Systems Compared qua Melodic Pitch Classes

This is merely a summary; please refer to my earlier posts for more
info.

I have given evidences that the octave is divided into the following
numbers of melodic pitch classes:

1. Just Intonation: 13 - 17
2. 12-tone Equal (and Pythagorean): 11 - 12
3. 31-tone Equal: 12 - 14
4. 19-tone Equal: 18 - 19


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From: "Paul H. Erlich"
Subject: RE: 22-tone equal & chromaticism
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Paul Erlich has doon a good job of proving how Gregg Gibson's
chromatic genera are not restricted to meantone temperaments.
Correspondents may recall that I "discovered" 6 genera that
Gregg omitted. I have (at time of writing) yet to receive
comment on this but, in addition to all 10 being distinct in
12 equal, they all work in JI.

I use the word "genera" because Gregg did. I would much
rather reserve the word for Euler genera. Anyway, here they
are again:

C D Eb F G Ab A C C Eb E F G A Bb C
C C# E F G G# A C C Eb Fb F G Ab Cb C
C C# E F F# G A C C Eb F Gb G Ab Cb C

These are grouped in otonal-utonal pairs. The graph for one
of a pair is the graph for the other rotated by 180?. Here,
then, are the graphs for the left-hand genera:


D---------A
\ / \
\ / \
\ / \
\ / \
F---------C---------G
\ / \ /
\ / \ /
\ / \ /
\ / \ /
Ab--------Eb

[Notice that there is an out of tune fifth G-D which does
not take part in any chords.]


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


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


These last two diagrams are interesting in that, if you combine
them, you get a single 8-note genus of which the four 7-note
genera are subsets.


I've also come across another pair:

C Eb E F G Ab B C C Db E F G Ab A C

E---------B
/ \ /
/ \ /
/ \ /
/ \ /
F---------C---------G
\ / \ /
\ / \ /
\ / \ /
\ / \ /
Ab--------Eb

Which brings the total to 12. Interestingly, my 8 all have
5 consonant chords, two of which share a root.


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From: gbreed@cix.compulink.co.uk (Graham Breed)
Subject: Re: More of Septimals
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