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[[Negri|Negri]] temperament has various extensions to the [[11-limit|11 limit]].
{{Breadcrumb|Negri}}


=Tuning Spectra=
[[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


==Spectrum of Negri Tunings by Eigenmonzos==
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]].


[[Gencom|Gencom]]: [2 15/14; 45/44 49/48 56/55 78/77]
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.


[[Gencom_map|Gencom map]]: [<1 2 2 3 4 4|, <0 -4 3 -2 -5 -3|]
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]].


{| class="wikitable"
== 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
|
|-
|-
! | Eigenmonzo
| 2
! | Supersecor
| 250.7
| 7/6, '''8/7''', 15/13
| 13/11
|
|
|  
|-
|-
| | 16/15
| 3
| | 111.731
| 376.1
| '''5/4''', '''16/13'''
| 11/9, 14/11
|
|
|  
|-
|-
| | 8/7
| 4
| | 115.587
| 501.4
| '''4/3'''
| 15/11
|  
| '''11/8'''
|  
|-
|-
| | 11/9
| 5
| | 115.803
| 626.8
| 10/7, 13/9
| '''16/11'''
|
| 22/15
|  
|-
|-
| | 15/14
| 6
| | 119.443
| 752.1
| 14/9, 20/13, 32/21
|
|  
| 11/7
|  
|-
|-
| | 16/13
| 7
| | 119.824
| 877.5
| 5/3
|
|
| 22/13
| 18/11
|-
|-
| | 1\10
| 8
| | 120.000
| 1002.8
| '''16/9'''
| 20/11
|  
| 11/6
|  
|-
|-
| | 7/5
| 9
| | 123.498
| 1128.2
| 35/18, 40/21, 52/27
| 64/33
|  
| 88/45
|  
|-
|-
| | 15/13
| 10
| | 123.871
| 53.5
| 25/24, 28/27, 50/49, 64/63
|  
| 33/32
|
| 45/44
|-
|-
| | 3\29
| 11
| | 124.138
| 178.9
| 10/9
|  
| 11/10
|  
| 12/11
|-
|-
| | 13/10
| 12
| | 124.298
| 304.3
| 25/21
|
|
| 11/9
| 13/11
|-
|-
| | 4/3
| 13
| | 124.511 (7 and 9 limit minimax)
| 429.6
| 35/27
|  
|  
|
| 14/11
|-
|-
| | 5\48
| 14
| | 125.000
| 555.0
| 25/18
|  
| '''11/8'''
|  
| 15/11
|-
|-
| | 30517578125/26795786661
| 15
| | 125.469 (7 limit least squares)
| 680.3
| 40/27
|
| 22/15
|  
| '''16/11'''
|-
|-
| | |0 63 -26 -1>
| 16
| | 125.579 (9 limit least squares)
| 805.7
| 100/63
|  
| 11/7
|
|  
|-
|-
| | 10/9
| 17
| | 125.673
| 931.0
| 140/81
|  
| 22/13
|
|  
|-
|-
| | 11337408/9765625
| 18
| | 126.238 (5 limit least squares)
| 1056.4
| 50/27
|  
| 11/6
|
|  
|-
|-
| | 2\19
| 19
| | 126.316
| 1181.7
| 125/63, 160/81
|
| 88/45
|  
| 64/33
|}
<nowiki/>* In 2.3.5.7.13-subgroup [[CWE tuning]]
 
== Tuning spectra ==
=== Undecimal negri ===
 
{| class="wikitable center-all left-4"
|-
|-
| | 6/5
! Edo<br>generator
| | 126.337 (5 limit minimax)
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
|-
| | |0 79 -40 -5 16 2&gt;
|  
| | 126.445 (13 limit least squares)
| 15/8
| 111.731
|  
|-
|-
| | |0 34 -17 -2 7&gt;
|  
| | 126.511 (11 limit least squares)
| 7/4
| 115.587
|  
|-
|-
| | |0 73 -46 -4 20 4&gt;
|  
| | 126.619 (15 limit least squares)
| 11/9
| 115.803
|  
|-
|-
| | 18/13
|  
| | 127.324
| 15/14
| 119.443
|  
|-
|-
| | 9/7
|  
| | 127.486 (11, 13 and 15 limit minimax)
| 13/8
| 119.824
|  
|-
|-
| | 5\47
| 1\10
| | 127.660
|  
| 120.000
|
|-
|-
| | 14/13
|  
| | 128.298
| 7/5
| 123.498
|  
|-
|-
| | 3\28
|  
| | 128.571
| 15/13
| 123.871
|  
|-
|-
| | 5/4
| 3\29
| | 128.771
|  
| 124.138
|  
|-
|-
| | 11/10
|  
| | 129.374
| 13/10
| 124.298
|  
|-
|-
| | 11/8
|  
| | 129.736
| 3/2
| 124.511
| 7- and 9-odd-limit minimax
|-
|-
| | 7/6
| 5\48
| | 133.435
|  
| 125.000
|  
|-
|-
| | 15/11
|  
| | 134.238
| {{monzo| 0 -13 15 -5 }}
| 125.469
| 7-odd-limit least squares
|-
|-
| | 13/12
|  
| | 138.573
| {{monzo| 0 63 -26 -1 }}
| 125.579
| 9-odd-limit least squares
|-
|-
| | 14/11
|  
| | 139.169
| 9/5
| 125.673
|  
|-
|-
| | 13/11
|  
| | 144.605
| {{monzo| 6 11 -10 }}
| 126.238
| 5-odd-limit least squares
|-
|-
| | 12/11
| 2\19
| | 150.637
|
| 126.316
|
|-
|
| 5/3
| 126.337
| 5-odd-limit minimax
|-
|  
| {{monzo| 0 79 -40 -5 16 2 }}
| 126.445
| 13-odd-limit least squares
|-
|
| {{monzo| 0 34 -17 -2 7 }}
| 126.511
| 11-odd-limit least squares
|-
|
| {{monzo| 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
|
|}
|}


==Spectrum of Negril Tunings by Eigenmonzos==
=== Negril ===


[[Gencom|Gencom]]: [2 15/14; 49/48 65/64 91/90 875/858]
{| class="wikitable center-all left-4"
 
|-
[[Gencom_map|Gencom map]]: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]
! Edo<br>generator
 
