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Some commas and Paul's chart

🔗genewardsmith <genewardsmith@juno.com>

12/14/2001 1:21:08 PM

I was wondering whether the secondary peak on Paul's chart had something to do with the ragisma, but that does not seem to be the case:

Comma 1/log2(comma)

25/24 16.97974789
28/27 19.05944685
36/35 24.60509736
49/48 33.61644777
50/49 34.30961893
64/63 44.01393598
81/80 55.79763048
2048/2025 61.37300833
245/243 84.56347925
126/125 86.98951088
4000/3969 89.09131854
1728/1715 91.78824531
1029/1024 142.3028203
225/224 155.6112748
3136/3125 197.2631833
5120/5103 208.4128371
6144/6125 223.7951436
2401/2400 1663.898452
4375/4374 3032.168156

It would be interesting to see the numeric values of the other peaks.

No apologies--deciding in advance that something is "useless" is *not* the way research should be done, and since our resources really don't require conservation, there is no point in acting like congresscritters deciding what to cut out of next year's NSF budget. There is something here we don't understand, and it could make a difference even to the relentlessly and prosaically "practical"--if that is what music is supposed to be.

🔗paulerlich <paul@stretch-music.com>

12/15/2001 8:18:31 AM

The reason the schisma and the breedsma may be showing up as so
important is that they're far smaller than any simpler commas in their
respective limits. So most of the good ETs go along for a while
treating them as if they don't exist, but once they are treated as a
finite number of steps in the ETs, a whole new standard of 5- or
7-limit accuracy is attained.

Perhaps we should look more closely at the lower periods in the FFT,
all of which are kind of squished toward the left in the graph that I
posted . . . the first example of this kind of phenomenon in 5-limit
would be 15:16, about 1/11 octave, producing high points in the
5-limit ET graph wherever it's a near-integer number of steps: 1
(12-tET), 2 (19-tET, 22-tET), 3 (31-tET, 34-tET), 4 (41-tET, 43-tET),
5 (53-tET, 55-tET).

Clearly we're on the path to a full (or fuller) understanding of the
patterns that have been known for decades, if not centuries, in the
graph of ET quality, and have typically been regarded as random noise,
aside from instances where the sum of two good ETs is a good ET.