An equal temperament uniquely articulates just intonation in some odd limit if all just intervals of that odd limit are approximated by different numbers of steps in the ET.
For example, 12-tET does not uniquely articulate the 7-limit because 7:5 and 10:7 are both approximated by 6 steps and 6:5 and 7:6 are both represented by 3 steps. The data for this case are tabulated in the last column of this table.
Although this definition is perfectly applicable to non-octave equal temperaments, uniqueness may also be defined with respect to an "integer limit". The data for this case are tabulated in the second-to-last column of this table.
Date: Fri, 17 Jul 1998. Subject: Equal Temperaments 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.)
The data for ETs thru 72 are plotted on the following chart:
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