Amity

Amity is a temperament for the 5, 7, 11, and 13 prime limits. EDOs that support amity include 46edo, 53edo, 99edo, 152edo, and 205edo.

In the 7-limit, amity tempers out 4375/4374 and 5120/5103, as well as 5-limit amity comma, 1600000/1594323. But extending it from the 7-limit to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: 11-limit amity (53&205) and hitchcock (46&53).

Tempering out 540/539 leads 11-limit amity, supported by 53, 152, and 205 edos. Tempering out 176/175 leads 11-limit hitchcock, supported by 46, 53, and 99 edos. They can be extended to the 13-limit by the 352/351 comma, and results in 625/624 and 729/728 being tempered out in 13-limit amity; 169/168 and 325/324 being tempered out in 13-limit hitchcock. In the 17-limit, 46&53 tempers out 154/153, 256/255, and 273/272.

See Amity family or Ragismic microtemperaments #Amity for more information.

Interval chain

Generators	Cents* (octave-reduced)	Approximate ratios
		Amity (53&205)	Hitchcock (46&53)
0	0.000	1/1
1	339.43		11/9
2	678.86	40/27
3	1018.30	9/5
4	157.73	35/32	12/11~35/32~11/10
5	497.16	4/3
6	836.59		21/13~13/8
7	1176.03	65/33, 77/39~160/81	63/32~128/65~65/33, 77/39~160/81
8	315.46	6/5
9	654.89		16/11~22/15
10	994.32	16/9
11	133.75	27/25	14/13~27/25~13/12
12	473.19	21/16
13	812.62	8/5
14	1152.05	35/18	64/33~35/18
15	291.48	13/11~32/27
16	630.92		13/9
17	970.35	7/4
18	109.78	16/15
19	449.21		13/10
20	788.64
21	1128.08	48/25~25/13	21/11~48/25
22	267.51	7/6
23	606.94
24	946.37		26/15
25	85.81	21/20
26	425.24	32/25	14/11~32/25
27	764.67	14/9
28	1104.10
29	243.53	15/13
30	582.97	7/5
31	922.40
32	61.83	28/27~27/26	28/27
33	401.26
34	740.69	20/13
35	1080.13	28/15
36	219.56	25/22
37	558.99	18/13

* in 7-limit POTE tuning

Tuning Spectra

Spectrum of Amity Tunings (53&205)

13-limit commas: 352/351, 540/539, 625/624, 729/728

Eigenmonzo	Generator	Comment
10/9	339.199
13/11	339.281
8/7	339.343
7/6	339.403
7/5	339.417	7 limit minimax
9/7	339.441	9 limit minimax
15/14	339.444
6/5	339.455
14/11	339.462	11 limit minimax
11/9	339.473
15/11	339.476
12/11	339.485
11/10	339.490
11/8	339.495	13, 15 limit minimax
14/13	339.505
5/4	339.514	5 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

Spectrum of Hitchcock Tunings (46&53)

13-limit commas: 121/120, 169/168, 176/175, 325/324

Eigenmonzo	Generator	Comment
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 limit minimax
9/7	339.441	9, 11, 13 limit minimax
15/14	339.444	15 limit minimax
6/5	339.455
5/4	339.514	5 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