source file: m1478.txt Date: Fri, 17 Jul 1998 14:04:58 -0400 Subject: Equal Temperaments From: "Paul H. Erlich" Here is a table showing the simplest equal temperaments with consistent representations of all just intervals through the m-limit _and_ unique representations of all just intervals through the n-limit (these are odd limits): m- 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 n | 3 3 3 5 5 22 26 29 58 80 94 94 282 282 282 311 311 311 5 9 9 12 22 26 29 58 80 94 94 282 282 282 311 311 311 7 27 27 31 41 41 58 80 94 94 282 282 282 311 311 311 9 41 41 41 41 58 80 94 94 282 282 282 311 311 311 11 58 58 58 58 80 94 94 282 282 282 311 311 311 13 87 87 94 94 94 94 282 282 282 311 311 311 15 111 111 111 111 282 282 282 282 311 311 311 17 149 217 217 282 282 282 282 311 311 311 19 217 217 282 282 282 282 311 311 311 21 282 282 282 282 282 311 311 311 23 282 282 282 282 311 311 311 25 388 388 388 388 388 388 27 388 388 388 388 388 This table cannot be extended without going beyond 650-tET. Notice that 58 is encountered in any progression from lower to higher limits. 282 is also, but that's more a curiosity than a musically important result. 7, 19, 46, 53, and 72 are conspicuous by their absence: there are simpler ETs that can "do" what they "do", just not always as accurately; namely, 3, 12, 41, 41, and 58 respectively. (Of course there can be other reasons besides accuracy to use 7, 19, 46, 53, or 72.) Source: ftp://ella.mills.edu/ccm/tuning/papers/consist_limits.txt (Manuel, can you update my e-mail address on that?)