source file: mills3.txt Date: Tue, 4 Nov 1997 19:45:00 +0100 Subject: RE: Erlich's Theory From: gbreed@cix.compulink.co.uk (Graham Breed) Fans of Paul Erlich will no doubt already have read his paper available through http://www-math.cudenver.edu/~jstarret/microtone.html . A brave attempt at a theory of diatonic, 7-limit harmony. Now, back to picking his posts apart paragraph by paragraph: >The harmonic entropy curve, like the Plomp-Levelt roughness curve for tones >with harmonic partials, has local minima at small-integer ratios. It is >standard in physics (and justified by calculus) to approximate any local >minimum with a parabola. Therefore, near a just ratio, the change in >dissonance is proportional to the squared detuning. I don't have Plomp & Levelt's curve to hand, but I do have Kameoka & Kuriyagawa's. The minima don't look much like parabolas to me. I question this generalisation for physics as well, but that needn't bother the rest of you. A Farey series would be like a Partch even limit, right? > These curves look remarkably like many of > the Helmholtz/Plomp curves that were derived from completely different > assumptions, I'd guess any function that bears some relation to the overtone series would produce a qualitatively equivalent curve. The entropy method then looks like a good way of going from functions of integer ratios to a continuous curve. There are lots of the former, of course -- the Partch limit, LCM, generalised harmonic distances, or counting the "filled" partials of the virtual pitch. > How to > weigh the various subsets' contributions to the probabilities of > particular fundamentals in an overall analysis is unclear. Even without > the consideration of subsets, there appears to be no mathematical theory > of ratios of three of more numbers analogous to Farey theory, and no > easy way to create one. Unlike roughness, tonalness is not merely > concerned with pairwise interactions of tones but three-way and higher > interactions as well. A mathematical model for it is out of my grasp at > the moment. It's crucial that a concordance theory be made to work for more than dyads. I think here that the entropy method will become rapidly computationally intensive the more notes are involved. I would try this myself, but I've got software to write, and I might get a job soon, so I don't really have the time ... As a composer, I feel a theory of chords is more useful than one of dyads. Most useful of all, though, is a theory of chord sequences. This is where lattices come into their own. As a programmer, speed and memory are important, so I'm planning to go for harmonic distances if I ever get around to dynamic tuning. I think a Euclidian distance on a parallelogram lattice should include approximations to both Partch limits and LCMs. BTW, I had a flanger on when I did my listening experiment before, which explains the unpleasentness of the 31TET 4:6:7. SMTPOriginator: tuning@eartha.mills.edu From: William Sethares Subject: consonance calculations PostedDate: 04-11-97 21:11:56 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 04-11-97 21:10:56-04-11-97 21:10:57,04-11-97 20:11:20-04-11-97 20:11:21 DeliveredDate: 04-11-97 20:11:21 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C1256545.006ED8BB; Tue, 4 Nov 1997 21:10:44 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02476; Tue, 4 Nov 1997 21:11:56 +0100 Date: Tue, 4 Nov 1997 21:11:56 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA02474 Received: (qmail 29649 invoked from network); 4 Nov 1997 12:11:51 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 4 Nov 1997 12:11:51 -0800 Message-Id: <199711042008.AA10088@eceserv0.ece.wisc.edu> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu