Quindromeda family: Difference between revisions

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Line 102: Line 102:
Mapping: [{{val| 1 2 0 -3 -5 -7 }}, {{val| 0 -5 28 70 102 129 }}]
Mapping: [{{val| 1 2 0 -3 -5 -7 }}, {{val| 0 -5 28 70 102 129 }}]


POTE generator: ~35/33 = 99.541
Optimal tunings:
* WE: ~2 = 1199.855¢, ~35/33 = 99.529¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.540¢


{{Optimal ET sequence|legend=0| 12f, 205df, 217 }}
{{Optimal ET sequence|legend=0| 12f, 205df, 217 }}


Badness: 0.052361
Badness (Sintel): 2.164


===== 17-limit =====
===== 17-limit =====
Line 115: Line 117:
Mapping: [{{val| 1 2 0 -3 -5 -7 5 }}, {{val| 0 -5 28 70 102 129 -11 }}]
Mapping: [{{val| 1 2 0 -3 -5 -7 5 }}, {{val| 0 -5 28 70 102 129 -11 }}]


POTE generator: ~18/17 = 99.540
Optimal tunings:
* WE: ~2 = 1199.890¢, ~18/17 = 99.531¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.539¢


{{Optimal ET sequence|legend=0| 12f, 205df, 217 }}
{{Optimal ET sequence|legend=0| 12f, 205df, 217 }}


Badness: 0.035653
Badness (Sintel): 1.816


===== 19-limit =====
===== 19-limit =====
Line 128: Line 132:
Mapping: [{{val| 1 2 0 -3 -5 -7 5 4 }}, {{val| 0 -5 28 70 102 129 -11 3 }}]
Mapping: [{{val| 1 2 0 -3 -5 -7 5 4 }}, {{val| 0 -5 28 70 102 129 -11 3 }}]


POTE generator: ~18/17 = 99.541
Optimal tunings:
* WE: ~2 = 1199.859¢, ~18/17 = 99.529¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.540¢


{{Optimal ET sequence|legend=0| 12f, 205df, 217 }}
{{Optimal ET sequence|legend=0| 12f, 205df, 217 }}


Badness: 0.025782
Badness (Sintel): 1.569


==== Quintoneoid ====
==== Quintoneoid ====
Line 141: Line 147:
Mapping: [{{val| 1 2 0 -3 -5 11 }}, {{val| 0 -5 28 70 102 -88 }}]
Mapping: [{{val| 1 2 0 -3 -5 11 }}, {{val| 0 -5 28 70 102 -88 }}]


POTE generator: ~35/33 = 99.537
Optimal tunings:
* WE: ~2 = 1199.901¢, ~35/33 = 99.529¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.537¢


{{Optimal ET sequence|legend=0| 12, 205d, 217 }}
{{Optimal ET sequence|legend=0| 12, 205d, 217 }}


Badness: 0.072826
Badness (Sintel): 3.009


===== 17-limit =====
===== 17-limit =====
Line 154: Line 162:
Mapping: [{{val| 1 2 0 -3 -5 11 5 }}, {{val| 0 -5 28 70 102 -88 -11 }}]
Mapping: [{{val| 1 2 0 -3 -5 11 5 }}, {{val| 0 -5 28 70 102 -88 -11 }}]


POTE generator: ~18/17 = 99.537
Optimal tunings:
* WE: ~2 = 1199.912¢, ~18/17 = 99.530¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.537¢


{{Optimal ET sequence|legend=0| 12, 205d, 217 }}
{{Optimal ET sequence|legend=0| 12, 205d, 217 }}


Badness: 0.042339
Badness (Sintel): 2.157


===== 19-limit =====
===== 19-limit =====
Line 167: Line 177:
Mapping: [{{val| 1 2 0 -3 -5 11 5 4 }}, {{val| 0 -5 28 70 102 -88 -11 3 }}]
Mapping: [{{val| 1 2 0 -3 -5 11 5 4 }}, {{val| 0 -5 28 70 102 -88 -11 3 }}]


POTE generator: ~18/17 = 99.537
Optimal tunings:
* WE: ~2 = 1199.889¢, ~18/17 = 99.528¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.537¢


{{Optimal ET sequence|legend=0| 12, 205d, 217 }}
{{Optimal ET sequence|legend=0| 12, 205d, 217 }}


Badness: 0.028983
Badness (Sintel): 1.763


=== Quinsandra ===
=== Quinsandra ===
Line 197: Line 209:
Mapping: [{{val| 1 2 0 -3 13 11 }}, {{val| 0 -5 28 70 -115 -88 }}]
Mapping: [{{val| 1 2 0 -3 13 11 }}, {{val| 0 -5 28 70 -115 -88 }}]


POTE generator: ~55/52 = 99.548
Optimal tunings:
* WE: ~2 = 1199.922¢, ~55/52 = 99.541¢
* CWE: ~2 = 1200.000¢, ~55/52 = 99.548¢


{{Optimal ET sequence|legend=0| 12e, 217, 446, 663c }}
{{Optimal ET sequence|legend=0| 12e, 217, 446, 663c }}


Badness: 0.067730
Badness (Sintel): 2.799


==== 17-limit ====
==== 17-limit ====
Line 210: Line 224:
Mapping: [{{val| 1 2 0 -3 13 11 5 }}, {{val| 0 -5 28 70 -115 -88 -11 }}]
Mapping: [{{val| 1 2 0 -3 13 11 5 }}, {{val| 0 -5 28 70 -115 -88 -11 }}]


POTE generator: ~18/17 = 99.548
Optimal tunings:
* WE: ~2 = 1199.932¢, ~18/17 = 99.542¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.548¢


{{Optimal ET sequence|legend=0| 12e, 217, 446, 663c }}
{{Optimal ET sequence|legend=0| 12e, 217, 446, 663c }}


Badness: 0.038153
Badness (Sintel): 1.944


==== 19-limit ====
==== 19-limit ====
Line 223: Line 239:
Mapping: [{{val| 1 2 0 -3 13 11 5 4 }}, {{val| 0 -5 28 70 -115 -88 -11 3 }}]
Mapping: [{{val| 1 2 0 -3 13 11 5 4 }}, {{val| 0 -5 28 70 -115 -88 -11 3 }}]


POTE generator: ~18/17 = 99.547
Optimal tunings:
* WE: ~2 = 1199.907¢, ~18/17 = 99.540¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.547¢


{{Optimal ET sequence|legend=0| 12e, 217, 446, 663ch }}
{{Optimal ET sequence|legend=0| 12e, 217, 446, 663ch }}


Badness: 0.026654
Badness (Sintel): 1.621


=== Quinsandric ===
=== Quinsandric ===
Line 251: Line 269:
Mapping: [{{val| 1 2 0 -3 -6 -8 }}, {{val| 0 -5 28 70 114 141 }}]
Mapping: [{{val| 1 2 0 -3 -6 -8 }}, {{val| 0 -5 28 70 114 141 }}]


POTE generator: ~55/52 = 99.578
Optimal tunings:
* WE: ~2 = 1199.730¢, ~55/52 = 99.555¢
* CWE: ~2 = 1200.000¢, ~55/52 = 99.575¢


{{Optimal ET sequence|legend=0| 12f, 217ef, 229, 241, 470cd, 711ccd }}
{{Optimal ET sequence|legend=0| 12f, 217ef, 229, 241, 470cd, 711ccd }}


Badness: 0.065701
Badness (Sintel): 2.715


===== 17-limit =====
===== 17-limit =====
Line 264: Line 284:
Mapping: [{{val| 1 2 0 -3 -6 -8 5 }}, {{val| 0 -5 28 70 114 141 -11 }}]
Mapping: [{{val| 1 2 0 -3 -6 -8 5 }}, {{val| 0 -5 28 70 114 141 -11 }}]


POTE generator: ~18/17 = 99.574
Optimal tunings:
* WE: ~2 = 1199.827¢, ~18/17 = 99.560¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.573¢


{{Optimal ET sequence|legend=0| 12f, 217ef, 229, 241, 470cd }}
{{Optimal ET sequence|legend=0| 12f, 217ef, 229, 241, 470cd }}


Badness: 0.046624
Badness (Sintel): 2.375


===== 19-limit =====
===== 19-limit =====
Line 277: Line 299:
Mapping: [{{val| 1 2 0 -3 -6 -8 5 4 }}, {{val| 0 -5 28 70 114 141 -11 3 }}]
Mapping: [{{val| 1 2 0 -3 -6 -8 5 4 }}, {{val| 0 -5 28 70 114 141 -11 3 }}]


POTE generator: ~18/17 = 99.575
Optimal tunings:
* WE: ~2 = 1199.804¢, ~18/17 = 99.559¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.574¢


{{Optimal ET sequence|legend=0| 12f, 217ef, 229, 241, 470cd }}
{{Optimal ET sequence|legend=0| 12f, 217ef, 229, 241, 470cd }}


Badness: 0.033145
Badness (Sintel): 2.016


==== Quinsandro ====
==== Quinsandro ====
Line 290: Line 314:
Mapping: [{{val| 1 2 0 -3 -6 11 }}, {{val| 0 -5 28 70 114 -88 }}]
Mapping: [{{val| 1 2 0 -3 -6 11 }}, {{val| 0 -5 28 70 114 -88 }}]


Optimal tunings:
* WE: ~2 = 1200.032¢, ~200/189 = 99.559¢
* CWE: ~2 = 1200.000¢, ~200/189 = 99.556¢
POTE generator: ~200/189 = 99.556
POTE generator: ~200/189 = 99.556


{{Optimal ET sequence|legend=0| 12, 217e, 229, 446e, 675ceef }}
{{Optimal ET sequence|legend=0| 12, 217e, 229, 446e, 675ceef }}


Badness: 0.100195
Badness (Sintel): 4.140


===== 17-limit =====
===== 17-limit =====
Line 303: Line 330:
Mapping: [{{val| 1 2 0 -3 -6 11 5 }}, {{val| 0 -5 28 70 114 -88 -11 }}]
Mapping: [{{val| 1 2 0 -3 -6 11 5 }}, {{val| 0 -5 28 70 114 -88 -11 }}]


POTE generator: ~18/17 = 99.556
Optimal tunings:
* WE: ~2 = 1200.030¢, ~18/17 = 99.559¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.556¢


{{Optimal ET sequence|legend=0| 12, 217e, 229, 446e, 675ceef }}
{{Optimal ET sequence|legend=0| 12, 217e, 229, 446e, 675ceef }}


Badness: 0.057851
Badness (Sintel): 2.947


===== 19-limit =====
===== 19-limit =====
Line 316: Line 345:
Mapping: [{{val| 1 2 0 -3 -6 11 5 4 }}, {{val| 0 -5 28 70 114 -88 -11 3 }}]
Mapping: [{{val| 1 2 0 -3 -6 11 5 4 }}, {{val| 0 -5 28 70 114 -88 -11 3 }}]


POTE generator: ~18/17 = 99.557
Optimal tunings:
* WE: ~2 = 1199.989¢, ~18/17 = 99.556¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.557¢


{{Optimal ET sequence|legend=0| 12, 217e, 229, 446e, 675ceefh }}
{{Optimal ET sequence|legend=0| 12, 217e, 229, 446e, 675ceefh }}


Badness: 0.040410
Badness (Sintel): 2.458


== Quintakwai ==
== Quintakwai ==
Line 567: Line 598:


== Hemiquindromeda ==
== Hemiquindromeda ==
The ''hemiquindromeda'' temperament (193&217) splits the fourth in ten, and tempers out the [[canousma]] (4802000/4782969) and the decovulture comma (67108864/66976875) in the 7-limit.
The ''hemiquindromeda'' temperament (193&217) splits the fourth in ten, and tempers out the [[canousma]] (4802000/4782969) and the [[decovulture comma]] (67108864/66976875) in the 7-limit.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7

Latest revision as of 22:17, 1 March 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 is also the temperament where the fourth (4/3) is identified by a stack of five generators, 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]]

Optimal tunings:

  • WE: ~2 = 1199.855¢, ~35/33 = 99.529¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.540¢

Optimal ET sequence: 12f, 205df, 217

Badness (Sintel): 2.164

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]]

Optimal tunings:

  • WE: ~2 = 1199.890¢, ~18/17 = 99.531¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.539¢

Optimal ET sequence: 12f, 205df, 217

Badness (Sintel): 1.816

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]]

Optimal tunings:

  • WE: ~2 = 1199.859¢, ~18/17 = 99.529¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.540¢

Optimal ET sequence: 12f, 205df, 217

Badness (Sintel): 1.569

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]]

Optimal tunings:

  • WE: ~2 = 1199.901¢, ~35/33 = 99.529¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.537¢

Optimal ET sequence: 12, 205d, 217

Badness (Sintel): 3.009

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]]

Optimal tunings:

  • WE: ~2 = 1199.912¢, ~18/17 = 99.530¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.537¢

Optimal ET sequence: 12, 205d, 217

Badness (Sintel): 2.157

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]]

Optimal tunings:

  • WE: ~2 = 1199.889¢, ~18/17 = 99.528¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.537¢

Optimal ET sequence: 12, 205d, 217

Badness (Sintel): 1.763

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]]

Optimal tunings:

  • WE: ~2 = 1199.922¢, ~55/52 = 99.541¢
  • CWE: ~2 = 1200.000¢, ~55/52 = 99.548¢

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

Badness (Sintel): 2.799

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]]

Optimal tunings:

  • WE: ~2 = 1199.932¢, ~18/17 = 99.542¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.548¢

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

Badness (Sintel): 1.944

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]]

Optimal tunings:

  • WE: ~2 = 1199.907¢, ~18/17 = 99.540¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.547¢

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

Badness (Sintel): 1.621

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]]

Optimal tunings:

  • WE: ~2 = 1199.730¢, ~55/52 = 99.555¢
  • CWE: ~2 = 1200.000¢, ~55/52 = 99.575¢

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

Badness (Sintel): 2.715

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]]

Optimal tunings:

