source file: mills2.txt Date: Mon, 20 Nov 1995 04:54:29 -0800 Subject: Pierce-Bohlen in JI, 12-tone scales From: "John H. Chalmers" It is rather difficult to generate a consistent JI for the 9-tone Pierce scale in JI. Using the matrix Pierce gives of four 1/1 5/3 7/3 major chords (3:5:7) and reduction by the 3/1 "tritave" (rather than the 2/1 octave), one obtains the following ratios: V IX III III VII I VI I IV IV VIII II 27/49 45/49 9/7 9/7 15/7 1/1(3/1) 9/5 1/1 7/5 7/5 7/3 49/15 Note the equivalence between 1/1 as the dominant of one chord and the mediant of the next rather than tonic (VII is not VI). When reduced to one "tritave," the ratios are 1/1 49/45 9/7 7/5 81/49 9/5 15/7 7/3 135/49 3/1. Alternate tunings for some tones are needed in some harmonic contexts, i.e, 5/3 for 81/49, 25/9 for 135/49, 35/27 for 9/7 etc. However, the scale is best construed as a subset of the 13th root of 3 temperament and hence has many just intonation interpretations. As for the 12-tone chromatic scale, the so-called "Pythagorean" or 3-limit cycle of 4ths and/or 5ths tuning can be traced back to the Babylonians in the 2nd millennium BCE, though some Chinese legends credit a court official whose title was Ling Lun with an even earlier date. There are many 5-limit chromatic tunings; I posted a couple of dozen to the list earlier) and also published them in 1/1. J. Murray Barbour's book Tuning and Temperament is a good source for the historical western ones and Mandelbaum's dissertation for some more theoretical and 7-limit varieties. --John Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 20 Nov 1995 19:19 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA03621; Mon, 20 Nov 1995 09:19:48 -0800 Date: Mon, 20 Nov 1995 09:19:48 -0800 Message-Id: <647.199511201717@altair> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu