Amity

For technical data, see Amity family #Amity. 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.

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

• For Amity (2023) – amity in 463edo tuning