locally concordant tetrads -- verifying Margo and George

In case you have no clue what harmonic entropy is, see
http://www.ixpres.com/interval/td/entropy.htm. It is, at this point, a
dyadid discordance measure, which certainly conforms to Pierre Lamothe's
partial ordering for simple ratios, but is a realistic, continuous function
that is well-defined for all intervals, simple ratios or otherwise. For this
exercise, I used a Farey series of order N=100 and an auditory resolution of
1%. Within the first two octaves, the resulting harmonic entropy curve shows
significant local minima at the ratios 5:6, 4:5, 7:9, 3:4, 7:5, 2:3, 5:8,
5:3, 4:7, 1:2, 4:9, 3:7, 2:5, 3:8, 4:11, 1:3, 3:10, 2:7, 3:11, as well as
hints of minima at 7:8, 6:7, 5:12, 5:13, 2:15, and 1026¢ (the last one
results from being less than 1% from two different simple ratios -- 5:9
below, and 11:6 above). I'll make a graph of this curve available soon.

Anyway, I calculated the total pairwise harmonic entropy for about 8 million
tetrads, allowing each of the 3 intervals between adjacent notes to vary
between 150¢ and 550¢ in 2¢ increments. By "total pairwise harmonic entropy"
I mean I added the six harmonic entropy values for the six possible pairs of
notes found within the tetrad. As I've mentioned before, this ignores
effects that arise from the synergy of three or more notes. I found 140
tetrads that were local minima (i.e., changing one, two, or three notes by
2¢ would necessarily increase the total pairwise harmonic entropy), and
ranked them according to total pairwise harmonic entropy (call it
discordance, since that's what it's a crude model of). Going from lowest
(most concordant) up the list:

First there were a bunch of chords which were really triads with the lower
note doubled an octave higher.

The next most concordant tetrad was:
0 498 886 1384¢
or 9:12:15:20 or 1/1:4/3:5/3:20/9. It is an open-voiced JI minor seventh
chord in third inversion, or an open-voiced JI major added sixth chord in
second inversion. So, in a sense, Margo agrees with this model (note,
HOWEVER, caveat in second-to-last paragraph below) that this "added sixth"
chord is the most concordant tetrad with distinct pitch classes, though
perhaps not in the voicing she would have suggested (of course, had I
increased the upper bound for adjacent intervals, some other inversion of
this chord, or some other chord, might have come out as most concordant).

The next most concordant tetrad was one that George Kahrimanis brought up
recently, and Margo and I have brought up in the past:
0 492 980 1472¢
It is the "stack of fourths" chord where the fourths are tempered narrow so
that the outer interval approaches 3:7, and the bass-alto and tenor-soprano
intervals approach 4:7. To the 2¢ accuracy of the calculation, this is the
same as the 22-tET tuning of the "stack of fourths". This clearly shows that
the relationship between concordance and JI is quite different when dealing
with dyads vs. dealing with larger chords, even when the model only takes
dyadic discordance, with local minima at JI ratios, into account. Also,
since the triadic analogue to this calculation showed a local minimum at 0
498 996¢, this model agrees with George K. that while a stack of two fourths
sounds best pure, a stack of three fourths can be improved with some
temperament.

The next most concordant tetrad was the JI major seventh chord, 0 386 702
1088¢ or 8:10:12:15 or 1/1:5/4:3/2:15/8.

The next most concordant tetrads were (tied)
0 502 1002 1390¢
and
0 388 888 1390¢
which have their fourths and major thirds tempered a bit wide, as in a very
mild meantone. The first one is the "Carole King" chord, the second is an
add6, add9 (6/9) chord with no 5th, in typical jazz fashion. Relative to
what would seem ideal just tunings of 9:12:16:20 (1/1:4/3:16/9:20/9) and
36:45:60:80 (1/1:5/4:5/3:20/9), respectively, the intervals are expanded a
bit, probably to bring the outer 9:20 closer to the much more concordant
4:9.

The next most concordant tetrad was the JI minor seventh chord, 0 316 702
1018¢ or 10:12:15:18 or 1/1:6/5:3/2:9/5.

The next most concordant tetrads were (tied) the second-inversion JI major
triad with the fourth in the bass, 0 204 702 1088¢ or 8:9:12:15 or
1/1:9/8:3/2:15/8, and the first-inversion JI minor triad with the ninth on
top, 0 386 884 1088¢ or 24:30:40:45 or 1/1:5/4:5/3:15/8.

The next most concordant tetrad was a surprise -- a very modern "augmented
octave" chord, 0 388 886 1274¢ or 12:15:20:25 or 1/1:5/4:5/3:25/12. It
contains two 4:5s, two 3:5s, and one 3:4s, all concordant enough to
counteract the great discordance of the 12:25.

While the usual, 5-prime-limit JI minor seventh chord has appeared in two
different voicings so far in our list, at this point the next most
concordant tetrad was the other JI minor seventh chord discussed at
http://www.cix.co.uk/~gbreed/erlichs.htm, 0 268 702 970¢ or 12:14:18:21 or
1/1:7/6:3/2:7/4.

The next most concordant tetrads (tied) were
0 318 818 1320¢
and
0 502 1002 1320¢
obtained by stacking a minor third and two fourths, or two fourths and a
minor third. In JI, these would be 15:18:24:32 (1/1:6/5:8/5:32/15) and
45:60:80:96 (1/1:4/3:16/9:32/15), respectively, but the intervals are
expanded a touch, probably to help alleviate the sharp discordance of the
outer interval.

