source file: mills2.txt Date: Tue, 7 Nov 1995 09:55:58 -0800 From: "John H. Chalmers" From: mclaren Subject: More about n-j n-e-t scales --- "As physicians have always their instruments and knives ready for cases which suddenly require their skill, so you must have principles ready for the understanding of things..." [Marcus Aurelius, "Meditations"] Because of the general lack of incomprehension and puzzlement at my mention of non-just non- equal-tempered scales, it's clear that some further info is required. In general, I'm not talking about linear or meantone tunings. While it's true that these are technically non-just non-equal-tempered scales, most of 'em are just one or another way of bending twelve tones to obtain a more consonant third or fifth in a traditional Western triad. This surely has its value, but meantone scales decidely represent the outermost extreme conservative side of non-just non-equal-tempered scales. Instead, the kind of n-j n-e-t tunings I'm concerned with--and have been since the git-go--are those tunings which break completely with the Western mold. Some of these kinds of tunings have octaves, others don't. In general, they're so wildly foreign to any conventional harmonic or scalar vocabulary that there is hardly any intelligible way to talk about such scales as yet, except as raw numbers. For example: One of the simplest ways of generating such a completely non-Western non-just non-equal- tempered scale is by taking the natural logarithm of the factorial of a set of integers: 1! = 1, ln(1) = 0 2! = 2, ln(2) = 0.693147 3! = 6, ln(6) = 1.7181759 4! = 24, ln(24) = 3.17805383 5! = 125, ln(125) = 4.82831373 6! = 720, ln(720) = 6.5792512 7! = 5040, ln(5040) = 8.52516136 and so on. Taking ratios so as to eliminate dependence on the base of the logarithm, we have: scale step 1: 1.0 scale step 2: 2.478808 scale step 3: 4.5849625 scale step 4: 6.9657842 scale step 5: 9.4918531 scale step 6: 12.299208 These values can be octave-reduced or not. If not, the scale will have no octaves. If octave-reduced, carrying out the procedure will produce ever larger numbers of scale- steps within the octave, never overlapping. This is an inherently fractal process, first described by Thorwald Kornerup in his Golden Section scale. In a sense the procedure is analagous to that of just intonation, in which successive addition and subtraction of various small- integer-ratio intervals produces an ever-larger profusion of unequal divisions of the octave. However, there are a number of differences. At this point it's useful to introduce the concept of the "inharmonic series." By analogy with the harmonic series, an inharmonic series serves as the backbone of a non-just non-equal-tempered scale. In this case, the inharmonic series is Log(N!) where N runs from 1...infinity. Of course choosing N by some other criterion (perhaps by some recurrence relation: say, N4 = N1 - sqrt(N2 + N3^2)*N1) would produce an entirely different inharmonic series. There are an infinite number of inharmonic series, each generated by choosing N by a different method and then applying some non-linear operation to N. By contrast, the familiar harmonic series is obtained by setting N = the class of integers and performing the simplest possible linear operation on them--namely, the unary operation (which leaves the operand unchanged). Inharmnic series are important as a source of modulation and of vertical structures in non-just non-equal-tempered scales. One of the most interesting implications of non-just non-equal-tempered tunings, however, is the prospect of generating a scale of note durations (read: rhythms) derived from the scale steps, by analogy with the comparable procedure in traditional Western music. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 7 Nov 1995 20:04 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id KAA19700; Tue, 7 Nov 1995 10:04:49 -0800 Date: Tue, 7 Nov 1995 10:04:49 -0800 Message-Id: <9511071803.AA09055@ ccrma.Stanford.EDU > Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu