septimal keys

You will recall Paul Erlich's blues scale:

9---------0---------13
\ / \
\ / \
\ / \
\ / \
6---------19--------10

I'll rewrite that with different names. Remember that the two
D's are different:

D---------A---------E
\ / \
\ / \
\ / \
\ / \
C---------G--------D

I played around with this trying to get the difference between the
D's to be two commas. Then I decided the commas could be switched
to a ration of 36/35, what I call a septimal diesis. I ended up
with the following scale:

/ \ / \ /
/ C\--------G\--------D\
/ \ / \ /
D---------A---------E
\ / \
\ / \
\ / \
\ / \
C---------G

This is a representation of a 3-D lattice, with D-C\ the subminor
seventh 7/4.

D and D\ are separated by a septimal comma which is too small.
Between C\ and C, and G\ and G, there are septimal dieses, though.
This looks like a septimal equivalent of a diatonic scale: 7 notes
plus a comma and 6 consonant triads. Unfortunately, two of those
triads share a root.

I've been playing this with my schismic mapping, and it sounds
fine melodically. D\ isn't used much and, for the loss of a
chord, you could remove it from the key. In 22 equal, the comma
vanishes anyway:

A C\ C D\ D E G\ G A
5 1 3 0 4 5 1 3 =22

This is a kind of minor. There's also a major:

A---------E
\ / \
\ / \
\ / \
\ / \
C---------G---------D
/ \ / \ /
D/-/---\--A/-/---\--E/ /
/ \ / \ /

C D D/ E E/ G A A/ C
4 0 3 1 5 3 1 5 =22


The following scales, and their inversions, also have 7 notes and
6 consonant triads in 22-equal.

/ \ / \ /
/ C\--------G\--------D\
/ \ / \ /
D---------A---------E
\ / \ /
\ / \ /
\ / \ /
\ / \ /
F---------C

/ \ / \ /
/ C\--------G\--------D\
/ \ / \ /
D---------A---------E
/ \ /
/ \ /
/ \ /
/ \ /
Bb--------F

This last one is a subset of a decatonic key:

A Bb C\ D\ D E F G\ A
2 3 4 0 4 2 3 4 =22

2 3 2+2 2+2 2 3 2+2 Alternate Pentachordal Major

The rest aren't because they include steps of a septimal diesis.
All of these scales have more than two step sizes, so they don't
fulfil Paul Erlich's melodic criteria. So far, I've only listened
to the first type.


A---------E
\ / \
\ / \
\ / \
\ / \
C---------G
/ \ / \
/ \ / \
---Gb\-------Db\---
\ \ / / \
-\--Eb-/- \
\ / \
\ / \
Bbb\------Fb\

In meantone temperaments, Fb\=E and Bbb\=A. So, this has 7 notes
and 10 7-limit triads, or 7 triads plus a tetrad. It includes the
more dissonant 7-limit chords, but it's still pretty good.

C C# Eb E F# G A C
2 6 2 5 3 5 8 =31


All these 7 note keys should work with diatonic harmony. I
thought it couldn't be done, but I was wrong.


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