The next most concordant tetrads (tied) are Margo's vote the JI added sixth,
0 388 702 886¢ or 12:15:18:20 or 1/1:5/4:3/2:5/3, and its second inversion 0
184 498 886¢ or 10:12:15:18 or 1/1:10/9:4/3:5/3.

The next most concordant tetrads (tied) were
0 434 820 1320¢
and
0 500 886 1320¢
which exploit the fact that a 7:9 stacked with a 4:5 very nearly produces a
5:8 (exploiting the 225:224 comma), and add a 3:4 on either the low end or
the high end (the 3:4 is adjacent to the 4:5, producing a 3:5).

The next most concordant tetrads (tied) were
0 302 502 1004¢
and
0 502 702 1004¢
which are two of the ways a stack of fourths can be transposed to within one
octave. A slight meantone-like tempering of the fourths is evident, in order
to bring the 27:32 closer to a more concordant 5:6.

The next most concordant tetrads (tied) were the first-inversion JI major
triad with the ninth on top, 0 318 816 1020¢ or 5:6:8:9 or 1/1:6/5:8/5:9/5,
and the second-inversion JI minor triad with the fourth in the bass, 0 204
702 1020¢ or 40:45:60:72 or 1/1:9/8:3/2:9/5.

The next most concordant tetrads (tied) were also of the augmented octave
type,
0 498 888 1282¢
0 394 784 1282¢
where the 3:4 is attached to either the low end or the high end of a stack
of two 4:5s (each slightly stretched to alleviate the discordance of the
25:16), producing a 3:5.

The next most concordant tetrads (tied) exploit the fact that a major third
plus two fifths is very nearly a 5:7 (i.e., exploiting the 225:224 "comma"):
0 384 588 1086
0 498 702 1086
The major third is compressed a bit to make this tempering magic happen.

The next most concordant tetrad is an open-voiced third-inversion major
seventh chord,
0 500 816 1316¢
which would be 15:20:24:32 or 1/1:4/3:8/5:32/15 in JI, but the fourths are
expanded a touch, probably to help alleviate the sharp discordance of the
outer interval.

The next most concordant tetrads (tied) are the JI major 7 (#5) chord, 0 388
776 1090¢ or 16:20:25:30 or 1/1:5/4:25/16:15/8, and the JI minor (maj7)
chord. 0 314 702 1090¢ or 40:48:60:75 or 1/1:6/5:3/2:15/8.

The next most concordant tetrads (tied) are the second-inversion JI major
triad with ninth on top, 0 498 702 886¢ or 6:8:9:10 or 1/1:4/3:3/2:5/3, and
the first-inversion JI minor triad with the fourth in the bass, 0 184 388
886¢ or 36:40:45:60 or 1/1:10/9:5/4:5/3.

The next most concordant tetrad is
0 442 884 1326
which stacks three 442-cent intervals, each of which approximates 7:9 from
the high side, and each pair of which approximates 3:5 very closely. In
other words, an equal division of the 3:5!

The next most concordant tetrads (tied) are a major triad with 5-prime-limit
minor seventh in the bass, 0 186 576 888¢ or 18:20:25:30 or
1/1:10/9:25/18:5/3, and a JI minor triad with 5-limit major sixth on top, 0
312 702 888¢ or 30:36:45:50 or 1/1:6/5:3/2:5/3. The 18:25s in these chords
are actually expanded relative to JI, to help them approximate the
concordant 5:7. These are thus pretty close to inversions of the 7-odd-limit
tetrads, but the major sixth between the outer voices is clearly 3:5 and not
7:12, and in fact the 7-limit versions do not show up as local minima of
their own.

The next most concordant tetrads (tied) are the "root position" 7-odd-limit
tetrads, the otonal 0 388 702 970¢ or 4:5:6:7 or 1/1:5/4:3/2:7/4, and the
utonal 0 268 582 970¢ or 60:70:84:105 or 1/1:7/6:7/5:7/4. Although for the
purposes of my 22-tET paper, and Pierre's recent post, it might have been
nice if these appeared on top of the list, they didn't. (HOWEVER: the order
here is pretty arbitrary. The Farey-series-based harmonic entropy curve
greatly favors large intervals over small ones, and no open voicings of
these chords had an opportunity to make it into the comparison, since I
limited the adjacent intervals to be under 550¢. Voicings such as 0 702 970
1588 and 0 618 886 1588 would handily defeat all the chords listed above,
while even 0 498 886 1468 and 0 582 970 1468 would have come out ahead of
the major seventh chord above. Other formulations of the model, though they
might completely change the order, would have very little effect on _what_
these local minima are, so that's what's important here). Experientially
speaking, I'm not too opposed to the ranking as regards the utonal 7-limit
tetrad, while the otonal 7-limit tetrad is clearly more concordant than this
ranking implies, which reflects the fact that this model ignores synergies
between three or more notes. A true triadic or tetradic harmonic entropy
model (rather than a sum of dyadic harmonic entropy models, which is what
this is) would take this into account.

I stop here. If anyone wants the full list of 140 local minima (cents only),
I'll e-mail it to them. One could spend a considerable period of time just
listening to the chords above, and tweaking the notes one by one to see how
the particular tuning (just or tempered) is a local optimum for that chord.
Many of the chords above come from a standard application of 5-limit JI to
tetrads found in the diatonic scale (including melodic minor and harmonic
minor variants), but many are tempered in a subtle way, and some are
downright xenharmonic. An excellent exercise for a composer looking for
points of stability in the infinite realm of microtonal harmony would be to
spend some time with the chords above.