Ptolemy's Malakon Diatonic and 22 tet

Some time ago I wrote a piece of music in Just Intonation. I wanted to
restrict myself to only seven pitches and get inside the scale. So,
because of the nature of the piece I had in mind, I sang through a few
of the more singable tetrachord based scales and chose the 'Phrygian'
sounding Ptolemy's Malakon or soft Diatonic which lays out as follows:-

1/1 21/20 7/6
4/3 3/2 63/40 7/4 2/1

Cents 0 84.47 266.87 498.04
701.96 786.42 968.83 1200

Steps between a la John Chalmers:-

(21/20) (10/9) (8/7) (9/8)
(21/20) (10/9) (8/7)

Cents 84.47 182.4 231.17 203.91
84.47 182.4 231.17

There are interesting compositional challenges with this scale as
regards harmony if you want a variety of consonant vertical structures
and counterpoint. I chose to use modal techniques and simple dronal and
contrapuntal methods to achieve my objectives. Then I wrote and scored
out my piece for harp, marimba and organ, though it would sound fine for
other combinations, perhaps strings.

It was suggested by Paul that I look at realising the piece in 22 tet
which I've managed to find time to do. I've been pleased to find a
satisfying cross-learning process going on as I've looked at how close
22 tet comes to Ptolemy's scale, then at other 22 tet choices and then
at other Just Intonation choices.

Going by ear and using a 22 tet guitar, my first choice of steps was

0 2 5 9
13 15 18 22

Cents 0 109.09 272.73 490.91 709.09 818.18
981.82 1200

Steps 9 and 13 are about - and + 7.14 cents from the just 4/3 and 3/2
respectively. There's not much I can do about this and besides the
exotic fifths and fourths gives different temperaments their
individuality. Step 5 is 5.86 cents sharp of the septimal minor third
7/6 (sensuous interval BTW) which is good to my ears. Step 18 is nearly
13 cents sharp of the harmonic seventh 7/4, not quite satisfactory to my
ears.

The two remaining steps, 2 and 15 are substantially off the mark. Step 2
is 24.6 cents sharp of 21/20 and step 15 is 31.8 cents sharp of 63/40
which is just TOO MUCH. Must make a note to do some ear training.

Replacing step 2 with step 1 ( 54.55 cents) isn't much better. This puts
me nearly 30 cents flat of 21/20. Replacing step 15 with step 14
(763.64 cents) puts me nearly 23 cents flat of 63/40. The just pitches
lie pretty much in between the 22 tet steps. But the upside of this is
that I now have four 22 tet seven note scales to experiment with as I
try to arrange my original piece for guitar. These are:-

0 2 5 9 13 15 18 22 my original approximation
0 2 5 9 13 14 18 22
0 1 5 9 13 15 18 22
0 1 5 9 13 14 18 22

The last of these I find to be of considerable melodic interest.
Possible uses for these could be the use of different variants
ascending and descending in fixed passages or as improvisational choices
in freer passages. Both possibilities present themselves in my original
JI piece.

Looking back from my new 22 tet variables to close JI pitches I find
that step 1 is only 1.28 cents sharp of 33/32 (11 limit and very sexy).
Step 2 is only 2.64 cents flat of 16/15, an eminently singable tone.
Step 14 is 1.28 cents flat of 14/9 and step 15 is 4.49 cents sharp of
14/9. These are accurate representations. Although I haven't done this
yet, I intend to explore some of the new JI tetrachords that these
pitches suggest. A trawl through "Divisions of the Tetrachord" has
proved to be edifying. I reckon they will all be diatonic though upper
and lower tetrachords will obviously differ in some cases.

One example:-

1/1 33/32 7/6 4/3 3/2 14/9 7/4 2/1

Ultimately, the proof of the pudding lies in the eating. Some of the new
JI scales might stretch the already fragile consonant structures of the
original composition too much. Others might completely transform the
piece. I hope to get round to putting up sound files of the original in
Ptolemy's Malakon realised using samples and then the 22 tet guitar
version (or possibly versions) which I haven't mastered quite yet (pun
intended).

Comments (and corrections) welcome.

Best Wishes

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:

> Replacing step 2 with step 1 ( 54.55 cents) isn't much better. This puts
> me nearly 30 cents flat of 21/20. Replacing step 15 with step 14
> (763.64 cents) puts me nearly 23 cents flat of 63/40. The just pitches
> lie pretty much in between the 22 tet steps. But the upside of this is
> that I now have four 22 tet seven note scales to experiment with as I
> try to arrange my original piece for guitar. These are:-
>
> 0 2 5 9 13 15 18 22 my original approximation
> 0 2 5 9 13 14 18 22
> 0 1 5 9 13 15 18 22
> 0 1 5 9 13 14 18 22
>
> The last of these I find to be of considerable melodic interest.

That's the one I would have suggested in the first place! I've talked about this scale quite a bit --
note it's omnitetrachordal and has three very good 6:7:9 triads. You can call it the "22-tET
Pythagorean Phrygian".