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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]]
Retrieved from "https://en.xen.wiki/w/Negri_extensions"