Path: utzoo!utgpu!watmath!att!rutgers!ucsd!usc!merlin.usc.edu!aludra.usc.edu!alves From: alves@aludra.usc.edu (William Alves) Newsgroups: comp.music Subject: Re: New tunings Keywords: Intonation systems, octaves, tuning systems Message-ID: <6540@merlin.usc.edu> Date: 17 Nov 89 04:02:56 GMT References: <3068@husc6.harvard.edu> <6335@merlin.usc.edu> <3113@husc6.harvard.edu> <6460@merlin.usc.edu> <3194@husc6.harvard.edu> Sender: news@merlin.usc.edu Reply-To: alves@aludra.usc.edu (Bill Alves) Organization: University of Southern California, Los Angeles, CA Lines: 126 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