Consistency limits

I have created a list of equal temperaments up to about 650 tones
per octave with their consistency limits as was explained by Paul
Erlich's post of 13 June 1996. However it's not restricted to
integer divisions, all scale step sizes that give a different
consistency limit were calculated. Initially I thought this could
make a nice graph, step size horizontally, limit vertically,
but the function behaves very wildly and there are very many points.
Not included in the file are points where both left and right side
limits are smaller than 5.
This is the filename:
ftp://ella.mills.edu/ccm/tuning/papers/consist_limits.txt

Another larger file
ftp://ella.mills.edu/ccm/tuning/papers/cons_limit_bounds.txt

contains the step size bounds for each consistency limit of 3 to 18,
also up to 650-tET.


The first list looks like this:

Limit at right side (larger step)
Step size in cents
Number of tones per octave
Limit at left side (smaller step)
Uniqueness limits for this temperament

4 589.572 2.03537 5 3 2
5 543.415 2.20825 2 3 2
2 422.656 2.83918 5 4 4
5 413.594 2.90139 6 4 4
6 387.530 3.09653 7 4 4
7 371.508 3.23007 4 4 4
3 320.000 3.75000 7 3 2
7 308.735 3.88682 8 3 2
8 295.424 4.06195 5 3 2
5 293.296 4.09142 4 3 2
4 242.288 4.95278 9 4 4
9 241.594 4.96699 10 4 4
10 233.004 5.15010 6 4 4
6 229.774 5.22251 5 4 4
5 228.571 5.25000 3 4 4
2 200.205 5.99383 5 4 4
5 200.126 5.99622 8 4 4
8 199.770 6.00691 7 4 4
7 199.218 6.02355 3 4 4
3 177.777 6.75000 5 5 4
5 177.254 6.76992 6 5 4
6 168.867 7.10616 4 5 4
4 150.611 7.96752 6 4 4
6 149.050 8.05098 7 4 4
7 145.454 8.25000 3 4 4
4 135.917 8.82887 7 6 6
7 135.843 8.83369 8 6 6
8 132.110 9.08328 6 6 6
6 131.998 9.09104 5 6 6
5 131.169 9.14848 2 6 6
2 122.706 9.77941 5 5 4
5 121.645 9.86475 8 5 4
8 119.717 10.0236 9 5 4
9 118.566 10.1209 4 5 4
3 110.676 10.8423 6 5 4
6 110.375 10.8719 4 5 4
4 101.320 11.8436 6 6 6
6 100.562 11.9329 10 6 6
10 99.2594 12.0895 11 6 6

The second list runs like this:

3 2400.00 0.50000 - 1403.91 0.85475
3 1267.97 0.94639 - 800.000 1.50000
3 760.782 1.57732 - 543.415 2.20825
3 480.000 2.50000 - 467.970 2.56426
3 422.656 2.83918 - 345.810 3.47011
3 342.857 3.50000 - 292.608 4.10104
etc.

Manuel Op de Coul coul@ezh.nl

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>In fact 19-TET is the most common,
>almost the closest in sound to 12-TET of the "teen"
>temperaments (22-TET is closer), and perhaps the
>easiest to use of the < 20 TET equal temperaments.

I agree completely, except that it's difficult to say that 22-TET is closer.
The standard tonal repertory sounds good - if not better - in 19-TET.
22-TET fails to preserve normal diatonic relationships, and cannot be
notated with the conventional symbols. Performing Bach or Mozart in 19 is
trivial, while getting them to sound good in 22 would be a difficult task.
That doesn't prevent 22 from being my favorite tuning!

I'll give Brian the benefit of the doubt and assume he was concerned stricly
with the sound of consonant triads. 19 and 22 are both much smoother than
12, but 19 is the smoothest. In that sense, 22 is closer than 19 to 12. But
in that sense, 15 is even closer to 12 (15 is slightly less smooth than 12).

Is 22 a teen? Is 24 the first "adult" temperament? If so, I've got two days
to grow up.


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