Yahoo Tuning Groups Ultimate Backup tuning-math Temperament finder update

I noticed a mistake in my temperament finder program. For alternative
equivalence intervals and timbres, I wasn't using these to select the
consistent ETs. So, although all the temperaments found are still valid,
some would previously have been missing.

Schismic temperament is now included in the standard 5-limit list. I
don't think that's because of anything I changed this time. It probably
came in when I increased the number of ETs considered from 20 to 21, so
that schismic would appear as a 5-limit microtemperament. To make this
appear less arbitrary, I've made the number of ETs 25 now. It doesn't
make much difference.

This looks like a good temperament from the "9-limit" nonoctave tubulong
list:

1/5, 46.9 cent generator (really 70.4 cents)

basis:
(0.16666666666666666, 0.03910923891316731)

mapping by period and generator:
[(6, 0), (10, -1), (13, -2), (15, -1)]

mapping by steps:
[(24, 6), (39, 10), (50, 13), (59, 15)]

unison vectors:
[[2, -1, 1, -1], [1, 8, -2, -4]]

highest interval width: 2
complexity measure: 12 (18 for smallest MOS)
highest error: 0.003877 (4.653 cents) (really 7.0 cents)

The period is 1/4 of an octave, and you need three notes within that
period. So it works out at 12 notes per octave.

The scripts have been updated at

<http://x31eq.com/temper.py>
<http://x31eq.com/writetemper.py>
<http://x31eq.com/tritave.py>
<http://x31eq.com/tritave.nonoct.py>

and the output files that have changed significantly

Graham

I kept on overhauling the temperament finding scripts at

http://x31eq.com/temper/

The latest change is that the unison vector searches are
using a more efficient algorithm. It follows from the
previous change to find unison vectors using an efficient
saturated kernel basis algorithm. The same function works
to tune unison vectors into mappings (vals). Then I search
for equal temperaments in the tempered space which is more
efficient than searching in the full prime-limit space and
filtering for tempering out the commas. It happens that
the code for finding the equal temperaments is all new
because the old one had properties of Cangwu badness of
prime limits wired in.

There may be bugs lurking in there. There are too many
possible inputs to do a reliable test. If you see
something that seems to be wrong, especially if it used to
be right, please let me know.

There are things I could do to the interface. It depends
if I'm so worn out that I stop working on this now. I
could have a link on the single temperament page to get you
to lower rank subsets/followers/whatever of it. That's
easy because it'll be reusing the unison vector code. So
if you want to find out some rank 2 temperaments consistent
with a rank 3 temperament you're looking at, there's one
button to do it, instead of you having to copy the unison
vectors into the search box.

After that, there are searches for higher-limit extensions
or higher-rank generalizations. I've been asked to get
ratios to factorize for non-integer "prime" limits. That
could be done, but it's another fiddly thing that I don't
plan to do for now. I think integer harmonics should be
enough to be getting on with, and you can always use vector
form.

Graham

Temperament finder update

🔗graham@microtonal.co.uk

🔗Graham Breed <gbreed@gmail.com>