alves@aludra.usc.edu (William Alves) (11/14/89)
In article <3113@husc6.harvard.edu> elkies@brauer.UUCP (Noam Elkies) writes: > >Of course you can easily retune the fundamentals of any number of existing >instruments to your favorite New Tuning. But this still leaves two >problems: First, the Pythagorean overtone structure would remain, whereas >you'd probably prefer to redo the entire harmonic series as well; that's where >you'd spend all this time constructing new instruments. [I realize that >I may be here implicitly invoking a controversial assumption about the >relation between the harmonic series and Western tuning---see below.(*)] Aha! Is there or should there be a connection between tuning systems and the harmonic series? As, I think, a previous posting of mine demonstrated, it is the 12-tone equal temperament system which bears little relationship to the intervals found in the harmonic series. That is why the lovely the ratio be- tween the fifth and fourth harmonics (5/4) has been tempered practically be- yond recognition to the modern major third for the sake of rendering all the keys equally useful (or equally out of tune, as Lou Harrison has said). Instead, it is JUST INTONATION which seeks to preserve the intervals found in the harmonic series for the sake of improved consonance (and at the expense of rendering some keys less "useful"). Virtually every non-percussion instrument as well as most mallet instruments have harmonic spectra (after the first few milliseconds of the attack on some). An exception to this that has been brought up is the piano, where the inharmonicity resulting from the stiffness of the strings necessitates stretched and compressed octaves. Here, admittedly, the question becomes more complex, and I'm not entirely satisfied by the "conso- nance formulas" in such acoustics texts as Dick Moore's that take into account the relative ratios of some of the partials, and not just the fundamentals. However, consider this: if the fundamentals have a low number integral ratio, so too will the lower-order harmonics. >Second, while the instrumentalist will then easily play the notes in >your score, (s)he will need a lot of coaching to play them musically; >I see no reason why musical intuition developed over years in the context >of Western tuning must directly transfer to your new tuning system. If you're talking about a fixed pitch instrument, such as a retuned piano, I don't think it would necessarily require extra training at all. It has been my experience that musicians who have a sense of the important issues of musicality (phrasing, dynamics, etc.) are not "thrown off" by having a 5/4 result from two keys which would normally give them a major third. It is somewhat more difficult if you're talking about continuous pitch instru- ments, such as unfretted strings, but because they have to learn to play in tune, not because of a cognitive dissonance [:-)] which goes against "musical intuition developed over years." >But the various systems that were used were all within a few cents of >each other and of equal temperament, and thus mutually compatible for >many purposes; furthermore, as several posters have noted, the differences >are academic except for rigidly pre-tuned instruments (mostly keyboards >and fretted strings). I'm sorry, but I don't think that's true at all. I will admit that the dif- ferences between the various "good" temperaments towards the end of the 18th century and equal temperament could be considered "academic" or "within a few cents." But that is not at all true before that period. The differences be- tween Pythagorean tuning, mean-tone temperaments, just tunings, and equal temperament are quite apparent and not simply "a few cents" off. For the de- tails see the reference I mentioned before "Tunings and Temperaments: A His- torical Survey" by J. Murray Barbour (even though it's awfully prejudiced to equal temperament). > >Why, then, did generations of composers struggle between a theory that >proclaimed the fourth a "perfect" consonance and the third an "imperfect" >one, and their ears that told them otherwise? Well, first of all, up until about 1200 many theorists called the fourth a con- sonance and the third a dissonance. As the thirds became more popular with composers, theorists begrudgingly labelled them imperfect consonances. The problem here was that if one used Pythagorean tuning (standard at that time because it was Boethius' fave) then the thirds are complex ratios (major third, or ditone, = 81/64 and minor third, or semiditone = 32/27 or 19/16). It was Walter Odington, I believe, who was the first to suggest that they were ac- tually consonances because of their proximity to 5/4 and 6/5 respectively. The fourth was considered a dissonance because they always measured intervals from the bottom voice up. If one used a fourth and a fifth up from the bottom voice, the dissonance would, of course, be the result of the second in the upper voices. And if one uses it in two-part writing, the dominant is in the bottom voice when the tonic is in the top, creating not a "dissonance" but a unstable sonority in the harmonic scheme. > >So, out of honest curiosity, I ask: why do you create tuning systems? > That's a big question, and I can only answer for myself. First of all, I work in just tuning systems, not microtonal or other tempered systems. I do so partly because of the increased consonance, but also from the enormously in- creased pallette that results from the inclusion of such intervals as 7/4, 11/8, or 7/6. The difference in clarity and sonority is striking and beautiful. Such chords as a 5/7/11, which when played on the corresponding keys in equal temperament would be jarringly dissonant, are actually quite consonant, not as consonant as a 4/5/6 (major chord), but somewhere in the middle of that vast spectrum. >(*) Something I've been wondering about intermittently, and reminded >of by your mention of gamelan music: Gamelan music is dominated by >instruments with an overtone series very different from the familiar >overtone series that dominates most Western music. It also uses >very different tunings, tunings which unlike their Western counterpart >developed (I assume) without knowledge of overtones. Thus it could >make an interesting test case for the perennial debate about the >naturalness of a system of tonality based on the overtone series. >Has any significant research been done into the relation or lack >thereof between gamelan tunings and gamelan overtones? > Funny you should ask, Noam, because I just happen to have done a number of spectral analyses on gamelan instruments. First of all, most of these are not at all "very different" from Western instruments in their spectra. As I said before, apart from their attacks, these instruments exhibit the same harmonic frequencies as most other instruments. Exceptions are: 1) The hanging gongs. In these instruments the harmonics have been (presumably) intentionally com- pressed by the flange, giving them the beats that characterize them as well as the stretched and compressed octaves in the other instruments. 2) Drums. 3) The bonang and other kettle-gongs (kempul, ketuk) have some inharmonic par- tials, but they also exhibit very strong harmonics, like the other instruments. However, I don't think such a study will necessarily shed any light on the relationship between the tuning system and the spectra, because there is no standard gamelan tuning system. Every set of instruments is tuned to itself and no other. See Mantle Hood's "Pelog and Slendro Redefined" in Selected Reports (in Ethnomusicology) 1#1. Bill Alves USC School of Music / Center for Scholarly Technology
elkies@walsh.harvard.edu (Noam Elkies) (11/17/89)
Well, Bill, I'm glad I asked >>So, out of honest curiosity, I ask: why do you create tuning systems? From your answer it is apparent that the tuning systems you're working with are not at all of the kind to which the obstacles I mentioned would apply. In fact, from your description of the ideas you're working with I'm very interested in finding out more details (either posted or e-mailed) and hearing the results. Meanwhile, some comments on other points raised in your post: >Aha! Is there or should there be a connection between tuning systems and the >harmonic series? As, I think, a previous posting of mine demonstrated, it is >the 12-tone equal temperament system which bears little relationship to the >intervals found in the harmonic series. That is why the lovely the ratio be- >tween the fifth and fourth harmonics (5/4) has been tempered practically be- >yond recognition to the modern major third for the sake of rendering all the >keys equally useful (or equally out of tune, as Lou Harrison has said). I can and did work out the powers of 2^(1/12) and their continued fractions. Personally I've never found the tempered third to be so horribly out of tune--- I guess different ears have different tolerances for such discrepancies. If you insist on exact rational ratios, though, the minor triad (10/12/15) should be even more dissonant-sounding than your 5/7/11 sonorities; apparently our ear accepts this as a variation of the major triad (10/12.5/15) [I'm not sure whether this is Hindemith's interpretation or his quoting an earlier theorist, but at any rate it's the most plausible explanation I've seen], and accepting the consonance of the tempered major triad (10/12.599/14.983) is rather less of a stretch. > [...] Virtually every non-percussion instrument >as well as most mallet instruments have harmonic spectra (after the first few >milliseconds of the attack on some). An exception to this that has been brought >up is the piano, where the inharmonicity resulting from the stiffness of the >strings necessitates stretched and compressed octaves. [...] I was the one who brought this up. I'm not sure where the statement about "virtually every non-percussion instrument..." comes from; it would seem to me that any physical oscillator (except a periodically forced one) would deviate from a harmonic spectrum, and mallet instruments would deviate considerably because they are not effectively one-dimensional. About "imperfect" thirds and "perfect" fourths---I find it hard to believe that the pre-1200 theoretical definition of a minor third was as complex a ratio as 19/16 (much harder to rationalize than a major third ratio of 81/64, which is just four perfect fifths less two octaves) in preference to 6/5, but I'll take your word that this was Boethius' Rx. Once the fifth partial is accepted, though, I have doubts about your explanation that > And if one uses [the fourth] in two-part writing, the dominant is in the >bottom voice when the tonic is in the top, creating not a "dissonance" but a >unstable sonority in the harmonic scheme. This rings true to modern ears used to common-practice harmonic schemes (though even there such a voicing need not be unstable---consider the final chord of the second movement of Beethoven's 7th), but it might be stretching it to apply such considerations to the period in question; indeed the downward tendency of "perfect" fourths must have contributed to the development of "harmonic schemes". Finally, my inquiry about gamelan tuning elicited some very interesting responses, both posted and e-mailed, which I'll summarize later. --Noam D. Elkies (elkies@zariski.harvard.edu) Department of Mathematics, Harvard University
alves@aludra.usc.edu (William Alves) (11/17/89)
In article <3194@husc6.harvard.edu> elkies@walsh.harvard.edu (Noam Elkies) writes: > >In fact, from your description of the ideas you're working with I'm very >interested in finding out more details (either posted or e-mailed) and hearing >the results. Thank you for your interest, but I'm not sure that I can give you any "re- sults" beyond the sound of my own compositions. By the way, for those in- terested in just tuning systems, might I suggest you contact the: Just Intonation Network 535 Stevenson St. San Francisco CA 94103 (415)864-8123 Among other things, they have a catalog of books, recordings, and software, and they have also released "Rational Music for an Irrational World," a compilation album of music in just tuning systems. (Including, I hope it is not too immodest to add, a work by me, and their catalog includes a cassette of mine. For more details, you can email me.) >I guess different ears have different tolerances for such discrepancies. There's no denying this, but a lot of these tolerances are culturally con- ditioned. I know a composer with dead-on perfect pitch who would visibly wince at my 7/4 harmonies, because he heard them as minor sevenths 30 cents out of tune. If one can get past these expectations, I think most people will discover that the distinction is significant and even differences of a few cents can make a great difference in the sonority. >If you insist on exact rational ratios, though, the minor triad (10/12/15) >should be even more dissonant-sounding than your 5/7/11 sonorities; apparently >our ear accepts this as a variation of the major triad (10/12.5/15) [I'm not >sure whether this is Hindemith's interpretation or his quoting an earlier >theorist, but at any rate it's the most plausible explanation I've seen], >and accepting the consonance of the tempered major triad (10/12.599/14.983) >is rather less of a stretch. I don't know whose theory that is, but many theorists seem to have been bo- thered by the fact that the minor triad does not occur naturally in the har- monic series, unlike the major. Zarlino, Rameau, and (least successfully) Hindemith went to elaborate lengths to find "justification" for the existence of the minor triad. Hindemith used a system of both overtones and "undertones" plus rampant fudging of intervals, but as far as I know, none of them suggested that its consonance was due to the fact that it was a "variation" of the major. Unlike some theorists, I haven't worked out a "formula" for finding the rela- tive consonance of a complex sonority, but intuitively, I would reduce it to its component intervals (6/5 and 3/2), which are both lower number ratios and hence more consonant than my example (7/5 and 11/7). But it's not a distinction I feel strongly enough to argue about. > >> [...] Virtually every non-percussion instrument >>as well as most mallet instruments have harmonic spectra (after the first few >>milliseconds of the attack on some). An exception to this that has been brought >>up is the piano, where the inharmonicity resulting from the stiffness of the >>strings necessitates stretched and compressed octaves. [...] > >I was the one who brought this up. I'm not sure where the statement about >"virtually every non-percussion instrument..." comes from; it would seem to me >that any physical oscillator (except a periodically forced one) would deviate >from a harmonic spectrum, and mallet instruments would deviate considerably >because they are not effectively one-dimensional. > Of course in a physical system, such frequencies are never exact. All strings have stiffness, no vibrating material is completely pure, etc. Even so, I think you would find that the frequency deviation even in the piano is very small. I have looked at the spectra of dozens of common and uncommon instruments, and the vast majority are perfectly harmonic within the resolution of my system (about 6 Hz). The vibraphone and marimba (after the initial attack) are not only harmonic but almost sinusoidal. (Not so the glockenspiel, chimes, or crotales). The reason, I believe, that we hear the partials of a piano as harmonic (as opposed to, say, the partials of a bell) has to do with the phenomenon of the "fusing" of those frequencies that lie within a certain small proximity to each other. It's been known for sometime that, while we can tell very small differences of frequencies when the tones are played separately, when they are played together, when hear an average frequency, plus beats. It's my hypothesis, and I don't know if this has ever been tested, that the same phe- nomenon also applies to intervals which are integral multiples of each other (harmonics). What I object to is those who use this so-called "critical band- width" to justify temperament. While a 5/4 sounds similar to a major third, the existence of beats makes it a distinctly different sonority to me. As I mentioned before, gongs in a gamelan seem to have their partials deli- berately detuned. According to Vetter's article referenced in my last posting, gamelan builders tune the partials so that beats (called "ombak," lit. waves) that result are at the desired frequency. >> And if one uses [the fourth] in two-part writing, the dominant is in the >>bottom voice when the tonic is in the top, creating not a "dissonance" but a >>unstable sonority in the harmonic scheme. > >This rings true to modern ears used to common-practice harmonic schemes >(though even there such a voicing need not be unstable---consider the final >chord of the second movement of Beethoven's 7th), but it might be stretching >it to apply such considerations to the period in question; indeed the downward >tendency of "perfect" fourths must have contributed to the development of >"harmonic schemes". > You bring up a good distinction which I should have made earlier, that is, consonance/dissonance is spoken of in two senses: 1) as an objective acoustic phenomenon, and 2) as a subjective musical parameter. The former is mainly defined by the proximity of the frequencies to a small whole number ratio, and the latter, well who knows? In the first sense I would definitely argue that the fourth is a consonance, and that theorists classification of it as a dissonance was the result of its place in contrapuntal rules, not its sound outside of musical context. I definitely think that medieval and renaissance composers were aware of what I will loosely call harmonic schemes. Certainly they bore little relationship to the triadic-based "common-practice period" practices, but neither did they think completely "linearly" despite what the textbooks say. I had a theory teacher once who played a major sixth and asked the class whether it was consonant. Of course we replied that it was. Then he wrote it on the chalkboard as a diminished seventh and asked. Of course in that context, we answered that it was "dissonant." I hope I haven't been too long-winded in this exchange, and, if so, I apolo- gize for taking up bandwidth in a newsgroup that's supposed to be dedicated to computers and music (or is that what comp stands for?) Bill Alves USC School of Music / Center for Scholarly Technology
ladasky@codon4.berkeley.edu (John Ladasky;1021 Solano No. 2;528-8666) (11/19/89)
In article <6540@merlin.usc.edu> alves@aludra.usc.edu (Bill Alves) writes: >have stiffness, no vibrating material is completely pure, etc. Even so, I think >you would find that the frequency deviation even in the piano is very small. >I have looked at the spectra of dozens of common and uncommon instruments, and >the vast majority are perfectly harmonic within the resolution of my system >(about 6 Hz). The vibraphone and marimba (after the initial attack) are not >only harmonic but almost sinusoidal. (Not so the glockenspiel, chimes, or >crotales). I'm taking a psychoacoustics and comuter music class from Dave Wessel here at U.C. Berkeley (Dave is formerly of IRCAM). My notes from the class indicate that the 29th harmonic of a typical piano has a frequency 30 times that of the first harmonic. He didn't say what string this was, but I would assume that it was a bass string, since the bass strings, so I've been told, exhibit more inharmonicity. BTW, I'm pretty sure that the spectra of the vibraphone and marimba have a lot of interesting inharmonic activity in the attack. Just my $.02. T CROSS POLICE LINE DO NOT CROSS POLICE LINE DO NOT CROSS POLICE LINE DO NOT CR _______________________________________________________________________________ "Do unto others as you would like - John J. Ladasky ("ii") to do unto them. " Richard Bach (ladasky@enzyme.berkeley.edu)
alves@aludra.usc.edu (William Alves) (11/20/89)
In article <1989Nov19.012518.23314@agate.berkeley.edu> ladasky@codon4.berkeley.edu.UUCP (John Ladasky) writes: > I'm taking a psychoacoustics and comuter music class from Dave Wessel >here at U.C. Berkeley (Dave is formerly of IRCAM). My notes from the class >indicate that the 29th harmonic of a typical piano has a frequency 30 times >that of the first harmonic. He didn't say what string this was, but I would >assume that it was a bass string, since the bass strings, so I've been told, >exhibit more inharmonicity. That would make the inharmonicity about 3%, within the range of my system to detect, but I must admit, I have never tried to measure a piano's inharmonicity systematically. Another disclaimer I should have added is that my and most spectral displays are not of much use looking at very low amplitude partials (such as those very high harmonics where the inharmonicity would become signi- ficant) because they get lost in the inherent noise and the amplitude resolu- tion of the system. However, they are still important to the timbre. My intui- tive statement that virtually all of the spectra of non-percussion instruments are harmonic (or very nearly so) is still meaningful, I think, especially in the context of the discussion on how they might influence tuning systems. >BTW, I'm pretty sure that the spectra of the >vibraphone and marimba have a lot of interesting inharmonic activity in the >attack. Absolutely! In fact, one long known principal of psychoacoustics is that attacks are crucial to our timbre perception. However, one interesting study which I believe originated from CCRMA (maybe someone could point me to the exact reference) found that the exact nature of the inharmonics in the attack were not important; it was their overall amplitude and duration that made a difference to the timbre. Incidentally, an interesting thing that I discovered in my own work is that bowing a vibraphone does not give you the timbre of a struck vibraphone without the attack. If it did, the timbre certainly would not be very interesting. Instead the scraping of the bow creates very high and ethereal inharmonic partials which give it its distinctive timbre. Bill Alves USC School of Music / Center for Scholarly Technology
elkies@walsh.harvard.edu (Noam Elkies) (11/23/89)
In article <6540@merlin.usc.edu> alves@aludra.usc.edu (Bill Alves) writes:
:In article <3194@husc6.harvard.edu> [I wrote:]
:>
:>In fact, from your description of the ideas you're working with I'm very
:>interested in finding out more details (either posted or e-mailed) and hearing
:>the results.
:
:Thank you for your interest, but I'm not sure that I can give you any
:"results" beyond the sound of my own compositions.
That's what I meant --- certainly not "results" in the mathematician's sense
of "theorems"...
:[the Just Intonation Network] also released
:"Rational Music for an Irrational World",
:-)
:>I guess different ears have different tolerances for such discrepancies.
:
:There's no denying this, but a lot of these tolerances are culturally con-
:ditioned. I know a composer with dead-on perfect pitch who would visibly
:wince at my 7/4 harmonies, because he heard them as minor sevenths 30 cents
:out of tune.
Ah. His (her?) problem is that "perfect" pitch is (no pun intended) relative---
presumably anybody's pitch is "perfect" enough to tell when a familiar piece
is played an octave up or down, and I doubt anyone can reliably pin down a
pitch played out of the blue to within say 5 cents. So we compensate by
digitizing pitch perception so an interval of (say) 782.5 cents is heard
as a quite flat minor sixth rather than 775+-15 cents. Then it transpires
that the digitization error completely obscured a near-perfect 11/7 interval...
:If one can get past these expectations, I think most people
:will discover that the distinction is significant and even differences of
:a few cents can make a great difference in the sonority.
True, on instruments where the pitch and overtones are defined that precisely
(thus not a piano each of whose 3-string courses is intentionally a few cents
out of tune with itself, nor an instrument like a violin as usually played
nowadays, with such fine gradations swamped by vibrato).
:>[Hindemith: we accept the consonance of a minor triad as a variant
:> of a major one]
:
:I don't know whose theory that is, but many theorists seem to have been bo-
:thered by the fact that the minor triad does not occur naturally in the har-
:monic series, unlike the major. Zarlino, Rameau, and (least successfully)
:Hindemith went to elaborate lengths to find "justification" for the existence
:of the minor triad. Hindemith used a system of both overtones and "undertones"
:plus rampant fudging of intervals, but as far as I know, none of them suggested
:that its consonance was due to the fact that it was a "variation" of the major.
Yes, Hindemith's attempted reconciliation of the chromatic scale with just
intonation succeeds only as a parody of earlier such attempts. I suspect
that Hindemith himself was unsatisfied with it and thus offered the "variation"
theory, which is of a different flavor entirely. I do not have chapter and
verse for this now; I'll look these up after Thanksgiving break.
:>[Hardly any physical instrument has an exact Pythagorean overtone series]
:
:Of course in a physical system, such frequencies are never exact. All strings
:have stiffness, no vibrating material is completely pure, etc. Even so, I think
:you would find that the frequency deviation even in the piano is very small.
:I have looked at the spectra of dozens of common and uncommon instruments, and
:the vast majority are perfectly harmonic within the resolution of my system
:(about 6 Hz).
6 Hz!? That's about 16 cents at A-440, enough to make the just and tempered
third indistinguishable. Aren't more precise measurements available? Even
much smaller deviations would have significant consequences for an intonation
system based on overtone matching.
:The vibraphone and marimba (after the initial attack) are not
:only harmonic but almost sinusoidal.
So indistinguishable after the initial attack? Interesting---I'll have to
remember this!
--Noam D. Elkies (elkies@zariski.harvard.edu)
Department of Mathematics, Harvard Univesrityalves@aludra.usc.edu (William Alves) (11/25/89)
In article <3246@husc6.harvard.edu> elkies@walsh.harvard.edu (Noam Elkies) writes: >In article <6540@merlin.usc.edu> alves@aludra.usc.edu (Bill Alves) writes: >:[account of a certain composer with perfect pitch who would wince at just >:intervals] > >Ah. His problem is that "perfect" pitch is (no pun intended) relative--- >presumably anybody's pitch is "perfect" enough to tell when a familiar piece >is played an octave up or down, and I doubt anyone can reliably pin down a >pitch played out of the blue to within say 5 cents. So we compensate by >digitizing pitch perception so an interval of (say) 782.5 cents is heard >as a quite flat minor sixth rather than 775+-15 cents. Then it transpires >that the digitization error completely obscured a near-perfect 11/7 interval.. > Perfect pitch is indeed relative, and I would be interested to find out exactly what the tolerances for different people are. This particular compo- ser has pretty darn good perfect pitch, and I've only met a few people I could say that about. That aside, I think it's important to add that one doesn't have to have dead-on perfect pitch to hear differences of 5 cents within a sonority. Then the presence or absence of beats will tell you if the interval is far from a just ratio. I am convinced that even for those not trained to notice the beats or when the beats are obscured by a greater number of simultaneous pitches, the small differences are still very impor- tant to the sonority, if at only a subliminal level. >:I have looked at the spectra of dozens of common and uncommon instruments, and >:the vast majority are perfectly harmonic within the resolution of my system >:(about 6 Hz). > >6 Hz!? That's about 16 cents at A-440, enough to make the just and tempered >third indistinguishable. Aren't more precise measurements available? Even >much smaller deviations would have significant consequences for an intonation >system based on overtone matching. > OK, I'll admit that 6Hz can be a significant difference at higher frequencies, but not enough for a percussion instrument to look harmonic by mistake. The distinction between harmonic (winds, strings) and inharmonic (most percussion) is important, even if the ideal harmonic spectrum never exists in nature. At what point should a spectrum be considered "inharmonic"? I don't know. The point was that harmonicity (apart from "small" deviations) is a common pheno- menon. One doesn't need a spectrum analyzer to tell you that. (By the way, the program that I've been using has a maximum FFT window length of 8192 samples. At a 50 kHz sampling rate, which gives a window time of about .17 seconds, that's the resolution you get. Of course the only reasons that that's a limi- tation is that I haven't programmed one myself yet or found another program. Anyway, it's better that DigiDesign's Sound Designer, which has a resolution of...is it 96 Hz? Something like that.) Also, I never looked at these spectra explicitly for their inharmonicity. Your question as to whether these devia- tions would or should influence the tuning system is interesting, but I don't have the answer. >:The vibraphone and marimba (after the initial attack) are not >:only harmonic but almost sinusoidal. > >So indistinguishable after the initial attack? Interesting---I'll have to >remember this! > Not at all. First of all, the attack is a vital part of what we hear as the timbre and is hard to separate from the rest of the sound. Secondly, I should qualify my use of "almost" by saying that very small deviations (low ampli- tude partials) from the sinusoidal can be very important to the timbre; i.e. by "almost" I didn't mean "indistinquishable." Bill Alves USC School of Music / Center for Scholarly Techology