Negri extensions

Negri has various competing extensions to the 11-limit. This is evidenced by the fact that its supporting equal temperaments, 10 and 19, do less well in the 11-limit. The extensions are:

Undecimal negri (10 & 19) – tempering out 45/44, 49/48, and 56/55
Negril (10e & 19) – tempering out 49/48, 100/99, and 225/224
Negric (10e & 19e) – tempering out 33/32, 49/48, and 77/75
Negroni (10 & 19e) – tempering out 49/48, 55/54, and 225/224

The most important of these is undecimal negri, in which the generator, representing 13/12, 14/13, 15/14, and 16/15, goes one step further to stand in for ~12/11. The poor accuracy comes with a not overly complex mapping, as you can find ~16/11 just five generator steps away. Its evil cousin, negric, finds ~11/8 where negri will find ~15/11, which is of course conflated with 4/3.

The other pair of extensions are of higher complexity, but are well rewarded with better intonation. Negril finds ~11/8 where negroni will find ~15/11 and vice versa. They unite in 29edo, which can be recommended as a tuning for both.

Another possible path which relates a sense of compromise is to temper out 121/120, leading to wilsec. This has the effect of slicing the generator in two, and is supported by 20, 29, and 38df.

Interval chain

In the following table, odd harmonics and subharmonics 1–13 are in bold.

#	Cents*	Approximate ratios
		2.3.5.7.13 subgroup	Full 13-limit extensions
		2.3.5.7.13 subgroup	Negri	Negril	Negric	Negroni
0	0.0	1/1
1	125.4	13/12, 14/13, 15/14, 16/15	12/11		11/10
2	250.7	7/6, 8/7, 15/13	13/11
3	376.1	5/4, 16/13	11/9, 14/11
4	501.4	4/3	15/11		11/8
5	626.8	10/7, 13/9	16/11		22/15
6	752.1	14/9, 20/13, 32/21			11/7
7	877.5	5/3			22/13	18/11
8	1002.8	16/9	20/11		11/6
9	1128.2	35/18, 40/21, 52/27	64/33		88/45
10	53.5	25/24, 28/27, 50/49, 64/63		33/32		45/44
11	178.9	10/9		11/10		12/11
12	304.3	25/21			11/9	13/11
13	429.6	35/27				14/11
14	555.0	25/18		11/8		15/11
15	680.3	40/27		22/15		16/11
16	805.7	100/63		11/7
17	931.0	140/81		22/13
18	1056.4	50/27		11/6
19	1181.7	125/63, 160/81		88/45		64/33

* In 2.3.5.7.13-subgroup CWE tuning

Tuning spectra

Undecimal negri

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	15/8	111.731
	7/4	115.587
	11/9	115.803
	15/14	119.443
	13/8	119.824
1\10		120.000
	7/5	123.498
	15/13	123.871
3\29		124.138
	13/10	124.298
	3/2	124.511	7- and 9-odd-limit minimax
5\48		125.000
	[0 -13 15 -5⟩	125.469	7-odd-limit least squares
	[0 63 -26 -1⟩	125.579	9-odd-limit least squares
	9/5	125.673
	[6 11 -10⟩	126.238	5-odd-limit least squares
2\19		126.316
	5/3	126.337	5-odd-limit minimax
	[0 79 -40 -5 16 2⟩	126.445	13-odd-limit least squares
	[0 34 -17 -2 7⟩	126.511	11-odd-limit least squares
	[0 73 -46 -4 20 4⟩	126.619	15-odd-limit least squares
	13/9	127.324
	9/7	127.486	11-, 13- and 15-odd-limit minimax
5\47		127.660
	13/7	128.298
3\28		128.571
	5/4	128.771
	11/10	129.374
	11/8	129.736
1\9		133.333
	7/6	133.435
	15/11	134.238
	13/12	138.573
	11/7	139.169
	13/11	144.605
	11/6	150.637

Negril

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	15/8	111.731
	7/4	115.587
	15/14	119.443
	13/8	119.824
1\10		120.000
	7/5	123.498
	15/13	123.871
	11/7	123.906
	11/10	124.091
3\29		124.138
	13/11	124.164
	15/11	124.203
	13/10	124.298
	3/2	124.511	7-, 9-, 11-, 13- and 15-odd-limit minimax
	[0 -149 8 -15 113 -23⟩	124.700	15-odd-limit least squares
	[0 -136 21 -14 98 -21⟩	124.764	13-odd-limit least squares
	[0 -125 15 -15 81⟩	124.854	11-odd-limit least squares
	11/9	124.882
	11/6	124.965
5\48		125.000
	11/8	125.094
	[0 -13 15 -5⟩	125.469	7-odd-limit least squares
	[0 63 -26 -1⟩	125.579	9-odd-limit least squares
	9/5	125.673
	[6 11 -10⟩	126.238	5-odd-limit least squares
2\19		126.316
	5/3	126.337	5-odd-limit minimax
	13/9	127.324
	9/7	127.486
5\47		127.660
	13/7	128.298
3\28		128.571
	5/4	128.771
1\9		133.333
	7/6	133.435
	13/12	138.573

Negric

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	15/8	111.731
	7/4	115.587
	15/14	119.443
	13/8	119.824
1\10		120.000
	7/5	123.498
	15/13	123.871
3\29		124.138
	13/10	124.298
	3/2	124.511	7- and 9-odd-limit minimax
5\48		125.000
	[0 -13 15 -5⟩	125.469	7-odd-limit least squares
	[0 63 -26 -1⟩	125.579	9-odd-limit least squares
	9/5	125.673
	[6 11 -10⟩	126.238	5-odd-limit least squares
2\19		126.316
	5/3	126.337	5-odd-limit minimax
	13/9	127.324
	9/7	127.486
	[0 106 -31 4 -38 11⟩	127.602	13-odd-limit least squares
5\47		127.660
	[0 -95 25 -5 31⟩	127.706	11 limit least squares
	[0 109 -28 5 -43 13⟩	127.718	15-odd-limit least squares
	13/7	128.298
3\28		128.571
	5/4	128.771
	11/9	128.951	11-, 13- and 15-odd-limit minimax
	13/11	130.113
	11/7	130.415
	11/6	131.170
	15/11	132.610
1\9		133.333
	7/6	133.435
	11/8	137.829
	13/12	138.573
	11/10	165.004

Negroni

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	15/8	111.731
	7/4	115.587
	15/14	119.443
	13/8	119.824
1\10		120.000
	11/9	121.799	11-, 13- and 15-odd-limit minimax
	11/6	122.785
	11/8	123.245
	7/5	123.498
	15/13	123.871
	15/11	124.068
	13/11	124.101
3\29		124.138
	11/10	124.166
	[0 19 -22 -7 32⟩	124.180	11-odd-limit least squares
	[0 33 -66 -14 90 -6⟩	124.186	15-odd-limit least squares
	[0 49 -50 -15 76 -8⟩	124.215	13-odd-limit least squares
	13/10	124.298
	11/7	124.424
	3/2	124.511	7- and 9-odd-limit minimax
5\48		125.000
	[0 -13 15 -5⟩	125.469	7-odd-limit least squares
	[0 63 -26 -1⟩	125.579	9-odd-limit least squares
	9/5	125.673
	[6 11 -10⟩	126.238	5-odd-limit least squares
	5/3	126.337	5-odd-limit minimax
	13/9	127.324
	9/7	127.486
5\47		127.660
	13/7	128.298
3\28		128.571
	5/4	128.771
1\9		133.333
	7/6	133.435
	13/12	138.573