The Muddle Way

Let me first suggest a notational convention--if n+m denotes the
generator of the n+m et defined by the ets n and m, then I suggest
s;n+m for scales of s generator steps if n and m are relatively
prime, meaning they have no common factor. For instance, 7;7+5 would
be the diatonic scale in the 12-et, 7;19+12 would be the same scale
in the 31-et, and 21;41+31 would be Blackjack. If n and m have a
common factor of d, then we could have a sum of d different terms
before the n+m. Thus 5+5;12+10 would be Paul's symmetrical decatonic,
while 6+4;12+10 would be Paul's standard decatonic. 3+2+1;8+4 would
be the scale C C# D E F G# in the 12-et. The notation is intended to
cover all modes and transpositions of the scale in question.

We can extend this notation to muddles, by which I mean scales in
temperaments of temperaments. Thus 7;19+12;41+31 would be the
diatonic muddle of 31 notes tuned to Canasta but used as if they were
31-equal; that is to say, a temperament of 31 notes. The 7;19+12;41+31
muddle would not be the same as the 7;19+12;31+22 muddle, despite
being diatonic scales patterns from 31 notes, since the first would
be tempered according to Miracle and the 72-et, and the second
according to Orwell and the 53-et.

To get an idea of how the Muddle Way might work let's look at the
7;19+12;41+31 Miracle Muddle. If we go around the circle of fifths,
we get the following scales:

0 [0 12 24 30 42 54 65] ~ 1-9/8-63/50-4/3-3/2-27/16-15/8

1,6,11,16,21,26 [0 12 23 30 42 54 65] ~ 1-9/8-5/4-4/3-3/2-27/16-15/8

2,7,12,17,22,27 [0 12 23 30 42 53 65] ~ 1-9/8-5/4-4/3-3/2-5/3-15/8

3,8,13,18,23,28 [0 11 23 30 42 53 65] ~ 1-10/9-5/4-4/3-3/2-5/3-15/8

4,9,14,19,24,29 [0 11 23 30 41 53 65] ~ 1-10/9-5/4-4/3-40/27-5/3-15/8

5,10,15,20,25 [0 12 24 31 42 54 65] ~ 1-9/8-63/50-27/20-3/2-27/16-15/8

We have five scales 2,7,12,17,22,27 which are close to JI diatonic
major, and another five, 3,8,13,18,23,28 close to JI diatonic minor;
moreover 1,6,11,16,21,26 closely approximates the Indian diatonic
scale. Modulating down a fifth would take us from minor to major, and
again to Indian. This suggests that some of the scales contained in
the 12;19+12;41+31 muddle would be good 12-tone temperaments, not for
common practice, but for advocates of certain kinds of JI.