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 | |||||
| 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 |