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'''Amity''' is a temperament for the 5, 7, 11, and 13 [[Harmonic limit|prime limits]]. EDOs that support amity include [[46edo]], [[53edo]], [[99edo]], [[152edo]], and [[205edo]].
'''Amity''' is a temperament for the 5, 7, 11, and 13 [[Harmonic limit|prime limits]]. Equal temperaments that support amity include [[46edo|46]], [[53edo|53]], [[99edo|99]], [[152edo|152]], and [[205edo|205]].


In the 7-limit, amity tempers out 4375/4374 and 5120/5103, as well as 5-limit [[amity comma]], 1600000/1594323. But extending it from the 7-limit to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: 11-limit amity (53&205) and hitchcock (46&53).
In the 7-limit, amity tempers out 4375/4374 and 5120/5103, as well as 5-limit [[amity comma]], 1600000/1594323. But extending it from the 7-limit to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: 11-limit amity (53&205) and hitchcock (46&53).


Tempering out 540/539 leads 11-limit amity, supported by 53, 152, and 205 edos. Tempering out 176/175 leads 11-limit hitchcock, supported by 46, 53, and 99 edos. They can be extended to the 13-limit by the 352/351 comma, and results in 625/624 and 729/728 being tempered out in 13-limit amity; 169/168 and 325/324 being tempered out in 13-limit hitchcock. In the 17-limit, 46&53 tempers out 154/153, 256/255, and 273/272.
Tempering out 540/539 leads 11-limit amity, supported by 53et, 152et, and 205et. Tempering out 121/120 leads 11-limit hitchcock, supported by 46et, 53et, and 99et. They can be extended to the 13-limit by the 352/351 comma, and results in 625/624 and 729/728 being tempered out in 13-limit amity; 169/168 and 325/324 being tempered out in 13-limit hitchcock. In the 17-limit, 46&53 tempers out 154/153, 256/255, and 273/272.


See [[Amity family #Amity]] or [[Ragismic microtemperaments #Amity]] for more information.
See [[Amity family #Amity]] or [[Ragismic microtemperaments #Amity]] for technical data.


== Interval chain ==
== Interval chain ==
{| class="wikitable center-all right-2"
{| class="wikitable center-1 right-2"
|-
|-
! rowspan="2"| Generators
! rowspan="2"| #
! rowspan="2"| Cents* <br>(octave-reduced)
! rowspan="2"| Cents*  
! colspan="2"| Approximate ratios
! colspan="3" | Approximate Ratios
|-
|-
! Amity <br>(53&amp;205)
! 7-limit
! Hitchcock <br>(46&amp;53)
! 13-limit Extensions<br>Amity Mapping (53 &amp; 152)
! 13-limit Extensions<br>Hitchcock Mapping (46 &amp; 53)
|-
|-
| 0
| 0
| 0.000
| 0.00
| colspan="2"| 1/1
| 1/1
|
|
|-
|-
| 1
| 1
| 339.43
| 339.43
|  
| 128/105
|
| 11/9
| 11/9
|-
|-
| 2
| 2
| 678.86
| 678.86
| colspan="2"| 40/27
| 40/27
|
|
|-
|-
| 3
| 3
| 1018.30
| 1018.30
| colspan="2"| 9/5
| 9/5
|
|
|-
|-
| 4
| 4
| 157.73
| 157.73
| 35/32
| 35/32
| 12/11~35/32~11/10
|
| 12/11, 11/10
|-
|-
| | 5
| | 5
| 497.16
| 497.16
| colspan="2"| 4/3
| '''4/3'''
|
|
|-
|-
| 6
| 6
| 836.59
| 836.59
|  
| 81/50
| 21/13~13/8
|
| '''13/8''', 21/13
|-
|-
| 7
| 7
| 1176.03
| 1176.03
| 65/33, 77/39~160/81
| 63/32, 160/81
| 63/32~128/65~65/33, <br>77/39~160/81
| 65/33, 77/39
| 65/33, 77/39, 128/65
|-
|-
| 8
| 8
| 315.46
| 315.46
| colspan="2"| 6/5
| 6/5
|
|
|-
|-
| 9
| 9
| 654.89
| 654.89
|  
| 35/24
| 16/11~22/15
|
| '''16/11''', 22/15
|-
|-
| 10
| 10
| 994.32
| 994.32
| colspan="2"| 16/9
| 16/9
|
| 39/22
|-
|-
| 11
| 11
| 133.75
| 133.75
| 27/25
| 27/25
| 14/13~27/25~13/12
|
| 13/12, 14/13
|-
|-
| 12
| 12
| 473.19
| 473.19
| colspan="2"| 21/16
| 21/16
|
|
|-
|-
| 13
| 13
| 812.62
| 812.62
| colspan="2"| 8/5
| '''8/5'''
|
|
|-
|-
| 14
| 14
| 1152.05
| 1152.05
| 35/18
| 35/18
| 64/33~35/18
|
| 39/20, 64/33, 88/45
|-
|-
| 15
| 15
| 291.48
| 291.48
| colspan="2"| 13/11~32/27
| 32/27
| 13/11
| 13/11
|-
|-
| 16
| 16
| 630.92
| 630.92
|  
| 36/25
|
| 13/9
| 13/9
|-
|-
| 17
| 17
| 970.35
| 970.35
| colspan="2"| 7/4
| '''7/4'''
|
|
|-
|-
| 18
| 18
| 109.78
| 109.78
| colspan="2"| 16/15
| 16/15
|
|
|-
|-
| 19
| 19
| 449.21
| 449.21
|  
| 35/27
|
| 13/10
| 13/10
|-
|-
| 20
| 20
| 788.64
| 788.64
|  
| 63/40
|  
|
| 52/33
|-
|-
| 21
| 21
| 1128.08
| 1128.08
| 48/25~25/13
| 48/25
| 21/11~48/25
| 25/13
| 21/11, 52/27
|-
|-
| 22
| 22
| 267.51
| 267.51
| colspan="2"| 7/6
| 7/6
|
|
|-
|-
| 23
| 23
| 606.94
| 606.94
|  
| 64/45
|
|  
|  
|-
|-
| 24
| 24
| 946.37
| 946.37
|  
| 81/70
|
| 26/15
| 26/15
|-
|-
| 25
| 25
| 85.81
| 85.81
| colspan="2"| 21/20
| 21/20
|
|
|-
|-
| 26
| 26
| 425.24
| 425.24
| 32/25
| 32/25
| 14/11~32/25
|
| 14/11
|-
|-
| 27
| 27
| 764.67
| 764.67
| colspan="2"| 14/9
| 14/9
|
|
|-
|-
| 28
| 28
| 1104.10
| 1104.10
|  
| 256/135
|
|  
|  
|-
|-
| 29
| 29
| 243.53
| 243.53
| 147/128
| 15/13
| 15/13
|  
|  
Line 155: Line 200:
| 30
| 30
| 582.97
| 582.97
| colspan="2"| 7/5
| 7/5
|
|
|-
|-
| 31
| 31
| 922.40
| 922.40
|  
| 128/75
|  
|
| 56/33
|-
|-
| 32
| 32
| 61.83
| 61.83
| 28/27~27/26
| 28/27
| 28/27
| 27/26
|
|-
|-
| 33
| 33
| 401.26
| 401.26
|  
| 63/50
|
|  
|  
|-
|-
| 34
| 34
| 740.69
| 740.69
| 49/32
| 20/13
| 20/13
|  
|  
Line 179: Line 230:
| 35
| 35
| 1080.13
| 1080.13
| colspan="2"| 28/15
| 28/15
|
|
|-
|-
| 36
| 36
| 219.56
| 219.56
| 256/225
| 25/22
| 25/22
|  
|  
Line 188: Line 242:
| 37
| 37
| 558.99
| 558.99
| 112/81
| 18/13
| 18/13
|  
|  
|-
| 38
| 898.42
| 42/25
|
|
|-
| 39
| 37.86
| 49/48
| 40/39, 45/44
|
|}
|}
<nowiki>*</nowiki> in 7-limit POTE tuning
<nowiki>*</nowiki> in 7-limit POTE tuning, octave reduced


== Tuning Spectra ==
== Tuning spectra ==
=== Spectrum of Amity Tunings (53&amp;205) ===
=== Spectrum of amity tunings ===


13-limit commas: 352/351, 540/539, 625/624, 729/728
13-limit commas: 352/351, 540/539, 625/624, 729/728
Line 201: Line 268:
! Eigenmonzo
! Eigenmonzo
! Generator
! Generator
! Comment
! Comments
|-
|-
| 10/9
| 10/9
Line 221: Line 288:
| 7/5
| 7/5
| 339.417
| 339.417
| 7 limit minimax
| 7-odd-limit minimax
|-
|-
| 9/7
| 9/7
| 339.441
| 339.441
| 9 limit minimax
| 9-odd-limit minimax
|-
|-
| 15/14
| 15/14
Line 237: Line 304:
| 14/11
| 14/11
| 339.462
| 339.462
| 11 limit minimax
| 11-odd-limit minimax
|-
|-
| 11/9
| 11/9
Line 257: Line 324:
| 11/8
| 11/8
| 339.495
| 339.495
| 13, 15 limit minimax
| 13- and, 15-odd-limit minimax
|-
|-
| 14/13
| 14/13
Line 265: Line 332:
| 5/4
| 5/4
| 339.514
| 339.514
| 5 limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
| 16/15
Line 296: Line 363:
|}
|}


=== Spectrum of Hitchcock Tunings (46&amp;53) ===
=== Spectrum of hitchcock tunings ===


13-limit commas: 121/120, 169/168, 176/175, 325/324
13-limit commas: 121/120, 169/168, 176/175, 325/324
Line 339: Line 406:
| 7/5
| 7/5
| 339.417
| 339.417
| 7 limit minimax
| 7-odd-limit minimax
|-
|-
| 9/7
| 9/7
| 339.441
| 339.441
| 9, 11, 13 limit minimax
| 9-, 11-, and 13-odd-limit minimax
|-
|-
| 15/14
| 15/14
| 339.444
| 339.444
| 15 limit minimax
| 15-odd-limit minimax
|-
|-
| 6/5
| 6/5
Line 355: Line 422:
| 5/4
| 5/4
| 339.514
| 339.514
| 5 limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
| 16/15
Line 399: Line 466:


== Scales ==
== Scales ==
* [[Amity7]] - proper [[4L 3s]]
* [[Amity7]] – proper [[4L 3s]]
* [[Amity11]] - improper [[7L 4s]]
* [[Amity11]] – improper [[7L 4s]]
* [[Amity18]] - improper [[7L 11s]]
* [[Amity18]] – improper [[7L 11s]]
* [[Amity25]] - improper [[7L 18s]]
* [[Amity25]] – improper [[7L 18s]]
* [[Amity32]] - improper [[7L 25s]]
* [[Amity32]] – improper [[7L 25s]]
* [[Amity39]] - improper [[7L 32s]]
* [[Amity39]] – improper [[7L 32s]]
* [[Amity46]] - proper [[7L 39s]]
* [[Amity46]] – proper [[7L 39s]]


[[Category:Amity family]]
[[Category:Amity family]]
Retrieved from "https://en.xen.wiki/w/Amity"