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Introduction
The study of just intonation usually employs constructs known as
"limits". Prime limits are infinite subsets, and odd limits
finite subsets under octave equivalence, of the rational numbers.
The term "limit" is due to the fact that rationals are included
in limit q if they can be somehow derived from numbers ≤ q. This
means that the q limit must be a superset of the p limit, p < q.
The near total reliance, dating back at least to Partch, on these
"limits" in the nomenclature of just intonation means that many
tuning systems have escaped study. When evaluating an 11-limit
temperament, for example, it may be useful to omit one identity
from the limit if the temperament's error can be greatly reduced.
The following is a series of attempts at finding systems like
this, which I first proposed in 2006.
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2008.03.26
In each ET between 5 and 99 notes per octave (95 ETs total), I
tune all triads, tetrads, pentads, and hexads that can be taken
from the set {2 3 5 7 11 13 17}. That's 35+35+21+7 = 98 chords.
For each ET, I kept only the most accurately approximated triad,
tetrad, etc. So 95 ETs * 4 chord sizes = 380 chords total. I
use TOP damage to assess tuning accuracy. TOP damage tells you
the maximum Tenney-weighted error of any interval that can be
formed from the notes of the chord (after the octave has been
optimally stretched/compressed). So it turns these chords into
something like prime limits.
I then throw out chords of more than ET/2 notes. So tetrads in
5-ET are thrown out, as are hexads in 7-ET, etc. For odd ETs I
round ET/2 up. So I keep triads in 5-ET, for example. This
leaves 368 chords.
Finally, I throw out chords which are not consistent their ET.
The remaining 287 chords are shown on the first sheet in this
Excel workbook:
SubgroupETs.xls
They are sorted first by chord size (triads, tetrads, etc.) and
then by increasing badness. Badness here is TOP damage times the
number of notes in the ET. Of course you can use Excel to sort
however you like.
The best triad is 3:5:17 in 72-ET. The best tetrad is 2:5:11:13
in 87-ET (Petr's temperament). The best pentad is 2:5:7:11:13 in
37-ET (Petr's temperament with 7 added). The best hexad is
2:3:5:7:11:17 in 72-ET.
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2008.04.05
By using "patent vals" and then casting out inconsistent ETs, I
missed some good mappings. So in this version, I check 9 vals
for each "ET" (rounded number of notes/octave) and take the one
with the lowest TOP damage. So there are a full 368 entries on
the second sheet in the workbook.
Here's a look at the best chords / prime limits...
pi "ET" primes val
3 41 (5 13 17) <96 153 169]
4 41 (5 11 13 17) <96 143 153 169]
5 41 (5 7 11 13 17) <96 116 143 153 169]
This looks 3 * 88CET.
6 83 (3 5 7 11 13 17) <131 192 232 286 306 338]
Looks suspiciously like 2 * the above.
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2008.05.04
You may argue that I shouldn't use weighted error, since it tends
to keep simpler chords from showing up (by allowing less error on
simple identities). And you'd be right.
Unfortunately, using unweighted error means we are truly dealing
with chords and not prime limits. So we have to use integers
2-17 instead of primes up to 17 as our master set, and then
eliminate chords whose notes are not coprime. There are 14,756
chords of 3-6 notes drawn from these integers, but considerably
fewer are coprime.
Our error function will be the maximum dyadic error in the chord,
in cents. See the third sheet in the workbook.
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2010.04.20
Our final approach to this can be a little different. Instead of
finding the best chord for each ET, we can find the best ET for
each chord. This allows us to return to Tenney weighted error
without fear of excluding simpler primes. I show the best and
2nd-best ETs 5-100 for each basis (according to logflat badness).
Results are available on the fourth sheet in the workbook.
Or, in this text file:
SubgroupETs4.txt
Each line shows:
(basis (best-ET (val) TOP-damage badness) (2nd-best ET ...))
The Scheme functions used to compute these are available here:
SubgroupETs.ss
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carl@lumma.org