Quindromeda family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''quindromeda family''' tempers out {{monzo| 56 -28 -5 }}, equating the [[Pythagorean comma]] with a stack of five [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]].  
The '''quindromeda family''' tempers out {{monzo| 56 -28 -5 }}, equating the [[Pythagorean comma]] with a stack of five [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]]. It does not split the octave but splits the fourth in five, making it a member of [[Ploidacot/Omega-pentacot|omega-pentacot]] temperaments.  


== Quindromeda ==
== Quindromeda ==
Line 11: Line 11:
[[Mapping]]: [{{val| 1 2 0 }}, {{val| 0 -5 28 }}]
[[Mapping]]: [{{val| 1 2 0 }}, {{val| 0 -5 28 }}]


[[POTE generator]]: ~4428675/4194304 = 99.526
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.854¢, ~4428675/4194304 = 99.514¢
* [[CWE]]: ~2 = 1200.000¢, ~4428675/4194304 = 99.523¢


{{Optimal ET sequence|legend=1| 12, 169, 181, 193, 205, 217, 422 }}
{{Optimal ET sequence|legend=1| 12, 169, 181, 193, 205, 217, 422 }}


[[Badness]]: 0.399849
[[Badness]] (Sintel): 9.380


=== Subgroup temperament ===
=== Subgroup temperament ===
Line 32: Line 34:
[[Mapping|Sval mapping]]: [{{val| 1 2 0 5 }}, {{val| 0 -5 28 -11 }}]
[[Mapping|Sval mapping]]: [{{val| 1 2 0 5 }}, {{val| 0 -5 28 -11 }}]


[[Tp tuning|POL2 generator]]: ~18/17 = 99.524
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.878¢, ~18/17 = 99.513¢
* [[CWE]]: ~2 = 1200.000¢, ~18/17 = 99.523¢


{{Optimal ET sequence|legend=1| 12, 169, 181, 193, 205, 422 }}
{{Optimal ET sequence|legend=1| 12, 169, 181, 193, 205, 422 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0757 cents
[[Badness]] (Sintel): 0.768


==== 2.3.5.17.19 ====
==== 2.3.5.17.19 ====
Line 49: Line 53:
[[Mapping|Sval mapping]]: [{{val| 1 2 0 5 4 }}, {{val| 0 -5 28 -11 3 }}]
[[Mapping|Sval mapping]]: [{{val| 1 2 0 5 4 }}, {{val| 0 -5 28 -11 3 }}]


[[Tp tuning|POL2 generator]]: ~18/17 = 99.524
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.852¢, ~18/17 = 99.512¢
* [[CWE]]: ~2 = 1200.000¢, ~18/17 = 99.523¢


{{Optimal ET sequence|legend=1| 12, 169, 181, 193, 205, 422 }}
{{Optimal ET sequence|legend=1| 12, 169, 181, 193, 205, 422 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0813 cents
[[Badness]] (Sintel): 0.348
 
[[Badness]]: 0.007140


== Quintagar ==
== Quintagar ==
Line 66: Line 70:
[[Mapping]]: [{{val| 1 2 0 -3 }}, {{val| 0 -5 28 70 }}]
[[Mapping]]: [{{val| 1 2 0 -3 }}, {{val| 0 -5 28 70 }}]


[[POTE generator]]: ~200/189 = 99.555
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.824¢, ~200/189 = 99.540¢
* [[CWE]]: ~2 = 1200.000¢, ~200/189 = 99.553¢


{{Optimal ET sequence|legend=1| 12, 217, 229, 446, 675c }}
{{Optimal ET sequence|legend=1| 12, 217, 229, 446, 675c }}


[[Badness]]: 0.142897
[[Badness]] (Sintel): 3.616


=== Quintoneum ===
=== Quintoneum ===
Line 81: Line 87:
Mapping: [{{val| 1 2 0 -3 -5 }}, {{val| 0 -5 28 70 102 }}]
Mapping: [{{val| 1 2 0 -3 -5 }}, {{val| 0 -5 28 70 102 }}]


POTE generator: ~35/33 = 99.539
Optimal tunings:
* WE: ~2 = 1199.864¢, ~35/33 = 99.528¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.538¢


{{Optimal ET sequence|legend=1| 12, 205d, 217 }}
{{Optimal ET sequence|legend=1| 12, 205d, 217 }}


Badness: 0.087157
Badness (Sintel): 2.881


==== 13-limit ====
==== 13-limit ====
Line 174: Line 182:
Mapping: [{{val| 1 2 0 -3 13 }}, {{val| 0 -5 28 70 -115 }}]
Mapping: [{{val| 1 2 0 -3 13 }}, {{val| 0 -5 28 70 -115 }}]


POTE generator: ~200/189 = 99.551
Optimal tunings:
* WE: ~2 = 1199.867¢, ~200/189 = 99.540¢
* CWE: ~2 = 1200.000¢, ~200/189 = 99.551¢


{{Optimal ET sequence|legend=1| 12e, 217, 446 }}
{{Optimal ET sequence|legend=1| 12e, 217, 446 }}


Badness: 0.109908
Badness (Sintel): 3.633


==== 13-limit ====
==== 13-limit ====
Line 226: Line 236:
Mapping: [{{val| 1 2 0 -3 -6 }}, {{val| 0 -5 28 70 114 }}]
Mapping: [{{val| 1 2 0 -3 -6 }}, {{val| 0 -5 28 70 114 }}]


POTE generator: ~200/189 = 99.570
Optimal tunings:
* WE: ~2 = 1199.770¢, ~200/189 = 99.551¢
* CWE: ~2 = 1200.000¢, ~200/189 = 99.568¢


{{Optimal ET sequence|legend=1| 12, 217e, 229, 470cd, 699cd }}
{{Optimal ET sequence|legend=1| 12, 217e, 229, 470cd, 699cd }}


Badness: 0.093971
Badness (Sintel): 3.107


==== 13-limit ====
==== 13-limit ====
Line 319: Line 331:
[[Mapping]]: [{{val| 1 2 0 -2 }}, {{val| 0 -5 28 58 }}]
[[Mapping]]: [{{val| 1 2 0 -2 }}, {{val| 0 -5 28 58 }}]


[[POTE generator]]: ~625/588 = 99.483
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.876¢, ~625/588 = 99.473¢
* [[CWE]]: ~2 = 1200.000¢, ~625/588 = 99.482¢


{{Optimal ET sequence|legend=1| 12, 169, 181, 193 }}
{{Optimal ET sequence|legend=1| 12, 169, 181, 193 }}


[[Badness]]: 0.155536
[[Badness]] (Sintel): 3.936


=== 11-limit ===
=== 11-limit ===
Line 332: Line 346:
Mapping: [{{val| 1 2 0 -2 -4 }}, {{val| 0 -5 28 58 90 }}]
Mapping: [{{val| 1 2 0 -2 -4 }}, {{val| 0 -5 28 58 90 }}]


POTE generator: ~35/33 = 99.472
Optimal tunings:
* WE: ~2 = 1199.906¢, ~35/33 = 99.464¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.470¢


{{Optimal ET sequence|legend=1| 12, 181, 193, 374, 567ce }}
{{Optimal ET sequence|legend=1| 12, 181, 193, 374, 567ce }}


Badness: 0.073158
Badness (Sintel): 2.419


==== 13-limit ====
==== 13-limit ====
Line 464: Line 480:
[[Mapping]]: [{{val| 1 2 0 15 }}, {{val| 0 -5 28 -147 }}]
[[Mapping]]: [{{val| 1 2 0 15 }}, {{val| 0 -5 28 -147 }}]


[[POTE generator]]: ~4428675/4194304 = 99.529
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.843¢, ~4428675/4194304 = 99.516¢
* [[CWE]]: ~2 = 1200.000¢, ~4428675/4194304 = 99.529¢


{{Optimal ET sequence|legend=1| 205, 217, 422, 639, 1061 }}
{{Optimal ET sequence|legend=1| 205, 217, 422, 639, 1061 }}


[[Badness]]: 0.316850
[[Badness]] (Sintel): 8.019


=== 11-limit ===
=== 11-limit ===
Line 477: Line 495:
Mapping: [{{val| 1 2 0 15 -5 }}, {{val| 0 -5 28 -147 102 }}]
Mapping: [{{val| 1 2 0 15 -5 }}, {{val| 0 -5 28 -147 102 }}]


POTE generator: ~17325/16384 = 99.529
Optimal tunings:
* WE: ~2 = 1199.839¢, ~17325/16384 = 99.515¢
* CWE: ~2 = 1200.000¢, ~17325/16384 = 99.529¢


{{Optimal ET sequence|legend=1| 205, 217, 422, 639, 1061e }}
{{Optimal ET sequence|legend=1| 205, 217, 422, 639, 1061e }}


Badness: 0.085219
Badness (Sintel): 2.817


=== 13-limit ===
=== 13-limit ===
Line 529: Line 549:
[[Mapping]]: [{{val|1 2 0 6}}, {{val|0 -10 56 -77}}]
[[Mapping]]: [{{val|1 2 0 6}}, {{val|0 -10 56 -77}}]


[[POTE generator]]: ~36/35 = 49.756
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.925¢, ~36/35 = 49.753¢
* [[CWE]]: ~2 = 1200.000¢, ~36/35 = 49.757¢


{{Optimal ET sequence|legend=1| 24, 169d, 193, 217, 410 }}
{{Optimal ET sequence|legend=1| 24, 169d, 193, 217, 410 }}


[[Badness]]: 0.244589
[[Badness]] (Sintel): 6.190


=== 11-limit ===
=== 11-limit ===
Line 542: Line 564:
Mapping: [{{val|1 2 0 6 4}}, {{val|0 -10 56 -77 -13}}]
Mapping: [{{val|1 2 0 6 4}}, {{val|0 -10 56 -77 -13}}]


POTE generator: ~36/35 = 49.757
Optimal tunings:
* WE: ~2 = 1199.834¢, ~36/35 = 49.750¢
* CWE: ~2 = 1200.000¢, ~36/35 = 49.757¢


{{Optimal ET sequence|legend=1| 24, 169d, 193, 217, 410e }}
{{Optimal ET sequence|legend=1| 24, 169d, 193, 217, 410e }}


Badness: 0.093130
Badness (Sintel): 3.079


=== 13-limit ===
=== 13-limit ===
Line 595: Line 619:


[[Mapping]]: [{{val| 2 4 0 -5 }}, {{val| 0 -5 28 64 }}]
[[Mapping]]: [{{val| 2 4 0 -5 }}, {{val| 0 -5 28 64 }}]
: Mapping generators: ~10125/7168, ~1323/1250


[[POTE generator]]: ~1323/1250 = 99.521
[[Optimal tuning]]s:
* [[WE]]: ~10125/7168 = 599.929¢, ~1323/1250 = 99.509¢
* [[CWE]]: ~10125/7168 = 600.000¢, ~1323/1250 = 99.519¢


{{Optimal ET sequence|legend=1| 12, 398, 410, 422, 832, 1254d, 2086bd }}
{{Optimal ET sequence|legend=1| 12, 398, 410, 422, 832, 1254d, 2086bd }}


[[Badness]]: 0.233140
[[Badness]] (Sintel): 5.900


=== 11-limit ===
=== 11-limit ===
Line 609: Line 636:
Mapping: [{{val| 2 4 0 -5 -10 }}, {{val| 0 -5 28 64 102 }}]
Mapping: [{{val| 2 4 0 -5 -10 }}, {{val| 0 -5 28 64 102 }}]


POTE generator: ~1323/1250 = 99.525
Optimal tunings:
* WE: ~99/70 = 599.922¢, ~1323/1250 = 99.512¢
* CWE: ~99/70 = 600.000¢, ~1323/1250 = 99.523¢


{{Optimal ET sequence|legend=1| 12, 410, 422 }}
{{Optimal ET sequence|legend=1| 12, 410, 422 }}


Badness: 0.093926
Badness (Sintel): 3.105


=== 13-limit ===
=== 13-limit ===

Revision as of 22:50, 23 January 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The quindromeda family tempers out [56 -28 -5⟩, equating the Pythagorean comma with a stack of five schismas, making it a member of the schismic–Pythagorean equivalence continuum. It does not split the octave but splits the fourth in five, making it a member of omega-pentacot temperaments.

