Tonality diamonds understood (at last!)

Dear Alternative Tuning Folk,

Paul Erlich recently explained to me what tonality diamonds are all about.
We are both rather astounded that I could be on the list as long as I have
and not really understand them. It may be that I'm dense, or it may be that
the existing web resources are inadequate (or both).

Monz has just put his dictionary page on the topic back on-line.
http://www.ixpres.com/interval/dict/tondiam.htm
I read it some time ago and I failed to understand (a) why it was called a
diamond (except in the 5-limit case) and (b) that in a 2N-1 limit diamond
there were N otonalities and N utonalities (e.g. for 11-limit N=6). This is
a serious local maximum in consonances per tone, for strict JI. However,
one can't blame Monz, since he is simply quoting Partch. And thanks Monz,
for your recent email explaining things further.

Perhaps someone else can benefit from the following development of the
11-limit diamond. This diamond is contained within all of Partch's tunings.

First make a 2 dimensional array of the fractions between all the odd
numbers up to our chosen limit.

den| numerator
om.| 1 3 5 7 9 11
---|-----------------------------
1 | 1/1 3/1 5/1 7/1 9/1 11/1
|
3 | 1/3 3/3 5/3 7/3 9/3 11/3
|
5 | 1/5 3/5 5/5 7/5 9/5 11/5
|
7 | 1/7 3/7 5/7 7/7 9/7 11/7
|
9 | 1/9 3/9 5/9 7/9 9/9 11/9
|
11 | 1/11 3/11 5/11 7/11 9/1111/11

Notice that every row is an otonal hexad and every column a utonal one. One
diagonal consists of 6 spellings of 1/1. And because 3 and 9 are not
relatively prime (i.e. they have a common factor) there are two spellings
of 3/1 and two of 1/3. So there are only 29 different tones
(octave-equivalent).

Now rotate it 45 degrees clockwise and distort it to a rhombus (a.k.a
"diamond").

1/1
1/3 3/1
1/5 3/3 5/1
1/7 3/5 5/3 7/1
1/9 3/7 5/5 7/3 9/1
1/11 3/9 5/7 7/5 9/3 11/1
3/11 5/9 7/7 9/5 11/3
5/11 7/9 9/7 11/5
7/11 9/9 11/7
9/11 11/9
11/11

The 1/1's now form the "spine".

Assume octave-equivalence and put all the tones in first-octave form. We
follow Partch and retain different spellings for the multiple 1/1's, 3/2's
and 4/3's. i.e. We don't reduce them all to lowest terms.

1/1
4/3 3/2
8/5 3/3 5/4
8/7 6/5 5/3 7/4
16/9 12/7 5/5 7/6 9/8
16/11 12/9 10/7 7/5 9/6 11/8
12/11 10/9 7/7 9/5 11/6
20/11 14/9 9/7 11/10
14/11 9/9 11/7
18/11 11/9
11/11

Now Partch shuffles the hexad lines (no hexads are broken), so that pitch
increases from left to right.

7/7
12/7 7/6
11/7 3/3 14/11
10/7 11/6 12/11 7/5
9/7 5/3 11/11 6/5 14/9
8/7 9/6 20/11 11/10 12/9 7/4
4/3 18/11 5/5 11/9 3/2
16/11 9/5 10/9 11/8
8/5 9/9 5/4
16/9 9/8
1/1

This corresponds to the diagram on p. 159 of 'Genesis of a music'. The
notes to the left of the central spine are to be taken as being in a lower
octave than those to the right of it, so all the hexads are now in the form
8:9:10:11:12:14 and 1/(8:9:10:11:12:14).

I think that's absolutely brilliant. Unlike our usual prime lattices, the
same pitch can occur more than once in strict JI, and there is no
one-to-one mapping of interval and vector, but it beautifully reduces the
dimensionality (from 4 down to 2) by breaking some of the graphical
connections and making them nominal ones (by name) instead. AND it orders
the pitches somewhat.

Notice however that it doesn't order the pitches as well as the following.
I've also reduced everything to lowest terms this time.

1/1
11/6 12/11
5/3 1/1 6/5
3/2 20/11 11/10 4/3
4/3 18/11 1/1 11/9 3/2
7/6 16/11 9/5 10/9 11/8 12/7
14/11 8/5 1/1 5/4 11/7
7/5 16/9 9/8 10/7
14/9 1/1 9/7
7/4 8/7
1/1

As well as ordering them primarily from left to right, this orders them
secondarily from back to front (on the right hand side) or front to back
(on the left hand side). But this gives hexads 7:8:9:10:11:12 where the
root is not octave-equivalent to the virtual fundamental.

In the 13-limit diamond, the best ordering of pitches also gives an
octave-equivalent root (8:9:10:11:12:13:14). But IMHO ratios of 13 are not
good value (being too close to the 11-limit ratios). Better to add other
11-limit ratios (as Partch usually did).

Many thanks to Paul Erlich for checking this for accuracy (and explaining
it to me in the first place).

Paul pointed out that we might have avoided the (possibly obscure)
shuffling step by starting with the array headings 8 9 10 11 12 14 rather
than 1 3 5 7 9 11, and pointing out that 8 9 10 11 12 14 is
octave-equivalent to 1 9 5 11 3 7. However, I feel that this is too
specific to the 11-limit case and would obscure the general principles that
apply to all tonality diamonds. Also we have seen that Partch's ordering of
the pitches is something of a compromise.

Regards,

-- Dave Keenan
http://dkeenan.com

>Paul pointed out that we might have avoided the (possibly obscure)
>shuffling step by starting with the array headings 8 9 10 11 12 14 rather
>than 1 3 5 7 9 11, and pointing out that 8 9 10 11 12 14 is
>octave-equivalent to 1 9 5 11 3 7. However, I feel that this is too
>specific to the 11-limit case

???