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'''Amity''' is a temperament for the 5, 7, 11, and 13 [[Harmonic limit|prime limits]]. EDOs that support amity include [[46edo]], [[53edo]], [[99edo]], [[152edo]], and [[205edo]].
{{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 }}.


See [[Ragismic_microtemperaments#Amity|Ragismic microtemperaments]] for more information.
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]].


=Interval chain=
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>.
{| class="wikitable"
 
{{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"| Generators
! rowspan="2" | 7-limit
! rowspan="2"| Cents* <br>(octave-reduced)
! colspan="2" | 13-limit extensions
! colspan="2"| Approximate ratios
|-
|-
! | Amity <br>(53&amp;205)
! Amity ({{nowrap| 53 & 152 }})
! | Hitchcock <br>(46&amp;53)
! Hitchcock ({{nowrap| 46 & 53 }})
|-
|-
| | 0
| 0
| style="text-align:right;" | 0.000
| 0.00
| colspan="2"| 1/1
| '''1/1'''
|
|
|-
|-
| | 1
| 1
| style="text-align:right;" | 339.43
| 339.43
| |  
| 128/105
| | 11/9
|
| 11/9
|-
|-
| | 2
| 2
| style="text-align:right;" | 678.86
| 678.87
| colspan="2"| 40/27
| 40/27
|
|
|-
|-
| | 3
| 3
| style="text-align:right;" | 1018.30
| 1018.30
| colspan="2"| 9/5
| 9/5
|
|
|-
|-
| | 4
| 4
| style="text-align:right;" | 157.73
| 157.74
| | 35/32
| 35/32
| | 12/11~35/32~11/10
|
| 12/11, 11/10
|-
|-
| | 5
| 5
| style="text-align:right;" | 497.16
| 497.17
| colspan="2"| 4/3
| '''4/3'''
|
|
|-
|-
| | 6
| 6
| style="text-align:right;" | 836.59
| 836.61
| |  
| 81/50
| | 21/13~13/8
|
| '''13/8''', 21/13
|-
|-
| | 7
| 7
| style="text-align:right;" | 1176.03
| 1176.04
| | 65/33, 77/39~160/81
| 63/32, 160/81
| | 63/32~128/65~65/33, <br>77/39~160/81
| 65/33, 77/39
| 65/33, 77/39, 128/65
|-
|-
| | 8
| 8
| style="text-align:right;" | 315.46
| 315.48
| colspan="2"| 6/5
| 6/5
|
|
|-
|-
| | 9
| 9
| style="text-align:right;" | 654.89
| 654.91
| |  
| 35/24
| | 16/11~22/15
|
| '''16/11''', 22/15
|-
|-
| | 10
| 10
| style="text-align:right;" | 994.32
| 994.35
| colspan="2"| 16/9
| '''16/9'''
|
| 39/22
|-
|-
| | 11
| 11
| style="text-align:right;" | 133.75
| 133.78
| | 27/25
| 27/25
| | 14/13~27/25~13/12
|
| 13/12, 14/13
|-
|-
| | 12
| 12
| style="text-align:right;" | 473.19
| 473.22
| colspan="2"| 21/16
| '''21/16'''
|
|
|-
|-
| | 13
| 13
| style="text-align:right;" | 812.62
| 812.65
| colspan="2"| 8/5
| '''8/5'''
|
|
|-
|-
| | 14
| 14
| style="text-align:right;" | 1152.05
| 1152.09
| | 35/18
| 35/18
| | 64/33~35/18
|
| 39/20, 64/33, 88/45
|-
|-
| | 15
| 15
| style="text-align:right;" | 291.48
| 291.52
| colspan="2"| 13/11~32/27
| 32/27
| 13/11
| 13/11
|-
|-
| | 16
| 16
| style="text-align:right;" | 630.92
| 630.96
| |  
| 36/25
| | 13/9
|
| 13/9
|-
|-
| | 17
| 17
| style="text-align:right;" | 970.35
| 970.39
| colspan="2"| 7/4
| '''7/4'''
|
|
|-
|-
| | 18
| 18
| style="text-align:right;" | 109.78
| 109.83
| colspan="2"| 16/15
| '''16/15'''
|
|
|-
|-
| | 19
| 19
| style="text-align:right;" | 449.21
| 449.26
| |  
| 35/27
| | 13/10
|
| 13/10
|-
|-
| | 20
| 20
| style="text-align:right;" | 788.64
| 788.70
| |  
| 63/40
| |
|
| 52/33
|-
|-
| | 21
| 21
| style="text-align:right;" | 1128.08
| 1128.13
| | 48/25~25/13
| 48/25
| | 21/11~48/25
| 25/13
| 21/11, 52/27
|-
|-
| | 22
| 22
| style="text-align:right;" | 267.51
| 267.57
| colspan="2"| 7/6
| 7/6
|
|
|-
|-
| | 23
| 23
| style="text-align:right;" | 606.94
| 607.00
| |  
| 64/45
| |  
|
|  
|-
|-
| | 24
| 24
| style="text-align:right;" | 946.37
| 946.44
| |  
| 81/70
| | 26/15
|
| 26/15
|-
|-
| | 25
| 25
| style="text-align:right;" | 85.81
| 85.87
| colspan="2"| 21/20
| 21/20
|
|
|-
|-
| | 26
| 26
| style="text-align:right;" | 425.24
| 425.31
| | 32/25
| 32/25
| | 14/11~32/25
|
| 14/11
|-
|-
| | 27
| 27
| style="text-align:right;" | 764.67
| 764.74
| colspan="2"| 14/9
| 14/9
|
|
|-
|-
| | 28
| 28
| style="text-align:right;" | 1104.10
| 1104.18
| |  
| 256/135
| |  
|
|  
|-
|-
| | 29
| 29
| style="text-align:right;" | 243.53
| 243.61
| | 15/13
| 147/128
| |  
| 15/13
|  
|-
|-
| | 30
| 30
| style="text-align:right;" | 582.97
| 583.05
| colspan="2"| 7/5
| 7/5
|
|
|-
|-
| | 31
| 31
| style="text-align:right;" | 922.40
| 922.48
| |  
| 128/75
| |
|
| 56/33
|-
|-
| | 32
| 32
| style="text-align:right;" | 61.83
| 61.92
| | 28/27~27/26
| 28/27
| | 28/27
| 27/26
|  
|-
|-
| | 33
| 33
| style="text-align:right;" | 401.26
| 401.35
| |  
| 63/50
| |  
|
|  
|-
|-
| | 34
| 34
| style="text-align:right;" | 740.69
| 740.79
| | 20/13
| 49/32
| |  
| 20/13
|  
|-
|-
| | 35
| 35
| style="text-align:right;" | 1080.13
| 1080.22
| colspan="2"| 28/15
| 28/15
|
|
|-
|-
| | 36
| 36
| style="text-align:right;" | 219.56
| 219.66
| | 25/22
| 256/225
| |  
| 25/22
|  
|-
|-
| | 37
| 37
| style="text-align:right;" | 558.99
| 559.09
| | 18/13
| 112/81
| |  
| 18/13
|  
|-
| 38
| 898.53
| 42/25
|
|
|-
| 39
| 37.96
| 49/48
| 40/39, 45/44
|
|}
|}
<span style="font-family: Arial,Helvetica,sans-serif;">*in 7-limit POTE tuning</span>
<nowiki/>* In 7-limit CWE tuning, octave reduced


=Tuning Spectra=
== Tunings ==
==Spectrum of Amity Tunings (53&amp;205)==
=== Tunings spectra ===
 
==== Amity ====
13-limit commas: 352/351, 540/539, 625/624, 729/728
{| class="wikitable center-all left-4"
{| class="wikitable"
|-
! 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
|-
|-
! | Eigenmonzo
|  
! | Generator
| 7/6
| 339.403
|
|-
|-
| | 10/9
|  
| | 339.199
| 7/5
| 339.417
| 7-odd-limit minimax
|-
|-
| | 13/11
|  
| | 339.281
| 9/7
| 339.441
| 9-odd-limit minimax
|-
|-
| | 8/7
|  
| | 339.343
| 15/14
| 339.444
|
|-
|-
| | 7/6
|  
| | 339.403
| 5/3
| 339.455
|
|-
|-
| | 7/5
|  
| | 339.417 (7 limit minimax)
| 11/7
| 339.462
| 11-odd-limit minimax
|-
|-
| | 9/7
|  
| | 339.441 (9 limit minimax)
| 11/9
| 339.473
|
|-
|-
| | 15/14
| 43\152
| | 339.444
|  
| 339.474
| 152f val
|-
|-
| | 6/5
|  
| | 339.455
| 15/11
| 339.476
|
|-
|-
| | 14/11
|  
| | 339.462 (11 limit minimax)
| 11/6
| 339.485
|
|-
|-
| | 11/9
|  
| | 339.473
| 11/10
| 339.490
|
|-
|-
| | 15/11
|  
| | 339.476
| 11/8
| 339.495
| 13- and 15-odd-limit minimax
|-
|-
| | 12/11
|  
| | 339.485
| 13/7
| 339.505
|
|-
|-
| | 11/10
| 58\205
| | 339.490
|  
| 339.512
|
|-
|-
| | 11/8
|  
| | 339.495 (13, 15 limit minimax)
| 5/4
| 339.514
| 5-odd-limit minimax
|-
|-
| | 14/13
|  
| | 339.505
| 15/8
| 339.541
|
|-
|-
| | 5/4
|  
| | 339.514 (5 limit minimax)
| 13/9
| 339.551
|
|-
|-
| | 16/15
|  
| | 339.541
| 13/12
| 339.558
|
|-
|-
| | 18/13
|  
| | 339.551
| 13/8
| 339.563
|
|-
|-
| | 13/12
|  
| | 339.558
| 15/13
| 339.577
|
|-
|-
| | 16/13
|  
| | 339.563
| 13/10
| 339.582
|
|-
|-
| | 15/13
|  
| | 339.577
| 3/2
| 339.609
|
|-
|-
| | 13/10
| 15\53
| | 339.582
|  
| 339.623
| Upper bound of 11-, 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
|-
|-
| | 4/3
| 17\60
| | 339.609
|  
| 340.000
| 60deee… val, upper bound of 7- and 9-odd-limit diamond monotone
|}
|}


==Spectrum of Hitchcock Tunings (46&amp;53)==
==== Hitchcock ====
 
{| class="wikitable center-all left-4"
13-limit commas: 121/120, 169/168, 176/175, 325/324
|-
{| class="wikitable"
! 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
|  
|-
|-
! | Eigenmonzo
| 13\46
! | Generator
|  
| 339.130
| Lower bound of 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
|-
|-
| | 12/11
|  
| | 337.659
| 11/7
| 339.135
|  
|-
|-
| | 11/8
|  
| | 338.742
| 9/5
| 339.199
|  
|-
|-
| | 14/13
|  
| | 338.936
| 13/11
| 339.281
|  
|-
|-
| | 14/11
|  
| | 339.135
| 7/4
| 339.343
|
|-
|-
| | 10/9
| 28\99
| | 339.199
|  
| 339.394
|
|-
|-
| | 13/11
|  
| | 339.281
| 7/6
| 339.403
|
|-
|-
| | 8/7
|  
| | 339.343
| 7/5
| 339.417
| 7-odd-limit minimax
|-
|-
| | 7/6
|  
| | 339.403
| 9/7
| 339.441
| 9-, 11-, and 13-odd-limit minimax
|-
|-
| | 7/5
|  
| | 339.417 (7 limit minimax)
| 15/14
| 339.444
| 15-odd-limit minimax
|-
|-
| | 9/7
|  
| | 339.441 (9, 11, 13 limit minimax)
| 5/3
| 339.455
|
|-
|-
| | 15/14
|  
| | 339.444 (15 limit minimax)
| 5/4
| 339.514
| 5-odd-limit minimax
|-
|-
| | 6/5
|  
| | 339.455
| 15/8
| 339.541
|
|-
|-
| | 5/4
|  
| | 339.514 (5 limit minimax)
| 3/2
| 339.609
|
|-
|-
| | 16/15
| 15\53
| | 339.541
|  
| 339.623
| Upper bound of 11-, 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
|-
|-
| | 4/3
|  
| | 339.609
| 15/13
| 339.677
|
|-
|-
| | 15/13
|  
| | 339.677
| 13/10
| 339.695
|
|-
|-
| | 13/10
|  
| | 339.695
| 13/9
| 339.789
|
|-
|-
| | 18/13
|  
| | 339.789
| 13/12
| 339.870
|
|-
|-
| | 13/12
| 17\60
| | 339.870
|  
| 340.000
| 60de val, upper bound of 7- and 9-odd-limit diamond monotone
|-
|-
| | 16/13
|  
| | 340.088
| 13/8
| 340.088
|
|-
|-
| | 15/11
|  
| | 340.339
| 15/11
| 340.339
|
|-
|-
| | 11/10
|  
| | 341.251
| 11/10
| 341.251
|
|-
|-
| | 11/9
|  
| | 347.408
| 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]]
[[Category:Amity| ]] <!-- main article -->
[[Category:Temperament]]
[[Category:Rank-2 temperaments]]
[[Category:Amity family]]
[[Category:Ragismic microtemperaments]]
[[Category:Aberschismic temperaments]]
Retrieved from "https://en.xen.wiki/w/Amity"