saturated harmonies (cont'd)

My bad, there was nothing wrong with Graham's page. So continuing:

15-limit:
ratios cents
8:9:10:11:12:13:14:15 0 204 386 551 702 841 969 1088
1/15:1/14:1/13:1/12:1/11:1/10:1/9:1/8 0 119 247 386 537 702 884 1088
12:14:15:21 (=1/35:1/30:1/28:1/20) 0 267 386 969
20:28:30:35 (=1/21:1/15:1/14:1/12) 0 583 702 969

I leave the five saturated 17-limit chords and the five saturated 19-limit
chords as an exercise to the reader (an easy one thanks to Graham who
already did the hard part).

However, I think that well before one reaches this point, there are much
more consonant "saturated" chords that cannot be expressed in just
intonation. Many of these have been discussed recently in the context of
12-tone or meantone temperament (e.g., the major chord with added 6th and
9th, the augmented triad, the diminished seventh chord); I'll review one
less familiar example (I once called it the "magic chord") here:

In 72-tone equal temperament, take the chord C E- F#- Bb-- (where - means
flattening by 1/72 octave relative to 12-tone equal temperament). The six
intervals in this chord are:

Interval Cents Approximate JI Interval
C E- 383 5:4 (386)
C F#- 583 7:5 (583)
C Bb-- 967 7:4 (969)
E- F#- 200 9:8 (204)
E- Bb-- 583 7:5 (583)
F#- Bb-- 383 5:4 (386)

One cannot tune the chord in JI and have all the intervals listed in the
rightmost column. The best one can do is to use 28:25 instead of 9:8; then
the other five intervals can be just as shown. 28:25 is a 25-limit interval
but is clearly a case where the limit is of far less importance than the
proximity to simpler ratios. It is 196 cents, and has a fairly clear
interpretation as an 8-cent flat 9:8, but it might also be heard as a
14-cent sharp 10:9. Tuning it closer to 9:8 would increase its consonance
and the 72-tET version shown here does so with minimal damage to the other
intervals.

So the "magic chord" is, allowing for intervals to be tempered by up to 4
cents, a saturated 9-limit chord. Similarly, the augmented triad in 12-tET
is, allowing intervals to be tempered by up to 14 cents, a saturated 5-limit
chord. Neither of these chords can be expressed adequately in just
intonation. So putting aside the difficult issue of fleshing out the
historical implications of Margo's conjecture, we see that even the
theoretical implications present some significant problems.