Phillips Just-Intonation 0 #/b Scale

Phillips Just-Intonation 0 #/b "C" Scale Copyright © 2002 by Pauline W.
Phillips

Keyboard:
Ratios: 1/1 135/128 9/8 75/64 76545/65536 5/4
21/16
Cents: 0 92 204 274 269 386
471
Notes: C C# D D# 7Eb E
7F
Split digitals:
Ratios: 35/32*
81/64 10935/8192**
Cents: 155
408 500
Notes: 7D
E F

45/32 3/2 25/16 399/256*** 27/16 7/4 225/128
15/8 2/1
590 702 772 768 906 969 976
1088 1200
F# G G# 7Ab A 7Bb A#

189/128 105/64*
243/128 63/32
675 857
1110 1173
7G 7A
B 7C

*Presently I've substituted 30/17 to reduce the tone generators needed for
very-rarely-used notes.

** Multiplied from E 386, making F 500 really E#. Thus there's no complete F
chord in the C scale. Its major 3rd is missing. So the F scale is also needed
to play most music in 0 #/b. As the A split digital is already used for B7.
And it would be too awkward to have two chord notes on split digitals. The G
scale is also needed some.

***Multiplied from 57/32 Bb 999. In harmonic-partial JI, the only natural
kind, 17/16 Db = 105, 19/16 Eb = 298, 51/32 Ab = 807. There's no Gb in the 0
#/b "C" natural chromatic scale, as major chords aren't flatted. Thus there's
no Cb nor Fb either. However, ii, iii, vi, and possibly vii are flatted as
major chords. Thus a flat scale is sometimes needed. So my JT organ has three
manuals to accommodate C, F, G and flatted scales.

To make this scale practical, and to eliminate notes being locked in phase,
the scale is microtempered +/- 1-2 cents the theoretical values, using
three-4 12-ET Devtronix kit-organ frequency-divider tone generators. This
causes the #/b notes to merge. This allows also playing in c septimal minor,
as well as tertian e minor.

Seven white momentary-switch split digitals are required.

Sincerely,
Pauline W. Phillips, Moderator, <A HREF="/JohannusOrgansSchool ">Johannus Organs eSchool</A>
Johannus Orgelbouw, Holland, builds pipe, pipe-digital, digital-sampled
organs.
Moderator, <A HREF="/JustIntonationOrganSchool/">Just Intonation Organ eSchool</A>

In-Reply-To: <11b.17e459ee.2ad463e8@aol.com>
I've converted this to Dave Keenan's lattice with template

7
|
|
5 |
| |
| |
1----3----9

It's like a Fokker lattice in the 5-limit, but some notes have a 7-limit
interpretation. Here's the result:

7D---7A---- ---- ---- ---- ---- ---7Eb
| | |
| | |
G#---D#---A#---|---7C---7G |
| | | | | | |
| | | | | | |
E5---B5---F#---C#---|----|---- ----F---- ----
| | | | | | |
| | | | | | |
C----G----D----A----E----B---- ----

7Bb is omitted because it occupies the same position as A# (meaning they
differ by 225:224). E5 was another E on the original diagram, and B5 was
unnamed. There are some other notes omitted because they lie outside the
7-limit.

The F is a schismic fourth above C. I don't know why it's where it is.
7Eb is schismically related to D#.

Also, here it is on my kleismic lattice:

7
\
---5
/ \ /
3---1

7Eb-- --- --- --- --- --7A--7D
\ \ / \
7G--7C--- ---A#--D#--G#
\ \ / \ / \ /
F--- --- --- ---C#--F#--B5--E5
/ \ / \ / \ /
B---E---A---D---G---C

Graham