source file: mills2.txt Date: Wed, 26 Feb 1997 19:21:59 -0800 Subject: Re: More not less From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: PAULE >>The whole point is that it doesn't matter what the exact form dissonance >>function is! The formulas that result from this assumption is, as I tried to >>make clear in words, >>minimize (error of 3:1)^2+(error of 5:1)^2+(error of 5:3)^2 and >>minimize (3*error of 3:1)^2+(5*error of 5:1)^2+(5*error of 5:3)^2. >>The results of applying these to meantone tunings I have already posted in >>both cents and exact formulae. If you want me to work you through the >>calculus please try to reach me offline, I don't think we need to scare any >>more musicians off this list. >I, for one, would certainly not be scared off the list by such a discussion. I think this is overkill, since any meantone tuning from 1/3 comma to 1/5 comma is fine by me, all sounding wonderfully smooth. But one is naturally curious. Since a meantone tuning is generated by the (flattened) perfect fifth, which we will represent by v, express these intervals in terms of that fifth. Since were are assuming a tuning with perfect octaves, it doesn't matter which octave inversion or extension of the consonant intervals we examine. Expressing everything in units of 1octave: (1) minimize (v-log(3/2)/log(2))^2+(4*v-log(5)/log(2))^2+(3*v-log(10/3)/log(2))^2 (2) minimize (3*(v-log(3/2)/log(2)))^2+(5*(4*v-log(5)/log(2)))^2+(5*((3*v-log(10/3)/ log(2)))^2 Calculus lesson 1: To find the value of a variable that maximizes or minimizes a function, differentiate the function with respect to the variable, set the derivative equal to zero, and solve for the variable. If you don't know how to differentiate a function, or why the above is true, please consult a calculus textbook. Anyway, the resulting derivatives are (1) 52*v-2*log(3/2)/log(2)-8*log(5)/log(2)-6*log(10/3)/log(2) which simplifies to -2*(-26*v*log(2)-2*log(3)+2*log(2)+7*log(5))/log(2) which simplifies to 52*v-4+(4*log(3)-14*log(5))/log(2) and (2) 1268*v-18*log(3/2)/log(2)-200*log(5)/log(2)-150*log(10/3)/log(2) which simplifies to 2*(634*v*log(2)+66*log(3)+66*log(2)-175*log(5))/log(2) which simplifies to 1268*v-132+(132*log(3)-350*log(5))/log(2) Setting these equal to zero and solving for v gives (1) v (2-(2*log(3)+7*log(5))/log(2))/26 or v .58013737 octaves or v 696.1648 cents and (2) v (66-(66*log(3)+175*log(5))/log(2))/634 or v .58001560 octaves [.58 octaves is of course 29/50 octaves or the 50-tET perfect fifth] or v 696.0187 cents. >I am particularly interested in the ideas you call "virtual pitch, residue >pitch, complex tone pitch, fundamental tracking, and possibly periodicity >pitch, I have been working on this for some time" and I'd love to see some of >these ideas posted to the list. I'm sure someone else can give you a fuller bibliography on these ideas than I. A good start might be Juan Roederer's "Introduction to the Psychophysics of Music." I posted some of my musings, complete with mathematical derivations, a few months ago. This procedure I extended from intervals to chords with a kludge. I would now like to make this more rigorous, but I need to drop the assumption that ratios are always expressed in lowest terms. (This was a nice assumption because it allowed be to use the theory of rational approximations. There is no theory of rational approximations for ratios of three of more numbers. Graphing x/y vs. x/z for all combinations of x, y, and z integers less than N (N fixed) will give you some idea why no such theory exists. There is evidence (including personal experience) that even a rich complex tone, played quietly and in the presence of noise, can sometimes be interpreted as harmonics 2, 4, 6, 8, . . ., or even 3, 6, 9, 12, . . . of the consciously perceived fundamental. The lowest-terms restriction would not allow for these phenomena.) Anyway, I'm sort of not working on this now, but J. Kukula has suggested that wavelet theory might be useful. Brian McLaren has made the observation that "scientific" models of perception and cognition are continously revised to reflect the mathematics and computer science in vogue at the time. Rather than take this as evidence for a Kuhnian attack on these models, I would suggest that they represent successively better approximations to the truth, just as Newtonian physics remains a good approximation though relativity and quantum mechanics are far more accurate. So, when I cite the results of Goldstein, J. L. 1973. "An optimum processor theory for the central formation of the pitch of complex tones." J. Acoust. Soc. Amer. Vol. 54 p. 1499, and I often do, I realize that there may be improvements to the model, but as a first-order approximation, it fit the data quite well. Where has Brian been lately? I sort of miss him in a weird way. I kind of wanted to respond to his assertion that neutral thirds are more consonant than tritones by saying that, for tones with strong third partials and weak fifth partials, he is certainly right, if a roughness-type definition of consonance (based on Plomp and Levelt's critical bands) is used. The reason many objected is that when gradually changing the size of the interval, the _local_ minimum of dissonance achieved around the 7:5 is much more distinct than that achieved at the 11:9. In an absolute sense, however, the 7:5 may be more dissonant. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 27 Feb 1997 04:23 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA12641; Thu, 27 Feb 1997 04:23:57 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA12638 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id TAA29517; Wed, 26 Feb 1997 19:22:26 -0800 Date: Wed, 26 Feb 1997 19:22:26 -0800 Message-Id: <009B07BF6D01BDBD.5C04@vbv40.ezh.nl> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu