Amity: Difference between revisions

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=== Tunings spectra ===
=== Tunings spectra ===
==== Amity ====
==== Amity ====
{| class="wikitable center-all left-3"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 13\46
|
| 339.130
| 46ef val
|-
|
| 9/5
| 9/5
| 339.199
| 339.199
|  
|  
|-
|-
|
| 13/11
| 13/11
| 339.281
| 339.281
|  
|  
|-
|-
|
| 7/4
| 7/4
| 339.343
| 339.343
|  
|  
|-
|-
| 28\99
|
| 339.394
| 99ef val
|-
|
| 7/6
| 7/6
| 339.403
| 339.403
|  
|  
|-
|-
|
| 7/5
| 7/5
| 339.417
| 339.417
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
|-
|
| 9/7
| 9/7
| 339.441
| 339.441
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|-
|
| 15/14
| 15/14
| 339.444
| 339.444
|  
|  
|-
|-
|
| 5/3
| 5/3
| 339.455
| 339.455
|  
|  
|-
|-
|
| 11/7
| 11/7
| 339.462
| 339.462
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
|
| 11/9
| 11/9
| 339.473
| 339.473
|  
|  
|-
|-
| 43\152
|
| 339.474
| 152f val
|-
|
| 15/11
| 15/11
| 339.476
| 339.476
|  
|  
|-
|-
|
| 11/6
| 11/6
| 339.485
| 339.485
|  
|  
|-
|-
|
| 11/10
| 11/10
| 339.490
| 339.490
|  
|  
|-
|-
|
| 11/8
| 11/8
| 339.495
| 339.495
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|
| 13/7
| 13/7
| 339.505
| 339.505
|  
|  
|-
|-
| 58\205
|
| 339.512
|
|-
|
| 5/4
| 5/4
| 339.514
| 339.514
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
|
| 15/8
| 15/8
| 339.541
| 339.541
|  
|  
|-
|-
|
| 13/9
| 13/9
| 339.551
| 339.551
|  
|  
|-
|-
|
| 13/12
| 13/12
| 339.558
| 339.558
|  
|  
|-
|-
|
| 13/8
| 13/8
| 339.563
| 339.563
|  
|  
|-
|-
|
| 15/13
| 15/13
| 339.577
| 339.577
|  
|  
|-
|-
|
| 13/10
| 13/10
| 339.582
| 339.582
|  
|  
|-
|-
|
| 3/2
| 3/2
| 339.609
| 339.609
|
|-
| 15\53
|
| 339.623
|  
|  
|}
|}


==== Hitchcock ====
==== Hitchcock ====
{| class="wikitable center-all left-3"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)*
! Eigenmonzo<br>(unchanged-interval)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|
| 11/6
| 11/6
| 337.659
| 337.659
|  
|  
|-
|-
| 11\39
|
| 338.462
|
|-
|
| 11/8
| 11/8
| 338.742
| 338.742
|  
|  
|-
|-
|
| 13/7
| 13/7
| 338.936
| 338.936
|  
|  
|-
|-
| 13\46
|
| 339.130
|
|-
|
| 11/7
| 11/7
| 339.135
| 339.135
|  
|  
|-
|-
|
| 9/5
| 9/5
| 339.199
| 339.199
|  
|  
|-
|-
|
| 13/11
| 13/11
| 339.281
| 339.281
|  
|  
|-
|-
|
| 7/4
| 7/4
| 339.343
| 339.343
|  
|  
|-
|-
| 28\99
|
| 339.394
|
|-
|
| 7/6
| 7/6
| 339.403
| 339.403
|  
|  
|-
|-
|
| 7/5
| 7/5
| 339.417
| 339.417
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
|-
|
| 9/7
| 9/7
| 339.441
| 339.441
| 9-, 11-, and 13-odd-limit minimax
| 9-, 11-, and 13-odd-limit minimax
|-
|-
|
| 15/14
| 15/14
| 339.444
| 339.444
| 15-odd-limit minimax
| 15-odd-limit minimax
|-
|-
|
| 5/3
| 5/3
| 339.455
| 339.455
|  
|  
|-
|-
|
| 5/4
| 5/4
| 339.514
| 339.514
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
|
| 15/8
| 15/8
| 339.541
| 339.541
|  
|  
|-
|-
|
| 3/2
| 3/2
| 339.609
| 339.609
|  
|  
|-
|-
| 15\53
|
| 339.623
|
|-
|
| 15/13
| 15/13
| 339.677
| 339.677
|  
|  
|-
|-
|
| 13/10
| 13/10
| 339.695
| 339.695
|  
|  
|-
|-
|
| 13/9
| 13/9
| 339.789
| 339.789
|  
|  
|-
|-
|
| 13/12
| 13/12
| 339.870
| 339.870
|  
|  
|-
|-
| 17\60
|
| 340.000
| 60de val
|-
|
| 13/8
| 13/8
| 340.088
| 340.088
|  
|  
|-
|-
|
| 15/11
| 15/11
| 340.339
| 340.339
|  
|  
|-
|-
|
| 11/10
| 11/10
| 341.251
| 341.251
|  
|  
|-
|-
|
| 11/9
| 11/9
| 347.408
| 347.408

Revision as of 13:25, 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

In the following table, odd harmonics 1–21 and their inversions are labeled in bold.

# 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

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
13\46 339.130 46ef val
9/5 339.199
13/11 339.281
7/4 339.343
28\99 339.394 99ef val
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
43\152 339.474 152f val
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
58\205 339.512
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
15\53 339.623

Hitchcock

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
11/6 337.659
11\39 338.462
11/8 338.742
13/7 338.936
13\46 339.130
11/7 339.135
9/5 339.199
13/11 339.281
7/4 339.343
28\99 339.394
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\53 339.623
15/13 339.677
13/10 339.695
13/9 339.789
13/12 339.870
17\60 340.000 60de val
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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