Archytas clan: Difference between revisions

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The '''archytas clan''' (or '''archy family''') tempers out the [[64/63|Archytas comma]], 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to [[81/80]] in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of [[22edo]].  
The '''archytas clan''' (or '''archy family''') tempers out the [[64/63|Archytas comma]], 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to [[81/80]] in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of [[22edo|22EDO]].  


Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, 2430/2401 gives quasisuper, and 4375/4374 gives modus.  
Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, 2430/2401 gives quasisuper, and 4375/4374 gives modus.  
Line 6: Line 6:


= Archy =
= Archy =
Subgroup: 2.3.7
Subgroup: 2.3.7


Line 26: Line 25:


== Supra ==
== Supra ==
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


Line 41: Line 39:
POTE generator: ~3/2 = 707.192
POTE generator: ~3/2 = 707.192


{{Val list|legend=1| 5, 12, 17, 39d, 56d }}
Vals: {{Val list| 5, 12, 17, 39d, 56d }}


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]


=== Supraphon ===
=== Supraphon ===
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 61: Line 58:
POTE generator: ~3/2 = 706.137
POTE generator: ~3/2 = 706.137


{{Val list|legend=1| 12f, 17 }}
Vals: {{Val list| 12f, 17 }}


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]


== Suhajira ==
== Suhajira ==
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


Line 86: Line 82:


=== 2.3.7.11.13 ===
=== 2.3.7.11.13 ===
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 120: Line 115:
{{Val list|legend=1| 5, 17, 22, 27, 49 }}
{{Val list|legend=1| 5, 17, 22, 27, 49 }}


[[Badness]]: 0.0323
[[Badness]]: 0.032318


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 132: Line 126:
POTE generator: ~3/2 = 710.175
POTE generator: ~3/2 = 710.175


{{Val list|legend=1| 22, 27e, 49 }}
Vals: {{Val list| 22, 27e, 49 }}


Badness: 0.0250
Badness: 0.024976


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 146: Line 139:
POTE generator: ~3/2 = 710.479
POTE generator: ~3/2 = 710.479


{{Val list|legend=1| 22, 27e, 49, 76bcde }}
Vals: {{Val list| 22, 27e, 49, 76bcde }}


Badness: 0.0247
Badness: 0.024673


== Suprapyth ==
== Suprapyth ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 160: Line 152:
POTE generator: ~3/2 = 709.495
POTE generator: ~3/2 = 709.495


{{Val list|legend=1| 17, 22 }}
Vals: {{Val list| 17, 22 }}


Badness: 0.0328
Badness: 0.032768


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 174: Line 165:
POTE generator: ~3/2 = 708.703
POTE generator: ~3/2 = 708.703


{{Val list|legend=1| 17, 22, 83cdf }}
Vals: {{Val list| 17, 22, 83cdf }}


Badness: 0.0363
Badness: 0.036336


= Quasisuper =
= Quasisuper =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 186: Line 176:
[[Mapping]]: [{{val| 1 0 23 6 }}, {{val| 0 1 -13 -2 }}]
[[Mapping]]: [{{val| 1 0 23 6 }}, {{val| 0 1 -13 -2 }}]


{{Multival|legend=1| 1 -13 -2 -23 -2 -6 32 }}
{{Multival|legend=1| 1 -13 -2 -23 -6 32 }}


[[POTE generator]]: ~3/2 = 708.328
[[POTE generator]]: ~3/2 = 708.328
Line 192: Line 182:
{{Val list|legend=1| 17c, 22, 61d }}
{{Val list|legend=1| 17c, 22, 61d }}


[[Badness]]: 0.0638
[[Badness]]: 0.063794


== Quasisupra ==
== Quasisupra ==
Line 201: Line 191:
Comma list: 64/63, 99/98, 121/120
Comma list: 64/63, 99/98, 121/120


Mapping: [{{val| 1 2 -3 2 1 }}, {{val| 0 -1 13 2 6 }}]
Mapping: [{{val| 1 0 23 6 13 }}, {{val| 0 1 -13 -2 -6 }}]


POTE generator: ~3/2 = 708.205
POTE generator: ~3/2 = 708.205


{{Val list|legend=1| 17c, 22, 39d, 61d }}
Vals: {{Val list| 17c, 22, 39d, 61d }}


Badness: 0.0322
Badness: 0.032203


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 219: Line 208:
POTE generator: ~3/2 = 708.004
POTE generator: ~3/2 = 708.004


{{Val list|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
Vals: {{Val list| 17c, 22, 39d, 61df, 100bcdf }}


Badness: 0.0302
Badness: 0.030219


== Quasisoup ==
== Quasisoup ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 233: Line 221:
POTE generator: ~3/2 = 709.021
POTE generator: ~3/2 = 709.021


{{Val list|legend=1| 22 }}
Vals: {{Val list| 5ce, 17ce, 22 }}


Badness: 0.0835
Badness: 0.083490


= Schism =
= Schism =
Line 252: Line 240:
{{Val list|legend=1| 12, 41d, 53d }}
{{Val list|legend=1| 12, 41d, 53d }}


[[Badness]]: 0.0566
[[Badness]]: 0.056648


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 264: Line 251:
POTE generator: ~3/2 = 702.136
POTE generator: ~3/2 = 702.136


{{Val list|legend=1| 12, 29de, 41de }}
Vals: {{Val list| 12, 29de, 41de }}


Badness: 0.0375
Badness: 0.037482


= Blacksmith =
= Blacksmith =
[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]]
[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]]


== 5-limit (blackwood) ==
== 5-limit (blackwood) ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: 256/243
[[Comma]]: 256/243


Mapping: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]
[[Mapping]]: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]


Mapping generators: ~9/8, ~5
Mapping generators: ~9/8, ~5


POTE generator: ~5/4 = 399.594
[[POTE generator]]: ~5/4 = 399.594


{{Val list|legend=1| 5, 10, 15 }}
{{Val list|legend=1| 5, 10, 15 }}


Badness: 0.0638
[[Badness]]: 0.063760


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 28/27, 49/48
[[Comma list]]: 28/27, 49/48


Mapping: [{{val| 5 8 0 14 }}, {{val| 0 0 1 0 }}]
[[Mapping]]: [{{val| 5 8 0 14 }}, {{val| 0 0 1 0 }}]


Mapping generators: ~7/6, ~5
Mapping generators: ~7/6, ~5
Line 300: Line 284:
{{Multival|legend=1| 0 5 0 8 0 -14 }}
{{Multival|legend=1| 0 5 0 8 0 -14 }}


POTE generator: ~5/4 = 392.767
[[POTE generator]]: ~5/4 = 392.767


{{Val list|legend=1| 5, 10, 15, 40b, 55b }}
{{Val list|legend=1| 5, 10, 15, 40b, 55b }}


Badness: 0.0256
[[Badness]]: 0.025640


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 316: Line 299:
POTE generator: ~5/4 = 394.948
POTE generator: ~5/4 = 394.948


{{Val list|legend=1| 5, 10, 15, 40be, 55be, 70bde, 85bcde}}
Vals: {{Val list| 5, 10, 15, 40be, 55be, 70bde, 85bcde}}


Badness: 0.0246
Badness: 0.024641


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 328: Line 310:
Mapping: [{{val| 5 8 0 14 29 7 }}, {{val| 0 0 1 0 -1 1 }}]
Mapping: [{{val| 5 8 0 14 29 7 }}, {{val| 0 0 1 0 -1 1 }}]


POTE generator: ~5/4 = 391.0367
POTE generator: ~5/4 = 391.037


{{Val list|legend=1| 5, 10, 15, 25e, 40bef}}
Vals: {{Val list| 5, 10, 15, 25e, 40bef}}


Badness: 0.0205
Badness: 0.020498


== Farrier ==
== Farrier ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 344: Line 325:
POTE generator: ~5/4 = 398.070
POTE generator: ~5/4 = 398.070


{{Val list|legend=1| 5e, 10e, 15 }}
Vals: {{Val list| 5e, 10e, 15 }}


Badness: 0.0292
Badness: 0.029200


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 358: Line 338:
POTE generator: ~5/4 = 396.812
POTE generator: ~5/4 = 396.812


{{Val list|legend=1| 5e, 10e, 15 }}
Vals: {{Val list| 5e, 10e, 15 }}


Badness: 0.0223
Badness: 0.022325


== Ferrum ==
== Ferrum ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 372: Line 351:
POTE generator: ~5/4 = 374.763
POTE generator: ~5/4 = 374.763


{{Val list|legend=1| 5e, 10 }}
Vals: {{Val list| 5e, 10 }}


Badness: 0.0309
Badness: 0.030883


= Beatles =
= Beatles =
== 5-limit ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: 524288/492075
[[Comma]]: 524288/492075


Mapping: [{{val| 1 1 5 }}, {{val| 0 2 -9 }}]
[[Mapping]]: [{{val| 1 1 5 }}, {{val| 0 2 -9 }}]


POTE generator: ~512/405 = 355.930
[[POTE generator]]: ~512/405 = 355.930


{{Val list|legend=1| 10, 17c, 27, 64b, 91bc, 118bc }}
{{Val list|legend=1| 10, 17c, 27, 64b, 91bc, 118bc }}


Badness: 0.3585
[[Badness]]: 0.358542


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 64/63, 686/675
[[Comma list]]: 64/63, 686/675


Mapping: [{{val| 1 1 5 4 }}, {{val| 0 2 -9 -4 }}]
[[Mapping]]: [{{val| 1 1 5 4 }}, {{val| 0 2 -9 -4 }}]


{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}
{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}


POTE generator: ~49/40 = 355.904
[[POTE generator]]: ~49/40 = 355.904


{{Val list|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
{{Val list|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}


Badness: 0.0459
[[Badness]]: 0.045872


; Music
; Music
Line 411: Line 387:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 420: Line 395:
POTE generator: ~49/40 = 356.140
POTE generator: ~49/40 = 356.140


{{Val list|legend=1| 27e, 37, 64be, 91bcde }}
Vals: {{Val list| 27e, 37, 64be, 91bcde }}


Badness: 0.0456
Badness: 0.045639


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 434: Line 408:
POTE generator: ~16/13 = 356.229
POTE generator: ~16/13 = 356.229


{{Val list|legend=1| 27e, 37, 64be }}
Vals: {{Val list| 27e, 37, 64be }}


Badness: 0.0302
Badness: 0.030161


== Ringo ==
== Ringo ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 448: Line 421:
POTE generator: ~11/9 = 355.419
POTE generator: ~11/9 = 355.419


{{Val list|legend=1| 10, 17c, 27e }}
Vals: {{Val list| 10, 17c, 27e }}


Badness: 0.0329
Badness: 0.032863


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 462: Line 434:
POTE generator: ~11/9 = 355.456
POTE generator: ~11/9 = 355.456


{{Val list|legend=1| 10, 17c, 27e }}
Vals: {{Val list| 10, 17c, 27e }}


Badness: 0.0226
Badness: 0.022619


= Passion =
= Passion =
== 5-limit ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: 262144/253125
[[Comma]]: 262144/253125


Mapping: [{{val| 1 2 2 }}, {{val| 0 -5 4 }}]
[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -5 4 }}]


Mapping generators: 2, 16/15
Mapping generators: 2, 16/15


POTE generator: ~16/15 = 98.670
[[POTE generator]]: ~16/15 = 98.670


{{Val list|legend=1| 12, 49, 61, 73 }}
{{Val list|legend=1| 12, 49, 61, 73 }}


Badness: 0.1686
[[Badness]]: 0.168550


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 64/63, 3125/3087
[[Comma list]]: 64/63, 3125/3087


Mapping: [{{val| 1 2 2 2 }}, {{val| 0 -5 4 10 }}]
[[Mapping]]: [{{val| 1 2 2 2 }}, {{val| 0 -5 4 10 }}]


Mapping generators: 2, 16/15
Mapping generators: 2, 16/15
Line 495: Line 464:
{{Multival|legend=1| 5 -4 -10 -18 -30 -12 }}
{{Multival|legend=1| 5 -4 -10 -18 -30 -12 }}


POTE generator: ~16/15 = 98.153
[[POTE generator]]: ~16/15 = 98.153


{{Val list|legend=1| 12, 37, 49, 110bcd }}
{{Val list|legend=1| 12, 37, 49, 110bcd }}


Badness: 0.0623
[[Badness]]: 0.062327


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 511: Line 479:
POTE generator: ~16/15 = 98.019
POTE generator: ~16/15 = 98.019


{{Val list|legend=1| 12, 37, 49 }}
Vals: {{Val list| 12, 37, 49 }}


Badness: 0.0408
Badness: 0.040809


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 525: Line 492:
POTE generator: ~16/15 = 97.910
POTE generator: ~16/15 = 97.910


{{Val list|legend=1| 12f, 25f, 37, 49f }}
Vals: {{Val list| 12f, 25f, 37, 49f }}


Badness: 0.0309
Badness: 0.030886


= Fervor =
= Fervor =
== 5-limit ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: 67108864/61509375
[[Comma]]: 67108864/61509375


Mapping: [{{val| 1 4 -2 }}, {{val| 0 -5 9 }}]
[[Mapping]]: [{{val| 1 4 -2 }}, {{val| 0 -5 9 }}]


POTE generator: ~64/45 = 577.705
[[POTE generator]]: ~64/45 = 577.705


{{Val list|legend=1| 25, 27 }}
{{Val list|legend=1| 2, 25, 27 }}


Badness: 0.8526
[[Badness]]: 0.852612


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 64/63, 9604/9375
[[Comma list]]: 64/63, 9604/9375


Mapping: [{{val| 1 4 -2 -2 }}, {{val| 0 -5 9 10 }}]
[[Mapping]]: [{{val| 1 4 -2 -2 }}, {{val| 0 -5 9 10 }}]


{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}
{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}


POTE generator: ~7/5 = 577.777
[[POTE generator]]: ~7/5 = 577.776


{{Val list|legend=1| 25, 27 }}
{{Val list|legend=1| 2, 25, 27 }}


Badness: 0.1085
[[Badness]]: 0.108455


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 570: Line 533:
POTE generator: ~7/5 = 577.850
POTE generator: ~7/5 = 577.850


{{Val list|legend=1| 25e, 27e }}
Vals: {{Val list| 2, 25e, 27e }}


Badness: 0.0521
Badness: 0.052054


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 584: Line 546:
POTE generator: ~7/5 = 578.060
POTE generator: ~7/5 = 578.060


{{Val list|legend=1| 27e }}
Vals: {{Val list| 2f, 25ef, 27e }}


Badness: 0.0397
Badness: 0.039705


= Progress =
= Progress =
== 5-limit ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: 32768/30375
[[Comma]]: 32768/30375


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


POTE generator: ~64/45 = 561.264
[[POTE generator]]: ~64/45 = 561.264


{{Val list|legend=1| 15, 32c, 47bc, 62bc }}
{{Val list|legend=1| 2, 13, 15, 32c, 47bc, 62bc }}


Badness: 0.2461
Badness: 0.246073


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 64/63, 392/375
[[Comma list]]: 64/63, 392/375


Mapping: [{{val| 1 0 5 6 }}, {{val| 0 3 -5 -6 }}]
[[Mapping]]: [{{val| 1 0 5 6 }}, {{val| 0 3 -5 -6 }}]


{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}
{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}


POTE generator: ~7/5 = 562.122
[[POTE generator]]: ~7/5 = 562.122


{{Val list|legend=1| 15, 32c, 79bcc, 111bcc }}
{{Val list|legend=1| 2, 13, 15, 32c, 79bcc, 111bcc }}


Badness: 0.0664
[[Badness]]: 0.066400


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 629: Line 587:
POTE generator: ~7/5 = 562.085
POTE generator: ~7/5 = 562.085


{{Val list|legend=1| 15, 32c, 47bc, 79bcce }}
Vals: {{Val list| 2, 13, 15, 32c, 47bc, 79bcce }}


Badness: 0.0310
Badness: 0.031036
 
=== 13-limit ===


== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 643: Line 600:
POTE generator: ~7/5 = 562.365
POTE generator: ~7/5 = 562.365


{{Val list|legend=1| 15, 17c, 32cf }}
Vals: {{Val list| 15, 17c, 32cf }}


Badness: 0.0262
Badness: 0.026214
 
=== Progressive ===


== Progressive ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 657: Line 613:
POTE generator: ~7/5 = 563.239
POTE generator: ~7/5 = 563.239


{{Val list|legend=1| 15f, 17c, 32c, 49c }}
Vals: {{Val list| 15f, 17c, 32c, 49c }}


Badness: 0.0327
Badness: 0.032721


= Sixix =
= Sixix =
:''See also: [[Dual-fifth temperaments#Dual-3 Sixix]]''
:''See also: [[Dual-fifth temperaments#Dual-3 Sixix]]''
== 5-limit ==


Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: 3125/2916
[[Comma]]: 3125/2916


Mapping: [{{val| 1 3 4 }}, {{val| 0 -5 -6 }}]
[[Mapping]]: [{{val| 1 3 4 }}, {{val| 0 -5 -6 }}]


POTE generator: ~6/5 = 338.365
[[POTE generator]]: ~6/5 = 338.365


{{Val list|legend=1| 7, 25, 32 }}
{{Val list|legend=1| 7, 25, 32 }}


Badness: 0.1531
[[Badness]]: 0.153088


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7


Subgroup: 2.3.5.7
[[Comma list]]: 64/63, 3125/2916


Comma list: 64/63, 3125/2916
[[Mapping]]: [{{val| 1 3 4 0 }}, {{val| 0 -5 -6 10 }}]


Mapping: [{{val| 1 3 4 0 }}, {{val| 0 -5 -6 10 }}]
{{Multival|legend=1| 5 6 -10 -2 -30 -40 }}


POTE generator: ~6/5 = 337.4419
[[POTE generator]]: ~6/5 = 337.442


{{Val list|legend=1| 7, 25, 32 }}
{{Val list|legend=1| 7, 25, 32 }}
[[Badness]]: 0.158903


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]

Revision as of 12:53, 28 May 2021

The archytas clan (or archy family) tempers out the Archytas comma, 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to 81/80 in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of 22EDO.

Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, 2430/2401 gives quasisuper, and 4375/4374 gives modus.

Discussed under their respective 5-limit families are mother, dominant, augene, porcupine, pajara, catalan, modus, and immunized. The rest are considered below.

Archy

Subgroup: 2.3.7

Comma list: 64/63

Sval mapping: [⟨1 0 6], ⟨0 1 -2]]

Sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 1 0 4], ⟨0 1 0 -2]]

Gencom: [2 3/2; 64/63]

POTE generator: ~3/2 = 709.321

Template:Val list

Scales: archy5, archy7, archy12

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]

Sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 1 0 4 7], ⟨0 1 0 -2 -6]]

Gencom: [2 3/2; 64/63 99/98]

POTE generator: ~3/2 = 707.192

Vals: Template:Val list

Scales: supra7, supra12

Supraphon

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 99/98

Sval mapping: [⟨1 0 6 13 18], ⟨0 1 -2 -6 -9]]

Sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 1 0 4 7 9], ⟨0 1 0 -2 -6 -9]]

Gencom: [2 3/2; 64/63 78/77 99/98]

POTE generator: ~3/2 = 706.137

Vals: Template:Val list

Scales: supra7, supra12

Suhajira

Subgroup: 2.3.7.11

Comma list: 64/63, 243/242

Sval mapping: [⟨1 1 4 2], ⟨0 2 -4 5]]

Sval mapping generators: ~2, ~11/9

Gencom mapping: [⟨1 1 0 4 2], ⟨0 1 0 -4 5]]

Gencom: [2 3/2; 64/63 99/98]

POTE generator: ~11/9 = 353.958

Vals: Template:Val list

Scales: suhajira7, suhajira10, suhajira17

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 144/143

Sval mapping: [⟨1 1 4 2 4], ⟨0 2 -4 5 -1]]

Sval mapping generators: ~2, ~11/9

Gencom mapping: [⟨1 1 0 4 2 4], ⟨0 1 0 -4 5 -1]]

Gencom: [2 3/2; 64/63 78/77 99/98]

POTE generator: ~11/9 = 353.775

Vals: Template:Val list

Scales: suhajira7, suhajira10, suhajira17

Superpyth

Main article: Superpyth

Subgroup: 2.3.5.7

Comma list: 64/63, 245/243

Mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]

Wedgie: ⟨⟨ 1 9 -2 12 -6 -30 ]]

POTE generator: ~3/2 = 710.291

Template:Val list

Badness: 0.032318

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 245/243

Mapping: [⟨1 0 -12 6 -22], ⟨0 1 9 -2 16]]

POTE generator: ~3/2 = 710.175

Vals: Template:Val list

Badness: 0.024976

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 100/99

Mapping: [⟨1 0 -12 6 -22 -17], ⟨0 1 9 -2 16 13]]

POTE generator: ~3/2 = 710.479

Vals: Template:Val list

Badness: 0.024673

Suprapyth

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 99/98

Mapping: [⟨1 0 -12 6 13], ⟨0 1 9 -2 -6]]

POTE generator: ~3/2 = 709.495

Vals: Template:Val list

Badness: 0.032768

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 65/63, 99/98

Mapping: [⟨1 0 -12 6 13 18], ⟨0 1 9 -2 -6 -9]]

POTE generator: ~3/2 = 708.703

Vals: Template:Val list

Badness: 0.036336

Quasisuper

Subgroup: 2.3.5.7

Comma list: 64/63, 2430/2401

Mapping: [⟨1 0 23 6], ⟨0 1 -13 -2]]

Wedgie: ⟨⟨ 1 -13 -2 -23 -6 32 ]]

POTE generator: ~3/2 = 708.328

Template:Val list

Badness: 0.063794

Quasisupra

Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament supra, with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).

Subgroup: 2.3.5.7.11

Comma list: 64/63, 99/98, 121/120

Mapping: [⟨1 0 23 6 13], ⟨0 1 -13 -2 -6]]

POTE generator: ~3/2 = 708.205

Vals: Template:Val list

Badness: 0.032203

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 121/120

Mapping: [⟨1 0 23 6 13 18], ⟨0 1 -13 -2 -6 -9]]

POTE generator: ~3/2 = 708.004

Vals: Template:Val list

Badness: 0.030219

Quasisoup

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 2430/2401

Mapping: [⟨1 0 23 6 -22], ⟨0 1 -13 -2 16]]

POTE generator: ~3/2 = 709.021

Vals: Template:Val list

Badness: 0.083490

Schism

See also: Schismatic family § Schism

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]

Wedgie: ⟨⟨ 1 -8 -2 -15 -6 18 ]]

POTE generator: ~3/2 = 701.556

Template:Val list

Badness: 0.056648

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 64/63, 99/98

Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]

POTE generator: ~3/2 = 702.136

Vals: Template:Val list

Badness: 0.037482

Blacksmith

blacksmith10.jpg
Lattice of blacksmith

5-limit (blackwood)

Subgroup: 2.3.5

Comma: 256/243

Mapping: [⟨5 8 0], ⟨0 0 1]]

Mapping generators: ~9/8, ~5

POTE generator: ~5/4 = 399.594

Template:Val list

Badness: 0.063760

7-limit

Subgroup: 2.3.5.7

Comma list: 28/27, 49/48

Mapping: [⟨5 8 0 14], ⟨0 0 1 0]]

Mapping generators: ~7/6, ~5

Wedgie: ⟨⟨ 0 5 0 8 0 -14 ]]

POTE generator: ~5/4 = 392.767

Template:Val list

Badness: 0.025640

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 49/48, 55/54

Mapping: [⟨5 8 0 14 29], ⟨0 0 1 0 -1]]

POTE generator: ~5/4 = 394.948

Vals: Template:Val list

Badness: 0.024641

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 49/48, 55/54

Mapping: [⟨5 8 0 14 29 7], ⟨0 0 1 0 -1 1]]

POTE generator: ~5/4 = 391.037

Vals: Template:Val list

Badness: 0.020498

Farrier

Subgroup: 2.3.5.7.11

Comma list: 28/27, 49/48, 77/75

Mapping: [⟨5 8 0 14 -6], ⟨0 0 1 0 2]]

POTE generator: ~5/4 = 398.070

Vals: Template:Val list

Badness: 0.029200

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 49/48, 66/65

Mapping: [⟨5 8 0 14 -6 7], ⟨0 0 1 0 2 1]]

POTE generator: ~5/4 = 396.812

Vals: Template:Val list

Badness: 0.022325

Ferrum

Subgroup: 2.3.5.7.11

Comma list: 28/27, 35/33, 49/48

Mapping: [⟨5 8 0 14 6], ⟨0 0 1 0 1]]

POTE generator: ~5/4 = 374.763

Vals: Template:Val list

Badness: 0.030883

Beatles

Subgroup: 2.3.5

Comma: 524288/492075

Mapping: [⟨1 1 5], ⟨0 2 -9]]

POTE generator: ~512/405 = 355.930

Template:Val list

Badness: 0.358542

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 686/675

Mapping: [⟨1 1 5 4], ⟨0 2 -9 -4]]

Wedgie: ⟨⟨ 2 -9 -4 -19 -12 16 ]]

POTE generator: ~49/40 = 355.904

Template:Val list

Badness: 0.045872

Music

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 686/675

Mapping: [⟨1 1 5 4 10], ⟨0 2 -9 -4 -22]]

POTE generator: ~49/40 = 356.140

Vals: Template:Val list

Badness: 0.045639

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 91/90, 100/99, 169/168

Mapping: [⟨1 1 5 4 10 4], ⟨0 2 -9 -4 -22 -1]]

