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Yale University Department of Music
The Number Seven in the Theory of IntonationAuthor(s): Eric
RegenerSource: Journal of Music Theory, Vol. 19, No. 1 (Spring,
1975), pp. 140-153Published by: Duke University Press on behalf of
the Yale University Department of MusicStable URL:
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140
THE NUMBER SEVEN
IN
1 Introduction
Since the acceptance of the senario of Zarlino, the number 7 has
been the stepchild of intonation theory. In theory textbooks the
seventh partial of the fundamental C is usually indicated as Bb,
with the disapproving footnote "out of tune" or "slightly flat."
The best-known theorists are not sure what names to use for ratios
involving the number 7. Hindemith [1940, p. 105] suggests that the
augmented fourth can be either of the ratios 7:5 or 10:7, implying
that the former ratio could be either an augmented fourth or a
diminished fifth. Barbour [1953, p. 129] speaks, similarly, of "the
augmented sixth (that is, the minor seventh, 7:4)."
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141
THE THEORY OF
INTONATION
ERIC REGENER
One source of the theorists' difficulties is a failure to
dissociate pitch notation from its now standard interpretation in
twelve- tone tempered intonation. The enharmonic equivalence of the
diminished second and the perfect prime is a unique feature of this
particular intonation system and is not inherent in the no- tation.
In talking about intonation systems in general, the as- sumption
that enharmonically equivalent notes are the same is therefore
bound to lead to confusion. Such difficulties are easily overcome
by means of a consistent and objective mathematical framework such
as that of Regener [1973], in which the product of two
frequency-ratios always corresponds to the sum of the corresponding
intervals without enharmonic "cheating," and in which the intervals
of the notation are considered as purely
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142
abstract entities without any a priori assumption as to their
pitch representation. Such a framework imparts a new useful- ness
to the notation, since it frees the notes from an invidious
identification with any particular intonation system.
In a consistent system of just intonation, it is easy to
determine what notated interval corresponds to a given
frequency-ratio, provided the ratio contains only powers of 2, 3,
and 5 as fac- tors. For example, the ratio
9 3 3 4 5 2 ' 2 ' 5
corresponds to an interval of two fifths up, followed by a major
third down, or an upward minor seventh (see Figure la). The
ratio
16 2 2 -~-
2 2 ? - 3
corresponds to the same interval (Figure lb); but the ratio
125 2 2 2 5 5 5 72 3 3 *4' 44
corresponds to an augmented sixth and to no other interval
(Figure Ic).
Even in a consistent system, the number 7 poses areal dilem- ma.
Given any ratio including 7 as a prime factor (and no other prime
factors other than 2, 3, and 5), the corresponding interval will be
uniquely determined, provided we can assign an interval with
certainty to any one ratio involving the number 7, say the ratio
7/4. The hesitancy of the theorists is based here on a real
uncertainty, not necessarily due to the confusion engendered by
enharmonic equivalence. It is not clear, as it is with the more
fundamental ratios 3/2 and 5/4, what interval is "meant" by the
ratio 7/4.
We propose to show that this dilemma is susceptible to a con-
vincing abstract solution using only the simplest assumptions as to
the nature of just intonation. The ratio 7/4 will be seen to
correspond best, in fact, to the interval of an augmented sixth.
Before assembling the evidence, we spend a section developing the
necessary mathematical background. This sec- tion is a digest of
the relevant technical portions of Regener [1973] with slight
differences in symbolism and, in the author's opinion, a
considerable gain in clarity.
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143
FIGURE
1
a :b c
iiT +t-i +L iF I ^
, -
^ | ^ "
^ | , ^ (~~~~~~~~~momumoo
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144
2 Background
Any notated interval can be described as a sum of k upward
octaves and q upwardperfect fifths, where k and q are integers
(which may be negative). For example, a downward major third is the
sum of 2 octaves and -4 upward (i.e., 4 downward) fifths, and
anupward major third is the sum of -2 octaves and 4 fifths. It will
be convenient to write these two numbers as an ordered pair (k, q)
representing the interval. For example, an upward octave is (1, 0),
an upward perfect fifth (0, 1), an upward major third (-2, 4), and
so forth. It is clear, con- versely, that every orderedpair of
integers determines a unique interval: thus, we can use the two
designations interchangeably.
