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:  
[[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


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
| 16/15, 15/14, 14/13, 13/12
| 12/11
|
| 11/10
|
|-
| 2
| 250.7
| '''8/7''', 7/6, 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, 32/21, 20/13
|
|
| 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 ==

Revision as of 18:38, 5 December 2024

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

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 16/15, 15/14, 14/13, 13/12 12/11 11/10
2 250.7 8/7, 7/6, 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, 32/21, 20/13 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
16/15 111.731
8/7 115.587
11/9 115.803
15/14 119.443
16/13 119.824
1\10 120.000
7/5 123.498
15/13 123.871
3\29 124.138
13/10 124.298
4/3 124.511 7- and 9-odd-limit minimax
5\48 125.000
30517578125/26795786661 125.469 7-odd-limit least squares
[0 63 -26 -1⟩ 125.579 9-odd-limit least squares
10/9 125.673
11337408/9765625 126.238 5-odd-limit least squares
2\19 126.316
6/5 126.337 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
18/13 127.324
9/7 127.486 11-, 13- and 15-odd-limit minimax
5\47 127.660
14/13 128.298
3\28 128.571
5/4 128.771
11/10 129.374
11/8 129.736
7/6 133.435
15/11 134.238
13/12 138.573
14/11 139.169
13/11 144.605
12/11 150.637

Negril

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
16/15 111.731
8/7 115.587
15/14 119.443
16/13 119.824
1\10 120.000
7/5 123.498
15/13 123.871
14/11 123.906
11/10 124.091
3\29 124.138
13/11 124.164
15/11 124.203
13/10 124.298
4/3 124.511 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
12/11 124.965
5\48 125.000
11/8 125.094
30517578125/26795786661 125.469 7-odd-limit least squares
[0 63 -26 -1⟩ 125.579 9-odd-limit least squares
10/9 125.673
11337408/9765625 126.238 5-odd-limit least squares
2\19 126.316
6/5 126.337 5-odd-limit minimax
18/13 127.324
9/7 127.486
5\47 127.660
14/13 128.298
3\28 128.571
5/4 128.771
7/6 133.435
13/12 138.573

Negric

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
16/15 111.731
8/7 115.587
15/14 119.443
16/13 119.824
1\10 120.000
7/5 123.498
15/13 123.871
3\29 124.138
13/10 124.298
4/3 124.511 7- and 9-odd-limit minimax
5\48 125.000
30517578125/26795786661 125.469 7-odd-limit least squares
[0 63 -26 -1⟩ 125.579 9-odd-limit least squares
10/9 125.673
11337408/9765625 126.238 5-odd-limit least squares
2\19 126.316
6/5 126.337 5-odd-limit minimax
18/13 127.324
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
14/13 128.298
3\28 128.571
5/4 128.771
11/9 128.951 11-, 13- and 15-odd-limit minimax
13/11 130.113
14/11 130.415
12/11 131.170
15/11 132.610
7/6 133.435
11/8 137.829
13/12 138.573
11/10 165.004

Negroni

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
16/15 111.731
8/7 115.587
15/14 119.443
16/13 119.824
1\10 120.000
11/9 121.799 11-, 13- and 15-odd-limit minimax
12/11 122.785
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
14/11 124.424
4/3 124.511 7- and 9-odd-limit minimax
5\48 125.000
30517578125/26795786661 125.469 7-odd-limit least squares
[0 63 -26 -1⟩ 125.579 9-odd-limit least squares
10/9 125.673
11337408/9765625 126.238 5-odd-limit least squares
6/5 126.337 5-odd-limit minimax
18/13 127.324
9/7 127.486
5\47 127.660
14/13 128.298
3\28 128.571
5/4 128.771
7/6 133.435
13/12 138.573
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