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See [[Ragismic_microtemperaments#Amity|Ragismic microtemperaments]].
'''Amity''' is a [[regular temperament|temperament]] that divides a [[8/3|perfect eleventh]] into 5 [[generator]]s of acute minor thirds. A stack of 13 generators [[octave reduction|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 {{EDOs| 46, 53, 99, 152, and 205 }}.
[[Category:soft_redirect]]
 
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 ({{nowrap| 53 & 152 }}), catamite ({{nowrap| 46 & 145 }}), and hitchcock ({{nowrap| 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''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2064.html Yahoo! Tuning Group | ''Kleismic & co'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3481.html Yahoo! Tuning Group | ''32 best 5-limit linear temperaments redux'']</ref>.
 
{{Tdhat|Amity family #Amity}}
 
== Interval chain ==
In the following table, odd harmonics 1–21 and their inversions are labeled in '''bold'''.
 
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | Cents*
! colspan="3" | Approximate ratios
|-
! rowspan="2" | 7-limit
! colspan="2" | 13-limit extensions
|-
! Amity ({{nowrap| 53 & 152 }})
! Hitchcock ({{nowrap| 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
|
|}
<nowiki/>* In 7-limit CWE tuning, octave reduced
 
== Tunings ==
=== Tunings spectra ===
==== Amity ====
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(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 ====
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Unchanged interval<br>(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
|
|}
<nowiki/>* Besides the octave
 
== Music ==
; [[User:Francium|Francium]]
* [https://www.youtube.com/watch?v=AsDaJXCBd_w ''For Amity''] (2023) – in 463edo tuning
 
== Notes ==
 
[[Category:Amity| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Amity family]]
[[Category:Ragismic microtemperaments]]
[[Category:Hemifamity temperaments]]
Retrieved from "https://en.xen.wiki/w/Amity"