 159. 19.evdvs 121. 9 ; 124. 21, 23.evKaTa<pp6yt]Tos 123. 14.

i 96. 12.

Za«ui/&or 127. 25.

JjOos 115. 10; 128. 7, 10; 181.13.jjkticta. 112. 20.ijHtpa 128. 10 ; 133. 13.•flluiMos 116.14; 117 ; 141; 148.

4. 5 ; 155. 7-tf/uovs 118. 10; 115. 14 ; 116. 1 ;

136. 14, 15 ; 146. 22; 147. 10,1 2 .

ijfuToviatos 142 ; 148. 4, 12.f)iur6viov passim.•fipijiia 104. 1 3 ; 105. 1, 12, 23;

118. 21.•rjpe/ia 105. 13, 16, 20, 25.•qpnoopivos (see note on 95. 3) 97.

23 ; 110. 8, 22, 23, 25 ; 111. 8 ;125. 25 ; 129. 14 ; 132 ; 133 ;184. 5 ; 145. 3 ; 151. 18.

/ f a i 151. 3 ; 158. 22.0avna<TT6s 99. 3 ; 122. I I ; 183. 9,

2 1 .Beats 99. 17 ; 145. 5 ; 156. I, 24;

159. 18.6ea>pt<v95. II, 14; 98. 4 ; 101.18;

111. 4 ; 112. 21; 124. 18; 127.12, 15.

$eaipnriic6s 95. 7.eioipla 95. 8; 101. 10; 123. 22 ;

124. 14 ; 180. 4.

130. 11, 12.

Elementos de Armonía - Aristóxeno de Tarento [m-a-000274-f3] ()

Documents

research libraries

tetrachords

successive

aristides

disjunct tetrachords

upper boundary

sharp diatonic

assumed notes