All-key Harmonic-Partial Septimal 19-Limit Just-Intonation Scales (Phillips)

All,

Since charts in Yahoo! posts are messed up, and sometimes in eMail, the scale
info below is alos attached as an RTF file.

Phillips All-key Harmonic-Partial Septimal 19-Limit (C) Just-Intonation
Scales

Copyright © by Pauline W. Phillips

22 notes to the octave, less Gb (in C) and the octave.

The key-signature root cents show why the scales need to have the octave
divided into 12 equal parts in the ET manner, so Db is 100 cents, D 200, etc.
Roots C, F, G and Bb indicate this tempering is the right procedure. As the
exact cents a key signature root begins on +/- a few cents in relation to an
adjacent key-signature root is musically immaterial. This greatly reduces the
number of notes needed!

It also enables one to use frequency-divider tone generators for both
key-signature roots and their slightly tempered 3rds, 5ths and harmonic minor
7ths, when the notes are created electronically.

Ratios 1/1 135/128 35/32 9/8 76545/65536 75/64 5/4 81/64 21/16
76545/65536 45/32
Keys/Notes C C# 7D D 7Eb D# E E+ 7F
F F#
C 1/1 0 92 155 204 269 274 386 408 471
500 590
+ 105
Db 17/16 105
+ 99
D 9/8 204
+ 94
Eb 19/16 298
+ 110
E 81/64 408
+ 92
F 76545/ 500
+ 65536 110
Gb 91/64* 610
+ 92
G 3/2 702
+ 105
Ab 51/32 807
+ 99
A 27/16 906
+ 93
Bb 57/32 999
+ 111
B 243/128 1110

3/2 399/256 25/16 51/32 105/64 27/16 7/4 225/128 15/8 243/128
63/32 2/1
G 7Ab G# Ab 7A A 7Bb A# B
B+ 7C C2
C 702 769 772 807 857 906 969 976 1088
1110 1173 1200

*Gb is the 13th harmonic's harmonic minor 7th. Thus the 13th harmonic is used
indirectly. The 11th harmonic isn't needed. The missing A 886 cents
382725/262144 belongs to the F scale.

The septimal c minor scale (7c) uses underlined 7Eb and 7Ab. The tertian e
minor scale (e) uses underlined A#. The above tuning provides for all three
scales--C, 7c and e. It this were not done, many additional scales would have
to be provided! My just-temperament version merges these notes with their #
or b counterparts.

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>