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The three {125/108, 135/128} Fokker blocks

🔗Gene Ward Smith <gwsmith@svpal.org>

8/12/2004 6:10:24 PM

These have val <12 19 27|, and when listed in the val ordering Scala
counts them as epimorphic and consant structure, but not otherwise.
They all have a scale step of 81/80, which in the mode I list is
between 10/9 and 9/8. They can usefully be tempered via meantone,
which of course reduces the 12 notes to 11. It also makes the 125/108
promo (I won't call it a comma :)) into a meantone 7/6, which makes
one of the "fifths" in the circle a 7/4 (via 126/125). It makes
135/128 into a meantone 21/20, which makes three other fifths into
10/7s (via 225/225).

Each has six major triads and six minor triads, which is excellent,
the same number reached by the Duodene. Under marvel, we can add two
supermajor and two subminor triads for pel2, and one less for the
other two. The 0-4 intervals are either pure 6/5s, or (in four cases)
25/18, which is a tritone equivalent to 7/5 under 126/125 of size 569
cents. The 0-3 intervals are stranger, two being a 27/25 and one a
128/125; seven are 5/4s and the other two 32/27, approximately (via
352/351) a 13/11.

Below I list each the scales in monotonic order, in val order, in
terms of the deviation from 3/2 in the circle of fifths, and as
reduced to eleven notes of meantone.

Pel1
[1, 25/24, 10/9, 9/8, 5/4, 4/3, 25/18, 3/2, 25/16, 8/5, 5/3, 15/8]
[1, 10/9, 9/8, 5/4, 25/18, 4/3, 25/16, 3/2, 5/3, 8/5, 15/8, 25/12]
[1, 1, 128/135, 125/108, 1, 1, 128/135, 1, 1, 1, 128/135, 1]
[-4, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]

Pel2
[1, 25/24, 10/9, 9/8, 6/5, 5/4, 4/3, 3/2, 25/16, 8/5, 5/3, 15/8]
[1, 10/9, 9/8, 5/4, 6/5, 4/3, 25/16, 3/2, 5/3, 8/5, 15/8, 25/12]
[1, 1, 128/135, 1, 125/108, 1, 128/135, 1, 1, 1, 128/135, 1]
[-4, -3, -1, 0, 1, 2, 3, 4, 5, 7, 8]

Pel3
[1, 10/9, 9/8, 6/5, 5/4, 4/3, 3/2, 25/16, 8/5, 5/3, 9/5, 15/8]
[1, 10/9, 9/8, 5/4, 6/5, 4/3, 25/16, 3/2, 5/3, 8/5, 15/8, 9/5]
[1, 1, 128/135, 1, 1, 125/108, 128/135, 1, 1, 1, 128/135, 1]
[-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 8]