Amity: Difference between revisions

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<nowiki />* In 7-limit POTE tuning, octave reduced
<nowiki />* In 7-limit POTE tuning, octave reduced


== Tuning spectra ==
== Tunings ==
=== Amity ===
=== Tunings spectra ===
Gencom: [2 128/105; 352/351 540/539 625/624 729/728]
==== Amity ====
 
Gencom mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]
 
{| class="wikitable center-all left-3"
{| class="wikitable center-all left-3"
|-
|-
! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Generator<br />(¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 10/9
| 9/5
| 339.199
| 339.199
|  
|  
Line 283: Line 280:
|  
|  
|-
|-
| 8/7
| 7/4
| 339.343
| 339.343
|  
|  
Line 303: Line 300:
|  
|  
|-
|-
| 6/5
| 5/3
| 339.455
| 339.455
|  
|  
|-
|-
| 14/11
| 11/7
| 339.462
| 339.462
| 11-odd-limit minimax
| 11-odd-limit minimax
Line 319: Line 316:
|  
|  
|-
|-
| 12/11
| 11/6
| 339.485
| 339.485
|  
|  
Line 331: Line 328:
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
| 14/13
| 13/7
| 339.505
| 339.505
|  
|  
Line 339: Line 336:
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
| 15/8
| 339.541
| 339.541
|  
|  
|-
|-
| 18/13
| 13/9
| 339.551
| 339.551
|  
|  
Line 351: Line 348:
|  
|  
|-
|-
| 16/13
| 13/8
| 339.563
| 339.563
|  
|  
Line 363: Line 360:
|  
|  
|-
|-
| 4/3
| 3/2
| 339.609
| 339.609
|  
|  
|}
|}


=== Hitchcock ===
==== Hitchcock ====
Gencom: [2 11/9; 121/120 169/168 176/175 325/324]
 
Gencom mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]
 
{| class="wikitable center-all left-3"
{| class="wikitable center-all left-3"
|-
|-
! Eigenmonzo<br />(unchanged-interval)
! Eigenmonzo<br>(unchanged-interval)*
! Generator<br>(¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 12/11
| 11/6
| 337.659
| 337.659
|  
|  
Line 387: Line 380:
|  
|  
|-
|-
| 14/13
| 13/7
| 338.936
| 338.936
|  
|  
|-
|-
| 14/11
| 11/7
| 339.135
| 339.135
|  
|  
|-
|-
| 10/9
| 9/5
| 339.199
| 339.199
|  
|  
Line 403: Line 396:
|  
|  
|-
|-
| 8/7
| 7/4
| 339.343
| 339.343
|  
|  
Line 423: Line 416:
| 15-odd-limit minimax
| 15-odd-limit minimax
|-
|-
| 6/5
| 5/3
| 339.455
| 339.455
|  
|  
Line 431: Line 424:
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
| 15/8
| 339.541
| 339.541
|  
|  
|-
|-
| 4/3
| 3/2
| 339.609
| 339.609
|  
|  
Line 447: Line 440:
|  
|  
|-
|-
| 18/13
| 13/9
| 339.789
| 339.789
|  
|  
Line 455: Line 448:
|  
|  
|-
|-
| 16/13
| 13/8
| 340.088
| 340.088
|  
|  
Line 471: Line 464:
|  
|  
|}
|}
<nowiki/>* Besides the octave


== Music ==
== Music ==

Revision as of 12:28, 5 June 2025

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.

For technical data, see Amity family #Amity.

Etymology

Origin: Paul Erlich (2002)

Meaning: A restructuring of the words "acute minor third." The ideal generator for Amity is between a minor third and neutral third.

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

Tunings

Tunings spectra

Amity

Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
9/5 339.199
13/11 339.281
7/4 339.343
7/6 339.403
7/5 339.417 7-odd-limit minimax
9/7 339.441 9-odd-limit minimax
15/14 339.444
5/3 339.455
11/7 339.462 11-odd-limit minimax
11/9 339.473
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
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

Hitchcock

Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
11/6 337.659
11/8 338.742
13/7 338.936
11/7 339.135
9/5 339.199
13/11 339.281
7/4 339.343
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
5/3 339.455
5/4 339.514 5-odd-limit minimax
15/8 339.541
3/2 339.609
15/13 339.677
13/10 339.695
13/9 339.789
13/12 339.870
13/8 340.088
15/11 340.339
11/10 341.251
11/9 347.408

* Besides the octave

Music

Francium
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