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{{Infobox regtemp
| Title = Amity
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[1600000/1594323]] (2.3.5); <br>[[4375/4374]], [[5120/5103]] (2.3.5.7)
| Edo join 1 = 46 | Edo join 2 = 53
| Mapping = 1; -5 -13 17
| Generators = 243/200
| Generators tuning = 339.4
| Optimization method = CWE
| MOS scales = [[7L&nbsp;4s]], [[7L&nbsp;11s]], [[7L&nbsp;18s]], [[7L&nbsp;25s]]
| Pergen = (P8, cP4/5)
| Color name = Saquinyoti
| Odd limit 1 = 5 | Mistuning 1 = 0.47 | Complexity 1 = 18
| Odd limit 2 = 9 | Mistuning 2 = 1.68 | Complexity 2 = 32
}}
'''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 }}.
'''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 }}.


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]].
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 }}), stalagmite ({{nowrap| 46 & 145 }}), and hitchcock ({{nowrap| 46 & 53 }}). Undecimal amity tempers out 540/539 and has the harmonic 11 mapped to −62 generator steps. Stalagmite 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 stalagmite, 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>.
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>.
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== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–21 and their inversions are labeled in '''bold'''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
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| 0
| 0
| 0.00
| 0.00
| 1/1
| '''1/1'''
|
|
|
|
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| 10
| 10
| 994.35
| 994.35
| 16/9
| '''16/9'''
|
|
| 39/22
| 39/22
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| 12
| 12
| 473.22
| 473.22
| 21/16
| '''21/16'''
|
|
|
|
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| 18
| 18
| 109.83
| 109.83
| 16/15
| '''16/15'''
|
|
|
|
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=== Tunings spectra ===
=== Tunings spectra ===
==== Amity ====
==== Amity ====
{| class="wikitable center-all left-3"
{| class="wikitable center-all left-4"
|-
|-
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! 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
| 9/5
| 339.199
| 339.199
|  
|  
|-
|-
|
| 13/11
| 13/11
| 339.281
| 339.281
|  
|  
|-
|-
|
| 7/4
| 7/4
| 339.343
| 339.343
|  
|  
|-
|-
| 28\99
|
| 339.394
| 99ef val, lower bound of 11-, 13-, 15-, and 13-limit 21-odd-limit diamond monotone
|-
|
| 7/6
| 7/6
| 339.403
| 339.403
|  
|  
|-
|-
|
| 7/5
| 7/5
| 339.417
| 339.417
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
|-
|
| 9/7
| 9/7
| 339.441
| 339.441
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|-
|
| 15/14
| 15/14
| 339.444
| 339.444
|  
|  
|-
|-
|
| 5/3
| 5/3
| 339.455
| 339.455
|  
|  
|-
|-
|
| 11/7
| 11/7
| 339.462
| 339.462
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
|
| 11/9
| 11/9
| 339.473
| 339.473
|  
|  
|-
|-
| 43\152
|
| 339.474
| 152f val
|-
|
| 15/11
| 15/11
| 339.476
| 339.476
|  
|  
|-
|-
|
| 11/6
| 11/6
| 339.485
| 339.485
|  
|  
|-
|-
|
| 11/10
| 11/10
| 339.490
| 339.490
|  
|  
|-
|-
|
| 11/8
| 11/8
| 339.495
| 339.495
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|
| 13/7
| 13/7
| 339.505
| 339.505
|  
|  
|-
|-
| 58\205
|
| 339.512
|
|-
|
| 5/4
| 5/4
| 339.514
| 339.514
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
|
| 15/8
| 15/8
| 339.541
| 339.541
|  
|  
|-
|-
|
| 13/9
| 13/9
| 339.551
| 339.551
|  
|  
|-
|-
|
| 13/12
| 13/12
| 339.558
| 339.558
|  
|  
|-
|-
|
| 13/8
| 13/8
| 339.563
| 339.563
|  
|  
|-
|-
|
| 15/13
| 15/13
| 339.577
| 339.577
|  
|  
|-
|-
|
| 13/10
| 13/10
| 339.582
| 339.582
|  
|  
|-
|-
|
| 3/2
| 3/2
| 339.609
| 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 ====
==== Hitchcock ====
{| class="wikitable center-all left-3"
{| class="wikitable center-all left-4"
|-
|-
! Eigenmonzo<br>(unchanged-interval)*
! Edo<br>generator
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|
| 11/6
| 11/6
| 337.659
| 337.659
|  
|  
|-
|-
| 11\39
|
| 338.462
| Lower bound of 7-, 9, and 11-odd-limit diamond monotone
|-
|
| 11/8
| 11/8
| 338.742
| 338.742
|  
|  
|-
|-
|
| 13/7
| 13/7
| 338.936
| 338.936
|  
|  
|-
|-
| 13\46
|
| 339.130
| Lower bound of 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
|-
|
| 11/7
| 11/7
| 339.135
| 339.135
|  
|  
|-
|-
|
| 9/5
| 9/5
| 339.199
| 339.199
|  
|  
|-
|-
|
| 13/11
| 13/11
| 339.281
| 339.281
|  
|  
|-
|-
|
| 7/4
| 7/4
| 339.343
| 339.343
|  
|  
|-
|-
| 28\99
|
| 339.394
|
|-
|
| 7/6
| 7/6
| 339.403
| 339.403
|  
|  
|-
|-
|
| 7/5
| 7/5
| 339.417
| 339.417
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
|-
|
| 9/7
| 9/7
| 339.441
| 339.441
| 9-, 11-, and 13-odd-limit minimax
| 9-, 11-, and 13-odd-limit minimax
|-
|-
|
| 15/14
| 15/14
| 339.444
| 339.444
| 15-odd-limit minimax
| 15-odd-limit minimax
|-
|-
|
| 5/3
| 5/3
| 339.455
| 339.455
|  
|  
|-
|-
|
| 5/4
| 5/4
| 339.514
| 339.514
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
|
| 15/8
| 15/8
| 339.541
| 339.541
|  
|  
|-
|-
|
| 3/2
| 3/2
| 339.609
| 339.609
|  
|  
|-
|-
| 15\53
|
| 339.623
| Upper bound of 11-, 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
|-
|
| 15/13
| 15/13
| 339.677
| 339.677
|  
|  
|-
|-
|
| 13/10
| 13/10
| 339.695
| 339.695
|  
|  
|-
|-
|
| 13/9
| 13/9
| 339.789
| 339.789
|  
|  
|-
|-
|
| 13/12
| 13/12
| 339.870
| 339.870
|  
|  
|-
|-
| 17\60
|
| 340.000
| 60de val, upper bound of 7- and 9-odd-limit diamond monotone
|-
|
| 13/8
| 13/8
| 340.088
| 340.088
|  
|  
|-
|-
|
| 15/11
| 15/11
| 340.339
| 340.339
|  
|  
|-
|-
|
| 11/10
| 11/10
| 341.251
| 341.251
|  
|  
|-
|-
|
| 11/9
| 11/9
| 347.408
| 347.408
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[[Category:Amity family]]
[[Category:Amity family]]
[[Category:Ragismic microtemperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Hemifamity temperaments]]
[[Category:Aberschismic temperaments]]
Retrieved from "https://en.xen.wiki/w/Amity"