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The 5-limit parent comma for the '''augmented family''' is 128/125, the [[128/125|diesis]]. Its [[monzo]] is {{Monzo|7 0 -3}}, and flipping that yields {{multival|3 0 -7}} for the [[wedgie]]. Hence the period is 1/3 octave, and this is what is used for a major third. The [[generator]] can be taken as a fifth or a semitone, and [[12 EDO]], with its excellent fifth, is an obvious tuning for [[5-limit]] augmented, though a sharper fifth might be preferred to go with the sharp third.
The 5-limit parent comma for the '''augmented family''' is [[128/125]], the diesis. Its [[monzo]] is {{monzo| 7 0 -3 }}, and flipping that yields {{multival| 3 0 -7 }} for the [[wedgie]]. Hence the period is 1/3 octave, and this is what is used for 5/4, the classical major third. The [[generator]] can be taken as a fifth or a semitone, and [[12edo]], with its excellent fifth, is an obvious tuning for [[5-limit]] augmented, though a sharper fifth might be preferred to go with the sharp third.


== Augmented ==
== Augmented ==
Subgroup: 2.3.5
Subgroup: 2.3.5


[[Comma]]: 128/125
[[Comma list]]: 128/125


[[Mapping]]: [{{val|3 0 7}}, {{val|0 1 0}}]
[[Mapping]]: [{{val| 3 0 7 }}, {{val| 0 1 0 }}]
 
Mapping generators: ~5/4, ~3


[[POTE generator]]: ~3/2 = 706.638
[[POTE generator]]: ~3/2 = 706.638
Line 14: Line 16:
[[Badness]]: 0.022315
[[Badness]]: 0.022315


=== Seven-limit children ===
=== Seven-limit extensions ===
The second comma of the [[Normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. August adds 36/35, augene 64/63, hexe 256/245, hemiaug 245/243, and triforce 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quartertones instead of semitones.
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. August adds 36/35, augene 64/63, hexe 256/245, hemiaug 245/243, and triforce 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quartertones instead of semitones.


== August ==
== August ==
Line 22: Line 24:
[[Comma list]]: 36/35, 128/125
[[Comma list]]: 36/35, 128/125


[[Mapping]]: [{{val|3 0 7 -1}}, {{val|0 1 0 2}}]
[[Mapping]]: [{{val| 3 0 7 -1 }}, {{val| 0 1 0 2 }}]


{{Multival|legend=1|3 0 6 -7 1 14}}
{{Multival|legend=1| 3 0 6 -7 1 14 }}


[[POTE generator]]: ~3/2 = 696.011
[[POTE generator]]: ~3/2 = 696.011
Line 37: Line 39:
Comma list: 36/35, 45/44, 56/55
Comma list: 36/35, 45/44, 56/55


Mapping: [{{val|3 0 7 -1 1}}, {{val|0 1 0 2 2}}]
Mapping: [{{val| 3 0 7 -1 1 }}, {{val| 0 1 0 2 2 }}]


POTE generator: ~3/2 = 692.514
POTE generator: ~3/2 = 692.514
Line 50: Line 52:
Comma list: 27/26, 36/35, 45/44, 56/55
Comma list: 27/26, 36/35, 45/44, 56/55


Mapping: [{{val|3 0 7 -1 1 -3}}, {{val|0 1 0 2 2 3}}]
Mapping: [{{val| 3 0 7 -1 1 -3 }}, {{val| 0 1 0 2 2 3 }}]


POTE generator: ~3/2 = 688.783
POTE generator: ~3/2 = 688.783
Line 63: Line 65:
Comma list: 26/25, 36/35, 45/44, 56/55
Comma list: 26/25, 36/35, 45/44, 56/55


Mapping: [{{val|3 0 7 -1 1 11}}, {{val|0 1 0 2 2 0}}]
Mapping: [{{val| 3 0 7 -1 1 11 }}, {{val| 0 1 0 2 2 0 }}]


POTE generator: ~3/2 = 685.356
POTE generator: ~3/2 = 685.356
Line 76: Line 78:
[[Comma list]]: 64/63, 126/125
[[Comma list]]: 64/63, 126/125


[[Mapping]]: [{{val|3 0 7 18}}, {{val|0 1 0 -2}}]
[[Mapping]]: [{{val| 3 0 7 18 }}, {{val| 0 1 0 -2 }}]


{{Multival|legend=1|3 0 -6 -7 -18 -14}}
{{Multival|legend=1|3 0 -6 -7 -18 -14}}
Line 91: Line 93:
Comma list: 56/55, 64/63, 100/99
Comma list: 56/55, 64/63, 100/99


Mapping: [{{val|3 0 7 18 20}}, {{val|0 1 0 -2 -2}}]
Mapping: [{{val| 3 0 7 18 20 }}, {{val| 0 1 0 -2 -2 }}]


POTE generator: ~3/2 = 711.177
POTE generator: ~3/2 = 711.177
Line 104: Line 106:
Comma list: 40/39, 56/55, 64/63, 66/65
Comma list: 40/39, 56/55, 64/63, 66/65


Mapping: [{{val|3 0 7 18 20 16}}, {{val|0 1 0 -2 -2 -1}}]
Mapping: [{{val| 3 0 7 18 20 16 }}, {{val| 0 1 0 -2 -2 -1 }}]


POTE generator: ~3/2 = 712.013
POTE generator: ~3/2 = 712.013


Optimal GPV sequence: {{Val list| 12f, 15, 27ef }}
Optimal GPV sequence: {{Val list| 12f, 15, 27eff }}


Badness: 0.020785
Badness: 0.020785
Line 117: Line 119:
Comma list: 56/55, 64/63, 91/90, 100/99
Comma list: 56/55, 64/63, 91/90, 100/99


Mapping: [{{val|3 0 7 18 20 -8}}, {{val|0 1 0 -2 -2 4}}]
Mapping: [{{val| 3 0 7 18 20 -8 }}, {{val| 0 1 0 -2 -2 4 }}]


POTE generator: ~3/2 = 712.609
POTE generator: ~3/2 = 712.609
Line 130: Line 132:
Comma list: 56/55, 64/63, 78/77, 100/99
Comma list: 56/55, 64/63, 78/77, 100/99


Mapping: [{{val|3 0 7 18 20 35}}, {{val|0 1 0 -2 -2 -5}}]
Mapping: [{{val| 3 0 7 18 20 35 }}, {{val| 0 1 0 -2 -2 -5 }}]


POTE generator: ~3/2 = 709.677
POTE generator: ~3/2 = 709.677
Line 143: Line 145:
Comma list: 55/54, 64/63, 77/75
Comma list: 55/54, 64/63, 77/75


Mapping: [{{val|3 0 7 18 -4}}, {{val|0 1 0 -2 3}}]
Mapping: [{{val| 3 0 7 18 -4 }}, {{val| 0 1 0 -2 3 }}]


POTE generator: ~3/2 = 714.150
POTE generator: ~3/2 = 714.150


Optimal GPV sequence: {{Val list| 15, 27, 42 }}
Optimal GPV sequence: {{Val list| 12e, 15, 27, 42 }}


Badness: 0.035554
Badness: 0.035554
Line 160: Line 162:
[[Comma list]]: 28/27, 128/125
[[Comma list]]: 28/27, 128/125


[[Mapping]]: [{{val|3 0 7 -6}}, {{val|0 1 0 3}}]
[[Mapping]]: [{{val| 3 0 7 -6 }}, {{val| 0 1 0 3 }}]


{{Multival|legend=1|3 0 9 -7 6 21}}
{{Multival|legend=1| 3 0 9 -7 6 21 }}


[[POTE generator]]: ~3/2 = 722.719
[[POTE generator]]: ~3/2 = 722.719
Line 175: Line 177:
Comma list: 28/27, 55/54, 128/125
Comma list: 28/27, 55/54, 128/125


Mapping: [{{val|3 0 7 -6 -4}}, {{val|0 1 0 3 3}}]
Mapping: [{{val| 3 0 7 -6 -4 }}, {{val| 0 1 0 3 3 }}]


POTE generator: ~3/2 = 722.663
POTE generator: ~3/2 = 722.663
Line 188: Line 190:
[[Comma list]]: 21/20, 128/125
[[Comma list]]: 21/20, 128/125


[[Mapping]]: [{{val|3 0 7 13}}, {{val|0 1 0 -1}}]
[[Mapping]]: [{{val| 3 0 7 13 }}, {{val| 0 1 0 -1 }}]


{{Multival|legend=1|3 0 -3 -7 -13 -7}}
{{Multival|legend=1| 3 0 -3 -7 -13 -7 }}


[[POTE generator]]: ~3/2 = 681.629
[[POTE generator]]: ~3/2 = 681.629
Line 203: Line 205:
Comma list: 21/20, 33/32, 128/125
Comma list: 21/20, 33/32, 128/125


Mapping: [{{val|3 0 7 13 15}}, {{val|0 1 0 -1 -1}}]
Mapping: [{{val| 3 0 7 13 15 }}, {{val| 0 1 0 -1 -1 }}]


POTE generator: ~3/2 = 680.042
POTE generator: ~3/2 = 680.042
Line 216: Line 218:
[[Comma list]]: 50/49, 128/125
[[Comma list]]: 50/49, 128/125


[[Mapping]]: [{{val|6 0 14 17}}, {{val|0 1 0 0}}]
[[Mapping]]: [{{val| 6 0 14 17 }}, {{val| 0 1 0 0 }}]


{{Multival|legend=1|6 0 0 -14 -17 0}}
Mapping generators: ~28/25, ~3
 
{{Multival|legend=1| 6 0 0 -14 -17 0 }}


[[POTE generator]]: ~3/2 = 710.963
[[POTE generator]]: ~3/2 = 710.963
Line 231: Line 235:
Comma list: 50/49, 56/55, 125/121
Comma list: 50/49, 56/55, 125/121


Mapping: [{{val|6 0 14 17 21}}, {{val|0 1 0 0 0}}]
Mapping: [{{val| 6 0 14 17 21 }}, {{val| 0 1 0 0 0 }}]


POTE generator: ~3/2 = 714.304
POTE generator: ~3/2 = 714.304
Line 244: Line 248:
Comma list: 50/49, 56/55, 66/65, 105/104
Comma list: 50/49, 56/55, 66/65, 105/104


Mapping: [{{val|6 0 14 17 21 13}}, {{val|0 1 0 0 0 1}}]
Mapping: [{{val| 6 0 14 17 21 13 }}, {{val| 0 1 0 0 0 1 }}]


POTE generator: ~3/2 = 710.005
POTE generator: ~3/2 = 710.005
Line 257: Line 261:
[[Comma list]]: 128/125, 245/243
[[Comma list]]: 128/125, 245/243


[[Mapping]]: [{{val|3 1 7 -1}}, {{val|0 2 0 5}}]
[[Mapping]]: [{{val| 3 1 7 -1 }}, {{val| 0 2 0 5 }}]


{{Multival|legend=1|6 0 15 -14 7 35}}
Mapping generators: ~5/4, ~14/9


[[POTE generator]]: ~28/27 = 45.118
{{Multival|legend=1| 6 0 15 -14 7 35 }}
 
[[POTE generator]]: ~14/9 = 754.882 (or ~36/35 = 45.118)


{{Val list|legend=1| 24, 27, 105bcc }}
{{Val list|legend=1| 24, 27, 105bcc }}
Line 272: Line 278:
Comma list: 56/55, 128/125, 245/243
Comma list: 56/55, 128/125, 245/243


Mapping: [{{val|3 1 7 -1 1}}, {{val|0 2 0 5 5}}]
Mapping: [{{val| 3 1 7 -1 1 }}, {{val| 0 2 0 5 5 }}]


POTE generator: ~28/27 = 45.788
POTE generator: ~14/9 = 754.212 (or ~36/35 = 45.788)


Optimal GPV sequence: {{Val list| 24, 27e, 51ce, 78cee }}
Optimal GPV sequence: {{Val list| 24, 27e, 51ce, 78cee }}
Line 285: Line 291:
Comma list: 56/55, 91/90, 128/125, 245/243
Comma list: 56/55, 91/90, 128/125, 245/243


Mapping: [{{val|3 1 7 -1 1 13}}, {{val|0 2 0 5 5 -1}}]
Mapping: [{{val| 3 1 7 -1 1 13 }}, {{val| 0 2 0 5 5 -1 }}]


POTE generator: ~28/27 = 46.250
POTE generator: ~14/9 = 753.750 (or ~36/35 = 46.250)


Optimal GPV sequence: {{Val list| 24, 27e, 51ce, 78cee }}
Optimal GPV sequence: {{Val list| 24, 27e, 51ce, 78cee }}
Line 300: Line 306:
[[Comma list]]: 49/48, 128/125
[[Comma list]]: 49/48, 128/125


[[Mapping]]: [{{val|3 0 7 6}}, {{val|0 2 0 1}}]
[[Mapping]]: [{{val| 3 0 7 6 }}, {{val| 0 2 0 1 }}]


{{Multival|legend=1|6 0 3 -14 -12 7}}
Mapping generators: ~5/4, ~7/4


[[POTE generator]]: ~7/6 = 247.049
{{Multival|legend=1| 6 0 3 -14 -12 7 }}
 
[[POTE generator]]: ~7/4 = 952.951


{{Val list|legend=1| 6, 9, 15, 24, 39, 63cd, 102ccdd }}
{{Val list|legend=1| 6, 9, 15, 24, 39, 63cd, 102ccdd }}
Line 313: Line 321:
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 77/75, 128/125
Comma list: 49/48, 56/55, 77/75


Mapping: [{{val|3 0 7 6 8}}, {{val|0 2 0 1 1}}]
Mapping: [{{val| 3 0 7 6 8 }}, {{val| 0 2 0 1 1 }}]


POTE generator: ~7/6 = 247.068
POTE generator: ~7/4 = 952.932


Optimal GPV sequence: {{Val list| 6, 9, 15, 24, 39, 63cd, 102ccdd }}
Optimal GPV sequence: {{Val list| 6, 9, 15, 24, 39, 63cd, 102ccdd }}
Line 332: Line 340:
Comma list: 49/48, 56/55, 66/65, 77/75
Comma list: 49/48, 56/55, 66/65, 77/75


Mapping: [{{val|3 0 7 6 8 4}}, {{val|0 2 0 1 1 3}}]
Mapping: [{{val| 3 0 7 6 8 4 }}, {{val| 0 2 0 1 1 3 }}]


POTE generator: ~7/6 = 248.313
POTE generator: ~7/4 = 951.687


Optimal GPV sequence: {{Val list| 6, 9, 15, 24, 63cdf, 87cddff }}
Optimal GPV sequence: {{Val list| 6, 9, 15, 24, 63cdf, 87cddff }}
Line 348: Line 356:
Comma list: 49/48, 56/55, 77/75, 507/500
Comma list: 49/48, 56/55, 77/75, 507/500


Mapping: [{{val|6 0 14 12 16 27}}, {{val|0 2 0 1 1 -1}}]
Mapping: [{{val| 6 0 14 12 16 27 }}, {{val| 0 2 0 1 1 -1 }}]
 
Mapping generators: ~44/39, ~7/4


POTE generator: ~7/6 = 246.642
POTE generator: ~7/4 = 953.358


Optimal GPV sequence: {{Val list| 6, 24, 54cd, 78cd, 102ccddf }}
Optimal GPV sequence: {{Val list| 6, 24, 54cd, 78cd, 102ccddf }}
Line 361: Line 371:
[[Comma list]]: 128/125, 1323/1250
[[Comma list]]: 128/125, 1323/1250


[[Mapping]]: [{{val|3 1 7 14}}, {{val|0 2 0 -3}}]
[[Mapping]]: [{{val| 3 1 7 14 }}, {{val| 0 2 0 -3 }}]
 
Mapping generators: ~5/4, ~32/21


{{Multival|legend=1|6 0 -9 -14 -31 -21}}
{{Multival|legend=1| 6 0 -9 -14 -31 -21 }}


[[POTE generator]]: ~21/20 = 52.093
[[POTE generator]]: ~32/21 = 747.907 (or ~21/20 = 52.093)


{{Val list|legend=1| 21, 24, 45c, 69cd }}
{{Val list|legend=1| 21, 24, 45c, 69cd }}
Line 376: Line 388:
Comma list: 56/55, 128/125, 1323/1250
Comma list: 56/55, 128/125, 1323/1250


Mapping: [{{val|3 1 7 14 16}}, {{val|0 2 0 -3 -3}}]
Mapping: [{{val| 3 1 7 14 16 }}, {{val| 0 2 0 -3 -3 }}]


POTE generator: ~21/20 = 51.655
POTE generator: ~32/21 = 748.345 (or ~33/32 = 51.655)


Optimal GPV sequence: {{Val list| 21, 24, 69cd, 93ccdd }}
Optimal GPV sequence: {{Val list| 21, 24, 69cd, 93ccdd }}
Line 389: Line 401:
Comma list: 56/55, 66/65, 105/104, 507/500
Comma list: 56/55, 66/65, 105/104, 507/500


Mapping: [{{val|3 1 7 14 16 13}}, {{val|0 2 0 -3 -3 -1}}]
Mapping: [{{val| 3 1 7 14 16 13 }}, {{val| 0 2 0 -3 -3 -1 }}]


POTE generator: ~21/20 = 51.548
POTE generator: ~32/21 = 748.452 (or ~21/20 = 51.548)


Optimal GPV sequence: {{Val list| 21, 24, 69cdf, 93ccddf }}
Optimal GPV sequence: {{Val list| 21, 24, 69cdf, 93ccddf }}
Line 402: Line 414:
[[Comma list]]: 128/125, 360/343
[[Comma list]]: 128/125, 360/343


[[Mapping]]: [{{val|3 1 7 6}}, {{val|0 3 0 2}}]
[[Mapping]]: [{{val| 3 1 7 6 }}, {{val| 0 3 0 2 }}]
 
Mapping generators: ~5/4, ~48/35


[[POTE generator]]: ~48/35 = 501.980
[[POTE generator]]: ~48/35 = 501.980
Line 415: Line 429:
[[Comma list]]: 128/125, 2401/2400
[[Comma list]]: 128/125, 2401/2400


[[Mapping]]: [{{val|3 3 7 8}}, {{val|0 4 0 1}}]
[[Mapping]]: [{{val| 3 3 7 8 }}, {{val| 0 4 0 1 }}]


{{Multival|legend=1|12 0 3 -28 -29 7}}
Mapping generators: ~5/4, ~28/25


[[POTE generator]]: ~8/7 = 223.354<!--223.3540873874507317-->
{{Multival|legend=1| 12 0 3 -28 -29 7 }}
 
[[POTE generator]]: ~28/25 = 176.646


{{Val list|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}
{{Val list|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}
Line 430: Line 446:
Comma list: 56/55, 128/125, 3773/3750
Comma list: 56/55, 128/125, 3773/3750


Mapping: [{{val|3 3 7 8 10}}, {{val|0 4 0 1 1}}]
Mapping: [{{val| 3 3 7 8 10 }}, {{val| 0 4 0 1 1 }}]


POTE generator: ~8/7 = 223.019
POTE generator: ~28/25 = 176.981


Optimal GPV sequence: {{Val list| 6, 21, 27e }}
Optimal GPV sequence: {{Val list| 6, 21, 27e }}
Line 443: Line 459:
Comma list: 56/55, 78/77, 128/125, 507/500
Comma list: 56/55, 78/77, 128/125, 507/500


Mapping: [{{val|3 3 7 8 10 12}}, {{val|0 4 0 1 1 -2}}]
Mapping: [{{val| 3 3 7 8 10 12 }}, {{val| 0 4 0 1 1 -2 }}]


POTE generator: ~8/7 = 223.449
POTE generator: ~28/25 = 176.551


Optimal GPV sequence: {{Val list| 6, 21, 27e }}
Optimal GPV sequence: {{Val list| 6, 21, 27e }}
Line 456: Line 472:
[[Comma list]]: 128/125, 12005/11664
[[Comma list]]: 128/125, 12005/11664


[[Mapping]]: [{{val|6 1 14 4}}, {{val|0 2 0 3}}]
[[Mapping]]: [{{val| 6 1 14 4 }}, {{val| 0 2 0 3 }}]


{{Multival|legend=1| 12 0 18 -28 -5 42}}
Mapping generators: ~54/49, ~45/28


[[POTE generator]]: ~15/14 = 144.515
{{Multival|legend=1| 12 0 18 -28 -5 42 }}
 
[[POTE generator]]: ~45/28 = 855.485 (or ~36/35 = 55.485)


{{Val list|legend=1| 18, 24, 42, 66c, 108bcc }}
{{Val list|legend=1| 18, 24, 42, 66c, 108bcc }}
Line 471: Line 489:
Comma list: 56/55, 128/125, 3773/3645
Comma list: 56/55, 128/125, 3773/3645


Mapping: [{{val|6 1 14 4 8}}, {{val|0 2 0 3 3}}]
Mapping: [{{val| 6 1 14 4 8 }}, {{val| 0 2 0 3 3 }}]


POTE generator: ~15/14 = 144.780
POTE generator: ~18/11 = 855.220 (or ~36/35 = 55.220)


Optimal GPV sequence: {{Val list| 24, 42e, 66ce, 108bccee }}
Optimal GPV sequence: {{Val list| 24, 42e, 66ce, 108bccee }}
Line 480: Line 498:


== Niner ==
== Niner ==
Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27EDO. Niner[18], therefore, has nine such tetrads.
Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 486: Line 504:
[[Comma list]]: 128/125, 686/675
[[Comma list]]: 128/125, 686/675


[[Mapping]]: [{{val|9 0 21 11}}, {{val|0 1 0 1}}]
[[Mapping]]: [{{val| 9 0 21 11 }}, {{val| 0 1 0 1 }}]
 
Mapping generators: ~49/45, ~3


{{Multival|legend=1|9 0 9 -21 -11 21}}
{{Multival|legend=1| 9 0 9 -21 -11 21 }}


[[POTE generator]]: ~3/2 = 707.167
[[POTE generator]]: ~3/2 = 707.167
Line 501: Line 521:
Comma list: 56/55, 128/125, 540/539
Comma list: 56/55, 128/125, 540/539


Mapping: [{{val|9 0 21 11 17}}, {{val|0 1 0 1 1}}]
Mapping: [{{val| 9 0 21 11 17 }}, {{val| 0 1 0 1 1 }}]


POTE generator: ~3/2 = 706.726
POTE generator: ~3/2 = 706.726
Line 514: Line 534:
Comma list: 56/55, 78/77, 91/90, 128/125
Comma list: 56/55, 78/77, 91/90, 128/125


Mapping: [{{val|9 0 21 11 17 19}}, {{val|0 1 0 1 1 1}}]
Mapping: [{{val| 9 0 21 11 17 19 }}, {{val| 0 1 0 1 1 1 }}]


POTE generator: ~3/2 = 706.889
POTE generator: ~3/2 = 706.889
Line 525: Line 545:
* [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]
* [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Augmented]]
[[Category:Augmented]]
[[Category:Rank 2]]


<!-- interwiki links -->
<!-- interwiki links -->
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[[de:Übermässige Temperaturen]]
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