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Augmented family: Difference between revisions

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{{Interwiki
| en = Augmented family
| de = Übermässige Temperaturen
}}
{{Technical data page}}
{{Technical data page}}
The '''augmented family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diesis a.k.a. augmented comma, [[128/125]], the amount by which three [[5/4]] major thirds fall short of an [[2/1|octave]], and so identifies the major third with the 1/3-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]].  
The '''augmented family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diesis a.k.a. augmented comma, [[128/125]], the amount by which three [[5/4]] major thirds fall short of an [[octave]], and so identifies the major third with the 1/3-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]].  


== Augmented ==
== Augmented ==
Line 58: Line 54:
Comma list: 56/55, 64/63, 100/99
Comma list: 56/55, 64/63, 100/99


Mapping: {{mapping| 3 0 7 18 20 | 0 1 0 -2 -2 }}
{{Mapping|legend=0| 3 0 7 18 20 | 0 1 0 -2 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 73: Line 69:
Comma list: 40/39, 56/55, 64/63, 66/65
Comma list: 40/39, 56/55, 64/63, 66/65


Mapping: {{mapping| 3 0 7 18 20 16 | 0 1 0 -2 -2 -1 }}
{{Mapping|legend=0| 3 0 7 18 20 16 | 0 1 0 -2 -2 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 88: Line 84:
Comma list: 56/55, 64/63, 91/90, 100/99
Comma list: 56/55, 64/63, 91/90, 100/99


Mapping: {{mapping| 3 0 7 18 20 -8 | 0 1 0 -2 -2 4 }}
{{Mapping|legend=0| 3 0 7 18 20 -8 | 0 1 0 -2 -2 4 }}


Optimal tunings:  
Optimal tunings:  
Line 103: Line 99:
Comma list: 56/55, 64/63, 78/77, 100/99
Comma list: 56/55, 64/63, 78/77, 100/99


Mapping: {{mapping| 3 0 7 18 20 35 | 0 1 0 -2 -2 -5 }}
{{Mapping|legend=0| 3 0 7 18 20 35 | 0 1 0 -2 -2 -5 }}


Optimal tunings:  
Optimal tunings:  
Line 118: Line 114:
Comma list: 55/54, 64/63, 77/75
Comma list: 55/54, 64/63, 77/75


Mapping: {{mapping| 3 0 7 18 -4 | 0 1 0 -2 3 }}
{{Mapping|legend=0| 3 0 7 18 -4 | 0 1 0 -2 3 }}


Optimal tunings:  
Optimal tunings:  
Line 152: Line 148:
Comma list: 36/35, 45/44, 56/55
Comma list: 36/35, 45/44, 56/55


Mapping: {{mapping| 3 0 7 -1 1 | 0 1 0 2 2 }}
{{Mapping|legend=0| 3 0 7 -1 1 | 0 1 0 2 2 }}


Optimal tunings:  
Optimal tunings:  
Line 167: Line 163:
Comma list: 27/26, 36/35, 45/44, 56/55
Comma list: 27/26, 36/35, 45/44, 56/55


Mapping: {{mapping| 3 0 7 -1 1 -3 | 0 1 0 2 2 3 }}
{{Mapping|legend=0| 3 0 7 -1 1 -3 | 0 1 0 2 2 3 }}


Optimal tunings:  
Optimal tunings:  
Line 182: Line 178:
Comma list: 26/25, 36/35, 45/44, 56/55
Comma list: 26/25, 36/35, 45/44, 56/55


Mapping: {{mapping| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}
{{Mapping|legend=0| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}


Optimal tunings:  
Optimal tunings:  
Line 214: Line 210:
Comma list: 28/27, 55/54, 128/125
Comma list: 28/27, 55/54, 128/125


Mapping: {{mapping| 3 0 7 -6 -4 | 0 1 0 3 3 }}
{{Mapping|legend=0| 3 0 7 -6 -4 | 0 1 0 3 3 }}


Optimal tunings:  
Optimal tunings:  
Line 246: Line 242:
Comma list: 21/20, 33/32, 128/125
Comma list: 21/20, 33/32, 128/125


Mapping: {{mapping| 3 0 7 13 15 | 0 1 0 -1 -1 }}
{{Mapping|legend=0| 3 0 7 13 15 | 0 1 0 -1 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 282: Line 278:
Comma list: 50/49, 56/55, 125/121
Comma list: 50/49, 56/55, 125/121


Mapping: {{mapping| 6 0 14 17 21 | 0 1 0 0 0 }}
{{Mapping|legend=0| 6 0 14 17 21 | 0 1 0 0 0 }}


Optimal tunings:  
Optimal tunings:  
Line 297: Line 293:
Comma list: 50/49, 56/55, 66/65, 105/104
Comma list: 50/49, 56/55, 66/65, 105/104


Mapping: {{mapping| 6 0 14 17 21 13 | 0 1 0 0 0 1 }}
{{Mapping|legend=0| 6 0 14 17 21 13 | 0 1 0 0 0 1 }}


Optimal tunings:  
Optimal tunings:  
Line 335: Line 331:
Comma list: 49/48, 56/55, 77/75
Comma list: 49/48, 56/55, 77/75


Mapping: {{mapping| 3 0 7 6 8 | 0 2 0 1 1 }}
{{Mapping|legend=0| 3 0 7 6 8 | 0 2 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 354: Line 350:
Comma list: 49/48, 56/55, 66/65, 77/75
Comma list: 49/48, 56/55, 66/65, 77/75


Mapping: {{mapping| 3 0 7 6 8 4 | 0 2 0 1 1 3 }}
{{Mapping|legend=0| 3 0 7 6 8 4 | 0 2 0 1 1 3 }}


Optimal tunings:  
Optimal tunings:  
Line 374: Line 370:
Comma list: 49/48, 56/55, 77/75, 507/500
Comma list: 49/48, 56/55, 77/75, 507/500


Mapping: {{mapping| 6 0 14 12 16 27 | 0 2 0 1 1 -1 }}
{{Mapping|legend=0| 6 0 14 12 16 27 | 0 2 0 1 1 -1 }}


: mapping generators: ~44/39, ~7/4
: mapping generators: ~44/39, ~7/4
Line 412: Line 408:
Comma list: 56/55, 128/125, 243/242
Comma list: 56/55, 128/125, 243/242


Mapping: {{mapping| 3 1 7 -1 1 | 0 2 0 5 5 }}
{{Mapping|legend=0| 3 1 7 -1 1 | 0 2 0 5 5 }}


Optimal tunings:  
Optimal tunings:  
Line 427: Line 423:
Comma list: 56/55, 91/90, 128/125, 243/242
Comma list: 56/55, 91/90, 128/125, 243/242


Mapping: {{mapping| 3 1 7 -1 1 13 | 0 2 0 5 5 -1 }}
{{Mapping|legend=0| 3 1 7 -1 1 13 | 0 2 0 5 5 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 463: Line 459:
Comma list: 56/55, 128/125, 1323/1250
Comma list: 56/55, 128/125, 1323/1250


Mapping: {{mapping| 3 1 7 14 16 | 0 2 0 -3 -3 }}
{{Mapping|legend=0| 3 1 7 14 16 | 0 2 0 -3 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 478: Line 474:
Comma list: 56/55, 66/65, 105/104, 507/500
Comma list: 56/55, 66/65, 105/104, 507/500


Mapping: {{mapping| 3 1 7 14 16 13 | 0 2 0 -3 -3 -1 }}
{{Mapping|legend=0| 3 1 7 14 16 13 | 0 2 0 -3 -3 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 514: Line 510:
Comma list: 56/55, 128/125, 2401/2400
Comma list: 56/55, 128/125, 2401/2400


Mapping: {{mapping| 3 3 7 8 10 | 0 4 0 1 1 }}
{{Mapping|legend=0| 3 3 7 8 10 | 0 4 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 529: Line 525:
Comma list: 56/55, 78/77, 128/125, 507/500
Comma list: 56/55, 78/77, 128/125, 507/500


Mapping: {{mapping| 3 3 7 8 10 12 | 0 4 0 1 1 -2 }}
{{Mapping|legend=0| 3 3 7 8 10 12 | 0 4 0 1 1 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 565: Line 561:
Comma list: 56/55, 128/125, 3773/3645
Comma list: 56/55, 128/125, 3773/3645


Mapping: {{mapping| 6 1 14 4 8 | 0 2 0 3 3 }}
{{Mapping|legend=0| 6 1 14 4 8 | 0 2 0 3 3 }}


Optimal tunings:  
Optimal tunings:  
Line 576: Line 572:


== Trisected ==
== Trisected ==
{{Main|Trisected}}
{{See also|Subgroup temperaments #Trisect}}
Trisected tempers out [[1029/1024]] and [[1029/1000]], so it is the intersection of augmented and [[slendric]]. It can be described as the {{nowrap| 15 & 36 }} temperament.
Trisected tempers out [[1029/1024]] and [[1029/1000]], so it is the intersection of augmented and [[slendric]]. It can be described as the {{nowrap| 15 & 36 }} temperament.


It was named by [[User:Overthink|Overthink]] in 2026, since it splits every Pythagorean interval in three.
It was named by [[User:Overthink|Overthink]] in 2026, since it splits every Pythagorean interval in three.
{{Todo|review|comment=Last digits in ~8/7 for WE may be wrong. Also, comma lists could potentially be simplified.}}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 585: Line 582:
[[Comma list]]: 128/125, 1029/1000
[[Comma list]]: 128/125, 1029/1000


{{Mapping|legend=1| 3 3 7 9 | 0 3 0 -1 }}
{{Mapping|legend=1| 3 0 7 10 | 0 3 0 -1 }}
 
: mapping generators: ~5/4, ~10/7
: mapping generators: ~5/4, ~8/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~5/4 = 399.520{{c}}, ~8/7 = 234.195{{c}}
* [[WE]]: ~5/4 = 399.5199{{c}}, ~10/7 = 633.7146{{c}}
: [[error map]]: {{val| -1.440 -0.811 10.325 -7.342 }}
: [[error map]]: {{val| -1.440 -0.811 +10.325 -7.342 }}
 
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~10/7 = 634.3393{{c}}
* [[CWE]]: ~5/4 = 400.000{{c}}, ~8/7 = 234.339{{c}}
: error map: {{val| 0.000 +1.063 +13.686 -3.165 }}
: error map: {{val| 0.000 +1.063 +13.686 -3.165 }}


Line 601: Line 596:


=== 11-limit ===
=== 11-limit ===
This extension also tempers out [[4000/3993]], equating 3 intervals of [[11/10]] with [[4/3]].


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 606: Line 602:
Comma list: 56/55, 128/125, 1029/1000
Comma list: 56/55, 128/125, 1029/1000


Mapping: {{Mapping| 3 3 7 9 11 | 0 3 0 -1 -1 }}
Mapping: {{Mapping| 3 0 7 10 12 | 0 3 0 -1 -1 }}
 
: mapping generators: ~5/4, ~8/7


Optimal tunings:  
Optimal tunings:  
* WE: ~5/4 = 399.305{{c}}, ~8/7 = 234.484{{c}}
* WE: ~5/4 = 399.3049{{c}}, ~10/7 = 633.7891{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~10/7 = 634.7695{{c}}


* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.769{{c}}
{{Optimal ET sequence|legend=0| 15, 36, 51ce }}


{{Optimal ET sequence|legend=1| 15, 36, 51ce }}
Badness (Sintel): 1.60
 
[[Badness]] (Sintel): 1.60


=== 13-limit ===
=== 13-limit ===
This extension is natural since it equates the [[10/7]], which is 1/3 of [[3/1]] since 1029/1000 is tempered out, with [[13/9]], thus tempering out [[91/90]] and [[2197/2187]].


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 625: Line 619:
Comma list: 56/55, 91/90, 128/125, 1029/1000
Comma list: 56/55, 91/90, 128/125, 1029/1000


Mapping: {{Mapping| 3 3 7 9 11 7 | 0 3 0 -1 -1 7 }}
Mapping: {{Mapping| 3 0 7 10 12 0 | 0 3 0 -1 -1 7 }}
 
: mapping generators: ~5/4, ~8/7


Optimal tunings:  
Optimal tunings:  
* WE: ~5/4 = 399.319{{c}}, ~8/7 = 234.745{{c}}
* WE: ~5/4 = 399.3194{{c}}, ~10/7 = 634.0637{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~10/7 = 634.9907{{c}}


* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.991{{c}}
{{Optimal ET sequence|legend=0| 15, 36, 51ce }}


{{Optimal ET sequence|legend=1| 15, 36, 51ce }}
Badness (Sintel): 1.70
 
[[Badness]] (Sintel): 1.70
 
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 56/55, 91/90, 128/125, 136/135, 1029/1000
 
Mapping: {{Mapping| 3 3 7 9 11 7 7 | 0 3 0 -1 -1 7 9 }}
 
: mapping generators: ~5/4, ~8/7
 
Optimal tunings:
* WE: ~5/4 = 399.317{{c}}, ~8/7 = 234.609{{c}}
 
* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.852{{c}}
 
{{Optimal ET sequence|legend=1| 15g, 36, 51ceg }}
 
[[Badness]] (Sintel): 1.83
 
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 1029/1000
 
Mapping: {{Mapping| 3 3 7 9 11 7 7 10 | 0 3 0 -1 -1 7 9 3 }}
 
: mapping generators: ~5/4, ~8/7
 
Optimal tunings:
* WE: ~5/4 = 399.328{{c}}, ~8/7 = 234.614{{c}}
 
* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.852{{c}}
 
{{Optimal ET sequence|legend=1| 15g, 36, 51ceg }}
 
[[Badness]] (Sintel): 1.62
 
=== 23-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 161/160, 1029/1000
 
Mapping: {{Mapping| 3 3 7 9 11 7 7 10 13 | 0 3 0 -1 -1 7 9 3 1 }}
 
: mapping generators: ~5/4, ~8/7
 
Optimal tunings:
* WE: ~5/4 = 399.353{{c}}, ~8/7 = 234.601{{c}}
 
* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.843{{c}}
 
{{Optimal ET sequence|legend=1| 15g, 36, 51ceg }}
 
[[Badness]] (Sintel): 1.54
 
=== 29-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 161/160, 203/200, 1029/1000
 
Mapping: {{Mapping| 3 3 7 9 11 7 7 10 13 14 | 0 3 0 -1 -1 7 9 3 1 1 }}
 
: mapping generators: ~5/4, ~8/7
 
Optimal tunings:
* WE: ~5/4 = 399.388{{c}}, ~8/7 = 234.581{{c}}
 
* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.819{{c}}
 
{{Optimal ET sequence|legend=1| 15g, 36, 51ceg }}
 
[[Badness]] (Sintel): 1.52


== Niner ==
== Niner ==
Line 740: Line 655:
Comma list: 56/55, 128/125, 540/539
Comma list: 56/55, 128/125, 540/539


Mapping: {{mapping| 9 0 21 11 17 | 0 1 0 1 1 }}
{{Mapping|legend=0| 9 0 21 11 17 | 0 1 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 755: Line 670:
Comma list: 56/55, 78/77, 91/90, 128/125
Comma list: 56/55, 78/77, 91/90, 128/125


Mapping: {{mapping| 9 0 21 11 17 19 | 0 1 0 1 1 1 }}
{{Mapping|legend=0| 9 0 21 11 17 19 | 0 1 0 1 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 809: Line 724:
[[Comma list]]: 76/75, 96/95
[[Comma list]]: 76/75, 96/95


{{Mapping|legend=2| 3 0 7 8 | 0 1 0 1 }}
{{Mapping|legend=3| 3 0 7 8 | 0 1 0 1 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 822: Line 737:
* [https://web.archive.org/web/20211201070113/https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]
* [https://web.archive.org/web/20211201070113/https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]


[[Category:Augmented family| ]] <!-- main article -->
[[Category:Augmented| ]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Augmented family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Augmented_family"