An important function of an interval i = (k, q) is its "diatonic
span" ds(i), defined as
ds(k, q) = 7k + 4q.
It is easily verified that ds(i) is the number of lines and
spaces (diatonic steps) traversed between the two notes of any
repre- sentation of the interval on the staff; it is positive if
the second note is above the first, negative if below, and zero if
the inter- val is a perfect prime (0, 0) or a sum of augmented
primes (-4, 7). The sum of two arbitrary intervals (k, q) and (1,
r) is defined in the obvious way:
(k, q) + (1, r) = (k + 1, q + r).
Note that the representation of an interval in Regener [1973] is
not (k, q) but (d, q), where d = ds(k, q) is its diatonic span. The
former representation is chosen here since it simplifies the
mathematics.
An "intonation system" is a function assigning a unique fre-
quency-ratio to every interval, in such a way that the sum of
two intervals always corresponds to the product of their fre-
quency-ratios. An intonation system in which the octave has the
frequency-ratio 2 is called "regular". It is convenient to take
logarithms to the base 2: an intonation system is then a function,
say "lf", such that the sum of two intervals always corresponds to
the sum of their log frequency-ratios.
(1) lf(il + i2) = lf(il) + lf(i2),
and such that the octave has log frequency-ratio 1:
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145
(2) lf(l, 0) = 1.
Regener [1973, pp. 90-93] proves that a regular system is
completely determined by specifying the log frequency-ratio of the
upward perfect fifth: this log frequency-ratio is called the
"determining constant" c. For example, in ordinary twelve- tone
intonation the fifth comprises 7 out of 12 equal steps whose sum is
the octave (log frequency-ratio 1), and the determining constant of
this system is therefore 7/12. To derive the ex- pression for the
log frequency-ratio of any interval (k, q), we split it into its
octave- and fifth-components:
lf(k, q) = lf(k, 0) + lf(O, q) = k lf(l, 0) + q- lf(, 1),
both by (1); then by (2) and the definition of c, we have
(3) lf(k, q) = k + qc.
The system of just intonation, as ordinarily defined (without
the number 7), consists of a set of rational ratios (quotients of
integers) together with a rule specifying what notated interval
corresponds to a given ratio. The ratios ordinarily used are all
those which involve only the numbers 2, 3, and 5 as factors, that
is, all ratios of the form
2u 3v 5W,
where u, v, and w are integers (which may be positive, nega-
tive, or zero). We shall designate the set of all ratios of this
form by the symbolQ5. The rule, based on the harmonic ser- ies,
specifies that the ratio 2/1 (u = 1, v = w = 0) corres- ponds to
the octave, the ratio 3/2 = 2-1 * 3 to the perfect fifth, and the
ratio 5/4 = 2-2 5 to the major third, and that the product of two
ratios corresponds to the sum of the two corres- ponding intervals.
Defining "int" to be the function which as- signs the interval (k,
q) to a frequency-ratio belonging toQ5, we can write the given
rules in mathematical terms as follows:
int(2) = (1, 0), the upward octave; int(3/2) = (0, 1), the
upward perfect fifth; int(5/4) = (-2, 4), the upward major
third;
and the "rule of consistency", int(r ? s) = int(r) + int(s).
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146
We derive the general expression for the notated interval
corresponding to a frequency-ratio in Q5 by finding out what
interval corresponds to the ratios 3 and 5 (that for 2 is given),
using the rule of consistency:
int(3) = int(3/2) + int(2) = (1, 1); int(5) = int(5/4) + int(2)
+ int(2) = (0, 4).
Therefore the general expression, again by the rule of consis-
tency, is
int(2u 3V 5W) = int(2u) + int (3V) + int(5W) = u ? int(2) + v
int(3) + w int(5) = (u, 0) + (v, v) + (0, 4w) or
(4) int(2u 3v 5w) = (u + v, v + 4w).