! Unchanged interval<br>(eigenmonzo)
{| class="wikitable"
! Generator (¢)
! Comments
|-
|-
! | Eigenmonzo
|  
! | Supersecor
| 15/8
| 111.731
|
|-
|-
| | 16/15
|  
| | 111.731
| 7/4
| 115.587
|  
|-
|-
| | 8/7
|  
| | 115.587
| 15/14
| 119.443
|  
|-
|-
| | 15/14
|  
| | 119.443
| 13/8
| 119.824
|
|-
|-
| | 16/13
| 1\10
| | 119.824
|  
| 120.000
|  
|-
|-
| | 1\10
|  
| | 120.000
| 7/5
| 123.498
|  
|-
|-
| | 7/5
|  
| | 123.498
| 15/13
| 123.871
|
|-
|-
| | 15/13
|  
| | 123.871
| 11/7
| 123.906
|
|-
|-
| | 14/11
|  
| | 123.906
| 11/10
| 124.091
|  
|-
|-
| | 11/10
| 3\29
| | 124.091
|  
| 124.138
|
|-
|-
| | 3\29
|  
| | 124.138
| 13/11
| 124.164
|
|-
|-
| | 13/11
|  
| | 124.164
| 15/11
| 124.203
|
|-
|-
| | 15/11
|  
| | 124.203
| 13/10
| 124.298
|
|-
|-
| | 13/10
|  
| | 124.298
| 3/2
| 124.511
| 7-, 9-, 11-, 13- and 15-odd-limit minimax
|-
|-
| | 4/3
|  
| | 124.511 (7, 9, 11, 13 and 15 limit minimax)
| {{monzo| 0 -149 8 -15 113 -23 }}
| 124.700
| 15-odd-limit least squares
|-
|-
| | |0 -149 8 -15 113 -23&gt;
|  
| | 124.700 (15 limit least squares)
| {{monzo| 0 -136 21 -14 98 -21 }}
| 124.764
| 13-odd-limit least squares
|-
|-
| | |0 -136 21 -14 98 -21&gt;
|  
| | 124.764 (13 limit least squares)
| {{monzo| 0 -125 15 -15 81 }}
| 124.854
| 11-odd-limit least squares
|-
|-
| | |0 -125 15 -15 81&gt;
|  
| | 124.854 (11 limit least squares)
| 11/9
| 124.882
|
|-
|-
| | 11/9
|  
| | 124.882
| 11/6
| 124.965
|
|-
|-
| | 12/11
| 5\48
| | 124.965
|  
| 125.000
|  
|-
|-
| | 5\48
|  
| | 125.000
| 11/8
| 125.094
|
|-
|-
| | 11/8
|  
| | 125.094
| {{monzo| 0 -13 15 -5 }}
| 125.469
| 7-odd-limit least squares
|-
|-
| | 30517578125/26795786661
|  
| | 125.469 (7 limit least squares)
| {{monzo| 0 63 -26 -1 }}
| 125.579
| 9-odd-limit least squares
|-
|-
| | |0 63 -26 -1&gt;
|  
| | 125.579 (9 limit least squares)
| 9/5
| 125.673
|
|-
|-
| | 10/9
|  
| | 125.673
| {{monzo| 6 11 -10 }}
| 126.238
| 5-odd-limit least squares
|-
|-
| | 11337408/9765625
| 2\19
| | 126.238 (5 limit least squares)
|  
| 126.316
|
|-
|-
| | 2\19
|  
| | 126.316
| 5/3
| 126.337
| 5-odd-limit minimax
|-
|-
| | 6/5
|  
| | 126.337 (5 limit minimax)
| 13/9
| 127.324
|  
|-
|-
| | 18/13
|  
| | 127.324
| 9/7
| 127.486
|
|-
|-
| | 9/7
| 5\47
| | 127.486
|  
| 127.660
|
|-
|-
| | 5\47
|  
| | 127.660
| 13/7
| 128.298
|  
|-
|-
| | 14/13
| 3\28
| | 128.298
|  
| 128.571
|
|-
|-
| | 3\28
|  
| | 128.571
| 5/4
| 128.771
|
|-
|-
| | 5/4
| 1\9
| | 128.771
|  
| 133.333
|  
|-
|-
| | 7/6
|  
| | 133.435
| 7/6
| 133.435
|
|-
|-
| | 13/12
|  
| | 138.573
| 13/12
| 138.573
|
|}
|}


==Spectrum of Negric Tunings by Eigenmonzos==
=== Negric ===


[[Gencom|Gencom]]: [2 15/14; 33/32 49/48 65/64 91/90]
{| class="wikitable center-all left-4"
 
|-
[[Gencom_map|Gencom map]]: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
! Edo<br>generator
 
