Swetismic temperaments: Difference between revisions

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{{Mapping|legend=1| 2 1 0 2 | 0 3 0 5 | 0 0 1 0 }}
{{Mapping|legend=1| 2 1 0 2 | 0 3 0 5 | 0 0 1 0 }}
: mapping generators: ~343/243, ~9/7, ~5


: mapping generators: ~343/243, ~9/7, ~5
[[Optimal tuning]]s:  
* [[WE]]: ~343/243 = 599.9938{{c}}, ~9/7 = 433.8840{{c}}, ~5/4 = 386.3383{{c}}
: [[error map]]: {{val| -0.012 -0.309 -0.000 +0.582 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~9/7 = 433.8851{{c}}, ~5/4 = 386.3279{{c}}
: error map: {{val| 0.000 -0.300 +0.014 +0.600 }}


{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 202, 224, 354 }}
{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 202, 224, 354 }}


[[Badness]]: 0.748 × 10<sup>-3</sup>
[[Badness]] (Sintel): 3.30


=== 11-limit ===
=== 11-limit ===
Line 43: Line 48:


{{Mapping|legend=1| 2 1 0 2 3 | 0 3 0 5 -1 | 0 0 1 0 1 }}
{{Mapping|legend=1| 2 1 0 2 3 | 0 3 0 5 -1 | 0 0 1 0 1 }}
: mapping generators: ~99/70, ~9/7, ~5
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 599.9486{{c}}, ~9/7 = 433.9084{{c}}, ~5/4 = 386.2290{{c}}
: [[error map]]: {{val| -0.103 -0.281 -0.290 +0.613 +0.643 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~9/7 = 433.9204{{c}}, ~5/4 = 386.1272{{c}}
: error map: {{val| 0.000 -0.194 -0.187 +0.776 +0.889 }}


{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 224, 354, 506e, 578, 730e, 1084de, 1308dee }}
{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 224, 354, 506e, 578, 730e, 1084de, 1308dee }}


[[Badness]]: 0.361 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.434


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


{{Optimal ET sequence|legend=1| 22f, 36, 58, 72, 130, 224, 354, 578 }}
Optimal tunings:
* WE: ~99/70 = 599.9598{{c}}, ~9/7 = 433.8891{{c}}, ~5/4 = 386.1680{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 433.9059{{c}}, ~5/4 = 386.1014{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 36ce, 58, 72, 130, 224, 354, 578 }}


Badness: 0.535 × 10<sup>-3</sup>
Badness (Sintel): 0.501


=== Cerberus ===
=== Cerberus ===
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Mapping: {{mapping| 2 1 0 2 3 1 | 0 3 0 5 -1 -4 | 0 0 1 0 1 2 }}
Mapping: {{mapping| 2 1 0 2 3 1 | 0 3 0 5 -1 -4 | 0 0 1 0 1 2 }}


{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130 }}
Optimal tunings:
* WE: ~55/39 = 599.9565{{c}}, ~9/7 = 433.8372{{c}}, ~5/4 = 387.1905{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 433.8475{{c}}, ~5/4 = 387.1030{{c}}


Badness: 0.766 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 8d, 14cf, 22, 36f, 44ceff, 50, 58, 72, 130 }}
 
Badness (Sintel): 0.716


== Minos ==
== Minos ==
Line 75: Line 95:
[[Comma list]]: 540/539, 12005/11979
[[Comma list]]: 540/539, 12005/11979


{{Mapping|legend=1| 1 0 2 1 2 | 0 2 4 3 4 | 0 0 -5 -1 -3 }}
{{Mapping|legend=1| 1 0 -3 0 -1 | 0 2 4 3 4 | 0 0 5 1 3 }}
: mapping generators: ~2, ~121/70, ~66/49


: mapping generators: ~2, ~121/70, ~49/33
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.8923{{c}}, ~121/70 = 950.8327{{c}}, ~66/49 = 516.4492{{c}}
: [[error map]]: {{val| -0.108 -0.290 -0.414 +0.121 +1.468 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~121/70 = 950.8712{{c}}, ~66/49 = 516.5067{{c}}
: error map: {{val| 0.000 -0.213 -0.295 +0.294 +1.687 }}


{{Optimal ET sequence|legend=1| 9, 49, 58, 72, 130, 193, 202, 467 }}
{{Optimal ET sequence|legend=1| 9, …, 49, 58, 72, 130, 193, 202, 323e, 395e, 525e }} *


[[Badness]]: 0.886 × 10<sup>-3</sup>
<nowiki>*</nowiki> [[optimal patent val]]: [[467edo|467]]
 
[[Badness]] (Sintel): 1.06


=== 13-limit ===
=== 13-limit ===
Line 88: Line 115:
Comma list: 364/363, 540/539, 676/675
Comma list: 364/363, 540/539, 676/675


Mapping: {{mapping| 1 0 2 1 2 1 | 0 2 4 3 4 7 | 0 0 -5 -1 -3 -5 }}
Mapping: {{mapping| 1 0 -3 0 -1 -4 | 0 2 4 3 4 7 | 0 0 5 1 3 5 }}


{{Optimal ET sequence|legend=1| 9, 54, 58, 72, 121, 130, 193 }}
Optimal tunings:
* WE: ~2 = 1199.7669{{c}}, ~26/15 = 950.9360{{c}}, ~66/49 = 516.3652{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0324{{c}}, ~66/49 = 516.4883{{c}}


Badness: 0.724 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 9, …, 49f, 58, 72, 121, 130, 193, 251e, 323e }}
 
Badness (Sintel): 0.677


== Persephone ==
== Persephone ==
Line 99: Line 130:
[[Comma list]]: 540/539, 88209/87500
[[Comma list]]: 540/539, 88209/87500


{{Mapping|legend=1| 1 0 4 -2 10 | 0 1 4 0 7 | 0 0 -5 3 -11 }}
{{Mapping|legend=1| 1 0 -6 4 -12 | 0 1 4 0 7 | 0 0 5 -3 11 }}
: mapping generators: ~2, ~3, ~33/25


: mapping generators: ~2, ~3, ~100/33
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0185{{c}}, ~3/2 = 700.9179{{c}}, ~33/25 = 476.7538{{c}}
: [[error map]]: {{val| +0.019 -1.019 +1.090 +0.987 -0.694 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.9220{{c}}, ~33/25 = 476.7393{{c}}
: error map: {{val| 0.000 -1.033 +1.071 +0.956 -0.732 }}


{{Optimal ET sequence|legend=1| 50, 53, 58, 103, 108, 111, 161, 214, 272, 375, 1022bcde, 1236bcde, 1397bcde }}
{{Optimal ET sequence|legend=1| 8d, …, 45e, 50, 53, 58, 103, 161, 214, 272, 375 }}


[[Badness]]: 2.015 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.42


=== 13-limit ===
=== 13-limit ===
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Comma list: 351/350, 540/539, 847/845
Comma list: 351/350, 540/539, 847/845


Mapping: {{mapping| 1 0 4 -2 10 7 | 0 1 4 0 7 5 | 0 0 -5 3 -11 -7 }}
Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 1 4 0 7 5 | 0 0 5 -3 11 7 }}


{{Optimal ET sequence|legend=1| 50, 53, 58, 103, 108, 111, 161, 214, 375, 700cdef, 861bcdef, 1075bcdef, 1236bcdef, 1450bcdef }}
Optimal tunings:
* WE: ~2 = 1199.9553{{c}}, ~3/2 = 700.9477{{c}}, ~33/25 = 476.6709{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.9378{{c}}, ~33/25 = 476.7054{{c}}


Badness: 0.997 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 8d, …, 45ef, 50, 53, 58, 103, 111, 161, 214, 375 }}
 
Badness (Sintel): 0.933


== Hypnos ==
== Hypnos ==
Line 124: Line 164:


{{Mapping|legend=1| 1 0 0 -3 8 | 0 3 0 11 -13 | 0 0 1 0 1 }}
{{Mapping|legend=1| 1 0 0 -3 8 | 0 3 0 11 -13 | 0 0 1 0 1 }}
: mapping generators: ~2, ~81/56, ~5


: mapping generators: ~2, ~81/56, ~5
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3393{{c}}, ~81/56 = 633.6837{{c}}, ~5/4 = 385.6892{{c}}
: [[error map]]: {{val| +0.339 -0.904 +0.054 +0.677 -0.124 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~81/56 = 633.5228{{c}}, ~5/4 = 385.9359{{c}}
: error map: {{val| 0.000 -1.387 -0.378 -0.075 -1.178 }}


{{Optimal ET sequence|legend=1| 19, 36, 53, 72, 161, 197e, 214, 233, 286, 358, 591be, 663be, 949bce }}
{{Optimal ET sequence|legend=1| 17c, 19, …, 36, 53, 72, 161, 214, 233, 286, 358, 591be }}


[[Badness]]: 1.629 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.96


=== 13-limit ===
=== 13-limit ===
Line 138: Line 183:
Mapping: {{mapping| 1 0 0 -3 8 0 | 0 3 0 11 -13 7 | 0 0 1 0 1 0 }}
Mapping: {{mapping| 1 0 0 -3 8 0 | 0 3 0 11 -13 7 | 0 0 1 0 1 0 }}


{{Optimal ET sequence|legend=1| 19, 36, 53, 72, 125f, 197ef }}
Optimal tunings:
* WE: ~2 = 1200.6306{{c}}, ~81/56 = 633.8505{{c}}, ~5/4 = 385.0539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~81/56 = 633.5546{{c}}, ~5/4 = 385.4668{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 19, 36ce, 53, 72, 125f, 197ef }}


Badness: 1.439 × 10<sup>-3</sup>
Badness (Sintel): 1.35


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Swetismic temperaments| ]] <!-- main article -->
[[Category:Swetismic temperaments| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 11:48, 24 December 2025

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.

This is a collection of rank-3 temperaments tempering out the swetisma, 540/539 = [2 3 1 -2 -1⟩. For the rank-4 swetismic temperament, see Catalog of rank-4 temperaments #Swetismic (540/539).

Temperaments discussed elsewhere are:

Considered below are hades, minos, persephone and hypnos.

Hades

7-limit (stearnsmic)

Subgroup: 2.3.5.7

Comma list: 118098/117649

Mapping: [⟨2 1 0 2], ⟨0 3 0 5], ⟨0 0 1 0]]

mapping generators: ~343/243, ~9/7, ~5

Optimal tunings:

  • WE: ~343/243 = 599.9938 ¢, ~9/7 = 433.8840 ¢, ~5/4 = 386.3383 ¢
error map: ⟨-0.012 -0.309 -0.000 +0.582]
  • CWE: ~343/243 = 600.0000 ¢, ~9/7 = 433.8851 ¢, ~5/4 = 386.3279 ¢
error map: ⟨0.000 -0.300 +0.014 +0.600]

Optimal ET sequence: 22, 50, 58, 72, 130, 152, 202, 224, 354

Badness (Sintel): 3.30

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993

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

mapping generators: ~99/70, ~9/7, ~5

Optimal tunings:

  • WE: ~99/70 = 599.9486 ¢, ~9/7 = 433.9084 ¢, ~5/4 = 386.2290 ¢
error map: ⟨-0.103 -0.281 -0.290 +0.613 +0.643]
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 433.9204 ¢, ~5/4 = 386.1272 ¢
error map: ⟨0.000 -0.194 -0.187 +0.776 +0.889]

Optimal ET sequence: 22, 50, 58, 72, 130, 152, 224, 354, 506e, 578, 730e, 1084de, 1308dee

Badness (Sintel): 0.434

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573

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

Optimal tunings:

  • WE: ~99/70 = 599.9598 ¢, ~9/7 = 433.8891 ¢, ~5/4 = 386.1680 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 433.9059 ¢, ~5/4 = 386.1014 ¢

Optimal ET sequence: 22f, 36ce, 58, 72, 130, 224, 354, 578

Badness (Sintel): 0.501

Cerberus

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 540/539

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

Optimal tunings:

  • WE: ~55/39 = 599.9565 ¢, ~9/7 = 433.8372 ¢, ~5/4 = 387.1905 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~9/7 = 433.8475 ¢, ~5/4 = 387.1030 ¢

Optimal ET sequence: 8d, 14cf, 22, 36f, 44ceff, 50, 58, 72, 130

Badness (Sintel): 0.716

Minos

Subgroup: 2.3.5.7.11

Comma list: 540/539, 12005/11979

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

mapping generators: ~2, ~121/70, ~66/49

Optimal tunings:

  • WE: ~2 = 1199.8923 ¢, ~121/70 = 950.8327 ¢, ~66/49 = 516.4492 ¢
error map: ⟨-0.108 -0.290 -0.414 +0.121 +1.468]
  • CWE: ~2 = 1200.0000 ¢, ~121/70 = 950.8712 ¢, ~66/49 = 516.5067 ¢
error map: ⟨0.000 -0.213 -0.295 +0.294 +1.687]

Optimal ET sequence: 9, …, 49, 58, 72, 130, 193, 202, 323e, 395e, 525e *

* optimal patent val: 467

Badness (Sintel): 1.06

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 540/539, 676/675

Mapping: [⟨1 0 -3 0 -1 -4], ⟨0 2 4 3 4 7], ⟨0 0 5 1 3 5]]

Optimal tunings:

  • WE: ~2 = 1199.7669 ¢, ~26/15 = 950.9360 ¢, ~66/49 = 516.3652 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.0324 ¢, ~66/49 = 516.4883 ¢

Optimal ET sequence: 9, …, 49f, 58, 72, 121, 130, 193, 251e, 323e

Badness (Sintel): 0.677

Persephone

Subgroup: 2.3.5.7.11

Comma list: 540/539, 88209/87500

Mapping: [⟨1 0 -6 4 -12], ⟨0 1 4 0 7], ⟨0 0 5 -3 11]]

mapping generators: ~2, ~3, ~33/25

Optimal tunings:

  • WE: ~2 = 1200.0185 ¢, ~3/2 = 700.9179 ¢, ~33/25 = 476.7538 ¢
error map: ⟨+0.019 -1.019 +1.090 +0.987 -0.694]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.9220 ¢, ~33/25 = 476.7393 ¢
error map: ⟨0.000 -1.033 +1.071 +0.956 -0.732]

Optimal ET sequence: 8d, …, 45e, 50, 53, 58, 103, 161, 214, 272, 375

Badness (Sintel): 2.42

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845

Mapping: [⟨1 0 -6 4 -12 -7], ⟨0 1 4 0 7 5], ⟨0 0 5 -3 11 7]]

Optimal tunings:

  • WE: ~2 = 1199.9553 ¢, ~3/2 = 700.9477 ¢, ~33/25 = 476.6709 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.9378 ¢, ~33/25 = 476.7054 ¢

Optimal ET sequence: 8d, …, 45ef, 50, 53, 58, 103, 111, 161, 214, 375

Badness (Sintel): 0.933

Hypnos

Subgroup: 2.3.5.7.11

Comma list: 540/539, 72171/71680

Mapping: [⟨1 0 0 -3 8], ⟨0 3 0 11 -13], ⟨0 0 1 0 1]]

mapping generators: ~2, ~81/56, ~5

Optimal tunings:

  • WE: ~2 = 1200.3393 ¢, ~81/56 = 633.6837 ¢, ~5/4 = 385.6892 ¢
error map: ⟨+0.339 -0.904 +0.054 +0.677 -0.124]
  • CWE: ~2 = 1200.0000 ¢, ~81/56 = 633.5228 ¢, ~5/4 = 385.9359 ¢
error map: ⟨0.000 -1.387 -0.378 -0.075 -1.178]

Optimal ET sequence: 17c, 19, …, 36, 53, 72, 161, 214, 233, 286, 358, 591be

Badness (Sintel): 1.96

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 540/539, 729/728

Mapping: [⟨1 0 0 -3 8 0], ⟨0 3 0 11 -13 7], ⟨0 0 1 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.6306 ¢, ~81/56 = 633.8505 ¢, ~5/4 = 385.0539 ¢
  • CWE: ~2 = 1200.0000 ¢, ~81/56 = 633.5546 ¢, ~5/4 = 385.4668 ¢

Optimal ET sequence: 17c, 19, 36ce, 53, 72, 125f, 197ef

Badness (Sintel): 1.35

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