For example, the interval corresponding to the ratio 9/8 is
int(9/8) = int(2-3 32 50) = (-3 + 2, 2) = (-1, 2),
an upward major second; corresponding to the ratio 10/9 we
have
int(10/9) = int(21 3-2 51) = (1 - 2, -2 + 4) = (-1, 2),
the same interval. The quotient of these two intervals is the
ratio
9 9 81 2-4 34 5-1 . 1-0 2-
the "syntonic comma". Any power of this ratio corresponds to the
notated interval of a perfect prime:
(5) int(81r/80r) = int(2-4 34r 5r) = (-4r + 4r, 4r - 4r) = (0,
0),
for any integer r.
Clearly, any two frequency-ratios in Q5 whose quotient is a
power of the syntonic comma will correspond to the same in- terval.
The ordinary theory of just intonation specifies, rough- ly, that
the more "musically useful" of two such ratios is that
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147
involving the lowest numbers when in lowest terms. For ex-
ample, the ratios 81/64, 5/4, and 100/81 all correspond to the
major third, but 5/4 is the most useful; of the ratios 729/640,
9/8, 10/9, and 800/729, all of which correspond to the major
second, 9/8 and 10/9 are the most useful.
Regener [1974] shows that, if considerations of musical context
are excluded, it is possible to define which is the most musi-
cally useful, in this sense, of all the frequency-ratios corres-
ponding to the same interval (k, q). If the ratios are close to 1,
the most useful, roughly speaking again, is one of the two ratios
whose value is closest to
2k 5q/4
This is the frequency-ratio for the interval (k, q) in strict
meantone intonation; it can be verified that the frequency-ratio
for the major third (-2, 4) is precisely 5/4. Meantone intona- tion
is a regular intonation system in which the determining constant c
= log2 5/4 = (log2 5)/4.
Let us defineQ7 to be the set of all rational ratios having no
prime factor greater than 7. Our task will be to investigate the
possible ways of extending the explicit correspondence be- tween
rational ratios and notated intervals fromQ5 to Q7: that is, to
ratios involving powers of the prime number 7 as well as of 2, 3,
and 5. As our standard of ambiguity in distinguish- ing different
ratios we shall use the syntonic comma: this de- cision
deliberately excludes perceptual and psychological fac- tors from
what is meant to be a purely abstract discussion.
3 Intervals corresponding to the ratio 7
We want to choose a notated interval (k, q) which best corres-
ponds to the ratio 7:
int(7) = (k, q).
The rule of consistency implies that this choice will determine
the interval corresponding to any ratio inQ7 by the obvious ex-
tension of (4):
(6) int(2u 3v 5w 7x) = (u + v + kx, v + 4w + qx)
for any integers u, v, w, and x.
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148
But the interval (k, q) already corresponds to a frequency-ratio
in Q5. In fact, it corresponds to a class of such ratios whose
quotients are powers of the syntonic comma. Suppose that s is one
of these ratios: that is, s is a ratio in Q5 such that
int(s) = (k, q).
This implies that
(7) int(s/7) = (0, 0),
the perfect prime, in analogy to the relation (5) for the
syntonic comma.
We shall use the name "seven-comma" for any ratio of the form
s/7, where s is a rational ratio involving only the num- bers 2, 3,
and 5 as prime factors. Making a seven-comma s/7 correspond to the
interval of the perfect prime, as in (7), automatically implies
that
int(7) = int(s).
We shall now see that a useful choice of the interval int(7) de-
pends on the properties of the "class" of seven-commas s/7 for
which int(7) = int(s), rather than on any particular ratio.
For a minimum of ambiguity in the recognition of intervals, we
want s/7 to be relatively close to 1 (that is, zero cents): by our
earlier assumption, it should not be much further from 1 than
81/80, the syntonic comma. Musical usefulness requires in addition
that the ratio itself involve numbers as low as pos- sible.
The latter condition implies that s must be among the most
useful ratios of its class. For example, the seven-comma
14348907/14336000 = 2-14 315 5-3/7 = 1.558 cents
is much closer to 1 than the syntonic comma of 21. 506 cents.
The corresponding interval is (-14 + 15, 15 - (4 ? 3)) = (1, 3), a
major sixth plus two octaves, for which the ratio involving the
lowest numbers is 4 - 5/3 = 20/3: thus the most useful seven-comma
corresponding to this interval is 20/21. This ratio, though it
involves smallnumbers, is about -84.467 cents, almost a semitone.
It will turn out that this interval is not one of the best choices
for int(7).
