Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!uwm.edu!rpi!gem.mps.ohio-state.edu!think!husc6!walsh!elkies From: elkies@walsh.harvard.edu (Noam Elkies) Newsgroups: comp.music Subject: Re: New tunings Keywords: Intonation systems, octaves, tuning systems Message-ID: <3246@husc6.harvard.edu> Date: 23 Nov 89 04:25:56 GMT References: <3068@husc6.harvard.edu> <6335@merlin.usc.edu> <3113@husc6.harvard.edu> <6460@merlin.usc.edu> <3194@husc6.harvard.edu> <6540@merlin.usc.edu> Sender: news@husc6.harvard.edu Reply-To: elkies@walsh.harvard.edu (Noam Elkies) Organization: Harvard Math Department Lines: 81 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 Univesrity