skidoo subgroup temperament (2.3.7.11.13.23)

Hello, all.

This is a brief description of a regular rank-2 temperament
called skidoo, a subset temperament for 2.3.7.11.13.23 where the
usual tritone or augmented fourth from +6 fifths is at or near
23/16 (628.274 cents), with a pure 23/16 tuning at 704.712 cents.
Thus the name skidoo, from the expression "23 skidoo."

A look at the Xenharmonic Wiki pages on subgroup temperaments
suggests that skidoo is basically leapfrog (2.3.7.11.13) plus
prime 23 as an explicit part of the mix. The POL2 for leapfrog at
704.745 cents, or 63-EDO at 704.762 cents (with 23/16 wide by
0.297 cents), may also be considered as tunings of skidoo if 23
is a relevant consideration.

<http://xenharmonic.wikispaces.com/Subgroup+temperaments>

Since 23/16 is a higher prime which may have a rather narrow "sphere
of influence," one way to get a sense of the domain of skidoo would
be to propose that this ratio should be within about a cent of
just. A rough guide might be 704.5-704.9 cents, maybe with 80-EDO
as a northern suburb (fifth 705 cents, augmented fourth 630 cents).

While skidoo is the name for this abstract temperament in
general, 23-Skidoo is the tuning with a pure 23/16 (704.712 cents).

The general mapping of this rank-2 system is like that of
leapfrog with prime 23 added.

For the brief, Xenwiki-like description that follows, I simply
list the commas, or at least some of them; but I would be glad to
explain them at more length, and know that if I were confronted
with a similar list, I might have lots of questions!

----------

Skidoo

Group: 2.3.7.11.13.23

Commas: 896/891, 352/351, 364/363, 10648/10647; 14680064/14348907;
108224/177147, 19712/19683; 6656/6561; 169/168;
736/729, 208/207, 5632/5589, 897/896

Gencom mapping: [<1 1 0 -6 -3 -1 1|, <0 1 0 15 11 8 6|]

EDOs: 63, 109, 172

Some tunings and eigenmonzos:

109-EDO 704.587c
pure 7/4 704.588c
e-based 704.607c
pure 23/22 704.609c
172-EDO 704.651c
pure 11/8 704.665c
pure 23/16 704.712c
pure 39/32 704.720c
leapfrog POL2 704.745c
63-EDO 704.762c
pure 7/6 704.776c

Scala files:

23-Skidoo (12-MOS):
<http://www.bestII.com/~mschulter/23-skidoo_12.scl>

23-Skidoo (17-MOS):
<http://www.bestII.com/~mschulter/23-skidoo_17.scl>

23-Skidoo (24-note set):
<http://www.bestII.com/~mschulter/23-skidoo_24.scl>

23-Skidoo (29-MOS):
<http://www.bestII.com/~mschulter/23-skidoo_29.scl>

-------------
Musical notes -------------

The skidoo temperament is near the upper end of the gentle
family, and shares the themes of this family: regular thirds at
or near 14/11 and 13/11; and a "paraschismic" mapping for neutral
or submajor/supraminor intervals from 7-11 fifths, with the
diminished fourth (-8 fifths) as a submajor or largish neutral
third, and the augmented second (+9 fifths) as a supraminor or
smallish neutral third.

Additionally, as a rank-2 system, skidoo shares with leapfrog an
apotome (+7 fifths) about midway between 14/13 and 13/12, with
two apotomata (+14 fifths) approximating 7/6. The augmented third
or "wolf fourth" (+11 fifths) approximates 11/8; and the
augmented fifth (+8 fifths) approximates 13/8, although in the
lower range of skidoo this interval is about halfway between the
more complex Phi (833.090 cents) and 13/8, and may have some
qualities of each.

In this upper region of gentle (which ranges roughly from 29-EDO
to 80-EDO, or 703.45 to 705.0 cents), 14/11 is a bit larger than
pure, while the minor third is closer to 33/28 than to 13/11. In
the heartland of skidoo, the diatonic semitone is close to 23/22
(76.956 cents).

For maqam music, the skidoo region or subregion of gentle offers
large and small neutral intervals about a comma apart, such as
thirds at 341/363 cents in the e-based temperament or 343/362
cents in 63-EDO. This may fit many Syrian and some Turkish and
Persian tastes, while being "beginner-friendly" in that the
difference is easier to appreciate from an ear-training
perspective than the more subtle shadings of some maqam tunings.

The basic heptad of skidoo could be 4:6:7:9:11:13:23, with a
17-MOS yielding two of these heptads, and a 24-note set, nine.
A 24-note set, or possibly a 29-MOS, is recommended to enjoy more
fully the septimal (2.3.7.9) resources of this temperament.
However, a basic 12-MOS permits much exploration of the regular
thirds near 14/11 and 33/28 (or 13/11), the neutral intervals,
and, of course, the augmented fourth at or near 23/16.

While I'm not sure how far back the history of skidoo goes, the
e-based tuning at 704.607 cents, with its near-just 23/16, may be
one of the earlier examples on this list.

</tuning/topicId_20573.html#20573>

Best,

Margo