sign in sign up
<<
Home , Beauty in the Beast, Harmonic Scale, Harry Partch, Pelog, Slendro

WendyGarloo ?O. Box1024 Tuning:At the New YorkCit, New York 10276USA Grcssroads

lntrodrciion planned the construction of instruments that per- formed within the new "tunitrg of choice," and all The arena o{ musical scales and tuning has cer_ publishedpapers or books demonstretingthe supe- tainly not been a quiet place to be for the past thlee dority of their new scales in at least some way over hundred yeals. But it might iust as well have beenif €qual temperament.The tradition has continued we iudge by the results: the same 12V2 equally with Yunik and Swi{t {1980),Blackwood (1982),and temperedscale established then as the best avail- the presentauthor (Milano 1986),and shows no able tuning compromise, by J. S. Bach and many sign of slowing down despite the apparent apathy otheis lHelrnholtz 1954j Apel 1972),remains to with which the musical mainstream has regularly this day essentially the only scale heard in Westem grceted eech new proposal. . That monopoly crossesall musical styles, of course therc's a perf€ctly reasonable explana- {rom the most contemporary of and av^rf,t' tion lor the mainstream's evident preferetrce to rc- "rut-bound" gardeclassical, and musical masteeieces from the main when by now there are at least past, to the latest technopop rock with fancy s)'n- a dozen clearly better-sounding ways to tune our thesizers,and everwvherein between.Instruments scales,i{ only for at least part of the time: it re' ol the symphonyorchestra a((empr with varyirrg quires a lot of effort ol several kinds. I'm typing this deSreesof successto live up ro lhe 100-centsemi manuscript using a Dvorak keyboard (lor the ffrst tone, even though many would find it inherently far time!), and I assureyou it's not easyto unlearn the easierto do otherwise: the stdngs to "lapse" into QWERTY habits of a lifetime, even though I can Pythagoieen tuning, the brass into several keys of akeady feel the actual superiodty of this unloved lust irtonation lBarbour 1953).And th€se easily but demonstrablv better kevboard. It's been around might do so w€re it not for the constant viligance since 1934,and only now with computers and speed "directory on the part oI performe$, add the rcadily available and accuratedata-entry tasks such as yerdsticks for prcvided by the assistance"is tlere th€ slightest chanceit may be woodwinds to some extent, but more so by the harp, resurrected as a long overdue replacement for the organ, or omnipresent {in€xact standards that deliberately slow (no kidding! ) layout developed by they may in truth be). Sholes over a century ago. part Yet this appeient lack of adventuiousness is not Musical keyboards are a {undamental of the due to any lack oI good altematives (olson 1967' west's music-makinSculture, so keyboerdsneed to Backus 1977j Lloyd atrdBoyle 1979i Batemen I980j be addrcssed with regard to compatibility with any Balzano1980) or their champions.Indeed an experi' existent or {orthcoming tuning scheme. cleerly the enced musician would have to be prepostercusly fretless strinS falnily is almost utuesticted as to naive, sheltered,and deaf (!lnot to have encoun_ the paticular scalesand tuning that can be used (the different tered at least a name or two like Yasser{19751 or musiciatr's ears and training are rather a Pertch (1979),or in an eerlier er4 Bosanquet,white, matterl. This is sufffcieot reasonlor the histodcal Brcwn, or Ceneral Thompson (Helmholtz 1954; concern given to new and modiffed keyboard designs Partch 1979).These pioneercwere certainly not by past proponents of tuni[g reform {Bosanquet, white, Yasser, etc.), although in ptactic€ Partch and known Ior the shy reticence on behalf oI their "made various turriDg re{orm proposals. Nearly all built or others havefrequently do" with the standard seven'white, ff ve-black. But as with Dvorak replacing QWERTY, it's diIff- Coplright@ 1986 by WendyCados- cult to challenge any sort of standaid, once thet

29 r

standardhas persisted for more than one genera- This unlortunate quality of,,sentenceffrst, verdict tion.We all tend ro torgetrhe precrriousn(ss wrrn afterwards"places Barbour,s otherwise laudable which all standardsare birrled, and grant rhoserhat achievemenrinro the samezealot,s DulDit with come beiore us a sadosanct status which is likely Partch, Yacser,Bosanquet, poole, Brown, perret, unjustifiable, and which the origin:l desisners Captain Herschel, ceneml Thompson, and others. mighr, if alive today, quite 6Jrd liughable Rerbo.jl's Tuning and Temperdm ena also suffels hom several snall ellors: "Furthermore, Turingand the Zealot Cx 3 and Eg 3 differed by only six c€nts . . ." {p. ll2). The secondnote ought be No doubt lrustrations do spawn a missiotrary zeal, and thereis ampleerrdence thar all of rhepioneeis ". . . ingenious mechenism by which DL and we've been talking about had th€ir Iair shareot At couldbe subsnruredtor Cdand DI . . frustration and zeal. It,s both instructive and amus- (p. 108).But cl is the ffJth aboveC{, not Dl. ing to read, back-to'back,the books on tuning by ". . . and(the Io8 ofl 2r3.1r, t7579t6lOO,... Partch {1979)and Barbour (19531.partch was argu- {p.119). But rhecorrect log is .1747916100. ably the greatestchampion of Iust intonauon ot thrscentury, and hr: writrnglrequenlly lapses into and, in lighr o{ many more as above, from olten iumorous sarcasm: hypocracy: "There Name-Your--Pay,your-pounds-and,Take, were, as usual with Kircher, maly It-Away lenninSs(Partch 1979, p.3941 eIIoIs. . ." {p.lI0). and emphatic impatience with those who seem and from sarcasm: most unwilling to acceptthe clear supeionry or "Only Just tuning: in the desiSn of the keyboards did the in- vento$ show their ingenuitt an . . . and the ear doesnot budgelor an instanr ingenuity rhst might better have beendevoted lrom its demand for e modicum o{ consonanc€ to something more piactical" (p. 113). in hamonic music nor enjoy being bilked by near-consonanceswhich it is told to rear as which quite ignores the ingenuity spent developinS- consonances.The eat Acceptssubstitutes th€ traditional seven-white,five,black kevboard ,given,,, againstits will lPartch 1979,p. 4t7). Barbouraccepts as a ro sav norhing of (he While understandable,rhe penasive undertone is menrsot many possiblegood new scalesrhat really eventually wearyinS(even if most often justiffed) oo requre ettonand inSenuiry ro find.Talk aboul and doeslittle to attract complete sympathy Irom a a closedmindset! And Barbourreprese[ts amorg neutral partch musiral theonsrsthe bestoI the knowledgeable reader.Barbour,s classic study (which "champions" generouslyprarseslis no Iesssubtlely dogrna(lc, al- on beba.lfof equal temperament. I his (his certajnlvseems (o be, lhoughin case(he a priori idealrs nor rus(in- subieclrhat al(racrspas- tonation, but equal temperament.He calculates sionatewords ls€eLloyd and Boyle 1979,Chapter g every alternative tuning's ,deviation,, from l2\az lor evenmorel from all sides! and argues most on behalf of those which exhibit the least mean and standard deviations from equal The Grossroads temperament.There's somethinsuncorfortably circular in the reasoning throughout the book, sort It mav seem unJairro \rngle our Barbouras abore, of like: although somebowI wince more from his and other theo sts' polemic on behalf oI ,,the haves,.or status Civen, equal temperament 'the quo versus ha\,eno(s,'ol anyone who mighrnot Therefore, equal temperament iorallv a&ee wllh rhe finalsenrenceof Barbour,s Q. E, D. book:

30 Computer Music lounal Perhapsthe philosophical Neidhardt should be equel temperament is not altogetheres benign as allowed to have the Iast word on the subject: all of us born into it commonly assumei "Thus equal temperament carrieswith itself Th€ ffrst oI these pieces contained a number its comJort atrd discomlot, like the holy estate o{ sustainedmajor thirds, which work perfectly ofnatrimony." well on an orSantuned to one of the unequal temperamentscommon in the seventeenth That's a cute quote, but its use here is not a little century but which frght unmercifully on to- smu& suggestingwhat was "good enoughfor day'sequally temperedinstruments. Dudng Grandpa"ought be good enough in peryetuity for the playing of it the audience stirred uneasily, the rest of us. There'sno thought I've read by even and, when I have played the tape, numeious the most me.*ianic of rurung reformisrs that rai'es musicians . . . have askedme what terrible my hackles as much. thing went wrong with the olgan. Most are in- The truth of the matter is that up until now credulouswh€n the explanation is given, cvcn there's simply b€en no way to investigate beyond when they listen to the piece by Bach played the standardscale within the limits of the precision the sameev€nin8 on the identical organ,sound- o{ the availabletechnology. Acoustic instruments rng. . .lrkerhe admjrable rnsuumenr rt rs, arealmost compatible with a i5 lo-cent pitch Bach . . . arrangedlor his thilds to come and toleranceat best. A ffne grandpiano cannot be ex- go, well disguisedby their musical context." pectedto perfom much better (Helmloltz 1954, (BenadeI976, pp. 312-313). pp.485-493), its particularly pro- not to overlook Computer-controlledsynthesis, on the other 1977,p. 292, nouncedoctave stretching {Backus hand, has no inherent need to respectthese here- pp.3I3 a phenomenonwe will Benede1976, 322), tofore inescapablelimitations. There's certainly lven rhebest anaJog examinelater. little n€ed here to provide any further motivation much better, and up until recently did a cannotdo for breaking out of the three-century "rut" that this (l shudderto rememberthe constant pitch lot worse long introductory material has documented-The drifts while reelizirg swttchetl on Bach on tn eaiy exponential growth in computers has ffnally ex- Moog!). pandedto include systemsdesigned expressly for More frustreting than that is the usual limit on music production, editin& and p€rformanceat low really the number of notes in an octave,which was elough cost to b€ as affordableas, say,a goodgrand "economic" musicians of Bach's wholly reasonfor piano. This is the first time instrumentation exists yield much preferablesounds of the sev- time to the that is both powedul enough and convenient enough of meantone tuning for I2V2 equal eral varieties to make practical the notion: any possibletimbre, Ellis describesthe results in his ffne temperament. in any possibletuning, with any possibletiming to Helmiolz's book p.434): appendixes 11954, sort of a "three Tb of music." That placesus at a crossroads,to ffgure out just how to use all of this ff canied out to 27 notes . . . it would probably newly availablecontrol. And we'll discoverthat the have still remaitredin use. . . . The only obiec' ttuee "T's" are really tied together. tion . . . was that the organ-build€rs,with rare exceptions. . . used only 12 notes to the octave MusicalTimbres . . . and hence this temperament {Meantonej was ffrst styled "unequal" (whereasthe organ, In Figs.1-3 we havespectral plots for three common not the temperament was not unequal, but types ofrnusical . Figure I depictspurely har- defectivel ^nd theI. ^b^ndoned. monic patials (the most comrnon, at leest in the west), such as heald on the horn and most wind and string instruments. The equationsof vibration here Not everyoneagr€ed with abandoningmeantone. are usually quite elementarY: The English held out on many instruments until the end ol the last century. Even now the truce of f, = nf'

Ca os 31 Fig. 1. Orisinal: pDrc haL Fi8. 2. Otiginal: sttetched Fis.3. Otisinal:nanhar

: , rat IE

IE tEl t5l

| | | r..------;;i;- rli-:I i I Itrsdo ';@ Fig.2 tor,of 2.76 tor everv ocrave&e rise above middle-C, and,srmttartydecreases by the samefactor as we go oown each t6l : octave(Benede 1976, p.3l5l. Figure3 depicts the nonlarmonic partials ,: head on more complexvibra(in8 I bodies.primarilv rhe lcl ,deoptones ot (he percussronfamilv, alrhouSh a fascinating world of new is now pos;ible ,,blown,,,,,bowe4" llr by combiniru properi€s oI and "sustained" ado i6-----iffi from strings and winds with themore complicatedparrial strucrureol ideophones.lThe be8inning5o[ Lhissort ol work whrch grvesthe Irequencyof rhe nth partral./". as on Umbral hybnds js discussed tMilano tne srmpleinieSer n mulriplesof rhe ,fundamenral in parrin I986l.Jtor a hinged bar the panials trequencr''/,. Fora perfec(lyrdealrzed vrbraring are described: string the expressionisi f" = n'F/ L')t[w{lb/ *A.1. = f" nlr/ Lr)\/lr/ dl\4u 4;) Here, as in the previous equation, yis the modulus wherelhe grven{roundl string has a lenSrhL. a ra af elasticity, ^rr L ^td d er€ the length and densir, dlus r. a densirvd. and iq underrensron I (Benade Justas wlth a stdng. 1976,p.313). We can seear once that this is merely Mor€ complex vibrating bodies have Iairly in, a more complete Iorm of the first equation. volved mathematical descriptionswhich we will . Figure 2 depicts the stretchedharmonic patials not go into here. For the interestedrcader an ex_ herrd on mosr,,realworld thickerstring., asthe cellent treatment can b e ionnd in The Theory ot' lowes(srrinSs of (he,.bur molr tamoLrslyon Soltrtd{Rayleigh 1945, pp. 255-432 of Vol. I). piano. the I his explains(he commenr mrde earlier In the caseoI r recranSularbar lBenade I976, about the piano being a less-than-idealDitch refer- p,511,l]vafies ence.The €quationsof motion of such;slrumcnts yield an overtonesequence described by: 'r - 1.00 - nft*.lr nz 2.68 f,= + In,) nt-373 which is the semeas the originalequation, cxcepr n4 - 5.23 that the /n, telm Sraduallyreis€s the succcssrve parrialfrequencies , aboverhe n/ | fundamental./ whrch,rt will be noLed.lorm a seriesof partrars rs a small coetficientgiven by: furiherrparr rhanan) of rheprevious timbres ln the case J = lt4Y/rL'lln/2)3 ot a vibrrung crrcularmembrane. the se- quence,here of well sndon aSoodpianowill a behavedumparu tBenade haveavalue close to.0O016 l9?6. p. 144, for "middle,C.,, alsosimilarlyRaylergh t94S, p. JJIJ, / increasesfairly uniformlyby a Iac- looks like:

32 Computer Music Jownal Fig.4. Supeimpased inter Fig.5. FDsionchart: pnx- val: lust maior thitd (hol imity ol intervals vercus rcughness,with examples af enlaryedbat sruphs.

n,-1000 n'-1504 \ - 1742 n4-2000 ns - 2 245 no - 2 494 n7 = 2 8O0 which, it will be note4 form a sedesof parrials closertogether than any of the previous timbres. {Notehow n r, nz, ra, and zu look when each tetm is multiplied by two, implying that the timpani has a "missing" theoretical fundamental.This inspired me to try synthesizinSa "basstimp" in which this @z implied partial was added,with a very agreeable e resultl) le

A Gonffuenceof Partials srPsnnrrPS tnLs<<(<<( urnv n0uoH ))>))) tutt0N,tMo0Tll

Whentwo (or more)musical timbres are soundcd Hz with musical pitci, while those somewhat at the sametime, at any random interyal between below are heard instead ^s loudnessfluctuations. their perceivedpitches, there is a high probability Oscillations of about 0-6 Hz arc usually descdbed that at least some of the various partials from each as choruslng or temolo, and those around 6 t6 as will overlap.Figure 4 depicts a "Fusion Chart," as lollg,riness.The remaining range, 16-30 Hz, can be wo such partials approach,begin ro with one quite subsonicand barely audible if soft, or at loud xnother/merge into one, and then repeatthe se- Ievelsperceived as a sot of "unpitched rumbling," quencein reverseorder as they separateand con- unless several(higher frequency) are tinue to move apart.A graphicaltepresentation of presentto give a genuinesense of pitch. Figure4 the rclAted citicd bandwidth function has been indicatesthe straightforwardgraphic conventions describedby Pierce11983, pp.74 8l)andWinckel we wiII be using for thc rest of this paper. {1967,pp. 134 148),and has its origins in Helm' Let's now play two simultaneous tones, both holtz'soutstanding work (1954,chapter VIIII oI a having the spectrum of Fig. I and tuned a Just centuryago. Helmholtz produceda chart of }armo, major third (of 386.313714- . . centsl apart.The niousnessof consonances(1967, p. 193)that also resulting combination spectrum looks like Fig. 4. formsthe balisfor Pdrrchs srmrlrrreprescntarion Both tones are shown with a grey crcss'hatchingso which he v,'himsically telll..ed the ane-t'ootedbide that where there is any overlappingit will appearas {1979,p. 155),and a simpler plot of coasonanceby black, with beating shown as in Fig. 5, by $aphi Pierce11983, p. 79)-Our rcpresentationhere takes cally descriptivealterneting bandsor stripes.In this "objectiffes" on a bell-shapedcurve which the casethe Justmajor third on hamonic pariials is normally subjectivequality of dissonanceinto the smooth, with no audible be3ting. morequantifrable phenomenon of beatsand their If the interval is now retuned to an equal- frequency-(In certain unusual conditions that will temperedmajor third {of 400.00cents) as in Fig. 6, not concernus here this simpliffcation may need there is only a modest chanSein the relative loca- qualiffcation. I tions of the lower partials. Bur between the fffth "" Surpdsingl, olrr earsassociate sound oscillations harmonic lhere the term is accurate) havinga frequencyof more than approximately30 of the lower tone and the fowth harmonic of the

33 Fig. 6. Supeimposedinter FiA. /. Supeimposed inteL Fig. 8. Superinpose.l inteL Fia. 9. Musical examples. va1: eqnal-tempered ma i o' val: pwe actave (harmonlc val: pue octave (s*etched (a) Very wide octove (1220 thir d (hat monic partials). cents). (b) vety wide oc- taves (1212cnts each). (c) Chotd and cllrstel (aqLol tempenment ver sus lust). ld) The h@manic

r-l

Fi8. 7

r6t

leadsdirectly to the stretchedoctaves mentioned earlier. There ere many instruments that have even grearerstretching of therrpartials than the pIano. and the effect on octavesis more audible. One Fig. 8 such instrument, which I call the "metimba," is a metal version ol the lwoodenlmarimba, and like the "basstimp" alluded to eailier, arosein my trmbre- tEt s)'nthesizingwork from one of those usually im- peitinent "what if?" questions.Sound Example 2 on the soundsheetcompares the metimba with tEi a clerinet performing the same ffguration {Fig. 9b): stretched'pure-stretchedon the {stretched- hamonicl metimba, the last being a chord, and then pure'strctched-pureon the (harmonic)clari upper one, sone rough beating ol 8 Hz occurs, net, again the last being a chord. Sincethe metrmba this for the 200'Hz ratrgewe are usin8. Juste little has also the addedcomplicetion of severalnon- higher the rete increasesinto the rcughest alea harmonic partials {which we'lI look at next), the around 12 Hz, and above that the rate can even smoothnessof the somewhat wide octaveis slightly climb up into the audio bend. disguisedby them, but, then, this is a much wider Now let's try a pure octave(of 1200.00cents)and octave than is {ound on a piatro. In any event, the the same harmonic wave, as in Fig. 7. The result metimba is better servedby a 6 l2-c€nt str€tched "fusion" has, not unexpectedlt evenmore distinct octaveas in Fig. 10, while the sounds than the Justmajor third, with all harmonics lined smootrherwith pure ,as in Fi8. 7. up and no beats. The more extreme case oI nonharmonia partiais II we retain the sameinterval of an octave,but is shown in Figs. l1 and 12, and heard as SoundEx- now substitute the stretchedharmonic spectrum ol ample I lseeth€ soundsheetwith this issuel.Figure Fig. 2, the result is as Fi8. L Notice that there now I I plots the rerult of one par(rculrrnoniarmonic appeermany fairly strong beatsbetween partials timbr€, that o{ Fig. 3, while SoundExample I com- that had been "fused together" in Fig 6 This is ex- paresa more t)'pical nonharmonic timbre, gam la ectly the situation that exists with a piano, and replica of the tuned'kettl€s in a gamelanorchestra),

34 Com1utet Music Iownal Fiq. 10. S'perimposed in- Fi8. 11.Supeimposed in. Fig. 12. Svperimposed in- Fie.13. Supeimposed in. terval : stetched oct ave tetval: pwe octave (non- tetval: vety wide octave terval: vety wicle octave (stetched hamani. (nonhamonic pat tlals). (hatmoni. pattials).

fd

tgl

Iig. 13

lrl

LgJ

with the samepassage played on the French horn, strument, and exactly vic€ versa.I say scalebecause andnotated in Fig. 9a. As Fig. 11 indicates, a pure even though the graphsare oI two simultaneous "rough octaresounds fairly on mostsuch rnstru. notes (andmelodic intewals are much less cdtical), ments.We can imagine a nonlrarmonic timbre that the soundsheetexamples are fust playedmelodi would insteadwork better the other way around, c.rlly,and some of the srmeproperries are repre- with smaller'than-purc, <12: l], octave,but most sentedmusically here as well, if only for the octave. "ac- histodcally dedved instrumental timbres have Pierce{1983, pp. 192-193) discussesa corre- cidently" been like the present example. spondingexample conceivedof from a very di{" on SoundExample I you will hear that the low- ferent direction. He includes a recordedexample estpitch, C4, is the referencelower note of the oc- (Ex.4.4, and also listen to 2.I 2.5, and seep. 86), tave,and the upper three notes, all nominally C5 which demonstratesthe awal illusion of a stretched- (theyare closer together in pitch than the sharps harmonic tone that soundsflatter when all of its anddouble-sharps rn Frg 9a vrsuallysuggest). are partials are exactly doubled! If insteadwe were to the tones of: moveup by a properlys(relched ocrave, as in our SoundExample l, it would then sound like a "real" l. A 1220-cent"pioperly" wide octave 2. A pure octaveol 1200cents 3. The 1220-centwide version again 4. A 1240 cent extrcmely wide octave TriadicGorffuence The differencesare subtle, but gam doessound smootheron the 1220-centwide version {like lig. Let's now add a third tone to the other two, to lorm l2), while the horn is best at the pure 2: I ratio oc' a triad. In Fig. 14 we have the simplest caseof a Just tave.The horn has very audible beatswhen heard maiortnad playedon a rimbre(hat hasharmonrc on the very wide octave,as in Fi8. 13. partials. The result is smooth, as we would expect, Clearlyrhe umbreof an rn:rrumenr"tronglv becausethere are quite a few patials that merge affectswhat tuning and scalesound best on that in- and "fuse."

35 r-

FlE.14. S:!Pe mPased in' Fig. 15. Snpdimqasettn' Fig.16. S petimposed1n tereal: ttad lhar tetval: I st ttiad (nonhar tetval: nanhatnoni. IDst "ttiad" lnonhamanic

t:: , , IMnNYMnICHftl 'fill svoolnrrMooIHrJ E rol Finfii-n:fl6;rft^n-ir;it |I rnLL I @ T- +r! : =; = I =: = = : : i I :: : il=lli=l-i.==.

Ii€. 15

SoundExample 3 ffnally plays the triad on the horn, and at last there's an audible difference:you f;l can clearly hear that the Iust version is a smoother, "consonant" more stable,clearly more tria4 even though the equal temperedversion is acceptableif LCI the chord is not long sustained(see the comment on Bach later). Although I have not included a sound example for it, Fig. 16 showsa type o{ harmony worth rnves_ tigating. Here we have a totally foreign and quite I{ we try the same triad on three tones with the nonharmonic 'triad," playedwith the samenonhar- nonharmonic spectrum of Fig. 3, the result is not monic partials as in Fig. 15, where the Just triad nearly so smooth. In {act, it's barely recognizableas soundsso diffuse.But the results here are surprising "triad" the samefundamental harmonic building block of to an extreme: the Peculiar is consonant, western musicl Figure 15 showswhat happens,and {usedand well focused,with very little roughness, you can listen to the effect on SoundExample 3. and many matches of partials in all their nonhar_ First we hear the triad with a traditional timbre monicglory. ill-equipped to handle harmonies,odd as it miSht This is the kind of harmony we find in the seem: the xylophone. You can judge the results lor gamelanmusic o{ and ,where the scalesare yourself, but while the xylophone has fewer con- and ,and the instruments are tuned flrctingnartrrl. th.ln our lrg. lc: nonharmonrc ,kettles, and metalphones.on my latest rc- timbre, the "fusion" we normally expect to hear on co:ldilrg,Beauty ln The Budst,is the composition you a Ju5t (flad\imply doe\nt occur. rerllv Lrn l (in homageto that maSicalisland) I call: Poemlor even hear much difference,never mind chooseone Bdli. In it these timbrcs and scalesare combined over the other, between the Iust 3nd equal_tempered with good effect, to producea music that is neither western nor Indoresian but is from both traditions "whatifs," By switchinS to another of my this Near the end a Concertofor Gameian and lsvm' a metal version of the xylophone (metal xylo), phony) Orchestra occllfs.In order to get the two time "replicas," which has somewhat greateramplitudes of the groups(both actually synthesized of "play" higher panials, and all havea much longer decay- colrrse)to be able to together,it was neces- "chea1' time than on a reSularxylophone {metal soaksup sary to the pelog scaleslishtly toward Just higher frequenciesmore slowly than wood), the intervals so that the (mostly) harmonic symphonic triad is slightly better defined.But the nonhar- timbres would not sound cacophonouswhen play_ partials allow Lrslittle preferencebe' ing it, and yet not so far that the gamelanensemble monic still "har' tween Just and equal-tempercdversions, although woutd suffer.The original pelog and modified some dif{erenceis at least audible. The last ol monic" versionsare (in cents):

36 Computer Music launal tL,. ORICINAL: blings. No\d I can use aII of the power for tunore 0 r23 530 683 8r4 1226 (: octave) sereration of the Alles card (Alles r979), while MODIFIED: working with total tuning flexibility ind prccision, 0 123 519 690 811 1216 (= octavc) a lar cry from the compromisesnecessary only a few yearsago lBlackwood I982), where the choice, "scales-or-sounds," The 6nal piecc with its "cheat" works very wel1. had to be made.I'm grateful to all the pioneerswho have made this possibleat Iong last. Partch'sFolly After working for a whilc within the boundsof traditional Iust intonation, I becamefrustrated with "plainncss" A tast thought on the examplesthus far is the real- both the {to earsbrought up olr Scravin- ization of one o{ Iife's funny ironies. I only under- skt Ligcti, and Bartok, at least)of the diatonic Just stoodit reccntly wlile researchingthis paper.ll's scalc lBlackwood 19851,and the often-describcd "problems" both tfagic and touching to think that , ore-comma of relegatingonc group o[ who almost single-handedlykcpt alive the spark notes gottcn from a seriesof 3/2 fifths with an- of Justintonation for half a century happenedto othcr gotten from a seriesof 5/4 major thirds lBar- chooseto build instruments most o{ which had borll 1953,Helmholtz 1954,Olson 1967;Lloyd and "twain rapiddecays and/or nonharmonic partials. As we've Boyle r979, Balzano 19801.That the never beenfinding out, thcsc are about the wo$t choices meer ha" beenan lr.lcliblc sou(e ol frx.rral on that can be madc to best show off the wondcrs of {seePartch 1979,pp. I90 194 especially)for Iust- lust tuning. J.S. Bach dcmonstrated,ard Bcnadeal tempted musician and theoretician alike. ludesto it in rhe quotc earlier on, when_youm st But if we arc willing to give up the ability ro useslightly defcctivetunings, like 12V2, it sounds modulatc, and solve that with the compu|er hard' a lot lessrough if you keep everything in motion: warc, we can tune all 12 stepsof our Syncrgykey dont lingcron rher-npertectroir. . .Terlpoi.v(rv boardsor whatever to a seriesof pitches from the much ticd up with tuning. Both cvolve togetherard overtoneseries of one particular tonic, as I have Br!ena bcLrerrunrng rhrn 12V2.we mrghr erp

Carlos ,7 Table l. Harmonic scaleon C table deference to the instrument manufacturers who drew the line at 12 notes per octave.You can heat the actual dedvation of the still,in-use approxlma- tion: two octavesdown and four (slighdy flat)per- cf t/l 0.000 Iec(fffths up lor in Lheopposire order) . a mrtor D[ t7/t6 r04_955 third, ffrst es it sounded in meantone, and then as Dq 9/8 203.910 it soundedin the then-new equal tempetament. D{I} t9/t6 297.5r3 There'slittle doubt 5/4 386.3I4 that earsaccustomed to the Fq 2t/t6 470.781 smoothnessof the former would havedetected the tI/8 551.318 falsity in the latter more acutely than we can toda, 3/2 70r.955 although the differenceis plain enough if you listen l3/8 840_528 attentively. 27/16 905.865 What is actually occurring is that we ar€ trying to 3, 7/4 968.826 Iorce l3/zlall/2l1to be = 5/4. The left tems total Bq I5/8 i088.269 8I/64, or 8I/80 higher than 5/4. In cents, the left side is {4 * 70r.9950) - (2 * 1200.01= 407.8200, and that is greaterthan 386.3137by the syr,onic posing it up and down by 100-centincrements, we comma of 2l.5063. Bvt musicians for centuries ardve at 144 distinct pitches to the octavefor out have notated the major tdad on C to be: C-E-c, "modulating" new version of the . not: C-E-flattened-by-one'comma c, that isi These are loaded into the Synergy frequency tables C-E r-G (Barbour1953, chapter III, Blackwood (in groups of 12)via an outboad Hewlett-Packard 1985,pp. 154-162;Helmloltz I954,pp.430-439). computer and a single'octavekeyboard which I've somethinghad to grve.and the besL-soundingway built lMilano 1986)to tdgger the computer each was meantone.There are four flfths to absorbthe time I key a change to a new chord {undamental excesscomma, so let/s give eacha quarter oI it: The result (which had classically been called im- 701.9950 |21.5063/4)= 696.5784forthe mean possible)is that we can now modulate completely tone ffIth. (T\doof theseup down an octavel "circle land around a of fiJths" while at all times retain- form a "tone"whichis exactly in betweenthe "II1a- "classic" ing not only the few Just intervals, but jor tone" of 9/8 and the "minor tone" of 10/9, all the other perfecdy tuned pitches of the har- hence the name, meaatone. . . .) Sincelow partials monic scale.And that's preciselywhat happens are involved, the resulting flat ffIths beat much twice in my rccent composition, /ust lmdairings slower than if some higher ratio were mistuned by lon ,Aud 200).An excerpt of the same 5.3766cents-if you will, the fffth can "circle" the fiist is heard in SoundExampre 6. "stand it" better. The major third that results from four of these ffJths is, as you can hear on trhefust half of SoundExample 7, pedect: 386.3137cents. TheBest and Worst To round out this capsuleexplanation, what oc- curs in the equaltempered version is easily shown. Earlier I suggestedabout meantonethat we have ex- By deffnition the size of an equal-tempered6fth is: cellent reasonsto believe that were it not lor modu- zt7tt't: 1.4983O7D8,which is ceitair y close to 3/2 "wolJ lation problems of tones" (when instruments land why we can get eway with it as the only devia- were tuned to only 12 m€antonepitches), this form tion, always melodicallt in the 144 notes per oc- of temperamentwould likely be in useright through tave modulating harmonic scale).ln cents we have to the present.In SoundExample 7 you can get the familiar fffth, 700.0.Now, go up four of these some small senseof the frustrations that must have and down two octaves:la * 700.01- l2 ,i 1200.0)= occurredfor musicians of J.S. Bach's day as the in, 400.0.And this is 400.0000 386.3I37 = 13.6863 "pollution" evitable of their triads ffrst occurred,in cents sharpfor a maior third. A 5/4 involves har-

38 Computer Music Jounal monics double those for a 3/2, so the beats of this division seemsto have arisen independentlywith sharpthird areworse than only four times the mean' severalmusical theorists, as eaily as the sixteenth tone fifth's error, as you can hear on the secondpart century in Italy. {zarlino and salinas both suggested of SoundExample 7. a division of 19 parts, although it's doubtful that By the wat to get a feel for the ratio form of de- their derivationswere exactly equal [Brrbour 1953, scribing,uorinlervals, thrnk of rhem asnaming p. l15l.l which two partials of the series form the In 1555Nicola Vicentino describedwhat wc row interval-for example,if it were the seventh and recognizeas the 3l'note division. lt was redis- the fourth (like B'flat7.6".and C)the ratio would coveredmathematically in HoIIand by Christian be 7/4. Huygen\in I724,who gavean e\rct descnption "harmonic The next brief SoundExample 8 placesthe ol his cycle" and how extrernely close smooth quality of meentone into a more musical it comesto standardquarter-comma meantone context. By employinS the samesuggested tech- intonation. As such it hAs all the merits of mcan- nique as the harmonic scaleabove, retuning as tone for pe omingmost westen music far more neededvia an external computer control, we can smoothly than 12V2, while also permitting un- maintain this tuning's benefftsto most of the exist limited modulations to an unprecedenteddegree ing repertoire,while avoiding both "woII tones" and (to the nearest1200/3I : 38.7097centsl). "oldest" any restdctions on modulation. I expect to seesuch But the distinction of genuinely €qual- an evolutionary version of meantonebecome more steppeddivision probably goesto the excellent common as digital synthesizingequipment of this s3-note division. A student of PythaSorusnamed kind becomesreadily available. Philolaus wrote a descriptionof a method for con- But for traditional Westernmusic the worst pos- structing inteNals which leadsdirectly to a measur- jng sible way to tune is probably the scaleof 13 equal scale of 53 comma'sized steps,in which form stepsin an octave {13V2), which is heard in the it is known as Merca,o t's cycle ll{ellJrhokz 1954, SoundExample 9 (with the samemusic as the last p.436). Justover a century ago this division was to aid in compaisons). As we will next seein equal championedby R. H. M. Bosanquet,who devel' divisions of the octave,riSht besideeach decent one opedthe ideal keyboardwith which all possible is a region of pretty awful ones.And the lowcJ the regular divisions becomereally playableby human numberof steps,ingeneral the lessgood the results. hand..the generahzedkeyboard (Helmholtz 1954. Sincel2\4 is the lowest decent division, right be pp.479-481; Yunik and Swift 1980,Fig. l). Even sideit somewherethere ought to be a "dillr" and Partch (1979,pp. 393 and 4381,who had littl€ love this one'sit! (Ycs,a new tool's strenSthworks both for any equal division, has kind words for Bosan- ways. , . ,) quet's generalizedkeyboard. There is scarcelya more worthwhile venture to pursue as soon as possiblethan adopting a standard for and then manufacturing at least a "Iirnited edi- SymmetricEqual Divisions Lion"of rhe*ekeyboards lor all of us now becoming involved with this ffeld. The presentauthor con- This could eventually be one of the most fruitful structed a generalizedkeyboard (which has subse- methodsof scaledevelopment for our new musical quently beenlostlback in the presynthesizerdays technologres.The norionof drvidingthe octaveinto of 1957and would love to hear of any seriouswork stepsof equal size also happensto have gotten a now being done in this direction. I have several greatdeal of attention over the past few hundrcd Iong-thought-out ideasand proposals(see Fig. I7), years,so theres reallya gooddedl of rnformation which I will gladly sharetowards the goal of "get- alreadyavailable about it (BarbourI953; Helmholtz ting the standardright," that we may avoid the fate 1954'Yasscr 1975, Partch I979r Yunik and Swift ofdigital taperecording.obviously it ought besome I980' Blackwood 1982,19851. As an idea,multiple variant of a MIDI general-purposeunit-