  • WE: ~2 = 1199.827¢, ~18/17 = 99.560¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.573¢

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

Badness (Sintel): 2.375

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]]

Optimal tunings:

  • WE: ~2 = 1199.804¢, ~18/17 = 99.559¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.574¢

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

Badness (Sintel): 2.016

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]]

Optimal tunings:

  • WE: ~2 = 1200.032¢, ~200/189 = 99.559¢
  • CWE: ~2 = 1200.000¢, ~200/189 = 99.556¢

POTE generator: ~200/189 = 99.556

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

Badness (Sintel): 4.140

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]]

Optimal tunings:

  • WE: ~2 = 1200.030¢, ~18/17 = 99.559¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.556¢

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

Badness (Sintel): 2.947

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]]

Optimal tunings:

  • WE: ~2 = 1199.989¢, ~18/17 = 99.556¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.557¢

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

Badness (Sintel): 2.458

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]]

Optimal tunings:

  • WE: ~2 = 1199.958¢, ~35/33 = 99.465¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.468¢

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

Badness (Sintel): 2.592

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]]

Optimal tunings:

  • WE: ~2 = 1199.929¢, ~18/17 = 99.463¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.469¢

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

Badness (Sintel): 1.928

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]]

Optimal tunings:

  • WE: ~2 = 1199.910¢, ~18/17 = 99.462¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.469¢

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

Badness (Sintel): 1.573

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]]

Optimal tunings:

  • WE: ~2 = 1199.964¢, ~35/33 = 99.453¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.455¢

Optimal ET sequence: 12f, 169e, 181

Badness (Sintel): 2.557

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]]

Optimal tunings:

  • WE: ~2 = 1199.901¢, ~18/17 = 99.450¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.458¢

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

Badness (Sintel): 2.216

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]]

Optimal tunings:

  • WE: ~2 = 1199.879¢, ~18/17 = 99.449¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.458¢

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

Badness (Sintel): 1.905

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]]

Optimal tunings:

  • WE: ~2 = 1199.844¢, ~35/33 = 99.471¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.483¢

Optimal ET sequence: 12f, 181f, 193

Badness (Sintel): 2.370

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]]

Optimal tunings:

  • WE: ~2 = 1199.838¢, ~18/17 = 99.471¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.483¢

Optimal ET sequence: 12f, 181f, 193

Badness (Sintel): 2.002

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]]

Optimal tunings:

  • WE: ~2 = 1199.826¢, ~18/17 = 99.470¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.484¢

Optimal ET sequence: 12f, 181f, 193

Badness (Sintel): 1.709

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]]

Optimal tunings:

  • WE: ~2 = 1199.825¢, ~143/135 = 99.514¢
  • CWE: ~2 = 1200.000¢, ~143/135 = 99.529¢

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

Badness (Sintel): 1.620

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]]

Optimal tunings:

  • WE: ~2 = 1199.843¢, ~18/17 = 99.516¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.529¢

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

Badness (Sintel): 1.228

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]]

Optimal tunings:

  • WE: ~2 = 1199.830¢, ~18/17 = 99.515¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.529¢

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

Badness (Sintel): 1.020

Hemiquindromeda

The hemiquindromeda temperament (193&217) splits the fourth in ten, and tempers out the canousma (4802000/4782969) and the decovulture comma (67108864/66976875) in the 7-limit.

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]]

Optimal tunings:

  • WE: ~2 = 1199.885¢, ~36/35 = 49.754¢
  • CWE: ~2 = 1200.000¢, ~36/35 = 49.759¢

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

Badness (Sintel): 2.013

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]]

Optimal tunings:

  • WE: ~2 = 1199.889¢, ~34/33 = 49.754¢
  • CWE: ~2 = 1200.000¢, ~34/33 = 49.759¢

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

Badness (Sintel): 1.434

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]]

Optimal tunings:

  • WE: ~2 = 1199.872¢, ~34/33 = 49.754¢
  • CWE: ~2 = 1200.000¢, ~34/33 = 49.759¢

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

Badness (Sintel): 1.178

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]]

Optimal tunings:

  • WE: ~99/70 = 599.927¢, ~1323/1250 = 99.511¢
  • CWE: ~99/70 = 600.000¢, ~1323/1250 = 99.522¢

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

Badness (Sintel): 2.205

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]]

Optimal tunings:

  • WE: ~99/70 = 599.938¢, ~18/17 = 99.512¢
  • CWE: ~99/70 = 600.000¢, ~18/17 = 99.521¢

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

Badness (Sintel): 1.766

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]]

Optimal tunings:

  • WE: ~99/70 = 599.925¢, ~18/17 = 99.511¢
  • CWE: ~99/70 = 600.000¢, ~18/17 = 99.522¢

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

Badness (Sintel): 1.547

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