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7-limit intervals of low Hahn badness

🔗Gene Ward Smith <gwsmith@svpal.org>

7/19/2004 12:24:05 PM

If we define Hahn badness as the size of an interval in cents times
the fourth power of the Hahn lattice distance from the origin, we can
produce a finite comma list by bounding badness and complexity. I
bounded badness by 10000 and Tenney distance (log2 of Tenney height)
by 100 for 7-limit intervals within an octave, and sorted them by
increasing badness. It seems there are 57 of them; I list the
interval, the Hahn distance, the size in cents, and the Hahn badness.
It's pretty much the same as things I've done before; note again that
2401/2400 is way down there in the badness basement, below the 7-limit
consonances; it's the ratio of two other commas (49/48 and 50/49)
which themselves have a very low Hahn badness. This comma just rules
the 7-limit.

2401/2400 4 .721197 184.626504
8/7 1 231.174094 231.174094
7/6 1 266.870906 266.870906
6/5 1 315.641287 315.641287
5/4 1 386.313714 386.313714
4/3 1 498.044999 498.044999
50/49 2 34.975615 559.609837
49/48 2 35.696812 571.148993
7/5 1 582.512193 582.512193
10/7 1 617.487807 617.487807
3/2 1 701.955001 701.955001
36/35 2 48.770381 780.326102
8/5 1 813.686286 813.686286
5/3 1 884.358713 884.358713
12/7 1 933.129094 933.129094
4375/4374 7 .395756 950.209846
7/4 1 968.825906 968.825906
126/125 3 13.794767 1117.376095
25/24 2 70.672427 1130.758830
21/20 2 84.467193 1351.475096
16/15 2 111.731285 1787.700564
15/14 2 119.442808 1911.084932
225/224 4 7.711523 1974.149886
250047/250000 9 .325441 2135.221096
1029/1024 4 8.432720 2158.776390
64/63 3 27.264092 2208.391436
35/32 2 155.139620 2482.233925
10/9 2 182.403712 2918.459394
28/25 2 196.198479 3139.175660
9/8 2 203.910002 3262.560028
128/125 3 41.058858 3325.767531
1728/1715 4 13.073569 3346.833747
6144/6125 5 5.362046 3351.278958
78125000/78121827 15 .070314 3559.669541
3136/3125 5 6.083244 3802.027259
420175/419904 8 1.116953 4575.040111
25/21 2 301.846520 4829.544326
28/27 3 62.960904 5099.833214
81/80 4 21.506290 5505.610137
875/864 4 21.902045 5606.923640
60/49 2 350.616902 5609.870429
49/40 2 351.338099 5621.409585
65625/65536 7 2.349477 5641.093458
5764801/5760000 8 1.442395 5908.048130
256/245 3 76.034473 6158.792329
360/343 3 83.745996 6783.425691
32/25 2 427.372572 6837.961156
9/7 2 435.084095 6961.345524
64/49 2 462.348187 7397.570993
21/16 2 470.780907 7532.494517
281484423828125/281474976710656 19 .058104 7572.206073
49/36 2 533.741811 8539.868979
48/35 2 546.815381 8749.046089
245/243 5 14.190522 8869.076548
25/18 2 568.717426 9099.478816
3955078125/3954653486 15 .185885 9410.418727
343/320 3 120.164006 9733.284449