Negri extensions: Difference between revisions

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[[Negri]] temperament 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:  
{{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
* '''Undecimal negri''' (10 & 19) – tempering out 45/44, 49/48, and 56/55
* '''Negril''' (10e & 19) – tempering out 49/48, 100/99, 225/224
* '''Negril''' (10e & 19) – tempering out 49/48, 100/99, and 225/224
* '''Negric''' (10e & 19e) – tempering out 33/32, 49/48, 77/75
* '''Negric''' (10e & 19e) – tempering out 33/32, 49/48, and 77/75
* '''Negroni''' (10 & 19e) – tempering out 49/48, 55/54, 225/224
* '''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]].  
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 13: 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 38: Line 222:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|  
|  
Line 68: Line 252:
|-
|-
|  
|  
| 4/3
| 3/2
| 124.511
| 124.511
| 7- and 9-odd-limit minimax
| 7- and 9-odd-limit minimax
Line 78: 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 103: 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 138: Line 322:
|-
|-
|  
|  
| 14/13
| 13/7
| 128.298
| 128.298
|  
|  
Line 160: Line 344:
| 11/8
| 11/8
| 129.736
| 129.736
|
|-
| 1\9
|
| 133.333
|  
|  
|-
|-
Line 178: Line 367:
|-
|-
|  
|  
| 14/11
| 11/7
| 139.169
| 139.169
|  
|  
Line 188: Line 377:
|-
|-
|  
|  
| 12/11
| 11/6
| 150.637
| 150.637
|  
|  
Line 198: 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 218: Line 407:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|  
|  
Line 238: Line 427:
|-
|-
|  
|  
| 14/11
| 11/7
| 123.906
| 123.906
|  
|  
Line 268: Line 457:
|-
|-
|  
|  
| 4/3
| 3/2
| 124.511
| 124.511
| 7-, 9-, 11-, 13- and 15-odd-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-odd-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 293: Line 482:
|-
|-
|  
|  
| 12/11
| 11/6
| 124.965
| 124.965
|  
|  
Line 308: Line 497:
|-
|-
|  
|  
| 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 333: 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 353: Line 542:
|-
|-
|  
|  
| 14/13
| 13/7
| 128.298
| 128.298
|  
|  
Line 365: Line 554:
| 5/4
| 5/4
| 128.771
| 128.771
|
|-
| 1\9
|
| 133.333
|  
|  
|-
|-
Line 383: 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 403: Line 597:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|  
|  
Line 433: Line 627:
|-
|-
|  
|  
| 4/3
| 3/2
| 124.511
| 124.511
| 7- and 9-odd-limit minimax
| 7- and 9-odd-limit minimax
Line 443: 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 468: 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 483: 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 493: 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 528: Line 722:
|-
|-
|  
|  
| 14/11
| 11/7
| 130.415
| 130.415
|  
|  
|-
|-
|  
|  
| 12/11
| 11/6
| 131.170
| 131.170
|  
|  
Line 540: Line 734:
| 15/11
| 15/11
| 132.610
| 132.610
|
|-
| 1\9
|
| 133.333
|  
|  
|-
|-
Line 568: 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 588: Line 787:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|  
|  
Line 603: Line 802:
|-
|-
|  
|  
| 12/11
| 11/6
| 122.785
| 122.785
|  
|  
Line 643: 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 663: 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 678: 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 718: Line 917:
|-
|-
|  
|  
| 14/13
| 13/7
| 128.298
| 128.298
|  
|  
Line 730: Line 929:
| 5/4
| 5/4
| 128.771
| 128.771
|
|-
| 1\9
|
| 133.333
|  
|  
|-
|-
Line 745: 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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