POTE generator: ~16/13 = 356.229

Vals: Template:Val list

Badness: 0.030161

Ringo

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 540/539

Mapping: [⟨1 1 5 4 2], ⟨0 2 -9 -4 5]]

POTE generator: ~11/9 = 355.419

Vals: Template:Val list

Badness: 0.032863

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 91/90

Mapping: [⟨1 1 5 4 2 4], ⟨0 2 -9 -4 5 -1]]

POTE generator: ~11/9 = 355.456

Vals: Template:Val list

Badness: 0.022619

Passion

Subgroup: 2.3.5

Comma: 262144/253125

Mapping: [⟨1 2 2], ⟨0 -5 4]]

Mapping generators: 2, 16/15

POTE generator: ~16/15 = 98.670

Template:Val list

Badness: 0.168550

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/3087

Mapping: [⟨1 2 2 2], ⟨0 -5 4 10]]

Mapping generators: 2, 16/15

Wedgie: ⟨⟨ 5 -4 -10 -18 -30 -12 ]]

POTE generator: ~16/15 = 98.153

Template:Val list

Badness: 0.062327

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 1375/1372

Mapping: [⟨1 2 2 2 2], ⟨0 -5 4 10 18]]

POTE generator: ~16/15 = 98.019

Vals: Template:Val list

Badness: 0.040809

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 196/195, 275/273

Mapping: [⟨1 2 2 2 2 2], ⟨0 -5 4 10 18 21]]

POTE generator: ~16/15 = 97.910

Vals: Template:Val list

Badness: 0.030886

Fervor

Subgroup: 2.3.5

Comma: 67108864/61509375

Mapping: [⟨1 4 -2], ⟨0 -5 9]]

POTE generator: ~64/45 = 577.705

Template:Val list

Badness: 0.852612

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 9604/9375

Mapping: [⟨1 4 -2 -2], ⟨0 -5 9 10]]

Wedgie: ⟨⟨ 5 -9 -10 -26 -30 2 ]]

POTE generator: ~7/5 = 577.776

Template:Val list

Badness: 0.108455

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 1350/1331

Mapping: [⟨1 4 -2 -2 3], ⟨0 -5 9 10 1]]

POTE generator: ~7/5 = 577.850

Vals: Template:Val list

Badness: 0.052054

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 507/500

Mapping: [⟨1 4 -2 -2 3 -4], ⟨0 -5 9 10 1 16]]

POTE generator: ~7/5 = 578.060

Vals: Template:Val list

Badness: 0.039705

Progress

Subgroup: 2.3.5

Comma: 32768/30375

Mapping: [⟨1 0 5], ⟨0 3 -5]]

POTE generator: ~64/45 = 561.264

Template:Val list

Badness: 0.246073

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 392/375

Mapping: [⟨1 0 5 6], ⟨0 3 -5 -6]]

Wedgie: ⟨⟨ 3 -5 -6 -15 -18 0 ]]

POTE generator: ~7/5 = 562.122

Template:Val list

Badness: 0.066400

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 77/75

Mapping: [⟨1 0 5 6 4], ⟨0 3 -5 -6 -1]]

POTE generator: ~7/5 = 562.085

Vals: Template:Val list

Badness: 0.031036

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 66/65, 77/75

Mapping: [⟨1 0 5 6 4 0], ⟨0 3 -5 -6 -1 7]]

POTE generator: ~7/5 = 562.365

Vals: Template:Val list

Badness: 0.026214

Progressive

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 64/63, 77/75

Mapping: [⟨1 0 5 6 4 9], ⟨0 3 -5 -6 -1 -10]]

POTE generator: ~7/5 = 563.239

Vals: Template:Val list

Badness: 0.032721

Sixix

See also: Dual-fifth temperaments#Dual-3 Sixix

Subgroup: 2.3.5

Comma: 3125/2916

Mapping: [⟨1 3 4], ⟨0 -5 -6]]

POTE generator: ~6/5 = 338.365

Template:Val list

Badness: 0.153088

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/2916

Mapping: [⟨1 3 4 0], ⟨0 -5 -6 10]]

Wedgie: ⟨⟨ 5 6 -10 -2 -30 -40 ]]

POTE generator: ~6/5 = 337.442

Template:Val list

Badness: 0.158903

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