source file: mills2.txt Date: Mon, 13 Nov 1995 21:45:55 -0800 From: "John H. Chalmers" From: mclaren Subject: n-j n-e-t vertical structures - part 5 --- "The safety of life is this: to examine everything all through, what is it of itself, what is its nature, what is its form..." [Marcus Aurelius, "Meditations"] Of the nature of vertical structures in a typical subinharmonic series, some has been said: more remains. By analogy with just intonation, the subinharmonic series formed from the vibrational modes of an ideal vibrating cylinder are obtained by inverting the values f0 = sqrt(4*9/5) = 2.68328/2.68328 = 1.0 f1 = sqrt(9*64/10) = 7.58946/2.68328 = 2.82842 f2 = sqrt(16*225/17) = 14.55197/2.68328 = 5.4320 f3 = sqrt(25*576/26) = 23.5339/2.68328 = 8.77057 f4 = sqrt(36*1225/37) = 34.5329/2.68328 = 12.8696 f5 = sqrt(49*2304/50) = 47.5173/2.68328 = 17.7086 f6 = sqrt(64*3969/65) = 62.5135/2.68328 = 23.297422 f7 = sqrt(81*6400/82) = 79.510699/2.68328 = 29.631905 f8 = sqrt(100*9801/101) = 98.508682/2.68328 = 36.71204 f9 = sqrt(121*14400/120) = 120.00417/2.68328 = 44.72294 f10 = sqrt(144*20449/143) = 143.49913/2.68328 = 53.47899 &c. to obtain fsub0 = 1/sqrt(4*9/5) = 2.68328/2.68328 = 1.0 fsub1 = 1/sqrt(9*64/10) = 7.58946/2.68328 = 0.353554 fsub2 = 1/sqrt(16*225/17) = 14.55197/2.68328 = 0.1840942 fsub3 = 1/sqrt(25*576/26) = 23.5339/2.68328 = 0.1140176 fsub4 = 1/sqrt(36*1225/37) = 34.5329/2.68328 = 0.0777024 fsub5 = 1/sqrt(49*2304/50) = 47.5173/2.68328 = 0.0564697 fsub6 = 1/sqrt(64*3969/65) = 62.5135/2.68328 = 0.0429232 fsub7 = 1/sqrt(81*6400/82) = 79.510699/2.68328 = 0.0337474 fsub8 = 1/sqrt(100*9801/101) = 98.508682/2.68328 = 0.027239 fsub9 = 1/sqrt(121*14400/120) = 120.00417/2.68328 = 0.0223598 fsub10 = 1/sqrt(144*20449/143) = 143.49913/2.68328 = 0.0186989 &c. Forming the subinharmonic series starting on inharmonic series member 10 produces: faleph = 53.47899*0.027239 = 1.4567142 fbeth = 53.47899*0.0223598 = 1.1957795 fgem = 53.47899*0.0186989 = 1.0 Reducing, this becomes faleph = 831.77662 cents fbeth = 309.52791 cents fgem = 0 cents Of particular interest here is the quasi-sixth formed by falph, which happens to identical with phi, the golden ratio. This vertical complex is very similar to one discussed by Thorwald Kornerup, and it is particularly interesting in this context to observe that this one arises *naturally* out of an ordinary physical process--namely, a subinharmonic series based on the modes of an ideal vibrating cylinder. Since Kornerup's Golden Scale is well known and discussed in detail in Mandelbaum's thesis (among other references), further discussion of this similar scale is of less interest than a consideration of general principles for organizing vertical progressions in n-j n-e-t scales. Thus, this example may serve to show the subtle links between relatively familiar non-just non-equal-tempered tunings and the purely mathematical derivation of n-j n-e-t scales from combinations of inharmonic and subinharmonic series. Clearly, both the inharmonic series vertical strucures *and* the subinharmonic series structures may be formed on any scale member. Equally clearly, one might imagine "modulating" between entirely different inharmonic or subinharmonic series. In that case, one could bring along the vertical complexes derived from one inharmonic series into another, entirely different, inharmonic series. For example, a series of vertical structures derived from the inharmonic series of the clamped metal bar might be played first in the n-n n-e-t scale derived from the modes of the clamped bar, then the same progression of vertical structures might continue to play while the tuning changed to that of an ideal vibrating sphere, and the tuning might then change into that of an ideal vibrating cylinder, and so on. In fact one could just as easily "modulate" between different modes of vibration of a sphere: zonal harmonics, torsional vibrations, etc., each giving rise to a different non-just non-equal-tempered scale. This is a process conceptually akin to "modulation" in JI and equal temperament, but more complex: for the subinharmonic series formed on a given n-j n-e-t are, as we have seen, in general not as closely related to the vertical structures formed from inharmonic series as is the minor triad to the harmonic series of the major triad in traditional Western harmony. Regardless, this set of posts may have given a glimpse of the universe of new harmonies and melodies awaiting the composer adventurous enough to dare composing in non-just non-equal-tempered tunings. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 14 Nov 1995 07:55 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id VAA08993; Mon, 13 Nov 1995 21:55:42 -0800 Date: Mon, 13 Nov 1995 21:55:42 -0800 Message-Id: <951114005507_105935759@mail04.mail.aol.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu