# consistency / consistent

[Paul Erlich, Tuning, Tonality, and Twenty-Two Tone Temperament, Xenharmonikon 17, footnote 8]

An equal temperament with an integer number of notes per octave is consistent with JI through some odd limit if a complete chord of that limit is constructed in that equal temperament in the same way no matter which intervals are approximated.

If for all odd integers a, b, c such that 1 <= a < b < c <= n, the ET's best approximation of b / a plus the ET's best approximation of c / b equals the ET's best approximation of c / a, then the ET is consistent in the n-limit.

For example, the smallest ETs consistent in the 11, 13, 15, and 17-limits are 22, 26, 29, and 58-tET, respectively.

Consistency may be defined for non-octave equal temperaments but then even as well as odd numbers must be considered, and the consistency will be through an "integer limit" rather than an odd limit.

The data for case are tabulated at http://www.xs4all.nl/~huygensf/doc/consist_limits.html and http://www.xs4all.nl/~huygensf/doc/cons_limit_bounds.html. The ordinary, odd-limit consistency of an integer ET can be read from this table as well: it is the largest odd number not exceeding the integer limit.

```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 73 are plotted on the following chart:

See Patrick Ozzard-Low, 21st Century Orchestral Instruments for a more in-depth exploration of consistency.

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