! Unchanged interval<br>(eigenmonzo)
{| class="wikitable"
! Generator (¢)
! Comments
|-
|-
! | Eigenmonzo
|  
! | Supersecor
| 15/8
| 111.731
|
|-
|-
| | 16/15
|  
| | 111.731
| 7/4
| 115.587
|  
|-
|-
| | 8/7
|  
| | 115.587
| 15/14
| 119.443
|  
|-
|-
| | 15/14
|  
| | 119.443
| 13/8
| 119.824
|
|-
|-
| | 16/13
| 1\10
| | 119.824
|  
| 120.000
|  
|-
|-
| | 1\10
|  
| | 120.000
| 7/5
| 123.498
|  
|-
|-
| | 7/5
|  
| | 123.498
| 15/13
| 123.871
|
|-
|-
| | 15/13
| 3\29
| | 123.871
|  
| 124.138
|  
|-
|-
| | 3\29
|  
| | 124.138
| 13/10
| 124.298
|
|-
|-
| | 13/10
|  
| | 124.298
| 3/2
| 124.511
| 7- and 9-odd-limit minimax
|-
|-
| | 4/3
| 5\48
| | 124.511 (7 and 9 limit minimax)
|  
| 125.000
|  
|-
|-
| | 5\48
|  
| | 125.000
| {{monzo| 0 -13 15 -5 }}
| 125.469
| 7-odd-limit least squares
|-
|-
| | 30517578125/26795786661
|  
| | 125.469 (7 limit least squares)
| {{monzo| 0 63 -26 -1 }}
| 125.579
| 9-odd-limit least squares
|-
|-
| | |0 63 -26 -1&gt;
|  
| | 125.579 (9 limit least squares)
| 9/5
| 125.673
|
|-
|-
| | 10/9
|  
| | 125.673
| {{monzo| 6 11 -10 }}
| 126.238
| 5-odd-limit least squares
|-
|-
| | 11337408/9765625
| 2\19
| | 126.238 (5 limit least squares)
|  
| 126.316
|
|-
|-
| | 2\19
|  
| | 126.316
| 5/3
| 126.337
| 5-odd-limit minimax
|-
|-
| | 6/5
|  
| | 126.337 (5 limit minimax)
| 13/9
| 127.324
|  
|-
|-
| | 18/13
|  
| | 127.324
| 9/7
| 127.486
|
|-
|-
| | 9/7
|  
| | 127.486
| {{monzo| 0 106 -31 4 -38 11 }}
| 127.602
| 13-odd-limit least squares
|-
|-
| | |0 106 -31 4 -38 11&gt;
| 5\47
| | 127.602 (13 limit least squares)
|  
| 127.660
|
|-
|-
| | 5\47
|  
| | 127.660
| {{monzo| 0 -95 25 -5 31 }}
| 127.706
| 11 limit least squares
|-
|-
| | |0 -95 25 -5 31&gt;
|  
| | 127.706 (11 limit least squares)
| {{monzo| 0 109 -28 5 -43 13 }}
| 127.718
| 15-odd-limit least squares
|-
|-
| | |0 109 -28 5 -43 13&gt;
|  
| | 127.718 (15 limit least squares)
| 13/7
| 128.298
|  
|-
|-
| | 14/13
| 3\28
| | 128.298
|  
| 128.571
|
|-
|-
| | 3\28
|  
| | 128.571
| 5/4
| 128.771
|
|-
|-
| | 5/4
|  
| | 128.771
| 11/9
| 128.951
| 11-, 13- and 15-odd-limit minimax
|-
|-
| | 11/9
|  
| | 128.951 (11, 13 and 15 limit minimax)
| 13/11
| 130.113
|  
|-
|-
| | 13/11
|  
| | 130.113
| 11/7
| 130.415
|
|-
|-
| | 14/11
|  
| | 130.415
| 11/6
| 131.170
|  
|-
|-
| | 12/11
|  
| | 131.170
| 15/11
| 132.610
|  
|-
|-
| | 15/11
| 1\9
| | 132.610
|  
| 133.333
|  
|-
|-
| | 7/6
|  
| | 133.435
| 7/6
| 133.435
|
|-
|-
| | 11/8
|  
| | 137.829
| 11/8
| 137.829
|
|-
|-
| | 13/12
|  
| | 138.573
| 13/12
| 138.573
|
|-
|-
| | 11/10
|  
| | 165.004
| 11/10
| 165.004
|
|}
|}


==Spectrum of Negroni Tunings by Eigenmonzos==
=== Negroni ===


[[Gencom|Gencom]]: [2 15/14; 49/48 55/54 65/64 91/90]
{| class="wikitable center-all left-4"
 
|-
[[Gencom_map|Gencom map]]: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]
! Edo<br>generator
 
! Unchanged interval<br>(eigenmonzo)
{| class="wikitable"
! Generator (¢)
! Comments
|-
|-
! | Eigenmonzo
|  
! | Supersecor
| 15/8
| 111.731
|
|-
|-
| | 16/15
|  
| | 111.731
| 7/4
| 115.587
|  
|-
|-
| | 8/7
|  
| | 115.587
| 15/14
| 119.443
|  
|-
|-
| | 15/14
|  
| | 119.443
| 13/8
| 119.824
|
|-
|-
| | 16/13
| 1\10
| | 119.824
|  
| 120.000
|  
|-
|-
| | 1\10
|  
| | 120.000
| 11/9
| 121.799
| 11-, 13- and 15-odd-limit minimax
|-
|-
| | 11/9
|  
| | 121.799 (11, 13 and 15 limit minimax)
| 11/6
| 122.785
|  
|-
|-
| | 12/11
|  
| | 122.785
| 11/8
| 123.245
|  
|-
|-
| | 11/8
|  
| | 123.245
| 7/5
| 123.498
|
|-
|-
| | 7/5
|  
| | 123.498
| 15/13
| 123.871
|
|-
|-
| | 15/13
|  
| | 123.871
| 15/11
| 124.068
|  
|-
|-
| | 15/11
|  
| | 124.068
| 13/11
| 124.101
|
|-
|-
| | 13/11
| 3\29
| | 124.101
|  
| 124.138
|
|-
|-
| | 3\29
|  
| | 124.138
| 11/10
| 124.166
|
|-
|-
| | 11/10
|  
| | 124.166
| {{monzo| 0 19 -22 -7 32 }}
| 124.180
| 11-odd-limit least squares
|-
|-
| | |0 19 -22 -7 32&gt;
|  
| | 124.180 (11 limit least squares)
| {{monzo| 0 33 -66 -14 90 -6 }}
| 124.186
| 15-odd-limit least squares
|-
|-
| | |0 33 -66 -14 90 -6&gt;
|  
| | 124.186 (15 limit least squares)
| {{monzo| 0 49 -50 -15 76 -8 }}
| 124.215
| 13-odd-limit least squares
|-
|-
| | |0 49 -50 -15 76 -8&gt;
|  
| | 124.215 (13 limit least squares)
| 13/10
| 124.298
|
|-
|-
| | 13/10
|  
| | 124.298
| 11/7
| 124.424
|
|-
|-
| | 14/11
|  
| | 124.424
| 3/2
| 124.511
| 7- and 9-odd-limit minimax
|-
|-
| | 4/3
| 5\48
| | 124.511 (7 and 9 limit minimx)
|  
| 125.000
|  
|-
|-
| | 5\48
|  
| | 125.000
| {{monzo| 0 -13 15 -5 }}
| 125.469
| 7-odd-limit least squares
|-
|-
| | 30517578125/26795786661
|  
| | 125.469 (7 limit least squares)
| {{monzo| 0 63 -26 -1 }}
| 125.579
| 9-odd-limit least squares
|-
|-
| | |0 63 -26 -1&gt;
|  
| | 125.579 (9 limit least squares)
| 9/5
| 125.673
|
|-
|-
| | 10/9
|  
| | 125.673
| {{monzo| 6 11 -10 }}
| 126.238
| 5-odd-limit least squares
|-
|-
| | 11337408/9765625
|  
| | 126.238 (5 limit least squares)
| 5/3
| 126.337
| 5-odd-limit minimax
|-
|-
| | 6/5
|  
| | 126.337 (5 limit minimax)
| 13/9
| 127.324
|  
|-
|-
| | 18/13
|  
| | 127.324
| 9/7
| 127.486
|
|-
|-
| | 9/7
| 5\47
| | 127.486
|  
| 127.660
|
|-
|-
| | 5\47
|  
| | 127.660
| 13/7
| 128.298
|  
|-
|-
| | 14/13
| 3\28
| | 128.298
|  
| 128.571
|
|-
|-
| | 3\28
|  
| | 128.571
| 5/4
| 128.771
|
|-
|-
| | 5/4
| 1\9
| | 128.771
|  
| 133.333
|  
|-
|-
| | 7/6
|  
| | 133.435
| 7/6
| 133.435
|
|-
|-
| | 13/12
|  
| | 138.573
| 13/12
| 138.573
|
|}
|}
[[Category:eigenmonzo]]
 
[[Category:negri]]
[[Category:Negri]]
[[Category:negric]]
[[Category:Temperament extensions]]
[[Category:negril]]
[[Category:Rank-2 temperaments]]
[[Category:negroni]]
[[Category:spectrum]]
Retrieved from "https://en.xen.wiki/w/Negri_extensions"