Boogiewoogiescale

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.