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149
In order to see just what the desirable choices are, we spread
our net relatively wide. We ask for all the seven-commas smaller
than a quarter-tone (+50 cents) and involving numbers of less than
six figures. A computer search reveals the possi- bilities listed
in Table 1. The seven-commas r = s/7 are given in fractional form,
in cents (1,200 times the\base-2 log- arithms; u, v, and w are the
powers of 2, 3, and 5 appearing in s = 2u 3v 5w; k and q are the
octave - and fifth-components of the interval int(s) (k, q). The
columns headed d and int display the diatonic span ds(int)s/4)) and
the name of the interval int(s/4) obtained by subtracting two
octaves from int(s); if s/ 7 is equated to a perfect prime as in
(6), this interval is int(7/4). The ratios corresponding to the
same interval are grouped together.
The last column, headed "meantone," presents the values t/7 in
cents for each interval, where
t = 2k 5q/4
is the frequency-ratio in meantone intonation corresponding to
the interval (k, q). Recalling that the most useful ratios s
corresponding to this interval are roughly those whose values are
nearest t, we see that the seven-commas r = s/7 involving the
lowest numbers are those near t/7. The quantity t/7 there- fore
shows where these lowest-numbered seven-commas are to be found,
even if they exceed a quarter-tone and therefore do not appear in
the table. It shows, for example, that the ratio 65536/65625 is the
most useful of the seven-commas s/7 for which int(s/4) is a triply
diminished octave, while 2187/ 2240 is far from being the most
useful of those corresponding to a major sixth (cf. the example
above). We are using the word "useful" here in its defined sense of
"involving the lowest numbers," of course: the most "useful" is not
necessarily the most useful for all purposes.
4 Superiority of the augmented sixth
Table 1 shows seven intervals which, within rather wide limits,
could be taken to correspond to the ratio 7/4. Considering this
ratio rather than 7 simplifies naming the intervals without
changing the relationships. To be precise, if a given seven- comma
r = s/7 is taken to correspond to the perfect prime, then the
interval i on the same line corresponds both to 7/4 and to s/4,
where the latter ratio is in Q5. The best intervals i have ratios
s/4 involving low numbers and close to 7/4. The column headed "r
(cents)" in fact displays the difference in cents between s/4 and
7/4, since
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150
s s/4 7 7/4'
We now examine the intervals in the table on their merits.
Though the ratio for the triply diminished octave is quite close
to 7/4 and involves the lowest numbers of its class, it involves
the highest numbers of all the ratios in the table. The other three
intervals possessing one ratio apiece combine undesirably large
numbers with considerable wideness of the mark.
The diminished and the minor seventh are both more reasonable
candidates. While their most useful seven-commas (512/525 and
36/35, respectively)are nearly as large as a quarter-tone, each
possesses another seven-comma (864/875 and 64/63, respectively)
which involves numbers of comparable size and which is closer to
zero cents (though still at least slightly larger than the syntonic
comma). Of these, the minor seventh seems preferable because the
numbers involved in its most useful seven-commas (64/63 and 36/35)
are even smaller than those of the syntonic comma.
Of the two most useful seven-commas for the augmented sixth, one
(125/126) is less than 1 and one (225/224) is greater. This means
that the corresponding ratios, 125/72 and 225/128, bracket the
value 7/4. In fact, as the last column shows, the meantone value
for the augmented sixth is different from 7/4 by only just over
three cents. The numbers involved are quite small as well,
considering that no other seven-comma com- parably close to 1
involves numbers less than 4,000. It seems reasonable, therefore,
to consider the augmented sixth as the best interval to represent
the ratio 7/4 in an extended system of just intonation.
