source file: mills2.txt Date: Tue, 26 Nov 1996 11:48:49 -0800 Subject: RE: Lost Post, Miscellany From: PAULE 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. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 27 Nov 1996 01:26 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA08917; Wed, 27 Nov 1996 01:27:52 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA09010 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id QAA12082; Tue, 26 Nov 1996 16:27:49 -0800 Date: Tue, 26 Nov 1996 16:27:49 -0800 Message-Id: <961126191702_1453270569@emout11.mail.aol.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu