Quintaleap family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''quintaleap family''' tempers out {{monzo| 37 -16 -5 }}, equating a stack of two [[Pythagorean comma]]s with a stack of five [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]]. It is also the temperament where [[4/3]] is identified by a stack of five [[135/128]]'s.  
The '''quintaleap family''' tempers out {{monzo| 37 -16 -5 }}, equating a stack of two [[Pythagorean comma]]s with a stack of five [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]]. It is also the temperament where [[4/3]] is identified by a stack of five [[135/128]]'s, making it a member of [[Ploidacot/Omega-pentacot|omega-pentacot]] temperaments.  


== Quintaleap ==
== Quintaleap ==
Line 11: Line 11:
[[Mapping]]: [{{val| 1 2 1 }}, {{val| 0 -5 16 }}]
[[Mapping]]: [{{val| 1 2 1 }}, {{val| 0 -5 16 }}]


[[POTE generator]]: ~135/128 = 99.267
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.359¢, ~135/128 = 99.214¢
* [[CWE]]: ~2 = 1200.000¢, ~135/128 = 99.256¢


{{Optimal ET sequence|legend=1| 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc }}
{{Optimal ET sequence|legend=1| 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc }}


[[Badness]]: 0.444506
[[Badness]] (Sintel): 10.427


=== Subgroup temperament ===
=== Subgroup temperament ===
Line 32: Line 34:
[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 }}, {{val| 0 -5 16 -11 }}]
[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 }}, {{val| 0 -5 16 -11 }}]


[[Tp tuning|POL2 generator]]: ~18/17 = 99.272
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.328¢, ~18/17 = 99.217¢
* [[CWE]]: ~2 = 1200.000¢, ~18/17 = 99.270¢


{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}
{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3051 cents
[[Badness]] (Sintel): 1.305


==== 2.3.5.17.19 ====
==== 2.3.5.17.19 ====
Line 49: Line 53:
[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 4 }}, {{val| 0 -5 16 -11 3 }}]
[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 4 }}, {{val| 0 -5 16 -11 3 }}]


[[Tp tuning|POL2 generator]]: ~18/17 = 99.276
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.446¢, ~18/17 = 99.230¢
* [[CWE]]: ~2 = 1200.000¢, ~18/17 = 99.273¢


{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}
{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3427 cents
[[Badness]] (Sintel): 0.712
 
Badness: 0.014623


== Quintupole ==
== Quintupole ==
Line 68: Line 72:
[[Mapping]]: [{{val| 1 2 1 0 }}, {{val| 0 -5 16 34 }}]
[[Mapping]]: [{{val| 1 2 1 0 }}, {{val| 0 -5 16 34 }}]


[[POTE generator]]: ~135/128 = 99.175
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.324¢, ~135/128 = 99.120¢
* [[CWE]]: ~2 = 1200.000¢, ~135/128 = 99.162¢


{{Optimal ET sequence|legend=1| 12, 97, 109, 121 }}
{{Optimal ET sequence|legend=1| 12, 97, 109, 121 }}


[[Badness]]: 0.111620
[[Badness]] (Sintel): 2.825


=== 11-limit ===
=== 11-limit ===
Line 81: Line 87:
Mapping: [{{val| 1 2 1 0 -1 }}, {{val| 0 -5 16 34 54 }}]
Mapping: [{{val| 1 2 1 0 -1 }}, {{val| 0 -5 16 34 54 }}]


POTE generator: ~35/33 = 99.156
Optimal tunings:
* WE: ~2 = 1199.342¢, ~35/33 = 99.101¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.144¢


Optimal ET sequence: {{Optimal ET sequence| 12, 109, 121, 351bde, 472bdee }}
Optimal ET sequence: {{Optimal ET sequence| 12, 109, 121, 351bde, 472bdee }}


Badness: 0.056501
Badness (Sintel): 1.868


=== 13-limit ===
=== 13-limit ===
Line 94: Line 102:
Mapping: [{{val| 1 2 1 0 -1 -2 }}, {{val| 0 -5 16 34 54 69 }}]
Mapping: [{{val| 1 2 1 0 -1 -2 }}, {{val| 0 -5 16 34 54 69 }}]


POTE generator: ~35/33 = 99.165
Optimal tunings:
* WE: ~2 = 1199.322¢, ~35/33 = 99.109¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.154¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 109, 121 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 109, 121 }}


Badness: 0.038431
Badness (Sintel): 1.588


=== 17-limit ===
=== 17-limit ===
Line 107: Line 117:
Mapping: [{{val| 1 2 1 0 -1 -2 5 }}, {{val| 0 -5 16 34 54 69 -11 }}]
Mapping: [{{val| 1 2 1 0 -1 -2 5 }}, {{val| 0 -5 16 34 54 69 -11 }}]


POTE generator: ~18/17 = 99.172
Optimal tunings:
* WE: ~2 = 1199.223¢, ~18/17 = 99.108¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.163¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 109, 121 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 109, 121 }}


Badness: 0.028721
Badness (Sintel): 1.463


=== 19-limit ===
=== 19-limit ===
Line 120: Line 132:
Mapping: [{{val| 1 2 1 0 -1 -2 5 4 }}, {{val| 0 -5 16 34 54 69 -11 3 }}]
Mapping: [{{val| 1 2 1 0 -1 -2 5 4 }}, {{val| 0 -5 16 34 54 69 -11 3 }}]


POTE generator: ~18/17 = 99.164
Optimal tunings:
* WE: ~2 = 1199.368¢, ~18/17 = 99.112¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.158¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 109, 121 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 109, 121 }}


Badness: 0.023818
Badness (Sintel): 1.449


== Quintapole ==
== Quintapole ==
Line 135: Line 149:
[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -5 16 22 }}]
[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -5 16 22 }}]


[[POTE generator]]: ~21/20 = 98.994
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.959¢, ~21/20 = 98.908¢
* [[CWE]]: ~2 = 1200.000¢, ~21/20 = 98.968¢


{{Optimal ET sequence|legend=1| 12, 73c, 85, 97d }}
{{Optimal ET sequence|legend=1| 12, 73c, 85, 97d }}


[[Badness]]: 0.192498
[[Badness]] (Sintel): 4.872


=== 11-limit ===
=== 11-limit ===
Line 148: Line 164:
Mapping: [{{val| 1 2 1 1 0 }}, {{val| 0 -5 16 22 42 }}]
Mapping: [{{val| 1 2 1 1 0 }}, {{val| 0 -5 16 22 42 }}]


POTE generator: ~21/20 = 98.954
Optimal tunings:
* WE: ~2 = 1198.982¢, ~21/20 = 98.870¢
* CWE: ~2 = 1200.000¢, ~21/20 = 98.931¢


Optimal ET sequence: {{Optimal ET sequence| 12, 73ce, 85, 97d }}
Optimal ET sequence: {{Optimal ET sequence| 12, 73ce, 85, 97d }}


Badness: 0.104353
Badness (Sintel): 3.450


==== Galileic ====
==== Galileic ====
Line 163: Line 181:
Mapping: [{{val| 1 2 1 1 0 -1 }}, {{val| 0 -5 16 22 42 57 }}]
Mapping: [{{val| 1 2 1 1 0 -1 }}, {{val| 0 -5 16 22 42 57 }}]


POTE generator: ~21/20 = 98.992
Optimal tunings:
* WE: ~2 = 1198.912¢, ~21/20 = 98.902¢
* CWE: ~2 = 1200.000¢, ~21/20 = 98.970¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceff, 85f, 97d }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceff, 85f, 97d }}


Badness: 0.078183
Badness (Sintel): 3.231


===== 17-limit =====
===== 17-limit =====
Line 176: Line 196:
Mapping: [{{val| 1 2 1 1 0 -1 5 }}, {{val| 0 -5 16 22 42 57 -11 }}]
Mapping: [{{val| 1 2 1 1 0 -1 5 }}, {{val| 0 -5 16 22 42 57 -11 }}]


POTE generator: ~18/17 = 99.002
Optimal tunings:
* WE: ~2 = 1198.798¢, ~18/17 = 98.903¢
* CWE: ~2 = 1200.000¢, ~18/17 = 98.986¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceffg, 85fg, 97dg }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceffg, 85fg, 97dg }}


Badness: 0.058826
Badness (Sintel): 2.997


===== 19-limit =====
===== 19-limit =====
Line 189: Line 211:
Mapping: [{{val| 1 2 1 1 0 -1 5 4 }}, {{val| 0 -5 16 22 42 57 -11 3 }}]
Mapping: [{{val| 1 2 1 1 0 -1 5 4 }}, {{val| 0 -5 16 22 42 57 -11 3 }}]


POTE generator: ~18/17 = 98.987
Optimal tunings:
* WE: ~2 = 1199.031¢, ~18/17 = 98.907¢
* CWE: ~2 = 1200.000¢, ~18/17 = 98.976¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceffg, 85fg, 97dg }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceffg, 85fg, 97dg }}


Badness: 0.045459
Badness (Sintel): 2.765


==== Catagali ====
==== Catagali ====
Line 202: Line 226:
Mapping: [{{val| 1 2 1 1 0 0 }}, {{val| 0 -5 16 22 42 45 }}]
Mapping: [{{val| 1 2 1 1 0 0 }}, {{val| 0 -5 16 22 42 45 }}]


POTE generator: ~21/20 = 98.882
Optimal tunings:
* WE: ~2 = 1199.021¢, ~21/20 = 98.801¢
* CWE: ~2 = 1200.000¢, ~21/20 = 98.860¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 73ce, 85 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ce, 85 }}


Badness: 0.074180
Badness (Sintel): 3.065


===== 17-limit =====
===== 17-limit =====
Line 215: Line 241:
Mapping: [{{val| 1 2 1 1 0 0 5 }}, {{val| 0 -5 16 22 42 45 -11 }}]
Mapping: [{{val| 1 2 1 1 0 0 5 }}, {{val| 0 -5 16 22 42 45 -11 }}]


POTE generator: ~18/17 = 98.905
Optimal tunings:
* WE: ~2 = 1198.792¢, ~18/17 = 98.805¢
* CWE: ~2 = 1200.000¢, ~18/17 = 98.887¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceg, 85g }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceg, 85g }}


Badness: 0.057924
Badness (Sintel): 2.951


===== 19-limit =====
===== 19-limit =====
Line 228: Line 256:
Mapping: [{{val| 1 2 1 1 0 0 5 4 }}, {{val| 0 -5 16 22 42 45 -11 3 }}]
Mapping: [{{val| 1 2 1 1 0 0 5 4 }}, {{val| 0 -5 16 22 42 45 -11 3 }}]


POTE generator: ~18/17 = 98.888
Optimal tunings:
* WE: ~2 = 1199.037¢, ~18/17 = 98.808¢
* CWE: ~2 = 1200.000¢, ~18/17 = 98.875¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceg, 85g }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceg, 85g }}


Badness: 0.044714
Badness (Sintel): 2.720


=== Quintain ===
=== Quintain ===
Line 241: Line 271:
Mapping: [{{val| 1 2 1 1 1 }}, {{val| 0 -5 16 22 30 }}]
Mapping: [{{val| 1 2 1 1 1 }}, {{val| 0 -5 16 22 30 }}]


POTE generator: ~21/20 = 98.816
Optimal tunings:
* WE: ~2 = 1198.804¢, ~21/20 = 98.718¢
* CWE: ~2 = 1200.000¢, ~21/20 = 98.784¢


Optimal ET sequence: {{Optimal ET sequence| 12, 61c, 73c, 85e }}
Optimal ET sequence: {{Optimal ET sequence| 12, 61c, 73c, 85e }}


Badness: 0.112830
Badness (Sintel): 3.730


==== 13-limit ====
==== 13-limit ====
Line 254: Line 286:
Mapping: [{{val| 1 2 1 1 1 0 }}, {{val| 0 -5 16 22 30 45 }}]
Mapping: [{{val| 1 2 1 1 1 0 }}, {{val| 0 -5 16 22 30 45 }}]


POTE generator: ~21/20 = 98.796
Optimal tunings:
* WE: ~2 = 1198.830¢, ~21/20 = 98.700¢
* CWE: ~2 = 1200.000¢, ~21/20 = 98.768¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 61cf, 73c, 85e }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 61cf, 73c, 85e }}


Badness: 0.079900
Badness (Sintel): 3.302


== Quinticosiennic ==
== Quinticosiennic ==
Line 269: Line 303:
[[Mapping]]: [{{val| 1 2 1 -1 }}, {{val| 0 -5 16 46 }}]
[[Mapping]]: [{{val| 1 2 1 -1 }}, {{val| 0 -5 16 46 }}]


[[POTE generator]]: ~135/128 = 99.345
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.352¢, ~135/128 = 99.291¢
* [[CWE]]: ~2 = 1200.000¢, ~135/128 = 99.334¢


{{Optimal ET sequence|legend=1| 12, 133, 145, 157, 302c, 459bcc }}
{{Optimal ET sequence|legend=1| 12, 133, 145, 157, 302c, 459bcc }}


[[Badness]]: 0.158041
[[Badness]] (Sintel): 4.000


=== 11-limit ===
=== 11-limit ===
Line 282: Line 318:
Mapping: [{{val| 1 2 1 -1 -2 }}, {{val| 0 -5 16 46 66 }}]
Mapping: [{{val| 1 2 1 -1 -2 }}, {{val| 0 -5 16 46 66 }}]


POTE generator: ~35/33 = 99.318
Optimal tunings:
* WE: ~2 = 1199.380¢, ~35/33 = 99.266¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.308¢


Optimal ET sequence: {{Optimal ET sequence| 12, 133, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 12, 133, 145 }}


Badness: 0.080674
Badness (Sintel): 2.667


=== 13-limit ===
=== 13-limit ===
Line 295: Line 333:
Mapping: [{{val| 1 2 1 -1 -2 -3 }}, {{val| 0 -5 16 46 66 81 }}]
Mapping: [{{val| 1 2 1 -1 -2 -3 }}, {{val| 0 -5 16 46 66 81 }}]


POTE generator: ~35/33 = 99.307
Optimal tunings:
* WE: ~2 = 1199.406¢, ~35/33 = 99.258¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.299¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 133, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 133, 145 }}


Badness: 0.052464
Badness (Sintel): 2.168


=== 17-limit ===
=== 17-limit ===
Line 308: Line 348:
Mapping: [{{val| 1 2 1 -1 -2 -3 5 }}, {{val| 0 -5 16 46 66 81 -11 }}]
Mapping: [{{val| 1 2 1 -1 -2 -3 5 }}, {{val| 0 -5 16 46 66 81 -11 }}]


POTE generator: ~18/17 = 99.308
Optimal tunings:
* WE: ~2 = 1199.389¢, ~18/17 = 99.257¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.302¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 133, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 133, 145 }}


Badness: 0.037108
Badness (Sintel): 1.890


=== 19-limit ===
=== 19-limit ===
Line 321: Line 363:
Mapping: [{{val| 1 2 1 -1 -2 -3 5 4 }}, {{val| 0 -5 16 46 66 81 -11 3 }}]
Mapping: [{{val| 1 2 1 -1 -2 -3 5 4 }}, {{val| 0 -5 16 46 66 81 -11 3 }}]


POTE generator: ~18/17 = 99.303
Optimal tunings:
* WE: ~2 = 1199.488¢, ~18/17 = 99.261¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.299¢


Optimal ET sequence: {{Optimal ET sequence| 12f, 133, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 133, 145 }}


Badness: 0.028440
Badness (Sintel): 1.730


== Decimaleap ==
== Decimaleap ==
Line 336: Line 380:
[[Mapping]]: [{{val|1 2 1 5}}, {{val|0 -10 32 -53}}]
[[Mapping]]: [{{val|1 2 1 5}}, {{val|0 -10 32 -53}}]


[[POTE generator]]: ~36/35 = 49.621
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.430¢, ~36/35 = 49.598¢
* [[CWE]]: ~2 = 1200.000¢, ~36/35 = 49.623¢


{{Optimal ET sequence|legend=1| 24, 97d, 121, 145, 266 }}
{{Optimal ET sequence|legend=1| 24, 97d, 121, 145, 266 }}


[[Badness]]: 0.314227
[[Badness]] (Sintel): 7.952


=== 11-limit ===
=== 11-limit ===
Line 349: Line 395:
Mapping: [{{val|1 2 1 5 4}}, {{val|0 -10 32 -53 -13}}]
Mapping: [{{val|1 2 1 5 4}}, {{val|0 -10 32 -53 -13}}]


POTE generator: ~36/35 = 49.622
Optimal tunings:
* WE: ~2 = 1199.337¢, ~36/35 = 49.594¢
* CWE: ~2 = 1200.000¢, ~36/35 = 49.625¢


Optimal ET sequence: {{Optimal ET sequence| 24, 97d, 121, 145, 266e }}
Optimal ET sequence: {{Optimal ET sequence| 24, 97d, 121, 145, 266e }}


Badness: 0.084612
Badness (Sintel): 2.797


=== 13-limit ===
=== 13-limit ===
Line 362: Line 410:
Mapping: [{{val|1 2 1 5 4 3}}, {{val|0 -10 32 -53 -13 17}}]
Mapping: [{{val|1 2 1 5 4 3}}, {{val|0 -10 32 -53 -13 17}}]


POTE generator: ~36/35 = 49.620
Optimal tunings:
* WE: ~2 = 1199.308¢, ~36/35 = 49.591¢
* CWE: ~2 = 1200.000¢, ~36/35 = 49.622¢


Optimal ET sequence: {{Optimal ET sequence| 24, 97d, 121, 266ef }}
Optimal ET sequence: {{Optimal ET sequence| 24, 97d, 121, 266ef }}


Badness: 0.041763
Badness (Sintel): 1.726


=== 17-limit ===
=== 17-limit ===
Line 375: Line 425:
Mapping: [{{val|1 2 1 5 4 3 5}}, {{val|0 -10 32 -53 -13 17 -22}}]
Mapping: [{{val|1 2 1 5 4 3 5}}, {{val|0 -10 32 -53 -13 17 -22}}]


POTE generator: ~34/33 = 49.621
Optimal tunings:
* WE: ~2 = 1199.289¢, ~34/33 = 49.591¢
* CWE: ~2 = 1200.000¢, ~34/33 = 49.623¢


Optimal ET sequence: {{Optimal ET sequence| 24, 97dg, 121, 266efg }}
Optimal ET sequence: {{Optimal ET sequence| 24, 97dg, 121, 266efg }}


Badness: 0.025552
Badness (Sintel): 1.302


=== 19-limit ===
=== 19-limit ===
Line 388: Line 440:
Mapping: [{{val|1 2 1 5 4 3 5 4}}, {{val|0 -10 32 -53 -13 17 -22 6}}]
Mapping: [{{val|1 2 1 5 4 3 5 4}}, {{val|0 -10 32 -53 -13 17 -22 6}}]


POTE generator: ~34/33 = 49.624
Optimal tunings:
* WE: ~2 = 1199.378¢, ~34/33 = 49.598¢
* CWE: ~2 = 1200.000¢, ~34/33 = 49.626¢


Optimal ET sequence: {{Optimal ET sequence| 24, 97dg, 121, 145, 266efg }}
Optimal ET sequence: {{Optimal ET sequence| 24, 97dg, 121, 145, 266efg }}


Badness: 0.021208
Badness (Sintel): 1.290


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Quintaleap family| ]] <!-- main article -->
[[Category:Quintaleap family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 23:50, 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 quintaleap family tempers out [37 -16 -5⟩, equating a stack of two Pythagorean commas with a stack of five schismas, making it a member of the schismic–Pythagorean equivalence continuum. It is also the temperament where 4/3 is identified by a stack of five 135/128's, making it a member of omega-pentacot temperaments.

Quintaleap

The name quintaleap comes from "quintans" (Latin for "one fifth") and "leapday", because the generator is 1/5 of the leapday fourth (~4/3, about 496 cents).

Subgroup: 2.3.5

Comma list: [37 -16 -5⟩ = 137438953472/134521003125

Mapping: [⟨1 2 1], ⟨0 -5 16]]

Optimal tunings:

  • WE: ~2 = 1199.359¢, ~135/128 = 99.214¢
  • CWE: ~2 = 1200.000¢, ~135/128 = 99.256¢

Optimal ET sequence: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc

Badness (Sintel): 10.427

Subgroup temperament

The quintaleap temperament works well for the 2.3.5.17.19 subgroup, tempering out 256/255 (equating 16/15 with 17/16), 361/360 (equating 19/18 with 20/19), and 4624/4617. 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 sixteen 18/17s with 5/2.

2.3.5.17

Subgroup: 2.3.5.17

Comma list: 256/255, 1419857/1417176

Gencom: [2 18/17; 256/255 1419857/1417176]

Gencom mapping: [⟨1 2 1 0 0 0 5], ⟨0 -5 16 0 0 0 -11]]

Sval mapping: [⟨1 2 1 5], ⟨0 -5 16 -11]]

Optimal tunings:

  • WE: ~2 = 1199.328¢, ~18/17 = 99.217¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.270¢

Optimal ET sequence: 12, 109, 121, 133

Badness (Sintel): 1.305

2.3.5.17.19

Subgroup: 2.3.5.17.19

Comma list: 256/255, 361/360, 4624/4617

Gencom: [2 18/17; 256/255 361/360 4624/4617]

Gencom mapping: [⟨1 2 1 0 0 0 5 4], ⟨0 -5 16 0 0 0 -11 3]]

Sval mapping: [⟨1 2 1 5 4], ⟨0 -5 16 -11 3]]

Optimal tunings:

  • WE: ~2 = 1199.446¢, ~18/17 = 99.230¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.273¢

Optimal ET sequence: 12, 109, 121, 133

Badness (Sintel): 0.712

Quintupole

Main article: Quintupole

The quintupole temperament tempers out the octagar comma (4000/3969) and the mistisma (458752/455625) in the 7-limit; 896/891 (pentacircle), 1375/1372 (moctdel), and 4375/4356 (fantares, luluzoquadyo) in the 11-limit. The word "quintupole" means five poles, but also a play on the words "quintuple" and "polypyth". It is so named because the generator is 1/5 of the polypyth fourth (~4/3, about 495.8 cents). Xenllium proposes the pronunciation of the word "quintupole" as /'kwɪntʊpəʊl/ or /'kwɪntʊpoʊl/, like as "quin-to-pole". Not to be confused with quintapole temperament (12&85).

Subgroup: 2.3.5.7

Comma list: 4000/3969, 458752/455625

Mapping: [⟨1 2 1 0], ⟨0 -5 16 34]]

Optimal tunings:

  • WE: ~2 = 1199.324¢, ~135/128 = 99.120¢
  • CWE: ~2 = 1200.000¢, ~135/128 = 99.162¢

Optimal ET sequence: 12, 97, 109, 121

Badness (Sintel): 2.825

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 1375/1372, 4375/4356

Mapping: [⟨1 2 1 0 -1], ⟨0 -5 16 34 54]]

Optimal tunings:

  • WE: ~2 = 1199.342¢, ~35/33 = 99.101¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.144¢

Optimal ET sequence: 12, 109, 121, 351bde, 472bdee

Badness (Sintel): 1.868

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 625/624, 2704/2695

Mapping: [⟨1 2 1 0 -1 -2], ⟨0 -5 16 34 54 69]]

Optimal tunings:

  • WE: ~2 = 1199.322¢, ~35/33 = 99.109¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.154¢

Optimal ET sequence: 12f, 109, 121

Badness (Sintel): 1.588

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 364/363, 375/374, 442/441

Mapping: [⟨1 2 1 0 -1 -2 5], ⟨0 -5 16 34 54 69 -11]]

Optimal tunings:

  • WE: ~2 = 1199.223¢, ~18/17 = 99.108¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.163¢

Optimal ET sequence: 12f, 109, 121

Badness (Sintel): 1.463

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374

Mapping: [⟨1 2 1 0 -1 -2 5 4], ⟨0 -5 16 34 54 69 -11 3]]

Optimal tunings:

  • WE: ~2 = 1199.368¢, ~18/17 = 99.112¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.158¢

Optimal ET sequence: 12f, 109, 121

Badness (Sintel): 1.449

Quintapole

The quintapole temperament (12&85) tempers out the marvel comma (225/224) and 7812500/7411887 (sepru-atritriyo). In the 11-limit, it tempers out the ptolemisma (100/99) as well as 85184/84035 (trilo-aquinru-agu). It is so named for the following reasons - it has the same commas as the apollo temperament, and its generator is a semitone five of which gives a flat fourth (~4/3, about 495 cents). Xenllium proposes the pronunciation of the word "quintapole" as /'kwɪntəpəʊl/ or /'kwɪntəpoʊl/, like as "quint-a-pole". Not to be confused with quintupole temperament (12&121).

Subgroup: 2.3.5.7

Comma list: 225/224, 7812500/7411887

Mapping: [⟨1 2 1 1], ⟨0 -5 16 22]]

Optimal tunings:

  • WE: ~2 = 1198.959¢, ~21/20 = 98.908¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.968¢

Optimal ET sequence: 12, 73c, 85, 97d

Badness (Sintel): 4.872

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 85184/84035

Mapping: [⟨1 2 1 1 0], ⟨0 -5 16 22 42]]

Optimal tunings:

  • WE: ~2 = 1198.982¢, ~21/20 = 98.870¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.931¢

Optimal ET sequence: 12, 73ce, 85, 97d

Badness (Sintel): 3.450

Galileic

The name galileic comes from "Vincenzo Galilei", because this temperament is strongly related to Galilei's tuning.

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 275/273, 12168/12005

Mapping: [⟨1 2 1 1 0 -1], ⟨0 -5 16 22 42 57]]

Optimal tunings:

  • WE: ~2 = 1198.912¢, ~21/20 = 98.902¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.970¢

Optimal ET sequence: 12f, 73ceff, 85f, 97d

Badness (Sintel): 3.231

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 120/119, 225/224, 275/273, 2431/2401

Mapping: [⟨1 2 1 1 0 -1 5], ⟨0 -5 16 22 42 57 -11]]

Optimal tunings:

  • WE: ~2 = 1198.798¢, ~18/17 = 98.903¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 98.986¢

Optimal ET sequence: 12f, 73ceffg, 85fg, 97dg

Badness (Sintel): 2.997

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 120/119, 225/224, 247/245, 275/273, 361/357

Mapping: [⟨1 2 1 1 0 -1 5 4], ⟨0 -5 16 22 42 57 -11 3]]

Optimal tunings:

  • WE: ~2 = 1199.031¢, ~18/17 = 98.907¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 98.976¢

Optimal ET sequence: 12f, 73ceffg, 85fg, 97dg

Badness (Sintel): 2.765

Catagali

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 847/845, 1040/1029

Mapping: [⟨1 2 1 1 0 0], ⟨0 -5 16 22 42 45]]

Optimal tunings:

  • WE: ~2 = 1199.021¢, ~21/20 = 98.801¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.860¢

Optimal ET sequence: 12f, 73ce, 85

Badness (Sintel): 3.065

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 120/119, 225/224, 442/441, 847/845

Mapping: [⟨1 2 1 1 0 0 5], ⟨0 -5 16 22 42 45 -11]]

Optimal tunings:

  • WE: ~2 = 1198.792¢, ~18/17 = 98.805¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 98.887¢

Optimal ET sequence: 12f, 73ceg, 85g

Badness (Sintel): 2.951

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 120/119, 209/208, 225/224, 361/357, 442/441

Mapping: [⟨1 2 1 1 0 0 5 4], ⟨0 -5 16 22 42 45 -11 3]]

Optimal tunings:

  • WE: ~2 = 1199.037¢, ~18/17 = 98.808¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 98.875¢

Optimal ET sequence: 12f, 73ceg, 85g

Badness (Sintel): 2.720

Quintain

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/242, 5000/4851

Mapping: [⟨1 2 1 1 1], ⟨0 -5 16 22 30]]

Optimal tunings:

  • WE: ~2 = 1198.804¢, ~21/20 = 98.718¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.784¢

Optimal ET sequence: 12, 61c, 73c, 85e

Badness (Sintel): 3.730

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 245/242, 275/273, 1040/1029

Mapping: [⟨1 2 1 1 1 0], ⟨0 -5 16 22 30 45]]

Optimal tunings:

  • WE: ~2 = 1198.830¢, ~21/20 = 98.700¢
  • CWE: ~2 = 1200.000¢, ~21/20 = 98.768¢

Optimal ET sequence: 12f, 61cf, 73c, 85e

Badness (Sintel): 3.302

Quinticosiennic

The quinticosiennic temperament (12&145) tempers out the hemifamity comma (5120/5103) and 395136/390625 (trizo-aquadbigu) in the 7-limit; 441/440 (werckisma), 896/891 (pentacircle), and 78408/78125 (lolosepgu) in the 11-limit. The word "quinticosiennic" means 5 (quintuple) × 29 (είκοσι εννέα) = 145, and so named because 1/5 of 29edo fourth, i.e. 12\145, is a possible generator.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 395136/390625

Mapping: [⟨1 2 1 -1], ⟨0 -5 16 46]]

Optimal tunings:

  • WE: ~2 = 1199.352¢, ~135/128 = 99.291¢
  • CWE: ~2 = 1200.000¢, ~135/128 = 99.334¢

Optimal ET sequence: 12, 133, 145, 157, 302c, 459bcc

Badness (Sintel): 4.000

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 78408/78125

Mapping: [⟨1 2 1 -1 -2], ⟨0 -5 16 46 66]]

Optimal tunings:

  • WE: ~2 = 1199.380¢, ~35/33 = 99.266¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.308¢

Optimal ET sequence: 12, 133, 145

Badness (Sintel): 2.667

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 78408/78125

Mapping: [⟨1 2 1 -1 -2 -3], ⟨0 -5 16 46 66 81]]

Optimal tunings:

  • WE: ~2 = 1199.406¢, ~35/33 = 99.258¢
  • CWE: ~2 = 1200.000¢, ~35/33 = 99.299¢

Optimal ET sequence: 12f, 133, 145

Badness (Sintel): 2.168

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 364/363, 3757/3750

Mapping: [⟨1 2 1 -1 -2 -3 5], ⟨0 -5 16 46 66 81 -11]]

Optimal tunings:

  • WE: ~2 = 1199.389¢, ~18/17 = 99.257¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.302¢

Optimal ET sequence: 12f, 133, 145

Badness (Sintel): 1.890

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475

Mapping: [⟨1 2 1 -1 -2 -3 5 4], ⟨0 -5 16 46 66 81 -11 3]]

Optimal tunings:

  • WE: ~2 = 1199.488¢, ~18/17 = 99.261¢
  • CWE: ~2 = 1200.000¢, ~18/17 = 99.299¢

Optimal ET sequence: 12f, 133, 145

Badness (Sintel): 1.730

Decimaleap

The decimaleap temperament (24&121) has a quarter-tone generator and tempers out the porwell comma (6144/6125) and 393379840/387420489 (sasaquadzo-ayo) in the 7-limit; 896/891 and 14700/14641 in the 11-limit. The name decimaleap comes from "decima" (Latin for "one tenth") and "leapday", because the generator is 1/10 of the leapday fourth.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 393379840/387420489

Mapping: [⟨1 2 1 5], ⟨0 -10 32 -53]]

Optimal tunings:

  • WE: ~2 = 1199.430¢, ~36/35 = 49.598¢
  • CWE: ~2 = 1200.000¢, ~36/35 = 49.623¢

Optimal ET sequence: 24, 97d, 121, 145, 266

Badness (Sintel): 7.952

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 6144/6125, 14700/14641

Mapping: [⟨1 2 1 5 4], ⟨0 -10 32 -53 -13]]

Optimal tunings:

  • WE: ~2 = 1199.337¢, ~36/35 = 49.594¢
  • CWE: ~2 = 1200.000¢, ~36/35 = 49.625¢

Optimal ET sequence: 24, 97d, 121, 145, 266e

Badness (Sintel): 2.797

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 676/675, 6144/6125

Mapping: [⟨1 2 1 5 4 3], ⟨0 -10 32 -53 -13 17]]

Optimal tunings:

  • WE: ~2 = 1199.308¢, ~36/35 = 49.591¢
  • CWE: ~2 = 1200.000¢, ~36/35 = 49.622¢

Optimal ET sequence: 24, 97d, 121, 266ef

Badness (Sintel): 1.726

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 364/363, 676/675, 1156/1155

Mapping: [⟨1 2 1 5 4 3 5], ⟨0 -10 32 -53 -13 17 -22]]

Optimal tunings:

  • WE: ~2 = 1199.289¢, ~34/33 = 49.591¢
  • CWE: ~2 = 1200.000¢, ~34/33 = 49.623¢

Optimal ET sequence: 24, 97dg, 121, 266efg

Badness (Sintel): 1.302

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 361/360, 364/363, 456/455, 665/663

Mapping: [⟨1 2 1 5 4 3 5 4], ⟨0 -10 32 -53 -13 17 -22 6]]

Optimal tunings:

  • WE: ~2 = 1199.378¢, ~34/33 = 49.598¢
  • CWE: ~2 = 1200.000¢, ~34/33 = 49.626¢

Optimal ET sequence: 24, 97dg, 121, 145, 266efg

Badness (Sintel): 1.290

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