This appeared in Tuning Digest 245, 10 Jul 1999.
[A typo has been corrected where the MA err for 227c was given as 22.4c]
Message: 10
Date: Sat, 10 Jul 1999 10:39:56 +1000
From: David C Keenan [address removed]
Subject: Good 7-limit generators
I've been doing some serious number crunching and can now announce that there
are only 3 single-chain generators of good (octave-based) 7-limit scales, and
only 6 double-chain (with a half-octave) generators.
To qualify, all they had to do was to have _more_ complete tetrads in a chain
of 10 notes, than meantone using augmented sixths (2), and have lower errors
(either RMS or max-absolute) than the best chain of fourths/fifths where the
dominant 7th chord is the 4:5:6:7 approximation. i.e. around a 702.5c fifth
(= 497.5c fourth).
Here's the info on these not-quite-good-enough generators for comparison:
Not good enough: No. generators in
Min Min interval
Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6
(+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7
-------------------------------------------------------------
503.4c 2 3.6c 5.4c -1 -4 3-10 -6 -9
497.5c 8 20.2c 25.4c -1 -4 3 2 6 3
Single chain: No. generators in
Min Min interval
Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6
(+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7
-------------------------------------------------------------
125c 6 12.2c 17.9c -4 3 -7 -2 -5 2
227c 4 16.5c 24.4c 3 7 -4 -1 -8 -4
317c 8 12.3c 17.9c 6 5 1 3 -2 -3
The generator sizes are only given to +-0.5c because the exact value will
depend on whether RMS error or Max-Absolute error or Max-otonal-beat-rate
(not shown) is the measure to be optimised.
Note that the minor-third generator, that we've been discussing recently, is
the best possible for a single chain. Note also that 227c is equivalent to
1200-227 = 973c, a 4:7 generator.
Double chain: No. generators in
Min Min interval
Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6
(+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7
-------------------------------------------------------------
71c 4 12.5c 18.2c 1 -3 -4 -3 0 4
230c 4 11.8c 17.5c 3 -1 4 -1 0 -4
380.5c 4 10.3c 17.5c -3 1 4 1 0 -4
491c 8 10.9c 17.5c -1 2 -3 2 0 3
506.5c 4 11.2c 17.5c -1 -4 3 -4 0 -3
521c 4 16.9c 23.1c -1 3 -4 3 0 4
In the table above, I haven't shown whether the half octave is included in an
interval or not, but that's not difficult to figure out.
Note that 230c is equivalent to 970c, an approximate 4:7. 380.5c is a major
third. The last three are fourths that correspond to fifths of 709c, 693.5c
and 679c. The first two of these last three were recognised by Paul Erlich as
being in the vicinity of the 22-tET and 26-tET fifths, and the last one is not
really good enough, with a 23.1c error in its fifths.
So far, 491c with a half-octave (22-tET) is the winner, with the (recently
discovered?) 317c minor-third a close second, and 125c in third place (based
on treating number-of-tetrads as more important than accuracy).
I wonder which of these have been discovered before? I wonder if I've missed
any? I'm 99% sure I haven't, but I'll be rechecking. Any journals likely to be
interested in this?
Does somone want to work out what ET's these embed in with sufficient
accuracy? Or how many notes in the largest 125c MOS with 12 notes or less?
I could look at higher multiple chains with the appropriate fraction of an
octave, if anyone cares. Does anyone want me to change my criteria in any way?
Regards,
-- Dave Keenan
http://users.bigpond.net.au/d.keenan