Perkins PNM Summer1965

7/30/2019 Perkins PNM Summer1965

1/12

Note ValuesAuthor(s): John Macivor PerkinsReviewed work(s):Source: Perspectives of New Music, Vol. 3, No. 2 (Spring - Summer, 1965), pp. 47-57Published by: Perspectives of New MusicStable URL: http://www.jstor.org/stable/832503 .

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2/12

FORUM: NOTATIONNOTE VALUES

JOHN MAC IVOR PERKINS

THE EX PLORATION of unconventionalrhythmic deas, and itsattendant roblems fnotation nd realization, as become a deepcon-cernformanyliving omposers. temming erhapsfrom new under-standing fthe experience nd perception ftime,1 uch ideas usuallyconstituten emancipation f ntricate r "irrational" uration elation-ships hroughmultiple imultaneousartificial ivision,"rregularmeter,"rhythmicmodulation,"ncommensurableempochangeand/or nalognotation.Emphasis is placed on the expansionof resources nd onflexibility,ftenat the expense of traditional ohesive,unifyingndorganizing orces,nd the musical results re thusroughly nalogoustothemusicalresults fthose arlierdevelopmentsn thearea ofpitch ela-tions alled by Schoenberg theemancipation fdissonance."But whilewe have nowreacheda temporary armonicplateau (despitepersistentand persuasive oicesofdissatisfaction)2n theacceptanceof welve-toneequal temperament, hether erially rganizedor not, t is evident hattheemancipation frhythmicissonance sfarfrom omplete. ew com-poserswoulddeny hat herhythmic,venmore urelyhan heharmonic,aspectof our musical anguage s currentlyn a stateofrapidtransition.In this ight, hefactthatrecent cores mploy confusingariety fnotation ystemssnotsurprising.he mostprominentystems3all ntotwo categories,reflecting two-pronged ssault on the mechanicalsymmetry,ndlessbipartite ivisions, tickiness,"4metrical igidity,ne-at-a-time empo imitationnd poverty f duration nd speedrelation-shipswhich ervedMozartsowell,but which eemnowsointolerablyndirrelevantlyestrictive.n analog notation, he horizontaldistancebe-tweenthe noteheads or other vent ymbols) s strictlyroportionalothe intended ime-differencef attack or other vent, uch as dynamic

1RobertErickson,Time-Relations,"ournalfMusicTheory,inter 963,pp. 174-92.2For a recent xamplesee BenJohnston,Scalar Orderas a Compositional esource,"PERSPECTIVES F NEWMUSIC, pring 1964, pp. 56-76. The ideas about rhythm n this articleand inErickson,p. it.,reclosely elevanto thepresentiscussion.3A convenienturveys included n Kurt Stone,"Problems nd MethodsofNotation,"PERSPECTIVES OF NEW MUSIC, Spring 963,pp.9-31.4Erickson, op. cit. . 47 *

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3/12

PERSPECTIVES OF NEW MUSICchange,imbrehange,rrelease). ence hename proportionateota-tion" or heseystems,namewhichwillnotbe usedhere ecause f hepossibilityf onfusion:onventionalotationresentsurationalropor-tions ore xplicitlyhandoes nalognotation,nd to thissduemanyfthe ractical isadvantagesf llanalog ystems.5he useof onventionalnotationwithnumerical,erbal, r other ymbolicxtensionsntailssome isadvantagesf ts wn,however,hemost bviousfwhichstheclutteringfthepagewith profusionf ignswhich reslow o read.Thiscompares oorlywith hebeautifulimplicityfthebetternalognotations-avisual nd conceptualimplicityhichmayormaynotlead to more fficienterformance.he lessobvious imitationsf on-ventionalotationremoremportant,6nd shouldfpossible e ana-lyzed nsomedetailbythose tudentsndyoung omposers hofindthemselvesonfrontedith choice.An easily ccomplishedutnotentirelyrivial irsttep n suchananalysiswouldbe an inventoryfavailable onventionalotevalues.In theory,n infiniteumber fdurationalues, orrespondingo theinfiniteumberf ational ractionsf heunit urationna givenempo,canofcourse e expressednconventionalotation, hen tissupple-mentedyproportionymbolsuch s thosemployed ytheDarmstadtcomposers.n practice, relativelymallfinite umberaboutfiftyerdurationctave-e.g.between quarter ote nd an eighth ote) s nfact vailable,wing othedifficultiesfreading largenumber fflagsand executing roportional odificationsnvolvingargenumbers. fthese,many reexceedinglyare.The limitationsfthe ccompanyingtable pp.50-51)havebeen hosen oapproximatehepracticalimita-tions t normalempi:dditivealuesnsixty-fourthsre ncluded,sareproportionalodificationsnvolvingumbersptoand ncludingifteen.Inaddition,few f hemoremportantalues esultingromhe ollow-ingnotationalroceduresavebeen isted:1) theaddition f rtificialdivisionsa tripletighthied o septupletighth,esultingna durationequal to thirteenwenty-firstsfa quarter ote; uchvalues refairlycommonnpractice,ut rerarelysed s beats r "counters"nrhyth-micmodulations);2) the imultaneousrotherwiseoordinatedmploy-ment f ompoundndsimplemetersthe ouble-dottedotted ote sedasa "counter"yCarternthe econdQuartet,mm. 38-39); 3) the seofartificialivisionsf higherrder hanfifteenthemost ommon fwhich sprobably 7:16); (4) thenestingf artificialivisionseyond5DiscussednStone, p. it.6Since at present onotationwillenable a "complex" hythmicatterno be readquicklyand easily, ven f, s Charles Wuorinen elieves, hedifficultiesf suchpatternsre largelycultural norigin ather han nherent;ee his"Noteson thePerformancefContemporaryMusic,"PERSPECTIVESF NEWMUSIC,Fall-Winter 964.* 48 *



4/12

FORUM: NOTATIONfive a triplet alfnotewithin septuplet, esultingn a duration f ix-teentwenty-firstsfa quarternote).At thispoint, word about the detailsof theparticularnotation on-vention employedin this list (and elsewhere) is unfortunatelyerynecessary, otonlybecausethepractice fcomposers as beenstrikinglyinconsistentn thisregard,but also because the citation f note valuesoutside a musical or metricalcontext-as forexample in note-valueequationsused nwriting hythmic odulations-imposes pecialcriteriainthe evaluation fnotation onventions.n particular,twould bemostdesirable hat, na given empo, ne and onlyoneduration alue berep-resented y each note-value ymbol, hesymbolbeingtakento includeits proportional r othermodifiers. his criterion oes notnecessarilyapplyinpracticalmusicnotation,' nd ofcourse tsconverses nottrue,for hevariety fsymbols vailable for heexpressionfeach durationsessential o theflexibilityfconventional otation.On the otherhand,the comparativeunexplicitnessf the Darmstadtproportion otation8does minimize henumber f uchequivalent ymbols, hich ssometimesadvantageous;and while it is undeniable that thisunexplicitness ayoccasionally ead toconfusionnpractice9and should nthese ituationsbe supplementedn the manner of Carter),there s no reasonwhyitshould lead in any situation o an actual ambiguityin the sensethatmore than one durationmayproperly e read for given ymbol). orthesereasons, he Darmstadtnotation s bestsuitedto thepresent ur-poses, nd isunderstoodnthefollowing ay:

duration(rx:Y' - durationn)n xwhere n) isanyconventionalincluding otted)notevalue.Forexample,theduration f quarternotewithin bracketmarked : 3 isalways qualto threefifthsfthe duration fan unmodifieduarternote nthe sametempo.For compactness,he number is frequentlymittedfrom hesymbol,nwhichcase it s assumed here)to be equal to thenextpowerof two smaller han x.10Thus:7The notation sedby EasleyBlackwood nMusic forFluteand Harpsichordsefficientand unambiguous ithoutonformingothis ule: thenumberwhich ppears fterhe olonin each case referso thenumber fbeats ccupiedbythebracketed otes. he meaning fparticular ymbolhusdepends n itsmetrical ontext. he same s true fHindemith's ota-tion f rtificialivisionsncompoundmeterseebelow,n. 10).8Stone, op. cit.9See Elliott arter, Letter o theEditor,"JournalfMusicTheory,inter 963,pp.270-73.10This abbreviation ulecorrespondsxactlywiththe methoddescribedn Hindemith'sElementaryrainingorMusicians,. 116,for imple uplemeters. arterhasalways mployed,and has recentlyrged hegeneral doption f, different ethoddescribednPrinciplesfMusicTheoryyLongy-Miquelle;eeCarter, p.cit.) ccording o which -7 = 7:8. The rulebehind hiswould eem obethatysassumed oequal thepower f wowhich sarithmetically* 49 *



5/12

duration,duration logarithmic- D Equivalent

.DOsymbols

- x00 1/1 2.000 1.000 1200.0 60 3-7

20/21 1.905 .9296 1115.5 6317 16/17 1.882 .9125 1095.0 4:5 8:1501 15/16 1.875 .9069 1088.3 64 "15:1415 5:7 6:2 r 14/151.867 .9004 1080.6 15 57 614:13 r r03 r 13/14 1.857 .8930 1071.7 7-' 7:6- r7.

04 13 12 12/13 1.846 .8846 1061.4 65 13--2:11 9:11 3-51211 11/12 1.833 .8745 1049.4 9 3 306 110 10/11 1.818 .8625 1035.0 66 1107 5"9 9/10 1.800 .8480 1017.6 53 5- 508 9' 8/9 1.778 .8301 996.1 3-F 3 109 7/8 1.750 .8074 968.8 6:7 4:7 4:520/23 1.739 .7984 958.1 69

15:13 15 3 r5 5 -70 1513 13/15 1.733 .7935 952.2 15 , 3r-7:6 6/7 1.714 .7776 933.1 70 7 7:91311 r~32 3 11/13 1.692 .7590 910.8 1313 5/6 1.667 .7370 884.4 72 9"10 3-

314 11 9/11 1.636 .7105 852.6 1112 11-15 ~ 13/16 1.625 .7004 840.5 12:13 13-5:6-516 5 4/5 1.600 .6781 813.7 75

5 315/19 1.579 .6589 790.7 767:11 7-- 717 11/14 1.571 .6521 782.518 107 7/9 1.556 .6374 764.9 9 9 36:7- T 319 10/13 1.539 .6215 745.8 78 13"3 16/21 1.524 .6076 729.220 1 3/4 1.500 .5850 702.0 80 23 3- 4315:11 10:11--, 3-r-- 5 15'1 11/15 1.467 .5525 663.0 r3 1511, 11 11 lO--r2 8/11 1.455 .5405 648.7 3 T s23 r 13/18 1.444 .5305 636.6 12:13 r9:5 /7 1.429 .5146 617.5 844 ( 5/7 1.429 .5146 617.5 84Table 1*50 *



6/12

duration,duration logarithmic8 D, .

EquivalentD 1c D 1 symbols5:7 1514- 5-,25 7/10 1.400 .4854 582.5 15 5

26 13 9/13 1.385 .4695 563.4 13":27 " 11/16 1.375 .4594 551.312:11 8:11

28 11"15 15/22 1.364 .4474 537.0 88 11:10 11 r 1129 3 2/3 1.333 .4150 498.0 90 9 r-3j 21/32 1.313 .3923 470.815/23 1.304 .3833 460.0 9210:13 15:13 5- - 5---30 r 13/20 1.300 .3785 454.2 "31 79 9/14 1.286 .3626 435.1 7:6- 7- -7r-r32 11:7 7/11 1.273 .3479 417.5 r11. 133 5/8 1.250 .3219 386.3 96 6:5 4:5 13:1234 13 8/13 1.231 .2996 359.5 13:12

9:1"211,9 r-- 1 195 r 11/18 1.222 .2895 347.4 9F 12:11 95:3 5 , 10:9, -536 5:3 3/5 1.200 .2630 315.6 100 5 10:9

37 1113 13/22 1.182 .2410 289.2 11- =1138 6:7 7/12 1.167 .2224 266.9 9:7- 3- 33-139 13:15 15/26 1.154 .2064 247.7 104 13:10-, 13:12 r13-40 7- 4/7 1.143 .1927 231.2 7:.6-7S2:3 r:91 9/16 1.125 .1699 203.9 23 6-8:9r r42 9:10 5/9 1.111 .1520 182.4 108 6:5 - 943 111 11/20 1.100 .1375 165.0 15:11 5 r 54412/11.091125550.61 1:91145 12:13 13/24 1.083 .1155 138.6 9:13 346 13 14 7/13 1.077 .1069 128.3 13-- r-1347 14 15 15/28 1.071 .0995 119.4 112 7:5 7 748 5~ 8/15 1.067 .0932 111.7 15-- 3r 5 15' 17/32 .063 0875105.015/29 .034 0489 58.7 11600 1/2 1.000 .0000 0.0 120

Table 1 (Cont.)

"51.



7/12

PERSPECTIVES OF NEW MUSICr 3--= 3:2-5-- = 5:4

r-7-- = 7:410r9- = 9:8-11-- = 11:8- 13-1 = 13:810r- 15--i = 15:810- 17--= 17:16etc.

(Even-numbered alues ofx are superfluousn out-of-contextitationsofnotevalues,and dangerouslymbiguous ven n contextftheyvalueis not also given.)"1The accompanying able, then,presents he symmetricalcale-likearrayof durationvalues available to the user of conventional otation.The axis of ymmetrynthisparticular uration-octavealls etween otevalues number24 and 25, and would be represented y an irrationaldurationequal to \F times hedurationofthe eighthnote.Thus,forexample, r"and - are symmetricallylaced around thisvalue (andwithin he "octave" r tor), formingn exact analogyto PythagoreanF andG (inthe cale ofC) which resymmetricallylacedaround qual-tempered (and within heoctaveC to c).The durations isted n the table may also, of course,be presentedgraphically,nd for number freasons logarithmiccale is best uitedto such a presentation.The value of each duration n "cents"-thelogarithmic nit mostfamiliar o musicians-has been listed.) n Ex. 1a slide rule is illustrated,he scales of whichhave been plotted n thismanner,with note values presented n the two slidesand metronomecalibrationsand durationsn seconds)on thefixed aces.The settingfclosest ox,whethert s arger r smaller.n thecaseof3,6,12, tc.,which reequallyplacedbetween owers f wo, he "older"rule s ofcourse pplied.The "logic"ofthismethod s ustas sound s thatofthe"older"rule, nd theresultingisual mage sprobablymoremusicaland easiertoread.Unfortunately,heillustrationsor his rticlewerepreparedbefore hepublication fCarter's Letter."As for heabbreviation fartificial ivisionsn triple ndcompoundmeters,heCarter-Miquellepproach, hought tends o a proliferationfdots,appearsdistinctlyuperioro themethod escribed yHindemith,othfor hereasons ivenin his etter ndbecause tavoids he ntroductionf mbiguitynout-of-contextitations.11A trulyxcellent ercussionist,ho sexperiencedntheperformancef hemostntricatecontemporaryusic, ecentlyailed epeatedlyoreadthefollowing igures a series fnineequal notes:

