source file: mills2.txt Date: Fri, 11 Jul 1997 18:41:13 +0200 Subject: Reply to David Finnamore From: "Paul H. Erlich" I thank you for your well-thought out comments on multidimensionality. >From a compositional perspective, the points you raise are all very relevant. But from a purely psychoacoustic perspective, it is unlikely that making exceptions to the prevailing odd-limit by allowing large powers of a small prime makes much sense. Using Kameoka & Kuriagawa, for example, there is no sense in which 27/16 will be more consonant that 5/3 for any listener. It may be more familiar due to melodic properties of the Pythagorean scale, but in terms of harmonic consonance, either the listener is sensitive to the 5-limit and will prefer 5/3 or the listener is not sensitive to the 5-limit and will simply find both intervals dissonant. The point (in the "evolution" toward greater dissonance) at which 9-limit intervals become consonant is somewhere between the point at which 7-limit intervals become consonant and the point at which 11-limit intervals become consonant. As for higher-limit intervals, the conditions under which they are important depend more on voicing to acheive certain virtual root and combination tone effects. The assumption of exact octave equivalence, though it has some psychoacoustic basis, should be taken more as a conveniece than an exact truth. In some sense, it is not the odd-limit but the integer-limit that really correlates with psychoacoustic consonance. However, in the traditional approach of pondering octave-repeating scales, the assumption of octave equivalence allows you the vast simplification of only considering what goes on within one octave. David, you may be interested to look into Georg Hajdu's paper in Interface, he cites some mysterious formula that gives a measure of the "complexity" of an integer; primality is a factor so that 7 is considered more complex than 9. As I have touched on above, I doubt such an approach is valid, but if you disagree with me, maybe you can find something useful in his formula. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 11 Jul 1997 19:01 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA22856; Fri, 11 Jul 1997 19:02:01 +0200 Date: Fri, 11 Jul 1997 19:02:01 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA22858 Received: (qmail 6791 invoked from network); 11 Jul 1997 16:39:33 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 11 Jul 1997 16:39:33 -0000 Message-Id: <199707111235_MC2-1AAE-A7A9@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu