Tuning: at the Crossroads Author(S): Wendy Carlos Reviewed Work(S): Source: Computer Music Journal, Vol

Home , Beauty in the Beast

Tuning: At the Crossroads Author(s): Wendy Carlos Reviewed work(s): Source: Computer Music Journal, Vol. 11, No. 1, Microtonality (Spring, 1987), pp. 29-43 Published by: The MIT Press Stable URL: http://www.jstor.org/stable/3680176 . Accessed: 04/01/2013 12:02

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This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions WendyCarlos At the P.O. Box 1024 Tuning: New YorkCity, New York 10276 USA Crossroads

Introduction planned the construction of instruments that per- formed within the new "tuning of choice," and all The arena of musical scales and tuning has cer- published papers or books demonstrating the supe- tainly not been a quiet place to be for the past three riority of their new scales in at least some way over hundred years. But it might just as well have been if equal temperament. The tradition has continued we judge by the results: the same 12/2 equally with Yunik and Swift (1980), Blackwood (1982), and tempered scale established then as the best avail- the present author (Milano 1986), and shows no able tuning compromise, by J.S. Bach and many sign of slowing down despite the apparentapathy others (Helmholtz 1954; Apel 1972), remains to with which the musical mainstream has regularly this day essentially the only scale heard in Western greeted each new proposal. music. That monopoly crosses all musical styles, Of course there's a perfectly reasonable explana- from the most contemporaryof jazz and avant- tion for the mainstream'sevident preferenceto re- garde classical, and musical masterpieces from the main "rut-bound"when by now there are at least past, to the latest technopop rock with fancy syn- a dozen clearly better-soundingways to tune our thesizers, and everywhere in between. Instruments scales, if only for at least part of the time: it re- of the symphony orchestra attempt with varying quires a lot of effort of several kinds. I'm typing this degrees of success to live up to the 100-cent semi- manuscript using a Dvorak keyboard(for the first tone, even though many would find it inherently far time!), and I assure you it's not easy to unlearn the easier to do otherwise: the strings to "lapse"into QWERTYhabits of a lifetime, even though I can Pythagoreantuning, the brass into several keys of already feel the actual superiority of this unloved Just intonation (Barbour1953). And these easily but demonstrably better keyboard.It's been around might do so were it not for the constant viligance since 1934, and only now with computers and speed on the part of performers,and the readily available and accurate data-entrytasks such as "directory yardsticks for equal temperament providedby the assistance" is there the slightest chance it may be woodwinds to some extent, but more so by the harp, resurrectedas a long overdue replacement for the organ, or omnipresent piano (inexact standardsthat deliberately slow (no kidding!)layout developed by they may in truth be). Sholes over a century ago. Musical are a fundamental of the Yet this apparentlack of adventurousness is not keyboards part so to due to any lack of good alternatives (Olson 1967; West'smusic-making culture, keyboardsneed Backus 1977; Lloyd and Boyle 1979; Bateman 1980; be addressedwith regardto compatibility with any Balzano 1980) or their champions. Indeed an experi- existent or forthcoming tuning schema. Clearly the enced musician would have to be preposterously fretless string family is almost unrestricted as to naive, sheltered, and deaf (!) not to have encoun- the particularscales and tuning that can be used tered at least a name or two like Yasser(1975) or (the musician's ears and training are rathera different Partch (1979), or in an earlier era, Bosanquet, White, matter).This is sufficient reason for the historical Brown, or General Thompson (Helmholtz 1954; concern given to new and modified keyboarddesigns Partch 1979). These pioneers were certainly not by past proponents of tuning reform (Bosanquet, known for their shy reticence on behalf of their White, Yasser,etc.), although in practice Partchand various tuning reform proposals. Nearly all built or others have frequently "made do" with the standard seven-white, five-black. But as with Dvorak replacing QWERTY,it's diffi- Copyright ? 1986 by WendyCarlos. cult to challenge any sort of standard,once that

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This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions standardhas persisted for more than one genera- This unfortunate quality of "sentence first, verdict tion. We all tend to forget the precariousness with afterwards"places Barbour'sotherwise laudable which all standardsare birthed, and grant those that achievement into the same zealot's pulpit with come before us a sacrosanct status which is likely Partch, Yasser,Bosanquet, Poole, Brown, Perret, unjustifiable, and which the original designers Captain Herschel, General Thompson, and others. might, if alive today, find quite laughable. Barbour'sTuning and Temperamentalso suffers from several small errors:

Tuningand the Zealot "Furthermore,Cx-3 and E -3 differedby only six cents .. ." (p. 112). The second note ought be No doubt frustrations do a spawn missionary zeal, E? -2 and there is evidence that all of the ample pioneers "... ingenious mechanism by which D6 and Ab we've been talking about had their fair share of could be substituted for C0 and D# .. ." frustration and zeal. It's both instructive and amus- (p. 108). But GOis the fifth above C0, not D#. to the books on ing read, back-to-back, tuning by "... and (the log of) 218/31,.1757916100, ..." Partch and Barbour Partchwas (1979) (1953). argu- (p. 119). But the correct log is .1747916100. ably the greatest champion of Just intonation of this century, and his writing frequently lapses into and, in light of many more as above, from often humorous sarcasm: hypocracy: Name-Your-Octave-Pay-Your-Pounds-and-Take- "There were, as usual with Kircher,many errors.. It-AwayJennings (Partch 1979, p. 394) .." (p. 110). and emphatic impatience with those who seem and from sarcasm: most unwilling to accept the clear superiority of Just tuning: "Only in the design of the keyboardsdid the in- ventors show their ingenuity, an ingenuity that and the ear does not for an instant ... budge might better have been devoted to something from its demand for a modicum of consonance more practical"(p. 113). in harmonic music nor enjoy being bilked by near-consonances which it is told to hear as which quite ignores the ingenuity spent developing consonances. The ear accepts substitutes the traditional seven-white, five-black keyboard against its will (Partch1979, p. 417). Barbouraccepts as a "given,"to say nothing of the merits of While the undertone is many possible good new scales that really understandable, pervasive do if most often requireeffort and ingenuity to find. Talk about eventually wearying (even justified) a closed mindset! And Barbour and does little to attract from a represents among complete sympathy musical theorists the best of the neutral reader.Barbour's classic study (which Partch knowledgeable "champions"on behalf of equal temperament. This generously praises) is no less subtlely dogmatic, al- certainly seems to be a subject that attracts though in this case the a priori ideal is not Just in- pas- sionate words (see Lloyd and 1979, 8 tonation, but equal temperament. He calculates Boyle Chapter for even more) from all sides! every alternative tuning's "deviation"from 12V2, and argues most on behalf of those which exhibit The Crossroads the least mean and standarddeviations from equal temperament. There's something uncomfortably It may seem unfair to single out Barbouras circular in the reasoning throughout the book, sort above, of like: although somehow I wince more from his and other theorists' polemic on behalf of "the haves" of status Given, equal temperament quo versus "the have nots" of anyone who might not Therefore, equal temperament totally agree with the final sentence of Barbour's Q.E.D. book:

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This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions Perhapsthe philosophical Neidhardt should be equal temperament is not altogether as benign as allowed to have the last word on the subject: all of us born into it commonly assume: "Thus equal temperament carries with itself The first of these pieces contained a number its comfort and like the estate discomfort, holy of sustained major thirds, which work perfectly of matrimony." well on an organ tuned to one of the unequal temperaments common in the seventeenth That's a cute quote, but its use here is not a little century, but which fight unmercifully on to- smug, suggesting what was "good enough for day's equally tempered instruments. During Grandpa"ought be good enough in perpetuity for the playing of it the audience stirred uneasily, the rest of us. There's no thought I've read by even and, when I have played the tape, numerous the most messianic of tuning reformists that raises musicians ... have asked me what terrible my hackles as much. thing went wrong with the organ. Most are in- The truth of the matter is that up until now credulous when the explanation is given, even there's simply been no way to investigate beyond when they listen to the piece by Bach played the standardscale within the limits of the precision the same evening on the identical organ, sound- of the available technology. Acoustic instruments ing ... like the admirableinstrument it is, are almost compatible with a ?+5-10-cent pitch Bach ... arrangedfor his thirds to come and tolerance at best. A fine grandpiano cannot be ex- go, well disguised by their musical context." pected to performmuch better (Helmholtz 1954, (Benade1976, pp. 312-313). pp. 485-493), not to overlook its particularlypro- Computer-controlledsynthesis, on the other nounced octave stretching (Backus 1977, p. 292; hand, has no inherent need to respect these here- Benade 1976, pp. 313-322), a phenomenon we will tofore inescapable limitations. There's certainly examine later. Even the best analog synthesizers little need here to provide any further motivation cannot do much better, and up until recently did a for breaking out of the three-century "rut"that this lot worse (I shudder to remember the constant pitch long introductory material has documented. The drifts while realizing Switched-on Bach on an early exponential growth in computers has finally ex- Moog!). panded to include systems designed expressly for More frustrating than that is the usual limit on music production, editing, and performanceat low number of notes in an octave, which really was the enough cost to be as affordableas, say, a good grand wholly "economic" reason for musicians of Bach's piano. This is the first time instrumentation exists time to yield the much preferablesounds of the sev- that is both powerful enough and convenient enough eral varieties of meantone tuning for 12/2 equal to make practical the notion: any possible timbre, temperament. Ellis describes the results in his fine in any possible tuning, with any possible timing, appendixes to Helmholz's book (1954, p. 434): sort of a "three T's of music." That places us at a crossroads,to figure out just how to use all of this If carried out to 27 notes . .. it would probably newly available control. And we'll discover that the have still remained in use. .... The only objec- three "T's"are really tied together. tion ... was that the organ-builders, with rare exceptions ... used only 12 notes to the octave MusicalTimbres ... and hence this temperament [Meantone] was first styled "unequal"(whereas the organ, In Figs. 1-3 we have spectralplots for three common not the temperament was-not unequal, but- types of musical timbre. Figure 1 depicts purely har- defective) and then abandoned. monic partials (the most common, at least in the West), such as heard on the horn and most wind and string instruments. The equations of vibration here Not everyone agreed with meantone. abandoning are usually quite elementary: The English held out on many instruments until the end of the last century. Even now the truce of fn = nf1

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This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions Fig. 1. Original: pure har- Fig. 2. Original: stretched Fig. 3. Original: nonharmonic partials. harmonic partials. monic partials.

100 dB 100 dB

0 o dBlo dB 200 Hz 400 800 1600 3200 Hz 200 Hz 400 800 1600 3200 Hz

Fig. 2 tor of 2.76 for every octave we rise above 10 0 d middle-C, and similarly decreases by the same factor as we go down each octave (Benade 1976, p. 315). Figure3 depicts the nonharmonic partials heard on more complex vibrating bodies, primarily the ideophones of the percussion family, although a fascinating world of new timbres is now possible by combining propertiesof "blown,""bowed," and Sd 400 200 Hz 400 800 160060 3200 Hz "sustained"from strings and winds with the more complicated partial structure of ideophones. (The beginnings of this sort of work on timbral hybrids which gives the frequency of the n th partial, as fn, is discussed in part in [Milano 1986].)For a hinged the simple integer n of the "fundamental multiples bar the partials are described: frequency,"f1. For a perfectly idealized vibrating the is: n 2(r/L2)NY/d)(IT/~2).( string expression fn= f, = n(1/Lr) /(T/d)/(1/4ir) Here, as in the previous equation, Y is the modulus and L where the given (round)string has a length L, a ra- of elasticity, and d are the length and density, dius r, a density d, and is under tension T (Benade just as with a string. More 1976, p. 313). We can see at once that this is merely complex vibrating bodies have fairly in- a more complete form of the first equation. volved mathematical descriptions which we will not into here. For Figure 2 depicts the stretched harmonic partials go the interested readeran excellent treatment can be found in heard on most "real world" thicker strings, as the The Theory of lowest strings of the 'cello, but most famously on Sound (Rayleigh 1945, pp. 255-432 of Vol. I). the piano. This explains the comment made earlier In the case of a rectangularbar (Benade 1976, n about the piano being a less-than-ideal pitch refer- p. 51), varies: ence. The equations of motion of such instruments yield an overtone sequence described by: n= 1.00 - n2 2.68 = * + - fn nfl (1 Jn2) n3 3.73 which is the same as the original equation, except n4 - 5.23 that the Jn2term graduallyraises the successive partial frequencies above the nfI "fundamental."J which, it will be noted, form a series of partials is a small coefficient given by: further apart than any of the previous timbres. In the case of a circular the se- = vibrating membrane, -J (r4Y/TL2)(7r/2)3 quence, here of a well-behaved timpani (Benade and on a good piano will have a value close to .00016 1976, p. 144; also similarly Rayleigh 1945, p. 331), for "middle-C."J increases fairly uniformly by a fac- looks like:

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This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions Fig.4. Superimposedinter- Fig. 5. Fusion chart: prox- val: Just major third (har- imity of intervals versus monic partials). roughness, with examples of enlarged bar graphs.

n, - 1.000 100d8 SMOOTH n2_ 1.504 n3 1.742 ? Just +3rd n4 2.000 - n5 2.245 n6 - 2.494 = n7 2.800 0 d8 which, it will be noted, form a series of partials 200 Hz 400 800 1600 3200 Hz closer than of the timbres. together any previous Fig. 5 (Note how n 1, n2, n4, and n6 look when each term is multiplied by two, implying that the timpani has b/s) (<6 b/s) a "missing" theoretical fundamental. This inspired 1025 b/s) (112 me to try synthesizing a "basstimp"in which this implied partial was added, with a very agreeable result!)

A Confluenceof Partials SEPRRATEPARTIRLS E<<<<< VERYROUGH >>>>>> FUSION, SMOOTH

When two (or more) musical timbres are sounded Hz with musical pitch, while those somewhat at the same time, at any random interval between below are heard instead as loudness fluctuations. their perceived pitches, there is a high probability Oscillations of about 0-6 Hz are usually described that at least some of the various partials from each as chorusing or tremolo, and those around 6-16 as will overlap. Figure 4 depicts a "Fusion Chart,"as roughness. The remaining range, 16-30 Hz, can be two such partials approach,begin to beat with one quite subsonic and barely audible if soft, or at loud another, merge into one, and then repeat the se- levels perceived as a sort of "unpitched rumbling," quence in reverse orderas they separate and con- unless several (higherfrequency) harmonics are tinue to move apart.A graphicalrepresentation of present to give a genuine sense of pitch. Figure4 the related critical bandwidth function has been indicates the straightforwardgraphic conventions described by Pierce (1983, pp. 74-81) and Winckel we will be using for the rest of this paper. (1967, pp. 134-148), and has its origins in Helm- Let's now play two simultaneous tones, both holtz's outstanding work (1954, chapter VIII)of a having the spectrum of Fig. 1 and tuned a Just century ago. Helmholtz produced a chart of harmo- major third (of 386.313714 ... cents) apart. The niousness of consonances (1967, p. 193) that also resulting combination spectrum looks like Fig. 4. forms the basis for Partch'ssimilar representation Both tones are shown with a grey cross-hatching so which he whimsically termed the one-footed bride that where there is any overlappingit will appearas (1979, p. 155), and a simpler plot of consonance by black, with beating shown as in Fig. 5, by graphi- Pierce (1983, p. 79). Our representationhere takes cally descriptive alternating bands or stripes. In this on a bell-shaped curve which "objectifies"the case the Just majorthird on harmonic partials is normally subjective quality of dissonance into the smooth, with no audible beating. more quantifiable phenomenon of beats and their If the interval is now retuned to an equal- frequency. (In certain unusual conditions that will tempered majorthird (of 400.00 cents) as in Fig. 6, not concern us here this simplification may need there is only a modest change in the relative loca- qualification.) tions of the lower partials. But between the fifth Surprisingly, our ears associate sound oscillations harmonic (here the term "harmonic"is accurate) having a frequency of more than approximately 30 of the lower tone and the fourth harmonic of the

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This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions Fig. 6. Superimposedinter- Fig. 7. Superimposedinter- Fig. 8. Superimposedinter- Fig. 9. Musical examples. val: equal-tempered major val: pure octave (harmonic val: pure octave (stretched (a) Very wide octave (1220 third (harmonic partials). partials). harmonic partials). cents). (b) Verywide octaves (1212 cents each). (c) Chord and cluster (equal temperament versus Just). (d) The harmonic scale on C.

100 d8 a (= 1220c.) = 1200c.) (S = 1220c.) (x - 1240c.) SOMEROUGH BEMTS I•l =( c.) E 8E.T.+3rL

alf b Note: = 12 c. flat, = 12 c. sharp etimba:"

Clan et: Note: 12 c. flat, 0 l2 c. sharp

c Horns: Strings:

Scale: 172 is8 19 2 21 22 24 26 27 28 2 0 32 200 Hz 400 800 1600 3200 Hz Harmonic 1 .1 - '- I 1 Fig. 8 leads directly to the stretched octaves mentioned NO BERTS earlier. There are many instruments that have even 0 dB9 H 200 Hz 400 800 1600 3200 Hz greater stretching of their partials than the piano, and the effect on octaves is more audible. One 8 Fig. such instrument, which I call the "metimba,"is a 100 dB . metal version of the and NY SLOW (wooden)marimba, like the i ! "basstimp"alluded to earlier, arose in my timbre- i . work from one of - synthesizing those usually im- - pertinent "what if?" questions. Sound Example 2 , on the soundsheet the metimba with . ... compares a clarinet performingthe same figuration (Fig. 9b): stretched-pure-stretchedon the (stretched- the last a and 200 Hz 400 800 1600 3200 Hz harmonic) metimba, being chord, then pure-stretched-pureon the (harmonic)clari- upper one, some rough beating of 8 Hz occurs, net, again the last being a chord. Since the metimba this for the 200-Hz range we are using. Just a little has also the added complication of several non- higher the rate increases into the roughest area harmonic partials (which we'll look at next), the around 12 Hz, and above that the rate can even smoothness of the somewhat wide octave is slightly climb up into the audio band. disguised by them, but, then, this is a much wider Now let's try a pure octave (of 1200.00 cents) and octave than is found on a piano. In any event, the the same harmonic wave, as in Fig. 7. The result metimba is better served by a 6-12-cent stretched has, not unexpectedly, even more distinct "fusion" octave as in Fig. 10, while the clarinet sounds than the Just major third, with all harmonics lined smoother with pure octaves, as in Fig. 7. up and no beats. The more extreme case of nonharmonic partials If we retain the same interval of an octave, but is shown in Figs. 11 and 12, and heard as Sound Ex- now substitute the stretched harmonic spectrum of ample 1 (see the soundsheet with this issue). Figure Fig. 2, the result is as Fig. 8. Notice that there now 11 plots the result of one particularnonharmonic appearmany fairly strong beats between partials timbre, that of Fig. 3, while Sound Example 1 com- that had been "fused together" in Fig. 6. This is ex- pares a more typical nonharmonic timbre, gam (a actly the situation that exists with a piano, and replica of the tuned-kettles in a gamelan orchestra),

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This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions Fig. 10. Superimposedin- Fig. 11. Superimposedin- Fig. 12. Superimposedin- Fig. 13. Superimposedin- terval: stretched octave terval: pure octave (non- terval: very wide octave terval: very wide octave (stretched harmonic harmonic partials). (nonharmonic partials). (harmonic partials). partials).

1- dB 100d8 SMOOTH 100 NOBERTS FRIRLY BERTS .. -..-...... -.....FEW SIMOO..TH1 -. ....- ..•. .=-.- .. l...... i

Oda- - --f 1 ' - 0 dB 200 Hz 400 800 1600 3200 Hz 200 Hz 400 800, 1600 3200 Hz

Fig. 11 Fig. 13

ooB OUGH.,FRST &ID:[?ROUGH FRSTBERTS SLOWBERTS ......

200Hz 400 800 1600 3200Hz 200Hz 400 800 1600 3200Hz

with the same passage played on the Frenchhorn, strument, and exactly vice versa. I say scale because and notated in Fig. 9a. As Fig. 11 indicates, a pure even though the graphsare of two simultaneous octave sounds fairly "rough"on most such instru- notes (andmelodic intervals are much less critical), ments. We can imagine a nonharmonic timbre that the soundsheet examples are first played melodi- would instead work better the other way around, cally, and some of the same propertiesare repre- with smaller-than-pure,<[2: 1], octave, but most sented musically here as well, if only for the octave. historically derived instrumental timbres have "ac- Pierce (1983, pp. 192-193) discusses a corre- cidently" been like the present example. sponding example conceived of from a very dif- On Sound Example 1 you will hear that the low- ferent direction. He includes a recordedexample est pitch, C4, is the reference lower note of the oc- (Ex. 4.4, and also listen to 2.1-2.5, and see p. 86), tave, and the upper three notes, all nominally C5 which demonstrates the auralillusion of a stretched- (they are closer together in pitch than the sharps harmonic tone that sounds flatter when all of its and double-sharpsin Fig. 9a visually suggest), are partials are exactly doubled! If instead we were to the tones of: move up by a properlystretched octave, as in our Sound Example 1, it would then sound like a "real" 1. A 1220-cent "properly"wide octave octave. 2. A pure octave of 1200 cents 3. The 1220-cent wide version again 4. A 1240-cent extremely wide octave TriadicConfluence The differences are subtle, but gam does sound smoother on the 1220-cent wide version (like Fig. Let's now add a third tone to the other two, to form 12), while the horn is best at the pure 2: 1 ratio oc- a triad. In Fig. 14 we have the simplest case of a Just tave. The horn has very audible beats when heard major triad played on a timbre that has harmonic on the very wide octave, as in Fig. 13. partials. The result is smooth, as we would expect, Clearly the timbre of an instrument strongly because there are quite a few partials that merge affects what tuning and scale sound best on that in- and "fuse."

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This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions Fig. 14. Superimposedin- Fig. 15. Superimposedin- Fig. 16. Superimposedin- terval: Just triad (har- terval: Just triad (nonhar- terval: nonharmonic monic partials). monic partials). "triad"(nonharmonic partials).

100dB 100dB

MANYMRTCHES MRNY MRTCHES ustMajor Triadj (LL SMOOTH) Nonhrmonic (ALL SMOOTH) ...... "T•id" 4, • i• i ::... i ...i::

0 dB 0d 200Hz 400 800 1600 3200 Hz 200 Hz 400, 800 1600 3200 Hz Fig. 15 100 dB Sound 3 the triad on the FEW MATCHES Example finally plays (RLL ROUGH) horn, and at last there's an audible difference:you ...... can clearly hear that the Just version is a smoother, more stable, clearly more "consonant"triad, even though the equal-temperedversion is acceptable if the chord is not long sustained (see the comment 0 dB 200 Hz 400 800 1600 3200 Hz on Bach later). Although I have not included a sound example for it, Fig. 16 shows a type of harmony worth inves- tigating. Here we have a totally foreign and quite If we try the same triad on three tones with the nonharmonic "triad,"played with the same nonhar- nonharmonic spectrum of Fig. 3, the result is not monic partials as in Fig. 15, where the Just triad nearly so smooth. In fact, it's barely recognizable as sounds so diffuse. But the results here are surprising the same fundamental harmonic building block of to an extreme: the peculiar "triad"is consonant, Western music! Figure 15 shows what happens, and fused and well focused, with very little roughness, you can listen to the effect on Sound Example 3. and many matches of partials in all their nonhar- First we hear the triad with a traditional timbre monic glory. ill-equipped to handle harmonies, odd as it might This is the kind of harmony we find in the seem: the xylophone. You can judge the results for gamelan music of Bali and Java,where the scales are yourself, but while the xylophone has fewer con- pelog and slendro, and the instruments are tuned flicting partials than our Fig. 15's nonharmonic gongs, kettles, and metalphones. On my latest re- timbre, the "fusion"we normally expect to hear on cording, Beauty In The Beast, is the composition a Just triad simply doesn't occur. You really can't (in homage to that magical island) I call: Poem for even hear much difference, never mind choose one Bali. In it these timbres and scales are combined over the other, between the Just and equal-tempered with good effect, to produce a music that is neither versions! Western nor Indonesian but is from both traditions. By switching to another of my "whatifs," this Near the end a Concerto for Gamelan and (Sym- time a metal version of the xylophone (metal-xylo), phony) Orchestra occurs. In orderto get the two which has somewhat greater amplitudes of the groups (both actually synthesized "replicas,"of higher partials, and all have a much longer decay- course) to be able to "play"together, it was neces- time than on a regularxylophone (metal soaks up sary to "cheat" the pelog scale slightly toward Just higher frequencies more slowly than wood), the intervals so that the (mostly) harmonic symphonic triad is slightly better defined. But the nonhar- timbres would not sound cacophonous when play- monic partials still allow us little preferencebe- ing it, and yet not so far that the gamelan ensemble tween Just and equal-temperedversions, although would suffer. The original pelog and modified "har- some difference is at least audible. The last of monic" versions are (in cents):

36 Computer Music Journal

This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions ORIGINAL: blings. Now I can use all of the power for timbre 0 123 530 683 814 1226 (=octave) generation of the Alles card (Alles 1979), while MODIFIED: working with total tuning flexibility and precision, 0 123 519 690 811 1216 (=octave) a far cry from the compromises necessary only a few years ago (Blackwood1982), where the choice, The final piece with its "cheat"works very well. "scales-or-sounds,"had to be made. I'm grateful to all the pioneers who have made this possible at long last. Partch'sFolly After working for a while within the bounds of traditional Just intonation, I became frustratedwith A last thought on the examples thus far is the real- both the "plainness"(to ears broughtup on Stravin- ization of one of life's funny ironies. I only under- sky, Ligeti, and Bartok, at least) of the diatonic Just stood it recently while researchingthis paper.It's scale (Blackwood 1985), and the often-described both tragic and touching to think that Harry Partch, one-comma "problems"of relegating one group of who almost single-handedly kept alive the spark notes gotten from a series of 3/2 fifths with an- of Just intonation for half a century, happened to other gotten from a series of 5/4 major thirds (Bar- choose to build instruments most of which had bour 1953; Helmholtz 1954; Olson 1967; Lloyd and rapid decays and/or nonharmonic partials. As we've Boyle 1979; Balzano 1980). That the "twain never been finding out, these are about the worst choices meet" has been an indelible source of frustration that can be made to best show off the wonders of (see Partch 1979, pp. 190-194 especially) for Just- Just tuning. J. S. Bach demonstrated, and Benade al- tempted musician and theoretician alike. ludes to it in the quote earlier on, when you must But if we are willing to give up the ability to use slightly defective tunings, like 12N/2, it sounds modulate, and solve that with the computer hard- a lot less rough if you keep everything in motion: ware, we can tune all 12 steps of our Synergy key- don't linger on the imperfections. .... Tempo is very boards or whatever to a series of pitches from the much tied up with tuning. Both evolve together and overtone series of one particulartonic, as I have given a better tuning than 12V/2, we might expect done with the harmonic scale (see Fig. 9[d]).Any slower or at least more sustained tempi to evolve selection of pitches, and indeed all of these pitches than we now use. But Partch tended to compose when sounded together fuse as consonantly as does music that generally moved right along, as we must the triad of Fig. 14 (at least for harmonically par- do in much equal-temperedmusic-which only tialed timbres). The particularchoice of 12 pitches further "hid" the beauty of his commendable 43- in Fig. 9(d) maintains the greatest continuity of suc- note-per-octave scale. Life plays tricks on us all. cessive prime partials, up to number 19, in fact. (Note that number 21 is a 7/4 above number 12, G an octave lower, and that number 27 is a 3/2 above The HarmonicScale number 18, i.e., they are not musically "prime.") These are shown in Table 1. Sound Example 4 of the soundsheet allows us to Figure 9(c) shows the notes played on Sound Ex- compare a chord played alternately on 122V2equal ample 4 and 5, both of which alternate 12V/2 and temperament, and on a sort of "super-Just"tuning harmonic-scale tuning, and give you a good com- I've been using called the harmonic scale. First I parison of the two. There's little doubt that the had to write a series of programswhich access the equal-temperedversions are inferior. You can hear tuning tables of the Synergy-Plusand GDS Digi- especially well in the high string-cluster example tal Synthesizers (Kaplan 1981) which I use, once the very low difference-tone fundamental, a natural Stoney Stockell (who developed the final software effect of everything being all in tune at last. and much of the hardware for both these systems) And by calculating the same sequence of harmon- found a way to open these up to my curious bum- ics for each of the 12 basic starting pitches, trans-

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This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions Table1. Harmonicscale on C table deference to the instrument manufacturerswho drew the line at 12 notes per octave. You can hear Note Ratio Cents the actual derivation of the still-in-use approxima- tion: two octaves down and four (slightly flat) per- CQ 1/1 0.000 fect fifths up (or in the opposite order)- a major Db 17/16 104.955 third, first as it sounded in meantone, and then as 9/8 203.910 D? it sounded in the then-new equal temperament. D#Eb 19/16 297.513 There's little doubt that ears accustomed to the 5/4 386.314 E? smoothness of the formerwould have detected the Fq 21/16 470.781 F 11/8 551.318 falsity in the latter more acutely than we can today, G? 3/2 701.955 although the differenceis plain enough if you listen Ab 13/8 840.528 attentively. A? 27/16 905.865 What is actually occurringis that we are trying to B6 7/4 968.826 force (3/2)4(1/2)2 to be = 5/4. The left terms total B? 15/8 1088.269 81/64, or 81/80 higher than 5/4. In cents, the left side is (4 * 701.9950) - (2 * 1200.0) = 407.8200, and that is greaterthan 386.3137 by the syntonic posing it up and down by 100-cent increments, we comma of 21.5063. But musicians for centuries arrive at 144 distinct pitches to the octave for our have notated the major triad on C to be: C-E-G, new "modulating"version of the harmonic scale. not: C-E-flattened-by-one-comma-G, that is: These are loaded into the Synergy frequency tables C-E_--G (Barbour1953, chapter III;Blackwood of via an outboardHewlett-Packard 154-162; Helmholtz 1954, pp. (in groups 12) 1985, pp. 43.0-439). computer and a single-octave keyboardwhich I've Something had to give, and the best-sounding way built (Milano 1986) to trigger the computer each was meantone. There are four fifths to absorbthe time I key a change to a new chord fundamental. excess comma, so let's give each a quarterof it: The result (which had classically been called im- 701.9950 - (21.5063/4) = 696.5784 for the mean- possible) is that we can now modulate completely tone fifth. (Two of these up [anddown an octave] around a "circle of fifths" while at all times retain- form a "tone" which is exactly in between the "ma- ing not only the few "classic" Just intervals, but jor tone" of 9/8 and the "minor tone" of 10/9, all the other perfectly tuned pitches of the har- hence the name, meantone ... .) Since low partials monic scale. And that's precisely what happens are involved, the resulting flat fifths beat much twice in my recent composition, Just Imaginings slower than if some higher ratio were mistuned by (on Beauty In The Beast, Aud 200). An excerpt of the same 5.3766 cents-if you will, the fifth can the first "circle" is heard in Sound Example 6. "stand it" better. The major third that results from four of these fifths is, as you can hear on the first half of Sound Example 7, perfect: 386.3137 cents. The Best and Worst To round out this capsule explanation, what occurs in the equal-temperedversion is easily shown. EarlierI suggested about meantone that we have ex- By definition the size of an equal-temperedfifth is: not for modu- = is close to 3/2 cellent reasons to believe that were it 217/12) 1.49830708, which certainly lation problems of "wolf tones" (when instruments (andwhy we can get away with it as the only devia- were tuned to only 12 meantone pitches), this form tion, always melodically, in the 144 notes per oc- of temperament would likely be in use right through tave modulating harmonic scale). In cents we have to the present. In Sound Example 7 you can get the familiar fifth, 700.0. Now, go up four of these some small sense of the frustrations that must have and down two octaves: (4 * 700.0) - (2 * 1200.0) = occurredfor musicians of J. S. Bach'sday as the in- 400.0. And this is 400.0000 - 386.3137 = 13.6863 evitable "pollution" of their triads first occurred,in cents sharp for a major third. A 5/4 involves har-

38 Computer Music Journal

This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions monics double those for a 3/2, so the beats of this division seems to have arisen independently with sharpthird are worse than only four times the mean- several musical theorists, as early as the sixteenth tone fifth's error,as you can hear on the second part century in Italy. (Zarlino and Salinas both suggested of Sound Example 7. a division of 19 parts, although it's doubtful that By the way, to get a feel for the ratio form of de- their derivations were exactly equal [Barbour1953, scribing Just intervals, think of them as naming p. 1151.) which two partials of the overtone series form the In 1555 Nicola Vicentino described what we now interval-for example, if it were the seventh and recognize as the 31-note division. It was redis- the fourth (like B-flat7-ha and C) the ratio would covered mathematically in Holland by Christian be 7/4. Huygens in 1724, who gave an exact description The next brief Sound Example 8 places the of his "harmonic cycle" and how extremely close smooth quality of meantone into a more musical it comes to standardquarter-comma meantone context. By employing the same suggested tech- intonation. As such it has all the merits of mean- nique as the harmonic scale above, retuning as tone for performingmost Western music far more needed via an external computer control, we can smoothly than 12V2, while also permitting un- maintain this tuning's benefits to most of the exist- limited modulations to an unprecedented degree ing repertoire,while avoiding both "wolf tones" and (to the nearest 1200/31 = 38.7097 cents!). any restrictions on modulation. I expect to see such But the distinction of "oldest" genuinely equal- an evolutionary version of meantone become more stepped division probablygoes to the excellent common as digital synthesizing equipment of this 53-note division. A student of Pythagorusnamed kind becomes readily available. Philolaus wrote a description of a method for con- But for traditional Western music the worst pos- structing intervals which leads directly to a measur- sible way to tune is probablythe scale of 13 equal ing scale of 53 comma-sized steps, in which form it is known as Mercator's steps in an octave (13/2), which is heard in the cycle (Helmholtz 1954, Sound Example 9 (with the same music as the last p. 436). Just over a century ago this division was to aid in comparisons).As we will next see in equal championed by R. H. M. Bosanquet, who devel- divisions of the octave, right beside each decent one oped the ideal keyboardwith which all possible is a region of pretty awful ones. And the lower the regulardivisions become really playable by human number of steps, in general the less good the results. hands, the generalized keyboard(Helmholtz 1954, Since 12N/2 is the lowest decent division, right be- pp. 479-481; Yunik and Swift 1980, Fig. 1). Even side it somewhere there ought to be a "dilly,"and Partch (1979, pp. 393 and 438), who had little love this one's it! (Yes,a new tool's strength works both for any equal division, has kind words for Bosan- ways ... .) quet's generalized keyboard. There is scarcely a more worthwhile venture to pursue as soon as possible than adopting a standard for and then manufacturingat least a "limited edi- SymmetricEqual Divisions tion" of these keyboardsfor all of us now becoming involved with this field. The present author con- This could eventually be one of the most fruitful structed a generalized keyboard (which has subse- methods of scale development for our new musical quently been lost) back in the presynthesizer days technologies. The notion of dividing the octave into of 1957 and would love to hear of any serious work steps of equal size also happens to have gotten a now being done in this direction. I have several great deal of attention over the past few hundred long-thought-out ideas and proposals (see Fig. 17), years, so there's really a good deal of information which I will gladly share towards the goal of "get- already available about it (Barbour 1953; Helmholtz ting the standard right," that we may avoid the fate 1954; Yasser 1975; Partch 1979; Yunik and Swift of digital tape recording. Obviously it ought be some 1980; Blackwood 1982, 1985). As an idea, multiple variant of a MIDI general-purpose unit.

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This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions Fig. 17. "Multiphonic"gen- a standard keyboard, ap- rals), black (for sharps), number of digits per oc- eralized keyboard. One oc- prox. 1/2" wide and 1/2" dark brown (forflats), light tave here, although it may tave, C to circa 31V_ note high, but only 1-3/4" long. greyish-blue (for double- be extended as necessary. names. This is a perspec- They are colored in five sharps), and golden tan This design is a modern tive drawing of one octave shades along the blue- (for double flats). The notes updated treatment of the from a proposed standard yellow axis of the CIE are named here for the original, conceived and of generalized keyboard by Chromaticity Diagram, to 31N/2 Division, but the key- built in 1875 by R. H. M. Wendy Carlos. The indi- facilitate proper use by the board is suitable for all Bosanquet. vidual notes are shaped color-blind. The suggested regular divisions of the oc- like the sharps and flats of shades are: white (fornatu- tave, up to 55 notes, the

FA 40.1 r

and if the task can be somewhat automated Which equal divisions look most useful? by such, Figure should we care? 18 is a computer plot I recently made to help nar- why Since harmonies from the natural seventh of 7/4 row down the decision. The inspiration came from and from the natural eleventh a simpler version by Yunik and Swift (1980, p. 63) (septimal), perhaps of 11/8 a role in in Computer Music Journal4(4). But I was curious (unidecimal) may play good any I've added these onto about what happened as one went continually and newly developing harmonies, the more common ratios as two gradually from integer step to integer step, and also "classic-Just" op- tions to be calculated and wanted to reflect the fact that with a computerized additionally plotted. ended different from frequency-table driver, there's little need to My algorithm up being quite "penal- Yunik and Swift's: ize" what may be a fine division that requires more than two dozen notes (although more than 60 could 1. Calculate and store the cents for each prime get unwieldy). The computer certainly doesn't care, ratio of interest. (I chose: 3/2, 4/3, 5/4, 6/5,

40 Computer Music Journal

This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions Fig. 18. Equal divisions of Fig. 19. Symmetric equal- the octave. I(deviations step divisions from 10 to from pure)2. 60 notes per octave.

53 65 /1# 1.9

If4i, 'TIT: If

ZII I1; I I Allri-rI L Is , ,,,iI , CLASSIC?JUST ,i; ~~CLA ST15TII < 41...... ABOVE+70,h HARM...... ABOVE7 uII1I:1111!I I131iIr~r11 1 Ii i II ---__ - ABOVE +llth -- - - ABOVE- 11thHARM. I------HA::?j

. j 11I11111111 ; ij Hi ITI in 1 0 n 1 l Im,NEQUAL DIVISIONm NOTESPER OCTAVE I N9N IIIm QUA01SISIrs -?OTM q eeInOC AVEm mO U -n 8 V

than 60 before for the elaborate 5/3, 8/5, plus 7/4 and 11/8. The rest can be steps heading really alternatives. derived from these and are not prime.) The solid line is of the deviations 2. Set up a loop with some small increment on "classic-Just" division size. only. The dotted line is that for septimal harmonies and it never above the 3. For each new division size, calculate the size added, obviously goes at times it can as of one step, and set up and clear the proper former, although get quite close, at the otherwise so-so 15 or at the remark- registers. division, able 31 harmonic of the 4. Find the nearest step to each given ratio and cycle Huygens. Similarly calculate its error. dashed line adds the harmonies from the 11th par- and fits well at those same two 5. Add the square of this errorto the "sum-of- tial, again divisions, 15 and 31. the-squares"register. The most of 13-24 6. Plot that value, increment the division size, thorough investigation cycles was that done Blackwood and loop to 3. likely recently by (1982 and 1985). He also took the extra step and com- The plot is carriedout three times: once for the posed a series of etudes, one in each cycle. It's very "classic-Just"ratios, once with the additional re- important that we do not fall into the trap (Lloyd strictions of satisfying the septimal ratio (7/4), and and Boyle) of remaining only theorists. We have to a third time adding also the unidecimal ratio (11/8). compose real music of many kinds within all and The resulting Fig. 18 fits very well my own and any of our new tuning schemes, if this work is to all other examined descriptions of the audible har- have any lasting value at all, or be taken seriously monic "merit" of particularequal divisions. Note by the music community (andby the public at large, the peaks singled out at several of the most impor- if it is to survive). More importantly, just as "Papa tant divisions: 12, 19, 31, 53, 65, and the one that is Bach" did with his generation'snew scale, this is a mean between 53 and 65, 118 (= 53 + 65, and is the only way we're going to learn how to control "nearly perfect" by all published reports and by our and use wisely these new gifts from the age of the diagram).Also notice that we are not searching in computer to us! particularfor diatonic scales (Blackwood 1985), another reason for omitting 9/8 and 10/9 from the list in step 1 of the previously mentioned algorithm. A AsymmetricDivisions much closer view over the range of 10 through 60 stepped divisions is given in Fig. 19. I think we will I've saved the most surprisingtuning concept for want initially to experience those divisions with less last. The previous divisions all made the same tacit

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This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions Fig. 20. Asymmetric equal- step divisions, locations of alpha, beta, and gamma.

E Ia CENTS PER can handle the even with an ex- STEP,•V octaving later, say ternal control computer as the modulating harmonic scale alreadyrequires, that may free up the compromise-screeningfunctions of our intensive- search programto find some really interesting equal- step specimens. This asymmetric division search uses the ratios: and 0 >I , program 3/2, 5/4, 6/5, 7/4, 11/8. The results are given in Fig. 20. There are three main peaks for the region between lri I I i I Ii I I iCLASSIC JUST 120-30 cents per step, the reciprocalto 10-40 equal ....ABOVE +7th HARM. octave. I call them beta and iinl#N~lll1115111Iiliii i I111 I II I I I I i I I 11ov:---ABOVE +clth HARM. steps per alpha (a), (P/), gamma (y). There occur "echo peaks" at each doub- ling of the number of steps from a past peak, some- EQUALSTEPSPER OCTAVE thing difficult to see in the symmetric division plots. The two in Fig. 20, a' and P', fall equally to either side of the essentially perfect(!)y (on the that each assumption: target ratio be available as classic-Just curve-for septimal harmonies the first an interval in all inversions. They were symmetric: "echo," a' is excellent). These happy discoveries oc- there was the ratio of the prime perfect fifth, 3/2, cur at: a = 78.0 cents/step = 15.385 steps/octave, but also the 4/3 perfect fourth, (given 2/1 and 3/2 it p = 63.8 cents/step = 18.809 steps/octave, y = 35.1 follows so is not directly, really "prime"in the sense cents/step = 34.188 steps/octave. The deviation been that we've using idea). Similarly,there was the axis's arbitraryunits are incompatible on this figure but also major third, 5/4, its inversion, the minor with those on the last two, but a quick comparison 8/5. Both 6/5 and 5/3 sixth, appeared.Only for the will show just how different (and also similar) the newer ratios a optional is nonredundantform ap- asymmetric and symmetric divisions are. we have 7/4 but not plied: 8/7, 11/8 but not 16/11. Sound Example 10 plays a "nearly"one-octave Since each of the redundantpairs is symmetric scale of alpha on the horn. Notice how there are with respect to the octave, the net result is sort of four steps to the minor third, five steps to the major an "over-representation"of this interval. Little won- third, and nine steps to the (this time no kidding) der that every one of the peaks to the plots in Fig. perfect fifth, but, of course, no octave (the final "at- 18 and 19 occurs at exact integer divisions of the tempt" at this is an awful 1170-cent version, the octave! But the octave is the ratio most common to next step to 1248 cents being even further away!). the "strategies"of most digital synthesizer architec- But that's the trade-offwe've requested: there's no tures, such as in the 16', 8', 4' octaving borrowed free lunch! Sound Example 11 is a brief chordal pas- from the pipe organ. Most timbre/instrument files sage in alpha. The harmonies are amazingly pure; include the similar designation of transpositions up the melodic motion amazingly exotic. or down by octaves. In my current Synergies the I've not included a similar example of beta or frequency of each note is stored in the previously gamma on the soundsheet but have experimented mentioned frequency-table,but only 12 main 16-bit with both (gammareally requires a "multiphonic" words control the middle-most octave, the other oc- generalized keyboard,like most <24 divisions). taves duplicating these frequencies by exact factors Beta is very like alpha in its harmonies, but with of two, a very common method of assuring all the five steps to the minor third, six to the major third, octaves can be made "beat-free."We have octave and eleven to the perfect fifth, melodic motions are possibilities all over the place. different, rather more diatonic in effect than alpha. So why not, as an experiment, omit the octave- Gamma (nine steps, eleven steps, twenty steps) is redundantratios from the first step of our algo- slightly smoother than these, having no palpable rithm? That will lose all octave symmetry, but if we difference from Just tuning in harmonies. But the

42 Computer Music Journal

This content downloaded on Fri, 4 Jan 2013 12:02:15 PM All use subject to JSTOR Terms and Conditions scale is yet a "thirdflavor," sort of intermediate to References a and p, although a diatonic scale is melodically available. I have searched but can find no previous Alles, H. G. 1979."An Inexpensive Digital Sound Syn- description of a, p, or y nor their asymmetric scale thesizer." ComputerMusic Journal3(3):28-37. family in any of the literature. Apel, W. 1972. HarvardDictionary of Music. Cambridge, Alpha has a musically interesting propertynot Mass.:Harvard University Press. 1977. The Acoustical Foundations Music. found in Western music: it splits the minor third Backus, J. of in half into This is what ini- New York:W. W. Norton & Co. exactly (also quarters). G. 1980. "The of led me to look for and I called it Balzano, S. Group-theoreticDescription tially it, merely 12-Foldand Microtonal Pitch Systems." minor-third scale of 78-cent Computer my "split steps. Beta, Music Journal4(4):66-84. like the 19 division it is does the symmetric near, M. 1953. Tuning and Temperament.East Lan- same to the fourth. This whole formal Barbour,J. thing perfect sing: Michigan State College Press. discovery came a few weeks after I had completed Bateman,W. A. 1980. Introduction To ComputerMusic. the album, Beauty In The Beast, which is wholly New York:John Wiley & Sons. in new tunings and timbres. The title cut from Benade, A. H. 1976. Fundamentals of Musical Acoustics. the album contains an extended study of some /, New York:Oxford University Press. but mostly a. An excerpt highlighting the proper- Blackwood, E. with J.Aikin. 1982. "Discoveringthe Mi- ties we've been discussing is the final Sound Ex- crotonal Resources of the Synthesizer." Keyboard ample 12. 8(5):26-38. Blackwood, E. 1985. The Structureof Recognizable Di- atonic Tunings. Princeton:Princeton University Press. Conclusion Helmholtz, H. 1954. On The Sensations Of Tone. Trans. A. J.Ellis. New York:Dover Publications. J.K. 1981. "Developing a Commercial Digital I Kaplan, don't pretend to be either a writer or theorist, so Sound Synthesizer." ComputerMusic Journal thank you for putting up with my attempts at both 5(3):62-73. in this lengthy report. But I have been fascinated by Lloyd, L. S., and H. Boyle. 1979. Intervals, Scales and scales and tuning for about thirty years, well before Temperaments.New York:St. Martins Press. discovering electronic music, with the single excep- Milano, D. 1986. "Interviewwith WendyCarlos." Key- tion of Olson's (1967, chapter 10) demonstration board 12(11):50-86. record of the RCA Synthesizer in 1953. Although Olson, H. F. 1967. Music, Physics and Engineering.New York:Dover Publications. my name is indelibly connected with the Moog since that accident of some Partch,H. 1979. Genesis Of a Music. New York:Da Capo Synthesizer unexpected Press. 18 it seems to me that now is our years ago, only Pierce, J. 1983. The Science of Musical Sound. New York: fledgling art starting to show healthy signs of grow- Scientific American Books. I'm ing up. passionately excited by the promise of: Rayleigh, J.W. S. 1945. The Theory Of Sound. 2 volumes. any possible timbre, any possible tuning. That's the New York:Dover Publications. reason for the work that went into this paper.It's Yasser,J. 1975. A Theory Of Evolving Tonality. New also a way to return the favor to Computer Music York:Da Capo Press. Journalfor a decade of ideas and inspiration that led Yunik, M., and G. Swift. 1980. "TemperedMusic Scales directly to my "very nearly" attaining this double- for Sound Synthesis." ComputerMusic Journal headed ability. For all those who helped and those 4(4):60-65. on whose shoulders I am to Winckel, F. 1967. Music, Sound and Sensation. New lucky stand, deepest York:Dover Publications. thanks. We've been treading water much too long, and I'm delighted that the real work now can begin.

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