Quindromeda

The name quindromeda comes from "quintans" (Latin for "one fifth") and "Andromeda", because the generator is 1/5 of the andromeda fourth (~4/3, about 497.6 cents).

Subgroup: 2.3.5

Comma: [56 -28 -5⟩ = 72057594037927936/71489976421753125

Mapping: [⟨1 2 0], ⟨0 -5 28]]

Optimal tunings:

  • WE: ~2 = 1199.854¢, ~4428675/4194304 = 99.514¢
  • CWE: ~2 = 1200.000¢, ~4428675/4194304 = 99.523¢

Optimal ET sequence: 12, 169, 181, 193, 205, 217, 422

Badness (Sintel): 9.380

Subgroup temperament

The quindromeda temperament works well for the 2.3.5.17.19 subgroup, tempering out the password comma (1216/1215), aureusma (1445/1444), and langwisma (6144/6137). An obvious 17-limit interpretation of the generator is ~18/17, equating three 18/17s with 19/16, five 18/17s with 4/3, and twenty-eight 18/17s with the fifth harmonic.

2.3.5.17

Subgroup: 2.3.5.17

Comma list: 24576/24565, 295936/295245

Gencom: [2 18/17; 24576/24565 295936/295245]

Gencom mapping: [⟨1 2 0 0 0 0 5], ⟨0 -5 28 0 0 0 -11]]

Sval mapping: [⟨1 2 0 5], ⟨0 -5 28 -11]]

Optimal tunings:

  • WE: ~2 = 1199.878¢, ~18/17 = 99.513¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.523¢

Optimal ET sequence: 12, 169, 181, 193, 205, 422

Badness (Sintel): 0.768

2.3.5.17.19

Subgroup: 2.3.5.17.19

Comma list: 1216/1215, 1445/1444, 6144/6137

Gencom: [2 18/17; 1216/1215 1445/1444 6144/6137]

Gencom mapping: [⟨1 2 0 0 0 0 5 4], ⟨0 -5 28 0 0 0 -11 3]]

Sval mapping: [⟨1 2 0 5 4], ⟨0 -5 28 -11 3]]

Optimal tunings:

  • WE: ~2 = 1199.852¢, ~18/17 = 99.512¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.523¢

Optimal ET sequence: 12, 169, 181, 193, 205, 422

Badness (Sintel): 0.348

Quintagar

The quintagar temperament (12&217) tempers out the hemimean comma (3136/3125) and the garischisma (33554432/33480783) in the 7-limit. In the 2.3.5.7.17.19 subgroup, 256/255 (the difference between 16/15 and 17/16), 289/288 (between 17/16 and 18/17), 324/323 (between 18/17 and 19/18), 361/360 (between 19/18 and 20/19), and 400/399 (between 20/19 and 21/20) are equated together, and 476/475 (between 28/25 and 19/17) is tempered out. Immediate 2.3.5.7.11.17.19 extensions include quintoneum (12&217, tempering out 441/440), quinsandra (217&229, equating 385/384 with 400/399), and quinsandric (12&229, equating 400/399 with 441/440). Full 19-limit extensions include quintoneum (12f&217), quintoneoid (12&217), quinsandra (217&229), quinsandric (12f&229), and quinsandro (12&229). The name quintagar is so named because the generator is 1/5 of the garibaldi fourth (~4/3, about 497.8 cents).

Subgroup: 2.3.5.7

Comma list: 3136/3125, 33554432/33480783

Mapping: [⟨1 2 0 -3], ⟨0 -5 28 70]]

Optimal tunings:

  • WE: ~2 = 1199.824¢, ~200/189 = 99.540¢
  • CWE: ~2 = 1200.000¢, ~200/189 = 99.553¢

Optimal ET sequence: 12, 217, 229, 446, 675c

Badness (Sintel): 3.616

Quintoneum

The name quintoneum is a play on the words "quintans" and "cotoneum".

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 7168000/7144929

Mapping: [⟨1 2 0 -3 -5], ⟨0 -5 28 70 102]]

Optimal tunings:

  • WE: ~2 = 1199.864¢, ~35/33 = 99.528¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.538¢

Optimal ET sequence: 12, 205d, 217

Badness (Sintel): 2.881

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3136/3125, 13720/13689

Mapping: [⟨1 2 0 -3 -5 -7], ⟨0 -5 28 70 102 129]]

POTE generator: ~35/33 = 99.541

Optimal ET sequence: 12f, 205df, 217

Badness: 0.052361

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 3136/3125, 3757/3750

Mapping: [⟨1 2 0 -3 -5 -7 5], ⟨0 -5 28 70 102 129 -11]]

POTE generator: ~18/17 = 99.540

Optimal ET sequence: 12f, 205df, 217

Badness: 0.035653

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 441/440, 476/475, 595/594, 1216/1215, 3757/3750

Mapping: [⟨1 2 0 -3 -5 -7 5 4], ⟨0 -5 28 70 102 129 -11 3]]

POTE generator: ~18/17 = 99.541

Optimal ET sequence: 12f, 205df, 217

Badness: 0.025782

Quintoneoid

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125, 59150/59049

Mapping: [⟨1 2 0 -3 -5 11], ⟨0 -5 28 70 102 -88]]

POTE generator: ~35/33 = 99.537

Optimal ET sequence: 12, 205d, 217

Badness: 0.072826

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 441/440, 595/594, 1001/1000, 2601/2600, 3136/3125

Mapping: [⟨1 2 0 -3 -5 11 5], ⟨0 -5 28 70 102 -88 -11]]

POTE generator: ~18/17 = 99.537

Optimal ET sequence: 12, 205d, 217

Badness: 0.042339

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 441/440, 476/475, 595/594, 1001/1000, 1216/1215, 2601/2600

Mapping: [⟨1 2 0 -3 -5 11 5 4], ⟨0 -5 28 70 102 -88 -11 3]]

POTE generator: ~18/17 = 99.537

Optimal ET sequence: 12, 205d, 217

Badness: 0.028983

Quinsandra

The name quinsandra is a play on the words "quintans" and "cassandra". This temperament tempers out 19712/19683 and 41503/41472 in the 2.3.7.11 subgroup as the cassandra temperament, but with the hemimean comma rather than the schisma tempered out.

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 19712/19683, 41503/41472

Mapping: [⟨1 2 0 -3 13], ⟨0 -5 28 70 -115]]

Optimal tunings:

  • WE: ~2 = 1199.867¢, ~200/189 = 99.540¢
  • CWE: ~2 = 1200.000¢, ~200/189 = 99.551¢

Optimal ET sequence: 12e, 217, 446

Badness (Sintel): 3.633

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3136/3125, 4096/4095, 19712/19683

Mapping: [⟨1 2 0 -3 13 11], ⟨0 -5 28 70 -115 -88]]

POTE generator: ~55/52 = 99.548

Optimal ET sequence: 12e, 217, 446, 663c

Badness: 0.067730

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 1156/1155, 1377/1375, 3136/3125, 4096/4095

Mapping: [⟨1 2 0 -3 13 11 5], ⟨0 -5 28 70 -115 -88 -11]]

POTE generator: ~18/17 = 99.548

Optimal ET sequence: 12e, 217, 446, 663c

Badness: 0.038153

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 476/475, 936/935, 1156/1155, 1216/1215, 1377/1375, 1729/1728

Mapping: [⟨1 2 0 -3 13 11 5 4], ⟨0 -5 28 70 -115 -88 -11 3]]

POTE generator: ~18/17 = 99.547

Optimal ET sequence: 12e, 217, 446, 663ch

Badness: 0.026654

Quinsandric

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 8019/8000, 15488/15435

Mapping: [⟨1 2 0 -3 -6], ⟨0 -5 28 70 114]]

Optimal tunings:

  • WE: ~2 = 1199.770¢, ~200/189 = 99.551¢
  • CWE: ~2 = 1200.000¢, ~200/189 = 99.568¢

Optimal ET sequence: 12, 217e, 229, 470cd, 699cd

Badness (Sintel): 3.107

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 2080/2079, 3136/3125, 10648/10647

Mapping: [⟨1 2 0 -3 -6 -8], ⟨0 -5 28 70 114 141]]

POTE generator: ~55/52 = 99.578

Optimal ET sequence: 12f, 217ef, 229, 241, 470cd, 711ccd

Badness: 0.065701

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 561/560, 3136/3125, 7744/7735

Mapping: [⟨1 2 0 -3 -6 -8 5], ⟨0 -5 28 70 114 141 -11]]

POTE generator: ~18/17 = 99.574

Optimal ET sequence: 12f, 217ef, 229, 241, 470cd

Badness: 0.046624

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 442/441, 476/475, 561/560, 627/625, 6144/6137

Mapping: [⟨1 2 0 -3 -6 -8 5 4], ⟨0 -5 28 70 114 141 -11 3]]

POTE generator: ~18/17 = 99.575

Optimal ET sequence: 12f, 217ef, 229, 241, 470cd

Badness: 0.033145

Quinsandro

Subgroup: 2.3.5.7.11.13

Comma list: 1573/1568, 3136/3125, 4096/4095, 4459/4455

Mapping: [⟨1 2 0 -3 -6 11], ⟨0 -5 28 70 114 -88]]

POTE generator: ~200/189 = 99.556

Optimal ET sequence: 12, 217e, 229, 446e, 675ceef

Badness: 0.100195

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 561/560, 715/714, 1701/1700, 3136/3125, 4096/4095

Mapping: [⟨1 2 0 -3 -6 11 5], ⟨0 -5 28 70 114 -88 -11]]

POTE generator: ~18/17 = 99.556

Optimal ET sequence: 12, 217e, 229, 446e, 675ceef

Badness: 0.057851

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 286/285, 476/475, 561/560, 627/625, 1216/1215, 1729/1728

Mapping: [⟨1 2 0 -3 -6 11 5 4], ⟨0 -5 28 70 114 -88 -11 3]]

POTE generator: ~18/17 = 99.557

Optimal ET sequence: 12, 217e, 229, 446e, 675ceefh

Badness: 0.040410

Quintakwai

The quintakwai temperament (12&193) tempers out the hemifamity comma (5120/5103) and the compass comma (9765625/9680832) in the 7-limit; 1375/1372 and 4375/4356 in the 11-limit. In the 2.3.5.7.17.19 subgroup, 225/224 (the difference between 15/14 and 16/15), 256/255 (between 16/15 and 17/16), 289/288 (between 17/16 and 18/17), 324/323 (between 18/17 and 19/18), and 361/360 (between 19/18 and 20/19) are equated together, and 400/399 (between 20/19 and 21/20) is tempered out. The name quintakwai is so named because the generator is 1/5 of the kwai fourth (~4/3, about 497.4 cents).

Subgroup: 2.3.5.7

Comma list: 5120/5103, 9765625/9680832

Mapping: [⟨1 2 0 -2], ⟨0 -5 28 58]]

Optimal tunings:

  • WE: ~2 = 1199.876¢, ~625/588 = 99.473¢
  • CWE: ~2 = 1200.000¢, ~625/588 = 99.482¢

Optimal ET sequence: 12, 169, 181, 193

Badness (Sintel): 3.936

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4356, 5120/5103

Mapping: [⟨1 2 0 -2 -4], ⟨0 -5 28 58 90]]

Optimal tunings:

  • WE: ~2 = 1199.906¢, ~35/33 = 99.464¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.470¢

Optimal ET sequence: 12, 181, 193, 374, 567ce

Badness (Sintel): 2.419

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 1375/1372, 1575/1573, 4096/4095

Mapping: [⟨1 2 0 -2 -4 10], ⟨0 -5 28 58 90 -76]]

POTE generator: ~35/33 = 99.468

Optimal ET sequence: 12, 181, 193, 374, 567ce, 941bce

Badness: 0.062737

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 375/374, 595/594, 1275/1274, 4096/4095

Mapping: [⟨1 2 0 -2 -4 10 5], ⟨0 -5 28 58 90 -76 -11]]

POTE generator: ~18/17 = 99.469

Optimal ET sequence: 12, 181, 193, 374, 567ce, 941bceg

Badness: 0.037855

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 375/374, 400/399, 595/594, 1216/1215, 1275/1274

Mapping: [⟨1 2 0 -2 -4 10 5 4], ⟨0 -5 28 58 90 -76 -11 3]]

POTE generator: ~18/17 = 99.469

Optimal ET sequence: 12, 181, 193, 374, 567ce, 941bcegh, 1508bccdeegghh

Badness: 0.025861

Quinkwai

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1375/1372, 4375/4356

Mapping: [⟨1 2 0 -2 -4 -5], ⟨0 -5 28 58 90 105]]

POTE generator: ~35/33 = 99.456

Optimal ET sequence: 12f, 169e, 181

Badness: 0.061873

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 375/374, 595/594, 833/832, 1375/1372

Mapping: [⟨1 2 0 -2 -4 -5 5], ⟨0 -5 28 58 90 105 -11]]

POTE generator: ~18/17 = 99.458

Optimal ET sequence: 12f, 169e, 181, 374ff, 555cff

Badness: 0.043506

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 352/351, 375/374, 400/399, 495/494, 595/594, 1375/1372

Mapping: [⟨1 2 0 -2 -4 -5 5 4], ⟨0 -5 28 58 90 105 -11 3]]

POTE generator: ~18/17 = 99.459

Optimal ET sequence: 12f, 169e, 181, 374ff, 555cff

Badness: 0.031313

Quintakwoid

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 625/624, 1375/1372, 5120/5103

Mapping: [⟨1 2 0 -2 -4 -6], ⟨0 -5 28 58 90 117]]

POTE generator: ~35/33 = 99.484

Optimal ET sequence: 12f, 181f, 193

Badness: 0.057357

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 442/441, 595/594, 5120/5103

Mapping: [⟨1 2 0 -2 -4 -6 5], ⟨0 -5 28 58 90 117 -11]]

POTE generator: ~18/17 = 99.484

Optimal ET sequence: 12f, 181f, 193

Badness: 0.039289

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215

Mapping: [⟨1 2 0 -2 -4 -6 5 4], ⟨0 -5 28 58 90 117 -11 3]]

POTE generator: ~18/17 = 99.485

Optimal ET sequence: 12f, 181f, 193

Badness: 0.028101

Quindro

The quindro temperament (205&217) is an extension of the quindromeda temperament which tempers out the ragisma (4375/4374), the vishdel comma (5632/5625), and the ibnsinma (2080/2079).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 72057594037927936/71489976421753125

Mapping: [⟨1 2 0 15], ⟨0 -5 28 -147]]

Optimal tunings:

  • WE: ~2 = 1199.843¢, ~4428675/4194304 = 99.516¢
  • CWE: ~2 = 1200.000¢, ~4428675/4194304 = 99.529¢

Optimal ET sequence: 205, 217, 422, 639, 1061

Badness (Sintel): 8.019

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 25165824/25109315

Mapping: [⟨1 2 0 15 -5], ⟨0 -5 28 -147 102]]

Optimal tunings:

  • WE: ~2 = 1199.839¢, ~17325/16384 = 99.515¢
  • CWE: ~2 = 1200.000¢, ~17325/16384 = 99.529¢

Optimal ET sequence: 205, 217, 422, 639, 1061e

Badness (Sintel): 2.817

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4375/4374, 5632/5625, 20480/20449

Mapping: [⟨1 2 0 15 -5 11], ⟨0 -5 28 -147 102 -88]]

POTE generator: ~4096/3861 = 99.529

Optimal ET sequence: 205, 217, 422, 639, 1061ef

Badness: 0.039203

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 1156/1155, 2080/2079, 2431/2430, 4375/4374, 5632/5625

Mapping: [⟨1 2 0 15 -5 11 5], ⟨0 -5 28 -147 102 -88 -11]]

POTE generator: ~18/17 = 99.529

Optimal ET sequence: 205, 217, 422, 639, 1061ef

Badness: 0.024110

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 1156/1155, 1216/1215, 1445/1444, 2080/2079, 2376/2375, 2431/2430

Mapping: [⟨1 2 0 15 -5 11 5 4], ⟨0 -5 28 -147 102 -88 -11 3]]

POTE generator: ~18/17 = 99.529

Optimal ET sequence: 205, 217, 422, 639h, 1061efh

Badness: 0.016764

Hemiquindromeda

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 67108864/66976875

Mapping: [⟨1 2 0 6], ⟨0 -10 56 -77]]

Optimal tunings:

  • WE: ~2 = 1199.925¢, ~36/35 = 49.753¢
  • CWE: ~2 = 1200.000¢, ~36/35 = 49.757¢

Optimal ET sequence: 24, 169d, 193, 217, 410

Badness (Sintel): 6.190

11-limit

Subgroup: 2.3.5.7.11

Comma list: 14700/14641, 16384/16335, 19712/19683

Mapping: [⟨1 2 0 6 4], ⟨0 -10 56 -77 -13]]

Optimal tunings:

  • WE: ~2 = 1199.834¢, ~36/35 = 49.750¢
  • CWE: ~2 = 1200.000¢, ~36/35 = 49.757¢

Optimal ET sequence: 24, 169d, 193, 217, 410e

Badness (Sintel): 3.079

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 676/675, 4096/4095, 19712/19683

Mapping: [⟨1 2 0 6 4 2], ⟨0 -10 56 -77 -13 41]]

POTE generator: ~36/35 = 49.759

Optimal ET sequence: 24, 169d, 193, 217, 410e, 627e

Badness: 0.048715

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 595/594, 676/675, 1156/1155, 4096/4095

Mapping: [⟨1 2 0 6 4 2 5], ⟨0 -10 56 -77 -13 41 -22]]

POTE generator: ~34/33 = 49.759

Optimal ET sequence: 24, 169d, 193, 217, 410e, 627e

Badness: 0.028155

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 595/594, 676/675, 1156/1155, 1216/1215, 1729/1728

Mapping: [⟨1 2 0 6 4 2 5 4], ⟨0 -10 56 -77 -13 41 -22 6]]

POTE generator: ~34/33 = 49.759

Optimal ET sequence: 24, 169d, 193, 217, 410e

Badness: 0.019364

Semiquindromeda

The semiquindromeda temperament (410&422) is an extension of the quindromeda temperament with a half-octave period.

Subgroup: 2.3.5.7

Comma list: 102760448/102515625, 1220703125/1219784832

Mapping: [⟨2 4 0 -5], ⟨0 -5 28 64]]

Mapping generators: ~10125/7168, ~1323/1250

Optimal tunings:

  • WE: ~10125/7168 = 599.929¢, ~1323/1250 = 99.509¢
  • CWE: ~10125/7168 = 600.000¢, ~1323/1250 = 99.519¢

Optimal ET sequence: 12, 398, 410, 422, 832, 1254d, 2086bd

Badness (Sintel): 5.900

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 9801/9800, 85937500/85766121

Mapping: [⟨2 4 0 -5 -10], ⟨0 -5 28 64 102]]

Optimal tunings:

  • WE: ~99/70 = 599.922¢, ~1323/1250 = 99.512¢
  • CWE: ~99/70 = 600.000¢, ~1323/1250 = 99.523¢

Optimal ET sequence: 12, 410, 422

Badness (Sintel): 3.105

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 5632/5625, 831875/830466

Mapping: [⟨2 4 0 -5 -10 -13], ⟨0 -5 28 64 102 123]]

POTE generator: ~1323/1250 = 99.523

Optimal ET sequence: 12f, 410, 422, 1254df, 1676bdff, 2098bcddff

Badness: 0.053361

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 1716/1715, 2080/2079, 2500/2499, 5632/5625, 15895/15876

Mapping: [⟨2 4 0 -5 -10 -13 10], ⟨0 -5 28 64 102 123 -11]]

POTE generator: ~18/17 = 99.522

Optimal ET sequence: 12f, 410, 422, 832, 1254df, 1676bdff

Badness: 0.034659

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 1216/1215, 1445/1444, 1716/1715, 2080/2079, 2376/2375, 2500/2499

Mapping: [⟨2 4 0 -5 -10 -13 10 8], ⟨0 -5 28 64 102 123 -11 3]]

POTE generator: ~18/17 = 99.523

Optimal ET sequence: 12f, 410, 422, 1254dfhh, 1676bdffhh

Badness: 0.025439

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