Pierce-Bohlen in JI, 12-tone scales

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


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