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Encyclopedia of Microtonal Music Theory

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unique

[Paul Erlich]

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.)
		
[Joe Monzo]

The data for ETs thru 72 are plotted on the following chart:

unique
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