Path: utzoo!yunexus!ists!jarvis.csri.toronto.edu!mailrus!uunet!samsung!gem.mps.ohio-state.edu!think!husc6!walsh!elkies From: elkies@walsh.harvard.edu (Noam Elkies) Newsgroups: comp.music Subject: Re: New tunings Summary: Why are we arguing?! Keywords: Intonation systems, octaves, tuning systems Message-ID: <3194@husc6.harvard.edu> Date: 16 Nov 89 19:27:21 GMT Article-I.D.: husc6.3194 References: <3068@husc6.harvard.edu> <6335@merlin.usc.edu> <3113@husc6.harvard.edu> <6460@merlin.usc.edu> Sender: news@husc6.harvard.edu Reply-To: elkies@walsh.harvard.edu (Noam Elkies) Organization: Harvard Math Department Lines: 66 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