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[1][2].
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
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 11\39 | 338.462 | 39ee… val, lower bound of 7- and 9-odd-limit diamond monotone | |
| 13\46 | 339.130 | 46ef val | |
| 9/5 | 339.199 | ||
| 13/11 | 339.281 | ||
| 7/4 | 339.343 | ||
| 28\99 | 339.394 | 99ef val, lower bound of 11-, 13-, 15-, and 13-limit 21-odd-limit diamond monotone | |
| 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 | ||
| 43\152 | 339.474 | 152f val | |
| 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 | ||
| 58\205 | 339.512 | ||
| 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 | ||
| 15\53 | 339.623 | Upper bound of 11-, 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone | |
| 17\60 | 340.000 | 60deee… val, upper bound of 7- and 9-odd-limit diamond monotone |
Hitchcock
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 11/6 | 337.659 | ||
| 11\39 | 338.462 | Lower bound of 7-, 9, and 11-odd-limit diamond monotone | |
| 11/8 | 338.742 | ||
| 13/7 | 338.936 | ||
| 13\46 | 339.130 | Lower bound of 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone | |
| 11/7 | 339.135 | ||
| 9/5 | 339.199 | ||
| 13/11 | 339.281 | ||
| 7/4 | 339.343 | ||
| 28\99 | 339.394 | ||
| 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\53 | 339.623 | Upper bound of 11-, 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone | |
| 15/13 | 339.677 | ||
| 13/10 | 339.695 | ||
| 13/9 | 339.789 | ||
| 13/12 | 339.870 | ||
| 17\60 | 340.000 | 60de val, upper bound of 7- and 9-odd-limit diamond monotone | |
| 13/8 | 340.088 | ||
| 15/11 | 340.339 | ||
| 11/10 | 341.251 | ||
| 11/9 | 347.408 |
* Besides the octave
Music
- For Amity (2023) – in 463edo tuning