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Kleismic family: Difference between revisions

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The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma. Its monzo is {{monzo| -6 -5 6 }}, and flipping that yields {{multival| 6 5 -6 }} for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)<sup>5</sup> = 5/2 × 15625/15552. This 5-limit temperament (virtually a [[microtemperament]]) is commonly called '''Hanson''', and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma. Its monzo is {{monzo| -6 -5 6 }}, and flipping that yields &lt;&lt;6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)<sup>5</sup> = 5/2 × 15625/15552. This 5-limit temperament (virtually a [[microtemperament]]) is commonly called '''Hanson''', and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].


= Hanson =
= Hanson =
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; Music:
; Music:
 
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Parizek/Hanson%20%20Improv.mp3 Hanson Improv] by [[Petr Parizek]]
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Parizek/Hanson%20%20Improv.mp3 Hanson Improv] by [[Petr Parizek]]
* [http://clones.soonlabel.com/public/micro/Hanson/daily20110127-in-hanson11.mp3 In Hanson11] by [[Chris Vaisvil]]
 
[http://clones.soonlabel.com/public/micro/Hanson/daily20110127-in-hanson11.mp3 In Hanson11] by [[Chris Vaisvil]]


; Analysis and diagrams:
; Analysis and diagrams:
 
* [http://dkeenan.com/Music/ChainOfMinor3rds.htm 11 note chain-of-minor-thirds scale], by David Keenan</span>
<span style="">[http://dkeenan.com/Music/ChainOfMinor3rds.htm 11 note chain-of-minor-thirds scale], by David Keenan</span>


== Seven limit extensions ==
== Seven limit extensions ==
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= Keemun =
= Keemun =
[[Comma]]s: 49/48, 126/125
[[Comma list]]: 49/48, 126/125


valid range: [300.000, 327.273] (4 to 11b)
[[POTE generator]]: ~6/5 = 316.473


nice range: [308.744, 322.942]
[[Mapping]]: [{{val| 1 0 1 2 }}, {{val| 0 6 5 3 }}]


strict range: [308.744, 322.942]
{{Multival|legend=1| 6 5 3 -6 -12 -7 }}


[[POTE tuning|POTE generator]]: ~6/5 = 316.473
[[Tuning ranges]]:  
* valid range: [300.000, 327.273] (4 to 11b)
* nice range: [308.744, 322.942]
* strict range: [308.744, 322.942]


Map: [&lt;1 0 1 2|, &lt;0 6 5 3|]
{{Val list|legend=1| 15, 19, 53d, 72dd, 91dd }}
 
[[Wedgie]]: &lt;&lt;6 5 3 -6 -12 -7||
 
EDOs: {{EDOs| 15, 19, 53d, 72dd, 91dd }}


[[Badness]]: 0.0274
[[Badness]]: 0.0274


== 11-limit ==
== 11-limit ==
Commas: 49/48, 56/55, 100/99
Comma list: 49/48, 56/55, 100/99


valid range: [315.789, 320.000] (19 to 15)
POTE generator: ~6/5 = 317.576
 
nice range: [308.744, 324.341]
 
strict range: [315.789, 320.000]


POTE generator: ~6/5 = 317.576
Mapping: [{{val| 1 0 1 2 4 }}, {{val| 0 6 5 3 -2 }}]


Map: [&lt;1 0 1 2 4|, &lt;0 6 5 3 -2|]
Tuning ranges:
* valid range: [315.789, 320.000] (19 to 15)
* nice range: [308.744, 324.341]
* strict range: [315.789, 320.000]


EDOs: {{EDOs| 4, 15, 19, 34 }}
{{Val list|legend=1| 4, 15, 19, 34 }}


Badness: 0.0274
Badness: 0.0274


=== 13-limit ===
=== 13-limit ===
Commas: 49/48, 56/55, 78/77, 100/99
Comma list: 49/48, 56/55, 78/77, 100/99


valid range: 315.789 (19)
POTE generator: ~6/5 = 316.611
 
nice range: [303.597, 324.341]
 
strict range: 315.789


POTE generator: ~6/5 = 316.611
Mapping: [{{val| 1 0 1 2 4 5 }}, {{val| 0 6 5 3 -2 -5 }}]


Map: [&lt;1 0 1 2 4 5|, &lt;0 6 5 3 -2 -5|]
Tuning ranges:
* valid range: 315.789 (19)
* nice range: [303.597, 324.341]
* strict range: 315.789


EDOs: {{EDOs| 4, 15f, 19, 53def, 72def }}
{{Val list|legend=1| 4, 15f, 19, 53def, 72def }}


Badness: 0.0297
Badness: 0.0297


=== Kema ===
=== Kema ===
Commas: 49/48, 56/55, 91/90, 100/99
Comma list: 49/48, 56/55, 91/90, 100/99


valid range: [315.789, 320.000] (19 to 15)
POTE generator: ~6/5 = 317.423


nice range: [308.744, 324.341]
Mapping: [{{val| 1 0 1 2 4 0 }}, {{val| 0 6 5 3 -2 14 }}]
 
strict range: [315.789, 320.000]
 
POTE generator: ~6/5 = 317.423


Map: [&lt;1 0 1 2 4 0|, &lt;0 6 5 3 -2 14|]
Tuning ranges:
* valid range: [315.789, 320.000] (19 to 15)
* nice range: [308.744, 324.341]
* strict range: [315.789, 320.000]


EDOs: {{EDOs| 15, 19, 34, 87ddee }}
{{Val list|legend=1| 15, 19, 34, 87ddee }}


Badness: 0.0227
Badness: 0.0227


=== Kumbaya ===
=== Kumbaya ===
Commas: 40/39, 49/48, 56/55, 66/65
Comma list: 40/39, 49/48, 56/55, 66/65


POTE generator: ~6/5 = 318.595
POTE generator: ~6/5 = 318.595


Map: [&lt;1 0 1 2 4 4|, &lt;0 6 5 3 -2 -1|]
Mapping: [{{val| 1 0 1 2 4 4 }}, {{val| 0 6 5 3 -2 -1 }}]


EDOs: {{EDOs| 4, 15, 19f, 34ff }}
{{Val list|legend=1| 4, 15, 19f, 34ff }}


Badness: 0.0316
Badness: 0.0316


== Qeema ==
== Qeema ==
Commas: 45/44, 49/48, 126/125
Comma list: 45/44, 49/48, 126/125


POTE generator: ~6/5 = 314.730
POTE generator: ~6/5 = 314.730


Map: [&lt;1 0 1 2 -1|, &lt;0 6 5 3 17|]
Mapping: [{{val| 1 0 1 2 -1 }}, {{val| 0 6 5 3 17 }}]


EDOs: {{EDOs| 4e, 19, 42bcd, 61bcdd }}
{{Val list|legend=1| 4e, 19, 42bcd, 61bcdd }}


Badness: 0.0401
Badness: 0.0401


=== 13-limit ===
=== 13-limit ===
Commas: 45/44, 49/48, 78/77, 126/125
Comma list: 45/44, 49/48, 78/77, 126/125


POTE generator: ~6/5 = 315.044
POTE generator: ~6/5 = 315.044


Map: [&lt;1 0 1 2 -1 0|, &lt;0 6 5 3 17 14|]
Mapping: [{{val| 1 0 1 2 -1 0 }}, {{val| 0 6 5 3 17 14 }}]


EDOs: {{EDOs| 4ef, 19 }}
{{Val list|legend=1| 4ef, 19 }}


Badness: 0.0294
Badness: 0.0294


== Darjeeling ==
== Darjeeling ==
Commas: 49/48, 55/54, 77/75
Comma list: 49/48, 55/54, 77/75


POTE generator: ~6/5 = 317.656
POTE generator: ~6/5 = 317.656


Map: [&lt;1 0 1 2 0|, &lt;0 6 5 3 13|]
Mapping: [{{val| 1 0 1 2 0 }}, {{val| 0 6 5 3 13 }}]


EDOs: {{EDOs| 15, 19e, 34e }}
{{Val list|legend=1| 15, 19e, 34e }}


Badness: 0.0276
Badness: 0.0276


=== 13-limit ===
=== 13-limit ===
Commas: 49/48, 55/54, 66/65, 77/75
Comma list: 49/48, 55/54, 66/65, 77/75


POTE generator: ~6/5 = 317.298
POTE generator: ~6/5 = 317.298


Map: [&lt;1 0 1 2 0 0|, &lt;0 6 5 3 13 14|]
Mapping: [{{val| 1 0 1 2 0 0 }}, {{val| 0 6 5 3 13 14 }}]


EDOs: {{EDOs| 15, 19e, 34e, 53dee }}
{{Val list|legend=1| 15, 19e, 34e, 53dee }}


Badness: 0.02144
Badness: 0.02144


= Catalan =
= Catalan =
Commas: 64/63, 15625/15552
Comma list: 64/63, 15625/15552


POTE generator: ~6/5 = 318.267
POTE generator: ~6/5 = 318.267


Map: [&lt;1 0 1 6|, &lt;0 6 5 -12|]
Mapping: [{{val| 1 0 1 6 }}, {{val| 0 6 5 -12 }}]


Wedgie: &lt;&lt;6 5 -12 -6 -36 -42||
{{Multival|legend=1| 6 5 -12 -6 -36 -42 }}


EDOs: {{EDOs| 15, 34d, 49, 132bcd }}
{{Val list|legend=1| 15, 34d, 49, 132bcd }}


Badness: 0.0949
Badness: 0.0949


== 11-limit ==
== 11-limit ==
Commas: 64/63, 100/99, 1331/1323
Comma list: 64/63, 100/99, 1331/1323


POTE generator: ~6/5 = 318.282
POTE generator: ~6/5 = 318.282


Map: [&lt;1 0 1 6 4|, &lt;0 6 5 -12 -2|]
Mapping: [{{val| 1 0 1 6 4 }}, {{val| 0 6 5 -12 -2 }}]


EDOs: {{EDOs| 15, 34d, 49 }}
{{Val list|legend=1| 15, 34d, 49 }}


Badness: 0.0369
Badness: 0.0369
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