source file: mills2.txt Date: Thu, 13 Jun 1996 13:30:29 -0700 Subject: Consistent tunings From: PAULE A typical way of evaluating equal temperaments is to measure the smallest possible tuning deviation of several just intervals of interest. The problem with this approach is that it does not guarantee that the approximating intervals are consistent with one another. For example, in 24tet, the calculation would approximate 5:4 as 8 steps, 7:5 as 12 steps, and 7:4 as 19 steps. But a 7:5:4 chord cannot be tuned with all three intervals as given! I have looked at all equal tunings up to 100 steps. For the 5-limit, the following tunings are INCONSISTENT: 1 2 11 13 14 17 20 21 32 33 40 51 52 54 64 66 67 85 86 98 For the 7-limit, the following tunings are INCONSISTENT: 1 2 3 7 8 11 13 14 17 20 21 23 24 25 28 30 32 33 34 38 39 40 44 47 48 51 52 54 55 61 64 65 66 67 69 71 74 75 79 85 86 92 96 97 98 100 For the 9-limit, the following tunings are CONSISTENT: 5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70 72 77 80 82 84 87 89 91 94 99 For the 11-limit, the following tunings are CONSISTENT: 22 26 29 31 41 46 58 72 80 87 89 94 For the 13-limit, the following tunings are CONSISTENT: 26 29 41 46 58 72 80 87 94 For the 15-limit, the following tunings are CONSISTENT: 29 41 58 72 80 87 94 For the 17-limit, the following tunings are CONSISTENT: 58 72 80 94 For the 19-limit, the following tunings are CONSISTENT: 80 94 For the 21-limit, the following tuning is CONSISTENT: 94 For the 23-limit, the following tuning is CONSISTENT: 94 NO tuning up to 100tet can consistently represent the 25-limit. So let's remove the inconsistent tunings from the table of "best" equal temperaments that I posted in TD 683 (Apr 8 '96) : pitch resolution -> .15% .25% .35% .45% .55% .65% .75% .85% .95% 1.05% 1.15% harmonic limit 3 41 29 12 12 12 12 12 12 12 7 5 5 53 34 34 19 19 12 12 12 12 12 12 7 72 31 31 31 31 22 22 15 12 12 10 9 53 41 41 31 31 19 19 12 12 12 12 11 72 41 31 31 31 22 22 22 22 22 13 87 41 15 87 41 17 72 72 19 21 23 94 Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 14 Jun 1996 08:29 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id XAA12697; Thu, 13 Jun 1996 23:29:09 -0700 Date: Thu, 13 Jun 1996 23:29:09 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu