Amity

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Amity is the rank-2 microtemperament on the 5-limit that tempers out the amity comma, 1600000/1594323. This page also assumes the canonical extension to the 7-limit, which means 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. In the 17-limit, hitchcock tempers out 154/153, 256/255, and 273/272.

For technical data, see Amity family #Amity.

Etymology

Origin: Paul Erlich (2002)

Meaning: A restructuring of the words "acute minor third." The ideal generator for Amity is between a minor third and neutral third.

Interval chain

# Cents* Approximate ratios
7-limit 13-limit Extensions
Amity Mapping (53 & 152) 		13-limit Extensions
Hitchcock Mapping (46 & 53)
0 0.00 1/1
1 339.43 128/105 11/9
2 678.86 40/27
3 1018.30 9/5
4 157.73 35/32 12/11, 11/10
5 497.16 4/3
6 836.59 81/50 13/8, 21/13
7 1176.03 63/32, 160/81 65/33, 77/39 65/33, 77/39, 128/65
8 315.46 6/5
9 654.89 35/24 16/11, 22/15
10 994.32 16/9 39/22
11 133.75 27/25 13/12, 14/13
12 473.19 21/16
13 812.62 8/5
14 1152.05 35/18 39/20, 64/33, 88/45
15 291.48 32/27 13/11 13/11
16 630.92 36/25 13/9
17 970.35 7/4
18 109.78 16/15
19 449.21 35/27 13/10
20 788.64 63/40 52/33
21 1128.08 48/25 25/13 21/11, 52/27
22 267.51 7/6
23 606.94 64/45
24 946.37 81/70 26/15
25 85.81 21/20
26 425.24 32/25 14/11
27 764.67 14/9
28 1104.10 256/135
29 243.53 147/128 15/13
30 582.97 7/5
31 922.40 128/75 56/33
32 61.83 28/27 27/26
33 401.26 63/50
34 740.69 49/32 20/13
35 1080.13 28/15
36 219.56 256/225 25/22
37 558.99 112/81 18/13
38 898.42 42/25
39 37.86 49/48 40/39, 45/44

* In 7-limit POTE tuning, octave reduced

Tuning spectra

Amity

Gencom: [2 128/105; 352/351 540/539 625/624 729/728]

Gencom mapping: [⟨1 3 6 -2 21 17], ⟨0 -5 -13 17 -62 -47]]

Eigenmonzo
(unchanged-interval) 		Generator
(¢) 		Comments
10/9 339.199
13/11 339.281
8/7 339.343
7/6 339.403
7/5 339.417 7-odd-limit minimax
9/7 339.441 9-odd-limit minimax
15/14 339.444
6/5 339.455
14/11 339.462 11-odd-limit minimax
11/9 339.473
15/11 339.476
12/11 339.485
11/10 339.490
11/8 339.495 13- and 15-odd-limit minimax
14/13 339.505
5/4 339.514 5-odd-limit minimax
16/15 339.541
18/13 339.551
13/12 339.558
16/13 339.563
15/13 339.577
13/10 339.582
4/3 339.609

Hitchcock

Gencom: [2 11/9; 121/120 169/168 176/175 325/324]

Gencom mapping: [⟨1 3 6 -2 6 2], ⟨0 -5 -13 17 -9 6]]

Eigenmonzo
(unchanged-interval) 		Generator
(¢) 		Comments
12/11 337.659
11/8 338.742
14/13 338.936
14/11 339.135
10/9 339.199
13/11 339.281
8/7 339.343
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
6/5 339.455
5/4 339.514 5-odd-limit minimax
16/15 339.541
4/3 339.609
15/13 339.677
13/10 339.695
18/13 339.789
13/12 339.870
16/13 340.088
15/11 340.339
11/10 341.251
11/9 347.408

Music

Francium
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