Non-octave tunings

"John A. deLaubenfels" <jadl@idcomm.com> wrote:
>> That's one of the neat properties of 88cET/87.745cET -
>> you get three very useful thirds - including both septimal
>> thirds.
>But, you DON'T get the "normal" major and minor thirds, of course...

18 steps of 88cents is 1200+384 cents, an almost perfect 5/2.

0 8 18 is an inversion of a major triad.

0 5 10 is a stack of 2 9/7's adding to a 5/3, implying a xenharmonic
bridge of 245/243.

0 4 8 is a neutral triad, implying a xenharmonic bridge of 243/242.

88CET approximates many just ratios. Some are < 2/1: 21/20 7/6 11/9 9/7
3/2 5/3 7/4.

Since 88CET is a n-Cents-Equal-Temperament, it repeats after a given
number of octaves. Since 88/1200 = 11/150,
we see that it has exactly 150 steps in the span of 11 octaves. There is
no "octave", we have to replace it with
2**11 = 2048. The 11-limit pitch factors (the H vector) become (log2048,
log3, log5, log7, log11)
the factors of the pitch vectors -- which normally is (2,3,5,7,11) -- to
(2048,3,5,7,11).

The following matrix has a determinant of 150:

+0001 000 00 0 00 : repetition after 11 octaves
-4/11 +04 +1 0 00 : syntonic comma
-5/11 -11 00 8 00 : 5 subminor thirds = 3 supramajor thirds
00000 -05 +1 2 00 : 9/7 * 9/7 ~= 5/3 comma=245/243
-0004 +07 -1 2 -3 : 3 neutral thirds ~= 7/4 * 21/20

Maybe there is a way to fit the ratio 243/242 in the matrix. This cute
superparticular ratio is the second discrete
derivative of 7/6 11/9 9/7.

-Kami

The "xenharmonic bridge" 245/243 as mentioned by Kami plays a role in the
theoretical treatment of the Bohlen-Pierce scale where it is called "minor BP
diesis"; it is the distance between many of its enharmonics (for instance
81/49 and 5/3). If it interests you, just have a look at "Development..." at
http://members.aol.com/bpsite/scales.html
and "Enharmonics" at .../intervals.html at the same site.

Heinz