Since for this choice we have
int(7) = (-3, 10),
the interval corresponding to any ratio in Q7 is shown by (6) to
be
(7) int(2u 3v 5w 7x) = (u + v - 3x, v + 4w + lOx)
for any integers u, v, w, and x. This function produces some
interesting small-number ratios for intervals with a relatively
wide span in fifths. For example,
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151
TABLE
1
r(cents) log2 r
-2.34948 -.00195789 16 -1 -5
-43.40834 -.03617361 9 -1 -2
-21.90205 -.01825170 5 3 -3
-.39576 -.00032979 1 7 -4
5.75780 .00479817 10 -6 1
27.26409 .02272008 6 -2 0
48.77038 .04064198 2 2 -1
-41.45461 -.03454551 -6 7 -1
-35.30106 -.02941755 3 -6 4
-13. 79477 -.01149564 -1 -2 3
7.71152 .00642627 -5 2 2
29.21781 .02434817 -9 6 1
35.37137 .02947614 0 -7 6
-33.34734 -.02778945 -12 2 5
u v w k q d int meantone (cents)
15 -21 7 ddd8 3.02709
8 -9 6 d7 -38.03177
4 -2 6 m7 38.01723
1 3 5 M6 -79.09062
-3 10 5 A6 -3.04163
-7 17 5 AA6 73.00737
-10 22 4AAA5 -44.10048
Ratio r
65536/65625
512/525
864/875
4374/4375
5120/5103
64/63
36/35
2187/2240
5000/5103
125/126
225/224
3645/3584
15625/15309
28125/28672
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152
int(12/7) = (6, -9), a diminished seventh; int(14/9) = (-4, 8),
an augmented fifth; int(7/5) = (-3, 6), an augmented fourth.
Svensson [1950] gives a diminished fifth for the last ratio.
It is easily verified that the class of small-number ratios in
Q7 corresponding to a given interval has a much narrower range
under equation (6) than under the analogous equations derived from
any of the other possibilities in Table 1. In fact, an examination
of all the small-number ratios in Q7 (up to a certain limit) and
their corresponding intervals turns up an interesting relationship
with 31-tone intonation, which willbe pursued in a later article.
We may remark, at least, that the augmented sixth has a ratio in
31-tone intonation remarkly close to'7/4: its logarithm is 25/31,
or roughly 0.80645161 (967.74194 cents), while log2 (7/4)is about
0.80735492 (968.82591 cents): the difference is only 0.00090331
(1.08397 cents), very nearly too small to be noticeable.
ACKNOWLEDGMENT
The author gratefully acknowledges the assistance of Grant
#S72-0087 and #S73-1210 from the Canada Council for the Arts,
and the stimulation of conversations with Bengt Hambraeus (who
kindly provided the reference to Svensson) and John Rea, both of
McGill University.
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153
BIBLIOGRAPHY
Barbour, J. Murray. TUNING AND TEMPERAMENT: A HISTORICAL SURVEY.
East Lansing, Michigan: Michigan State College Press.
Hindemith, Paul. UNTERWEISUNG IM TONSATZ. Mainz: B. Schott's
Sohne, 1940.
Regener, Eric. PITCH NOTATION AND EQUAL TEMPERAMENT: A FORMAL
STUDY. Occasional Papers, No. 6: Music. Berkeley: University of
California Press, 1973.
Regener, Eric. "A 'Lucky' System of Intonation. " Unpublished
article, 1974.
Svensson, Sven E. "Vart Tonssystem och dess Temperaturer 1."
SVENSK TID- SKRIFT FOR MUSIKFORSKNING 32:150-186.
Article Contentsp. 140p. 141p. 142p. 143p. 144p. 145p. 146p.
147p. 148p. 149p. 150p. 151p. 152p. 153
Issue Table of ContentsJournal of Music Theory, Vol. 19, No. 1
(Spring, 1975), pp. 1-182Front Matter [pp. 1 - 139]Some Models of
Unity in Musical Form [pp. 2 - 30]A Sixteenth-Century Hexachord
Composition [pp. 32 - 45]Analysis Symposium: Webern, Orchestra
Pieces (1913) Movement I ("Bewegt") [pp. 47 - 64]Some Comparisons
between Intuitive and Scientific Descriptions of Music [pp. 66 -
110]Choron, Ftis, and the Theory of Tonality [pp. 112 - 138]The
Number Seven in the Theory of Intonation [pp. 140 - 153]Book
Reviewsuntitled [pp. 154 - 160]untitled [pp. 161 - 171]
Letters: The Structure of Atonal Music [pp. 172 - 173]Books and
Articles [pp. 174 - 179]Corrections to Seay, Gilbert, Morris and
Cogan [p. 180]Back Matter [pp. 181 - 182]