Odd prime sum limit

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The n-odd-prime-sum-limit (abbreviated n-OPSL) is the collection of all just ratios where the no-twos sum of prime factors with repetition of both the numerator and the denominator does not exceed the integer n.

This concept was noted by Tristan Bay as a way to measure how accurately an edo approximates just intonation with lower primes weighted more heavily. Specifically, the idea is to use OPSLs as an alternative metric for consistency limit either instead of or alongside odd limits.

Comparison with odd limit

The 1- and 2-odd-prime-sum-limit are equivalent to the 1-odd-limit, which only contains a single interval pair {1/1, 2/1}. The 3- and 4-odd-prime-sum-limit are equivalent to the 3-odd-limit, which adds {3/2, 4/3}. All edos are consistent in those limits.

The 5-odd-prime-sum-limit is also equivalent to the 5-odd-limit, adding {5/4, 8/5} and {5/3, 6/5} to the 4-OPSL, and the 6-odd-prime-sum-limit adds {9/8, 16/9} and {9/5, 10/9}. The 7-odd-prime-sum-limit is equivalent to the 9-odd-limit, so it is the first OPSL that differs from the corresponding odd limit. It adds {7/4, 8/7}, {7/6, 12/7}, {7/5, 10/7}, and {9/7, 14/9} to the 6-OPSL. The 8-odd-prime-sum-limit adds {15/8, 16/15} and {15/14, 28/15}, the 9-odd-prime-sum-limit adds {27/16, 32/27}, {27/14, 28/27}, and {27/20, 40/27}, and the 10-odd-prime-sum-limit adds {21/16, 32/21}, {21/20, 40/21}, {25/16, 32/25}, {25/24, 48/25}, {25/14, 28/25}, {25/18, 36/25}, {25/21, 42/25}, and {27/25, 50/27}.

Minimal OPSL-consistent edos

OPSL	odd numbers newly added by each limit	Smallest Consistent Edo*
1	(none)	1
2	(none)	1
3	3	1
4	(none)	1
5	5	3
6	9	3
7	7	5
8	15	12
9	27	12
10	21 and 25	12
11	11 and 45	31
12	35 and 81	72
13	13, 63 and 75	72
14	33, 49 and 135	130
15	105, 125 and 243	270
16	39, 55, 189 and 225	270
17		954
18		1236
19		1578
20		1578
21		3395
22		3395
23		6079
24		8539
25		8539
26		8539
27		8539
28		102557
29		102557
30		102557
31		102557
32		102557
33		258008
34		258008
35		258008
36		258008

*apart from 0edo

Whole-interval OPSL

The n-whole-interval-OPSL, or n-WOPSL, is slightly different from the n-OPSL. This is the collection of all just ratios with a no-twos Wilson height that does not exceed the integer n. When using it to measure consistency in the same way as odd limits, lower primes are favored even more strongly than for OPSLs. It was confused with the original definition for n-OPSL (where the numerator and denominator are compared with n separately) at the time of this Wiki article's creation, but has since been corrected.

Comparison between odd-limit and WOPSL

Just like with OPSLs, the 1- and 2-WOPSL are equivalent to the 1-odd-limit, which only contains a single interval pair {1/1, 2/1}, and the 3- and 4-WOPSL are equivalent to the 3-odd-limit, which adds {3/2, 4/3}.

The 5-WOPSL adds {5/4, 8/5} without {5/3, 6/5} from the 5-odd-limit, so it is the first WOPSL that differs from the corresponding odd limit. The 6-WOPSL adds {9/8, 16/9}. The 7-WOPSL adds {7/4, 8/7} without {7/6, 12/7}, and the 8-WOPSL adds {5/3, 6/5} as well as {15/8, 16/15}. The 9-WOPSL adds {27/16, 32/27}, and the 10-WOPSL adds {7/6, 12/7}, {21/16, 32/21}, and {25/16, 32/25}. The 11-WOPSL adds {11/8, 16/11}, {9/5, 10/9}, and {45/32, 64/45}. The 12-WOPSL adds {7/5, 10/7}, {35/32, 64/35} and {81/64, 128/81}.