Boogiewoogiescale

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Boogie Woogie Scale

In this posting of the Yahoo tuning list, Paul G. Hjelmstad wrote:

Take the standard 12-bar boogie-woogie. Let's use F major:
F  A  C  Eb
Bb D  F  Ab
C  E  G  Bb

Tune to the seven-limit and keep fifths. You get
12 15 18 21
 4  5  6  7
36 45 54 63

Fit into one octave (F,G,Ab,A,Bb,C,D,Eb,E)
24,27,28,30,32,36,40,42,45 and 63 (extra Bb)

Taking all the ratios, we find that they are all superparticular
(n/n-1)
9/8, 28/27, 15/14, 16/15, 9/8, 10/9, 21/20, 15/14, 16/15
(and the schisma for Bb/Bb 64/63)

You also get 8/7, 7/6, 6/5, 4/3, 3/2, 1/1, with multiple scale
steps..

The first seven triangular numbers are used; 1/1, 3/2, 6/5, 10/9,
15/14, 21/20, 28/27

Five of the squares are used: 1/1, 4/3, 9/8, 16/15 and 64/63

8/7 and 7/6 are the only ratios which are not squared or triangular
superparticular ratios but they are still superparticular!

All from the simple boogie woogie!

Gene Ward Smith described some additional properties (in this posting):

Here it is in Scala format:

! boogie.scl
Paul Hjelmstad's boogie woogie scale
10
!
9/8
5/4
21/16
45/32
3/2
27/16
7/4
15/8
63/32
2/1

Three otonal tetrads, no utonal tetrads, not CS or epimorphic,
superparticular ratios as noted.

I found a number of ten-note seven limit epimorphic scales with four
tetrads; here's one Paul Erlich found first:

! cx1.scl
First 10/4 scale = erlich11 <10 16 23 28| epimorphic
10
!
15/14
7/6
5/4
4/3
10/7
3/2
5/3
7/4
15/8
2
! [0, -1, -1], [0, -1, 0], [0, 0, 0], [0, 0, 1]

Quite a lot of musical possibilities in these relatively small 7-limit
JI scales, I think.