Negri extensions: Difference between revisions

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[[Negri]] temperament has various extensions to the [[11-limit]].
{{Breadcrumb|Negri}}
 
[[Negri]] has various competing [[extension]]s to the [[11-limit]]. This is evidenced by the fact that its [[support]]ing equal temperaments, [[10edo|10]] and [[19edo|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 [[20edo|20]], [[29edo|29]], and [[38edo|38df]].
 
== Interval chain ==
In the following table, odd harmonics and subharmonics 1–13 are in '''bold'''.
 
{| class="wikitable center-1 right-2"
! rowspan="3" | #
! rowspan="3" | Cents*
! colspan="5" | Approximate ratios
|-
! rowspan="2" | 2.3.5.7.13 subgroup
! colspan="4" | 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
|}
<nowiki/>* In 2.3.5.7.13-subgroup [[CWE tuning]]


== Tuning spectra ==
== Tuning spectra ==
Line 7: Line 197:
|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval]])
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 16/15
| 15/8
| 111.731
| 111.731
|  
|  
|-
|-
|  
|  
| 8/7
| 7/4
| 115.587
| 115.587
|  
|  
Line 32: Line 222:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|  
|  
Line 62: Line 252:
|-
|-
|  
|  
| 4/3
| 3/2
| 124.511
| 124.511
| 7- and 9-odd-limit minimax
| 7- and 9-odd-limit minimax
Line 72: Line 262:
|-
|-
|  
|  
| 30517578125/26795786661
| {{monzo| 0 -13 15 -5 }}
| 125.469
| 125.469
| 7-odd-limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 63 -26 -1}}
| {{monzo| 0 63 -26 -1 }}
| 125.579
| 125.579
| 9-odd-limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 10/9
| 9/5
| 125.673
| 125.673
|  
|  
|-
|-
|  
|  
| 11337408/9765625
| {{monzo| 6 11 -10 }}
| 126.238
| 126.238
| 5-odd-limit least squares
| 5-odd-limit least squares
Line 97: Line 287:
|-
|-
|  
|  
| 6/5
| 5/3
| 126.337
| 126.337
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 79 -40 -5 16 2}}
| {{monzo| 0 79 -40 -5 16 2 }}
| 126.445
| 126.445
| 13-odd-limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 34 -17 -2 7}}
| {{monzo| 0 34 -17 -2 7 }}
| 126.511
| 126.511
| 11-odd-limit least squares
| 11-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 73 -46 -4 20 4}}
| {{monzo| 0 73 -46 -4 20 4 }}
| 126.619
| 126.619
| 15-odd-limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| 18/13
| 13/9
| 127.324
| 127.324
|  
|  
Line 132: Line 322:
|-
|-
|  
|  
| 14/13
| 13/7
| 128.298
| 128.298
|  
|  
Line 154: Line 344:
| 11/8
| 11/8
| 129.736
| 129.736
|
|-
| 1\9
|
| 133.333
|  
|  
|-
|-
Line 172: Line 367:
|-
|-
|  
|  
| 14/11
| 11/7
| 139.169
| 139.169
|  
|  
Line 182: Line 377:
|-
|-
|  
|  
| 12/11
| 11/6
| 150.637
| 150.637
|  
|  
Line 192: Line 387:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 16/15
| 15/8
| 111.731
| 111.731
|  
|  
|-
|-
|  
|  
| 8/7
| 7/4
| 115.587
| 115.587
|  
|  
Line 212: Line 407:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|  
|  
Line 232: Line 427:
|-
|-
|  
|  
| 14/11
| 11/7
| 123.906
| 123.906
|  
|  
Line 262: Line 457:
|-
|-
|  
|  
| 4/3
| 3/2
| 124.511
| 124.511
| 7, 9, 11, 13 and 15 limit minimax
| 7-, 9-, 11-, 13- and 15-odd-limit minimax
|-
|-
|  
|  
| {{vector|0 -149 8 -15 113 -23}}
| {{monzo| 0 -149 8 -15 113 -23 }}
| 124.700
| 124.700
| 15-odd-limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 -136 21 -14 98 -21}}
| {{monzo| 0 -136 21 -14 98 -21 }}
| 124.764
| 124.764
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 -125 15 -15 81}}
| {{monzo| 0 -125 15 -15 81 }}
| 124.854
| 124.854
| 11-odd-limit least squares
| 11-odd-limit least squares
Line 287: Line 482:
|-
|-
|  
|  
| 12/11
| 11/6
| 124.965
| 124.965
|  
|  
Line 302: Line 497:
|-
|-
|  
|  
| 30517578125/26795786661
| {{monzo| 0 -13 15 -5 }}
| 125.469
| 125.469
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 63 -26 -1}}
| {{monzo| 0 63 -26 -1 }}
| 125.579
| 125.579
| 9-odd-limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 10/9
| 9/5
| 125.673
| 125.673
|  
|  
|-
|-
|  
|  
| 11337408/9765625
| {{monzo| 6 11 -10 }}
| 126.238
| 126.238
| 5-odd-limit least squares
| 5-odd-limit least squares
Line 327: Line 522:
|-
|-
|  
|  
| 6/5
| 5/3
| 126.337
| 126.337
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
|  
|  
| 18/13
| 13/9
| 127.324
| 127.324
|  
|  
Line 347: Line 542:
|-
|-
|  
|  