Carlos 39 Fig. 17. "M\hiphonic gen- a standardkeyboaa, ap- rcls), btack (far shatps). n\mbet ol disits pet oc- erclized keyboad. one oc- ptox. 1/t wide anil1/2" da* brcw (fatflats),li9ht tave hete, ahholrgh it ndy tave,C to ctuca31'vA note hi8h,b\t only 1 3/4" long. (for dotble. be extendedas necessary. seyish-blDe 'fhis names.This is a perspec rhey arc calorcd in live sha4s), and golden tan ilesign is a modem nve &awins ol oneo.tave shadatalons the blue- (fot double flats). The nates updated teatment of the fram a prcposedstandad yellow axis ol the CIE are named herc lot the aiginal, conceived and ol Eenerulizedkeyboad by ChrcmaticityDio+tum, ta 31\A Division,but thekey built in 187s by R. H. M. WendyC arlos.'the indi- facilitate ptopet use by the bood is suitablelot all vid\al notesarc shapea colot -blind. The suggested rcgular divisions ol the oc- like the shatpsand flats of shadesarc: white (fotnatu- tave,up to 55notes, the

andil thetaskcanbe somewhat automated by such, which equal divisions look most useIul?Figure why shouldwe care? 18 is a computer plot I recently mad€ to help nar- Sinceharmonies from the naturalseventh of 7/4 row down the decision.The inspiration came from andperhaps from the naturaleleventh a simpler ve$ion by Yunik and Swift p. 631 lseptimal), {1980, oI I I 8 mayplay a goodrole rn ar|y tn Computet ll4usiclournal4141. Bu( | wa5curiou\ lunidecimal) newly developingharmonies, I've addedthese onto about what happenedes one werit continually and the morecommon "classic-lust" ratios as two op- gradually from integer step to integer step,and also tions to be additionallycalculated and plotted. wanted to reflect the fact that with a computedzed My algorithmended up beingquite differentfrom frequency-tabledriver, there'slittle n€edto "penal- Yunik andSwift's: ize" what may be a ffne division that requiresmore than two dozen notes lalthough more than 60 could I. Calculateand store the centsfor eachprime get unwieldyl. The computer certainly doesn'tcare, ratiool interest.(I choset 3/2,4/3,5/4,6/5,

40 ComputerMusic Jownal Fig. 18. Eqaal divisions of Fit. 19.Symmetic equal the oc tav e. > (dev iation s step divisions lrcm 10 to 60 notespet octave.

5/3, 8/5, plus 7/4 and I l/8. The rest can be than 60 stepsbefore heading {oi the ieally eleborate altertratives. derived f:om these and are not prime. ) "classic-lust" 2. Set up a loop with some small increment on The solid line is of the deviations division size. only. The dotted line is that for septimal harmo- goes 3. For each new division size, calculate the size nies adde4 and it obviously never abovethe get quite of one step,and set up alrd clear the proper former, elthough at times it can close, as registers. at trheotherwise so-so15 divisio4 or at the remark' 4. Iind the nearest step to eech given ratio and able 3l hamonic cycle oI Huygens. Similarly tlte par- calculete its efior. dashedline addsthe harmomesfrom the l lih 5. Add the squareof this error to the "sum-of- tial, and againffts well at those sametwo divisions, the-squares"register. 15and 31. investigation oI cycles 13-24 6. Plot that value, iflcrement the division size, The most thorough atrd loop to 3. waslikely that recentlydone by Blackwood{1982 and 19851.He alsotook the extrastep and com- The plot is canied out tfuee times: once for the poseda seriesoI etudes,one itr each cycle. It's very "classic-Tust" ratios, oncewith the additional rc- important that we do not Iall into the uap llloyd strictions of satisfying the septimal ratio {7/4), and and Boylel of remeining only theodsts. We have to a third time adding also the unidecimal mtio (I I /8). composereal music of maly kinds within all and The resulting Fig. I8 ffts very well my own end any of our new tuning schemes,iJ this work is to all other examined descriptions of the audible her- have any lasting value at all, or be taken sedously monic "merit" of particular equal divisions. Note by the music commu[ity (arldby the public at large, "Papa the peaks singled out at several of the most impor- iI it is to survive).More importantlt iust as tfilt divisions: 12, 19, 31, 53, 65, end the one that is Bach" did with his generation'snew scele,this is a mean between 53 atrd 65, 118 (= 5a + 65, and is the only way we'r€ going to learn how to control "nearly perfect" by all published reports and by our and use uisely theserew giJtsfrom the egeof the diagram). Also notice that we are not searching in computerto usl particular Ior diatonic scales{Blackwood 19851, an" other reasonfor omitting 9/8 and I0/9 frcm the list in step I of the prcviously mentioied algorithm. A AsymmetricDivisions much closer view over the raage oI l0 tbrough 60 stepped divisions is given in Fig. 19. I think we will I've saved the most surpdsing tuning concept for warlt initially to experience those divisions with less last. The previous divisions all made the same tacit

41 Fig.20. Asymmetic equal- step divisions, lacations al a|pha, beta, and gamma.

can handle the octaving later, say even with an ex- ternal control computer as the modulating har- monic scale already requires, that may free up the compromise-screeningfu nctioos of our inrensive- search program to 6nd some really interesting equal' step specimens.This asymmetric division search program usesthe ratiost 3/2,5/ 4,6/5,7 / 4, al:,d ll/8. The results are given in Fig. 20. There are three main peaks for the region between 120 30 centsp€r step,the reciprocalto 10-40 equal stepsp€r octave.I call them alpha (n), beta (BJ,and "echo gamma l7). There occur peaks" at each doub- ling of the nurnber of steps frcm a past peek, some- thing dr6culr ro .ee m ihe slTlJnetric di!ision plots. Th€ two in Fi8. 20, a' and B', fall equally to either side o{ the essentiallyperfect{l) 7 (on the assumption: that eachtarget ratio be availableas classic-fust curve-for s€ptimal harmonies the first an inLervalin all inversrons,They were svmmetlic: "echo,"a is excellentl.These happy discoveries oc- there was the prime iatio of the perfect fiItlt 3/2, cur at: d = 78.0 cents/step = 15.385steps/octave, but also the perfect loulj-h, 4/ 3 lglven 211 a\d 3 / 2 i. p = = y : "pdme" 63.8 cents/step 18.809steps/octave, 35.I follows directly, so is not really in the sense cents/step = 34.188steps/octave. The deviation we'v€ been usidg that ideal. Similarty, therc was the exis's arbitrary units are incompatible on this ffgure major third, 5/4, but also its inversion, thc mrnor with those on the last two, but a quick comparison sixth, 8/5. Both 6/5 and 5/3 app€ared.Ody for the will show just how different (and also similarl the newer optional €tios is a nonredundantforn ap- asymmetdc and symmetric divisions are. plied: we have 7/4 but not 8/7, 1l /8 but not l51 11. SoundExample l0 plays 4 "neerly" one-octave Since each of the redundant pairs is symmetic scale of alpha on the hom. Notice how there are with respectto the octave,the net result is sort of four stepsto the minor third, ffve steps to urc rDator "over-representation" an o{ this inte al. Little won- third, and nine steps to the {this time no kidding) der that every one of the peaksto th€ plots in Fig. periect ffIth, but, of course, no octave (the fitral "at- 18 and l9 occurs at exact integer divisions oI the tempt" et trhisis an awful Il7o-cent ve$ion, the octave!But the octaveis the ratio most common to next step to 1248cents being even fufiher away!). the "stntegies" of most digital erchitec- But that's the trade-offwe'v€requested: there's no tuies, such as in tle 16', 8', 4'octaving borrowed ftee lunch! SoundExample 1l is a bdel chodal pas- from the pipe organ.Most timbre/instiument ffles sAgein alpha. The harmonies are amazingly pure, include the similai designation of transpositions up the melodic motion amazingly exotic. or down by octaves. In my curent Synergies the I've not included a similar example o{ beta or {requency of each note is stored in the previously gamma on the soundsheet but have expe mented "multiphonic" mentioned frequency-table, but only 12 main l6-bit with both {gammareally requirese words control the midille-most octave, the other oc- generalized keyboar4 like most <24 divisions). taves duplicating these {requencies by exact lactors Beta is v€ry like alpha in its harmonies, but with of two, a v€ry common method of assuring all the ffve st€ps to the minor third, six to the major thir4 octavescan be made //beat-free."We hav€ octav€ and eleven to the perfect fiJtb, melodic motions are possibilities all over the place. different, rather more diatonic in effect than alpha. So why not, as an expeiment, ornit the octave- Gemme {nine steps, el€ven st€ps, twenty st€psl is redundant retios from the fust step of our algo- slightly smoother than these,having no palpable thm? That will lose all octavesymmedt but if we differencefrom fust tuning in harmonies.But the

42 ComDuterMusic lounal "third scaleis vet e flavor." sort of intermediate to References a and B, although a diatonic scaleis melodically ,vailable.I havesearched bur can find no previous Alles, H. G. 1979."An lnexpensiveDisltal SoundSyn descfiption of a, B, or 7 nor their asymmetric scale thesizeL't Computet Music loumal 3lSl :28-37. family in any oI the literature. ApeI, W 1972. Harva Dicnonary of Music. Cambidae, Alpha has a musically interesting property not Mass.:Harvard University Press. {ound in Westernmusic: it splits the minor third B^ckts, l. 1977. The Acoustical Foundationsol Music. exactly in hall (alsointo quarteisl. This is what ini- New York: W. W: Norton & Co. Balzmo, S. 1980."The tially led me to look for it, and I merely called it C. Cmup-th€oretic Description of r2-Fold and Microtonal Pitch Systems-"Computei my '/split minor-third scaleof 78-centsteps. Beta, Music loun al 4l4l | 66- 84. like the symmetric 19 division it is near. do€sth€ Barbour M. 19s3. ?rri, and Tempenment. East La'- samething to the pedect I. I fourth. This whole formal sing: Michigan StateCollese Press. discovery came 4 few weeks aJter I had completed Bateman,W. A. 1980./nltoduction To Computer Music. the alb\tm, Beauty In The Beast, which is wholly New York: lohn wiley €r Sons. in new tunings and timbres. Th€ title cut fmm Benade,A. H. 1976.Fu'dame'tals of Musical AcaDstics. the contains an extendedstudy o{ some p, New Yorki Oxfod University Prcss. "Discovering but mostly d. AIr excerpthighlighting the proper- Blackwood,E. with l. Aikin. 1982. the Mi ties we've been discussingis the ffnal SoundEx- c]otolral Resources of the Synthesizer." Keyboad ampl€ 12. 8ls)r26-38. Blackwood,E.1985. T,hr ,Sttu.tue of RecosnizableDi dfonicTlrnrnas. Pinceton: Princeton Univeisity Press. Conclusion Helmholtz,H. 1954.O, "n e sensotonsOf Tane.'rr^ns. A.I. Ellis.New York:Dover Publications. Krplm,l. K. r98r. "Developnga CommercialDigital I don't pr€tend to be either a wrirer or rheorist, so SoundSynthesizer." Com putet Music Jawnol thank you for putting up with my attempts at both sl3):62-73. in this lengthy report. But I havebeen fascinatedby Lloyd,L. S.,srd H. Boyle.1979. Intervak, Scalesan.l scales and tuning for about thifty years, well before Temperamenrs.New York:St. Martins Press. discoveringelectronic music, with the single excep- Milano,D. 1986."Interview with wendyCarlos." ,I(ey- tion of Olson's {1967,chapter I0) demonstration bodrdI211ll:50 86. recordof the RCA S,'nthesizerin 1953.Although Olson,H. F. 1967.MDsic, Physi.s and Engineeing.New my name is indeiibly connectedwith the Moog Music.New York:Da Capo Synthesizersince that unexpectedaccident of some P^ttch,H.1979. Genesis Old l8 yearsago, ir seemsto me that only now js our Pierce, 1983.fte SciencsolMusicai Sourd.New York: grow- l. fl€dgling art starting to show healthy signs of Scientiffc Americd Books. ing up. I'm passionatelyexcited promise by the o{: Rayleigh,I. w S. 1945.ft e Theoty Ol sound. 2 .lolnmes. any possibletimbre, any possibletuninS. That's the NewYork: Dover Publications. reason for the work trhst went into this paper. It's Ylsser, l. r975. A Theoty Ol Evolvins'. New also a way to return the lavor to Computet Music York:DaCapo Press. "Tempeied lournal lor a decadeoI ideas and inspiration that led Yunik, M., andG. Swift. 1980- MusicScales directly to my "very nearly" attaining this double- for SoundSynthesis-" Computer M sic Jounal headedability. For all those who helped and those 4l4J,60-65. on whose shoulde$ I am lucky to stan4 deepest Winckel,F. 1967.Music, Sou,d aDdSe,satio.. New thanks.We've been treadingwater much too long, YorkrDover Publications. and I'm delighted that the real work now can begin.

Carlos 43