RE: Lost Post, Miscellany

Kraehenbuehl & Schmidt deal strictly with just intonation, but don't look at
any chords other than the otonal chord on the first degree of the scale.
Their ratios sometimes mysteriously change from one system to the next.
Although it is true that 22-equal and 41-equal are good 7-limit and 11-limit
tunings, respectively, continuing the K&S construction to the 13-limit gives
you 78 tones, and 78-equal is not a good 13-limit tuning. So interpreting
their ideas as applying to temperament just doesn't work.

However, without intending to, their work argues for the virtues of
22-equal. Their 'simplest hyperchormatic system' is as follows, with all
'inflections' shown. When a ratio is more complex than all its inflections,
and its octave inversion is not present in K&S's list, the ratio is shown in
parentheses and given half weight in the reported mean. When K&S specified
an inflection of a ratio but did not include the octave inversion of the
inflection in their list, I have added the octave inversion as an inflection
of the octave inversion of the original ratio, unless this ratio turns out
to be more complex than all its inflections. These are shown in brackets and
contribute 1/2 weight to the reported mean:

ratio=cents mean cents 22-equal
1/1=0 0 0
25/24=71 (49/48=36) 36/35=49 [28/27=63] 56 55
16/15=112 15/14=119 21/20=84 105 109
35/32=155 [10/9=182] 164 164
8/7=231 (9/8=204) 222 218
7/6=267 (25/21=302) 279 273
6/5=316 (128/105=343) 325 327
5/4=386 (49/40=351) 375 382
9/7=435 435 436
4/3=498 21/16=471 (75/56=506) 489 491
48/35=547 27/20=520 533 545
10/7=617 7/5=583 (45/32=590) 598 600
35/24=653 653 655
3/2=702 32/21=729 716 709
25/16=773 49/32=738 (54/35=751)* [14/9=765] 756 764
8/5=814 45/28=821 (63/40=786) 811 818
5/3=884 884 873
12/7=933 (27/16=906) 924 927
7/4=969 (25/14=1004) [16/9=996] 984 982
64/35=1045 9/5=1018 (175/96=1039) 1033 1036
15/8=1088 40/21=1116 28/15=1081 1095 1091
27/14=1137 1137 1145

Close, huh?

*originally 56/35 in text -- must have been a typo since 56/35=8/5.


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