| 14/13
| 13/7
| 128.298
| 128.298
|  
|  
Line 359: Line 554:
| 5/4
| 5/4
| 128.771
| 128.771
|
|-
| 1\9
|
| 133.333
|  
|  
|-
|-
Line 377: Line 577:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 16/15
| 15/8
| 111.731
| 111.731
|  
|  
|-
|-
|  
|  
| 8/7
| 7/4
| 115.587
| 115.587
|  
|  
Line 397: Line 597:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|  
|  
Line 427: Line 627:
|-
|-
|  
|  
| 4/3
| 3/2
| 124.511
| 124.511
| 7- and 9-odd-limit minimax
| 7- and 9-odd-limit minimax
Line 437: Line 637:
|-
|-
|  
|  
| 30517578125/26795786661
| {{monzo| 0 -13 15 -5 }}
| 125.469
| 125.469
| 7-odd-limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 63 -26 -1}}
| {{monzo| 0 63 -26 -1 }}
| 125.579
| 125.579
| 9-odd-limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 10/9
| 9/5
| 125.673
| 125.673
|  
|  
|-
|-
|  
|  
| 11337408/9765625
| {{monzo| 6 11 -10 }}
| 126.238
| 126.238
| 5-odd-limit least squares
| 5-odd-limit least squares
Line 462: Line 662:
|-
|-
|  
|  
| 6/5
| 5/3
| 126.337
| 126.337
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
|  
|  
| 18/13
| 13/9
| 127.324
| 127.324
|  
|  
Line 477: Line 677:
|-
|-
|  
|  
| {{vector|0 106 -31 4 -38 11}}
| {{monzo| 0 106 -31 4 -38 11 }}
| 127.602
| 127.602
| 13-odd-limit least squares
| 13-odd-limit least squares
Line 487: Line 687:
|-
|-
|  
|  
| {{vector|0 -95 25 -5 31}}
| {{monzo| 0 -95 25 -5 31 }}
| 127.706
| 127.706
| 11 limit least squares
| 11 limit least squares
|-
|-
|  
|  
| {{vector|0 109 -28 5 -43 13}}
| {{monzo| 0 109 -28 5 -43 13 }}
| 127.718
| 127.718
| 15-odd-limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| 14/13
| 13/7
| 128.298
| 128.298
|  
|  
Line 522: Line 722:
|-
|-
|  
|  
| 14/11
| 11/7
| 130.415
| 130.415
|  
|  
|-
|-
|  
|  
| 12/11
| 11/6
| 131.170
| 131.170
|  
|  
Line 534: Line 734:
| 15/11
| 15/11
| 132.610
| 132.610
|
|-
| 1\9
|
| 133.333
|  
|  
|-
|-
Line 562: Line 767:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 16/15
| 15/8
| 111.731
| 111.731
|  
|  
|-
|-
|  
|  
| 8/7
| 7/4
| 115.587
| 115.587
|  
|  
Line 582: Line 787:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|  
|  
Line 597: Line 802:
|-
|-
|  
|  
| 12/11
| 11/6
| 122.785
| 122.785
|  
|  
Line 637: Line 842:
|-
|-
|  
|  
| {{vector|0 19 -22 -7 32}}
| {{monzo| 0 19 -22 -7 32 }}
| 124.180
| 124.180
| 11-odd-limit least squares
| 11-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 33 -66 -14 90 -6}}
| {{monzo| 0 33 -66 -14 90 -6 }}
| 124.186
| 124.186
| 15-odd-limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 49 -50 -15 76 -8}}
| {{monzo| 0 49 -50 -15 76 -8 }}
| 124.215
| 124.215
| 13-odd-limit least squares
| 13-odd-limit least squares
Line 657: Line 862:
|-
|-
|  
|  
| 14/11
| 11/7
| 124.424
| 124.424
|  
|  
|-
|-
|  
|  
| 4/3
| 3/2
| 124.511
| 124.511
| 7- and 9-odd-limit minimax
| 7- and 9-odd-limit minimax
Line 672: Line 877:
|-
|-
|  
|  
| 30517578125/26795786661
| {{monzo| 0 -13 15 -5 }}
| 125.469
| 125.469
| 7-odd-limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| {{vector|0 63 -26 -1}}
| {{monzo| 0 63 -26 -1 }}
| 125.579
| 125.579
| 9-odd-limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 10/9
| 9/5
| 125.673
| 125.673
|  
|  
|-
|-
|  
|  
| 11337408/9765625
| {{monzo| 6 11 -10 }}
| 126.238
| 126.238
| 5-odd-limit least squares
| 5-odd-limit least squares
|-
|-
|  
|  
| 6/5
| 5/3
| 126.337
| 126.337
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
|  
|  
| 18/13
| 13/9
| 127.324
| 127.324
|  
|  
Line 712: Line 917:
|-
|-
|  
|  
| 14/13
| 13/7
| 128.298
| 128.298
|  
|  
Line 724: Line 929:
| 5/4
| 5/4
| 128.771
| 128.771
|
|-
| 1\9
|
| 133.333
|  
|  
|-
|-
Line 739: Line 949:
[[Category:Negri]]
[[Category:Negri]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]

Latest revision as of 11:57, 6 August 2025

< Negri

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 		Unchanged interval
(eigenmonzo) 		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 		Unchanged interval
(eigenmonzo) 		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 		Unchanged interval
(eigenmonzo) 		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 		Unchanged interval
(eigenmonzo) 		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
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