Amity

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.

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

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

• For Amity (2023) – in 463edo tuning