source file: mills2.txt Date: Fri, 21 Jun 1996 13:18:33 -0700 Subject: Self-Similar scales From: John Chalmers I might say a few words about self-similar scales myself. Brian is talking about several different kinds of self-similarity. The type exhibited by the Harrison/Lucy system is true of any cyclic tuning, whether equally-tempered, linearly tempered (meantone, Helmholtzian, Sabat-Garibaldi's, etc.), or just (Pythagorean, cycles of 5/4's, etc.). Each new tone has a constant relation to the one preceding: either the cyclic interval or its complement re the interval of equivalence (octave inversion if the IE is 2/1). The Golden Ratio or Section tuning (Kornerup, Dudon) has another type of self-similarity. Many of the small intervals of the scale are in the GS relation to each other. In this case, the chromatic semitone/ diatonic semitone, the diatonic semitone/whole tone, the whole tone/ minor third, and the minor third/perfect fourth are in the GS relation. From these relations and the derivation of these intervals by a cycle of fifths, the "golden" fifth may be found. From this fifth (and/or its complement) all the intervals may be generated. A third type is described in Brian's Xenharmoniko^n papers. In this case, a given relation is iteratively applied within the the IE. For example, iterating the Golden Ratio relation (.618034..) in the octave yields 1200, 742, 459, 283....cents. A possible problem with this kind of scale is that the terms quickly converge to inaudible intervals. A variation on this last type may be obtained by limiting the iterations before the intervals become unusably small and applying the technique to the larger intervals already generated. This procedure yields scales with more evenly distributed intervals while preserving the sense of proportion. Hahn's question about the perceptual character of these scales is still relevant. I can only say that some successful pieces have been written using unusual relations, e.g., John Chowning's STRIA, which uses the 9th root of PHI (aka the GS). I believe Brian answered my own question by making a distinction between the acoustic and compositional levels of organization. I think the only way to address these concerns is to compose, record, and distribute more music and let the listeners decide. --John Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 22 Jun 1996 00:28 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id PAA19878; Fri, 21 Jun 1996 15:28:22 -0700 Date: Fri, 21 Jun 1996 15:28:22 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu