Amity
Amity is the microtemperament tempering out the amity comma, 1600000/1594323, in the 5-limit. This article also assumes the canonical mapping for 7, which means tempering out 4375/4374 and 5120/5103 in the 7-limit. 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.
See Amity family #Amity for more technical data.
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
- For Amity (2023) – amity in 463edo tuning