Amity: Difference between revisions

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m Cleanup
→Interval chain: adopt CWE tuning
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{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! rowspan="2" | #
! rowspan="3" | #
! rowspan="2" | Cents*  
! rowspan="3" | Cents*  
! colspan="3" | Approximate ratios
! colspan="3" | Approximate ratios
|-
|-
! 7-limit
! rowspan="2" | 7-limit
! 13-limit extensions, <br>amity mapping ({{nowrap| 53 & 152 }})
! colspan="2" | 13-limit extensions
! 13-limit extensions, <br>hitchcock mapping ({{nowrap| 46 & 53 }})
|-
! Amity ({{nowrap| 53 & 152 }})
! Hitchcock ({{nowrap| 46 & 53 }})
|-
|-
| 0
| 0
Line 31: Line 33:
|-
|-
| 2
| 2
| 678.86
| 678.87
| 40/27
| 40/27
|
|
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|-
|-
| 4
| 4
| 157.73
| 157.74
| 35/32
| 35/32
|
|
Line 49: Line 51:
|-
|-
| 5
| 5
| 497.16
| 497.17
| '''4/3'''
| '''4/3'''
|
|
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|-
|-
| 6
| 6
| 836.59
| 836.61
| 81/50
| 81/50
|
|
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|-
|-
| 7
| 7
| 1176.03
| 1176.04
| 63/32, 160/81
| 63/32, 160/81
| 65/33, 77/39
| 65/33, 77/39
Line 67: Line 69:
|-
|-
| 8
| 8
| 315.46
| 315.48
| 6/5
| 6/5
|
|
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|-
|-
| 9
| 9
| 654.89
| 654.91
| 35/24
| 35/24
|
|
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|-
|-
| 10
| 10
| 994.32
| 994.35
| 16/9
| 16/9
|
|
Line 85: Line 87:
|-
|-
| 11
| 11
| 133.75
| 133.78
| 27/25
| 27/25
|
|
Line 91: Line 93:
|-
|-
| 12
| 12
| 473.19
| 473.22
| 21/16
| 21/16
|
|
Line 97: Line 99:
|-
|-
| 13
| 13
| 812.62
| 812.65
| '''8/5'''
| '''8/5'''
|
|
Line 103: Line 105:
|-
|-
| 14
| 14
| 1152.05
| 1152.09
| 35/18
| 35/18
|
|
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|-
|-
| 15
| 15
| 291.48
| 291.52
| 32/27
| 32/27
| 13/11
| 13/11
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|-
|-
| 16
| 16
| 630.92
| 630.96
| 36/25
| 36/25
|
|
Line 121: Line 123:
|-
|-
| 17
| 17
| 970.35
| 970.39
| '''7/4'''
| '''7/4'''
|
|
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|-
|-
| 18
| 18
| 109.78
| 109.83
| 16/15
| 16/15
|
|
Line 133: Line 135:
|-
|-
| 19
| 19
| 449.21
| 449.26
| 35/27
| 35/27
|
|
Line 139: Line 141:
|-
|-
| 20
| 20
| 788.64
| 788.70
| 63/40
| 63/40
|
|
Line 145: Line 147:
|-
|-
| 21
| 21
| 1128.08
| 1128.13
| 48/25
| 48/25
| 25/13
| 25/13
Line 151: Line 153:
|-
|-
| 22
| 22
| 267.51
| 267.57
| 7/6
| 7/6
|
|
Line 157: Line 159:
|-
|-
| 23
| 23
| 606.94
| 607.00
| 64/45
| 64/45
|
|
Line 163: Line 165:
|-
|-
| 24
| 24
| 946.37
| 946.44
| 81/70
| 81/70
|
|
Line 169: Line 171:
|-
|-
| 25
| 25
| 85.81
| 85.87
| 21/20
| 21/20
|
|
Line 175: Line 177:
|-
|-
| 26
| 26
| 425.24
| 425.31
| 32/25
| 32/25
|
|
Line 181: Line 183:
|-
|-
| 27
| 27
| 764.67
| 764.74
| 14/9
| 14/9
|
|
Line 187: Line 189:
|-
|-
| 28
| 28
| 1104.10
| 1104.18
| 256/135
| 256/135
|
|
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|-
|-
| 29
| 29
| 243.53
| 243.61
| 147/128
| 147/128
| 15/13
| 15/13
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|-
|-
| 30
| 30
| 582.97
| 583.05
| 7/5
| 7/5
|
|
Line 205: Line 207:
|-
|-
| 31
| 31
| 922.40
| 922.48
| 128/75
| 128/75
|
|
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|-
|-
| 32
| 32
| 61.83
| 61.92
| 28/27
| 28/27
| 27/26
| 27/26
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|-
|-
| 33
| 33
| 401.26
| 401.35
| 63/50
| 63/50
|
|
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|-
|-
| 34
| 34
| 740.69
| 740.79
| 49/32
| 49/32
| 20/13
| 20/13
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|-
|-
| 35
| 35
| 1080.13
| 1080.22
| 28/15
| 28/15
|
|
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|-
|-
| 36
| 36
| 219.56
| 219.66
| 256/225
| 256/225
| 25/22
| 25/22
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|-
|-
| 37
| 37
| 558.99
| 559.09
| 112/81
| 112/81
| 18/13
| 18/13
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|-
|-
| 38
| 38
| 898.42
| 898.53
| 42/25
| 42/25
|
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|-
|-
| 39
| 39
| 37.86
| 37.96
| 49/48
| 49/48
| 40/39, 45/44
| 40/39, 45/44
|
|
|}
|}
<nowiki />* In 7-limit POTE tuning, octave reduced
<nowiki/>* In 7-limit CWE tuning, octave reduced


== Tunings ==
== Tunings ==

Revision as of 13:07, 5 June 2025

Amity is a temperament that divides a perfect eleventh into 5 generators of acute minor thirds. A stack of 13 generators octave reduced represents 8/5, tempering out the amity comma, 1600000/1594323. This article also assumes the canonical extension to the 7-limit, where a stack of 17 generators octave reduced represents 7/4, tempering out 4375/4374 and 5120/5103. Equal temperaments that support amity include 46, 53, 99, 152, and 205.

Extending amity from the 7-limit to the 11-limit is not so simple. There are three mappings that are comparable in complexity and error: undecimal amity (53 & 152), catamite (46 & 145), and hitchcock (46 & 53). Undecimal amity tempers out 540/539 and has the harmonic 11 mapped to −62 generator steps. Catamite tempers out 441/440 and has the harmonic 11 mapped to +37 generators steps. Hitchcock tempers out 121/120 and has the harmonic 11 mapped to −9 steps. They can be extended to the 13-limit through 352/351, and results in 625/624 and 729/728 being tempered out in 13-limit amity, 196/195 and 364/363 being tempered out in catamite, and 169/168 and 325/324 being tempered out in hitchcock. Hitchcock has an extra extension to the 17-limit where it tempers out 154/153, 256/255, and 273/272.

Amity was named by Gene Ward Smith in 2001–2002 as a restructuring of the phrase acute minor third[1][2].

For technical data, see Amity family #Amity.

Interval chain

# Cents* Approximate ratios
7-limit 13-limit extensions
Amity (53 & 152) Hitchcock (46 & 53)
0 0.00 1/1
1 339.43 128/105 11/9
2 678.87 40/27
3 1018.30 9/5
4 157.74 35/32 12/11, 11/10
5 497.17 4/3
6 836.61 81/50 13/8, 21/13
7 1176.04 63/32, 160/81 65/33, 77/39 65/33, 77/39, 128/65
8 315.48 6/5
9 654.91 35/24 16/11, 22/15
10 994.35 16/9 39/22
11 133.78 27/25 13/12, 14/13
12 473.22 21/16
13 812.65 8/5
14 1152.09 35/18 39/20, 64/33, 88/45
15 291.52 32/27 13/11 13/11
16 630.96 36/25 13/9
17 970.39 7/4
18 109.83 16/15
19 449.26 35/27 13/10
20 788.70 63/40 52/33
21 1128.13 48/25 25/13 21/11, 52/27
22 267.57 7/6
23 607.00 64/45
24 946.44 81/70 26/15
25 85.87 21/20
26 425.31 32/25 14/11
27 764.74 14/9
28 1104.18 256/135
29 243.61 147/128 15/13
30 583.05 7/5
31 922.48 128/75 56/33
32 61.92 28/27 27/26
33 401.35 63/50
34 740.79 49/32 20/13
35 1080.22 28/15
36 219.66 256/225 25/22
37 559.09 112/81 18/13
38 898.53 42/25
39 37.96 49/48 40/39, 45/44

* In 7-limit CWE tuning, octave reduced

Tunings

Tunings spectra

Amity

Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
9/5 339.199
13/11 339.281
7/4 339.343
7/6 339.403
7/5 339.417 7-odd-limit minimax
9/7 339.441 9-odd-limit minimax
15/14 339.444
5/3 339.455
11/7 339.462 11-odd-limit minimax
11/9 339.473
15/11 339.476
11/6 339.485
11/10 339.490
11/8 339.495 13- and 15-odd-limit minimax
13/7 339.505
5/4 339.514 5-odd-limit minimax
15/8 339.541
13/9 339.551
13/12 339.558
13/8 339.563
15/13 339.577
13/10 339.582
3/2 339.609

Hitchcock

Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
11/6 337.659
11/8 338.742
13/7 338.936
11/7 339.135
9/5 339.199
13/11 339.281
7/4 339.343
7/6 339.403
7/5 339.417 7-odd-limit minimax
9/7 339.441 9-, 11-, and 13-odd-limit minimax
15/14 339.444 15-odd-limit minimax
5/3 339.455
5/4 339.514 5-odd-limit minimax
15/8 339.541
3/2 339.609
15/13 339.677
13/10 339.695
13/9 339.789
13/12 339.870
13/8 340.088
15/11 340.339
11/10 341.251
11/9 347.408

* Besides the octave

Music

Francium

Notes

  1. ↑ Yahoo! Tuning Group | Kleismic & co
  2. ↑ Yahoo! Tuning Group | 32 best 5-limit linear temperaments redux
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