Hemiwuerschmidt, the Porcupine Complex, and transformations

These turn out to have some interesting properties.

Hemiwuerschmidt is the 7-limit linear temperament with wedgie [16, 2, 5,
6, 37, -34] and commas 2401/2400,
3136/3125, 6144/6125 and of course the wuerschmidt, 393216/390625.

13-note Fokker blocks

[1, 50/49, 28/25, 8/7, 5/4, 32/25, 7/5, 10/7, 25/16, 8/5, 7/4, 25/14,
49/25]
commas [15/14, 3136/3125, 2401/2400]
intervals 24 triads 12 tetrads 0

[1, 128/125, 28/25, 8/7, 5/4, 32/25, 7/5, 10/7, 25/16, 8/5, 7/4, 25/14,
125/64]
commas [15/14, 3136/3125, 6144/6125]
intervals 22 triads 10 tetrads 0

Both of these involve only the primes 2,5 and 7; and so may be thought of
as blocks for what I've called the "hemithirds no threes" 7-limit planar
temperament, whose defining comma is the 3-less 3136/3125. I suppose
"hemiwuerschmidt no threes" might be a better name.

[1, 49/48, 28/25, 8/7, 5/4, 32/25, 7/5, 10/7, 25/16, 8/5, 7/4, 25/14,
96/49]
[15/14, 2401/2400, 6144/6125]
intervals 22 triads 10 tetrads 0

If we temper by the 31, 68, 99, 130 or even (as I've done here) by the
[427, 677, 992, 1199] val, we get a 13-note MOS, which here I represent
by the 69/427 MOS of the 427-et:

[0, 13, 69, 82, 138, 151, 207, 220, 276, 289, 345, 358, 414]
intervals 31 triads 16 tetrads 0

This is almost, but not quite, the same as the "hemithirds no threes" 13
note MOS; it can't *exactly* be regarded as that since it has a couple of
poor but honest subminor thirds, though the actual notes are the same.

[1, 50/49, 35/32, 28/25, 8/7, 49/40, 5/4, 125/98, 343/250, 7/5, 10/7,
500/343, 196/125, 8/5, 80/49, 7/4, 25/14, 64/35,49/25]
commas [21/20, 3136/3125, 2401/2400]
intervals 38 triads 20 tetrads 0

The "0" does not count the four complete tetrads off by less than a cent
(by 2401/2400.)

[1, 128/125, 35/32, 28/25, 8/7, 49/40, 5/4, 32/25, 175/128, 7/5, 10/7,
256/175, 25/16, 8/5, 80/49, 7/4, 25/14,
64/35, 125/64]
comma [21/20, 3136/3125, 6144/6125]
intervals 38 triads 20 tetrads 0

Here the chord count (counting *only* theoretically exact chords) is the
same, and in fact the two characteristic polynomials are the same,
suggesting there is an isomorphism between the two graphs of the two
scales. This turns out to be the case: both scales are in the {2,3,7}
subgroup, and the mapping 5-->7, 7-->56/5 sends the first scale to the
second, and the inverse map 5->40/7, 7->5 sends it back again. This
transformation is of order 6, and induces a permutation of degree six on
the MOS. This MOS mapping is an automorphism of the "no threes" graph,
but not of the full hemiwuerschmidt graph.

[1, 49/48, 35/32, 28/25, 8/7, 49/40, 5/4, 245/192, 48/35, 7/5, 10/7,
35/24, 384/245, 8/5, 80/49, 7/4, 25/14, 64/35, 96/49]
commas [21/20, 2401/2400, 6144/6125];
intervals 42 triads 26 tetrads 2

The tetrad count grows to four if we allow 2401/2400 relationships.

All of these scales have the same hemiwuerschmidt 19-tone MOS:

[0, 13, 56, 69, 82, 125, 138, 151, 194, 207, 220, 233, 276, 289,
302, 345, 358, 371, 414]
intervals 63 triads 50 tetrads 6

The 13 and 19 tone hemiwuerschmidt MOS can both be considered as "no
threes" MOS, and in this way a part of the Porcupine Complex of
transformations between the 250/243, 3125/3087 and 3136/3125 planar
temperaments I wrote about a while back; unfortunately Yahoo won't let me
use the archives just now, so I don't have the message number.