The JI harmonic diminished, or Messiaen octatonic, scale, with alternates

Hi all,

If you've tried to follow my earlier meanderings on
this subject, I apologise for all the wrong turnings
my path has taken. However, I hope to provide a
clear map, below, to a method of realising this tuning
on a wide range of keyboards.

That includes the one I'm stuck with for now, my
aging Roland E-28. It has many lovely sounds - or so
_I_ think! - but has limited tuning capacity. It will
change each of the 12 pitches per octave, in the
range of 64 cents below to 63 cents above the
standard pitch, changing in increments of 1 cent.
The tuning of the same note is identical in all octaves.

I've set out my reasoning in a spreadsheet, which is
available to anyone who requests it by private email.
I extract that reasoning below; I also include the
final tuning instructions below, tabulated with dots,
for anyone to try out for themselves. However, the
full table of calculations is too tedious to format
and include in a plain text message (even with dots!)

The reasoning:
----------------

(Constructive C&C welcome, of course!)

The octatonic scale consists of:
C, Db, Eb, E, Gb, G, A, Bb.

There are four chains of four minor thirds 6/5
implied in this scale:

o ... The first is A 5/3, C 1/1, Eb 6/5, Gb 36/25.
These are exact ratios from the scale.

o ... The second is a perfect fifth 3/2 higher than
the first: E 5/4, G 3/2, Bb 9/5, Db 27/25. These
are exact ratios from the scale.

o ... The third is a major third 5/4 higher than the first:
C# 25/24, E 5/4, G 3/2, Bb 9/5. All are exact ratios
from the scale, except for the C#, which is 648/625
flatter than Db.

o ... The fourth is a minor third 6/5 higher than the
first: C 1/1, Eb 6/5, Gb 36/25, Bbb 216/125. All
are exact ratios from the scale, except for the Bbb,
which is 648/625 sharper than A.

To complete these four chains, we add C# as an
alternate for Db, and Bbb as an alternate for A.
Extending the chains in either direction produces
more non-scale degrees; Bbb and C# are just the
first two of these.

Using just the scale degrees and these two alternates:

o ... The interval C to C# is a (small) semitone 25/24,
just 71 cents.

o ... The interval Eb to E is a (small) semitone 25/24,
just 71 cents.

o ... We assign the symbol # to mean "raise by a (small)
semitone" and b to mean "lower by a (small) semitone".

o ... Using these symbols we can assign an unequivocal
meaning to every normal note spelling, eg Fb, G# or Cx.

o ... There is a major triad 1/1, 5/4, 3/2 on every other
degree: C, Eb, Gb, A.

... o ... The major triad on C consists of scale notes
entirely: C 1/1, E 5/4, G 3/2.

... o ... The major triad on Eb consists of scale notes
entirely: 6/5 * (1/1, 5/4, 3/2): Eb, G, Bb.

... o ... The major triad on Gb consists of scale notes
entirely: 36/25 * (1/1, 5/4, 3/2): Gb, Bb, Db.

... o ... The major triad on A consists of the notes:
5/3 * (1/1, 5/4, 3/2): A, C#, E.

... These four chords are spelled normally.

o ... There is no major triad rooted on G, Bb, Db or E.
... The major third of the A major triad is C#, which
lies between two scale degrees. It is an alternate
for Db, which is the major third of Bbb major. The
major triad on Bbb consists of the notes:
216/125 * (1/1, 5/4, 3/2): Bbb, Db, Fb+. The last of
these, Fb+, is 162/125, just 81/80 sharper than Fb.
It is not a note of the octatonic scale.

o ... There is a minor triad 1/1, 6/5, 3/2 on every other
degree: C, Eb, Gb, A.

... o ... The minor triad on C consists of scale notes
entirely: C 1/1, Eb 6/5, G 3/2.

... o ... The minor triad on Eb consists of scale notes
entirely: 6/5 * (1/1, 6/5, 3/2): Eb, Gb, Bb.

... o ... The minor triad on Gb consists of the notes:
36/25 * (1/1, 6/5, 3/2): Gb, Bbb, Db.

... o ... The minor triad on A consists of scale notes
entirely: 5/3 * (1/1, 6/5, 3/2): A, C, E.

... These four chords are spelled normally.

o ... There is no minor triad rooted on G, Bb, Db or E.
... To create them would require adding their perfect
fifths 3/2 above: D 9/8, F 27/20, Ab 81/50 or B 15/8.

o ... There is a diminished triad 1/1, 6/5, 36/25 on A, C
and Eb, but not on Gb.

... o ... The diminished triad on A consists of scale notes
entirely: 5/3 * (1/1, 6/5, 36/25): A, C, Eb.

... o ... The diminished triad on C consists of scale notes
entirely: C 1/1, Eb 6/5, Gb 36/25.

... o ... The diminished triad on Eb consists of the notes:
6/5 * (1/1, 6/5, 36/25): Eb, Gb, Bbb.

... These three chords are spelled normally.

o ... There is no diminished triad rooted on Gb.
... To create it would require adding the minor third 6/5
above Bbb 216/125: Dbb 648/625.

--- End of ---

Tuning instructions:
----------------------

Only the notes with a number 1, 2, 3 ... 8) are
scale notes. The notes 2-k and 7+k are a comma
of size 648/625 below and above notes 2 and 7;
these are the two alternate notes C# and Bbb.
All other named notes are shown only for
completeness, however, most double sharps and
double flats are not shown.

To tune the octatonic scale with two alternates,
use the table below. Tune the note with number
"Note No." and name "Note Name" by using the
12-EDO key normally used as "Use note" and
adjusting its tuning by "adjust note by" cents.

As will be seen from the table, the tuning is 37
cents sharp relative to A=440 Hertz.

The "Use note" column also tells you what note
you must play in order to produce the note named
in "Note Name". For example, to play a C major
triad, C-E-G, in Just Intonation, you will need to
play the notes normally used for C-F-G# in 12-EDO;
it will seem that you are fingering an F minor triad!
All notes except C, C# and A are played on the note
that would be one semitone higher in 12-EDO.

Note .. Note .. Use ..... adjust ...... Frequency
No. ...... Name . note ... note by ... (hertz)
1 ........... C ........... C .......... +53 c ......... 269.76
2-k ..... C# ....... C# ...... +24 c ......... 281.00
2 .......... Db ........ D .......... -14 c ......... 291.34
.............. D ........... .............. -43 c ......... 303.48
.............. D# ....... .............. +28 c ......... 316.12
3 .......... Eb ......... E .......... -31 c ......... 323.71
4 .......... E ............ F .......... -61 c ......... 337.20
.............. Fb ......... .............. -20 c ......... 345.29
.............. E# ........ ............. +10 c .......... 351.25
.............. F ............ .............. -49 c ......... 359.68
.............. F# ........ ............. +22 c ......... 374.67
5 .......... Gb ......... G ......... -16 c ......... 388.45
6 .......... G ............ G# ..... -45 c ........ 404.64
.............. G# ........ .............. +26 c ........ 421.50
.............. Ab ......... .............. -33 c ........ 431.61
7 .......... A ............ A ......... +37 c ........ 449.60
7+k .... Bbb ....... Bb ....... +0 c ........... 466.14
.............. A# ........ .............. +8 c ........... 468.33
8 .......... Bb .......... B .......... -29 c ........ 485.57
.............. B ............ .............. -59 c ........ 505.80
.............. Cb .......... .............. -18 c ......... 517.94
.............. B# ......... .............. +12 c ........ 526.87
(1) ...... C .............. .............. -47 c ........ 539.52

--- End of tuning instructions ---

Regards,
Yahya
--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.371 / Virus Database: 267.14.1/204 - Release Date: 15/12/05