3 6* 52 *



8/12

FORUM: NOTATIONtheslides llustrates he common emporatio3:4, as for xample nthefollowing emposhift:J = 108; -- ==3--; J= 72. In Ex. 2, a moreelaborateform f the samedevice illustrateshe unusualratio14:15; asubtle hythmic odulationnvolvinghis atiooccurs n Carter'sDoubleConcerto,m. 105: J= 105 (J.= 70); --7= -->;J 98. Simplecal-culations elating otevalues,tempi nd durations of ndividual vents,sections rcompositions),speciallyfthedata and requirednformationare approximations, ayoften e accomplishedmorequicklywith uchdevices thanbymentalarithmetic. he discovery f note-value quiva-lents, r nearequivalents,n twogiventempi s particularlyonvenient.The term irrational" s sometimes sed to designatebracketednotevalues. This is unfortunatend misleading,because all conventional(includingbracketed) urations re rationalfractionsin themathemat-ical senseoftheword)ofthe unitduration,which s a functionf empo.(A better name for the more intricatefractional alues is "artificialdivisions," ut it, too, is unfortunate.)12ruly rrational uration ela-tionshipsre ofcourseconceivable nd practical, venin a fixed empo,but in conventional otation heycan be expressed nly by using uchcrude, xtraneous evices s thefermata,r thevagueinstructiontemporubato".Truly rrational empoelationshipsre mostfrequentlyncoun-tered nconstant lowaccelerations, here,fmetronomicndications regiven, lose rationalapproximationso the intended rrational elation-ships regenerally otated.a3In analog notation, owever, otevalues will ingeneral e trulyrra-tional, rulyncommensurable.he attempt,n practice, o express on-ventional, ational,proportional,ommensurable ote values in purelyspacialnotationystemsnevitablyeadstogrossnaccuraciesnperform-ance. Ifunmodifiednalognotations held tobe useful t all, it mustbeassumed hatthetolerance orperformancerrornmusicwhich mployschieflyrrationalduration and temporelationshipss greater han inrationallyproportionedmusic. This assumption s probablycorrectn

12Why s3/5more rtificialhan3/4?The useof heword artificial"nthis ontextmpliesthatonlybinary and perhaps ernary) ivisions re natural;hismaybe true none sense fconventional otationand hence thequasi-justificationftheterm), utitsacoustic, sycho-logical, rmusical ruth asonly ccasionally eenassertedndnever rovenseeWuorinen,op. cit.).13There sa very imple xample nCarter'sVariations orOrchestraVariation ),wherethefollowing etronomeettingsppearonsuccessivemeasuresunder hegeneralnstruction"Accel.molto"):J = 80,96, 115,139, 166,201,J.= J= 80 etc.;theratios etweendjacentnumberspproximate3-.Anequal-temperedempo cale snotated,s a means f uggestinga steady empo lissando. oints epresentinguch series f empiwouldof ourse e equallyspaced on the logarithmicmetronomecales of Exx. 1 and 2. More complex nd subtlerapplications f he ameapproach otempo elationships-an pproachwhich scomplemen-tary nd antitheticalo "metricmodulation"-are notuncommonn Carter's ecentmusic;see,for xample, heSecondQuartet, . 55.. 53 *



9/12


10/12

I I I I0I I I IO10 1 0sec./32 .M.M. 3 69 84 92 1L610'uTM96T 19211 1312 1:109i 1513 3:11 119 r5-1 97 . rl, 7:5 13:9 11:15 10:13 1:7 :r 53 67 r7 910 1142 1 r15-r ri3r r r r r l rr 3 9 l 3 51 r "1 , r"1 ,.51, , ,27

r16:15 r137r r r28-7 r77 r-3 8:l3 7nI 31r5 9 r5 8:11 r3 r7 4: rI,-il ri3- 8R9 r drIr7 3 1r -S 15-e r.3 1--ir0 C,

r3,

rl -131 5-

14113:12 11:105 1 59133 : 10951 9:7 .1 S75 r3: 577-99; 1 11:13 1 01 I:1r3r: ipj, 7:5-1. r 15r "I5:117 53 R 7 ...r7]I:i r:i r -1"r% r3r76746 : 5 7:1 13:101511 5: 57 J15 7 9:11 1:13713 10112:13 16::5

sec.o io WgoImin:sec48sot+pe IpS T 6 5 4Ex. 2

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11/12

PERSPECTIVES OF NEW MUSICmanycases. Severalcompromise otations aveofcoursebeendevelopedand applied n an efforto derive he benefits fbothanalogandconven-tionalsystems,nd, althoughnoneofthesenotations as as yet ttracteda significantumber fcomposers,here s no doubtthat hebestof hemoffernprecedentedlexibilitynd versatility.The capacityto express rrational urationrelationshipsirectlyndsimply snotin itself he chiefvirtue fanalog notations. ny rrationalnote value can be approximated yone of theconventional alues istedin theabove table,with n error hat s almost ertainlyolerablemusi-cally,and veryprobablybelow the threshold fperceptibility,n mosttempi.On the otherhand,conventional otation oes in practice mposesubstantial imitationsn thevariety ffeasible atternsfduration, ndthese limitationsmay be whollycircumvented y recourseto analognotation.One simpleexample should illustratewhat is meant.The followingpitch-durationattern osesno notational roblems:

3 5Ex.3

A semitone ranspositionf thispresentationf a twelve-toneeries e-sults n a permutationr scramblingf tspitches:

Ex.4Iftheduration aluesoftheoriginalpatternweretobe scramblednthesameway sothat achpitchnthetranspositionould be associatedwiththesame duration s in theoriginal attern), heresultingeries fdura-tionswould be exceedingly ifficultopresentnconventional otation:14

14 his is not uggesteds an interestingariationechnique, or ompositionalpplication!The device f ranspositionsmentionedolely oprovide n impersonal asisfor ermutationofa series fdurations,omething hich tselfs interestingotas a musical esource ut as atheoretical ossibilityr tool.Objections o a durationbasisforrhythmicerializationrestronglyoiced in MiltonBabbitt, Twelve Tone Rhythmic tructure nd theElectronicMedium," PERSPECTIVESF NEW MUSIC, all 1962, pp. 49-79. Essentially,such a basis will failbecause ofthe absence in the realm ofduration f any phenomenon nalogousto octaveequivalence n the realm ofpitch, nd theabsenceofdurational qual-temperament.on-ventional hythmotation,with tsbinary ivisions,uggestsn octaveequivalence fdura-tionwhich imply oes notcorrespondo a perceptual eality omparable o thepitch ctave(which, trangely,snotat all reflectedn staffotation).

? 56



12/12

FORUM: NOTATIONr-15:11 - 5 3-= 5--n6:7-- 5---

II II3 5 L3

Analog notation ould conveyto a humanperformerhisnewpatternas easily s theoriginal ne,and with s little recision.The pattern-limitationsf conventionalnotation are complex andsubtle.Their detaileddefinition,nd a study ftheways nwhich om-posersnthepasthave attacked nd overcome hem,might rove eward-ing and valuable. The fact thatmanycomposerswho employ onven-tional notation n writing or conventionalnon-electronic)media arenot acutelyand constantlyware of their xistencemaybe due to theeducationalprocess:we are conditioned o think n termsofmusicalpatternswhich are notverydifficultr awkwardto express n conven-tionalnotation.Blindsubmission o suchconditionings notnecessarilyconducive otheconception f vitalrhythmicdeas: we havethewitnessofStravinskyo the fact hat he DanseSacralewas conceived, nd playedon thepiano,before t could be notated.But analog notation, or ll itseasy attractions,s not theonlyavailable escape from hose imitationswhich remain n modern onventional otation, nd it entails acrificesmanyare unwilling o make.To use BenJohnston'serminology,nsofaras analog notation mplies musicalorganization ased solelyupon alinear interval) cale ofduration,t denies hosehigherevels f rganiza-tionbasedupona proportionalratio)scale,and theresultmaybe a lossof integrativeowerand intelligibility.15urther xtensions frationalnotation repossiblewhichmayeventuallyancel all patternimitations,or reduce them to insignificance.he music of Babbitt,Carter andShapey,tonameonlythreedissimilar mericans, oint n thisdirection.

15Johnston,p. it., . 60. . 57 0

Perkins PNM Summer1965

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