Quintile family: Difference between revisions

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The '''pental family''' tempers out the pental comma, 847288609443/838860800000 = {{Monzo|-28 25 -5}}.
The '''pental family''' tempers out the [[pental comma]], 847288609443/838860800000 = {{monzo| -28 25 -5 }}.


== Pental ==
== Pental ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 847288609443/838860800000
[[Comma list]]: 847288609443/838860800000


[[Mapping]]: [{{val| 5 0 -28 }}, {{val| 0 1 5 }}]
{{Mapping|legend=1| 5 0 -28 | 0 1 5 }}


Mapping generators: ~59049/51200, ~3
: mapping generators: ~59049/51200, ~3


[[POTE generator]]: ~3/2 = 701.210
[[Optimal tuning]] ([[POTE]]): ~59049/51200 = 1\5, ~3/2 = 701.210


{{Optimal ET sequence|legend=1| 5, 60, 65, 190, 255, 575, 830b, 1405b }}
{{Optimal ET sequence|legend=1| 5, 60, 65, 190, 255, 575, 830b, 1405b }}
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The pental temperament can be described as 5&60 temperament (also known as ''pentacloud''), tempering out the [[cloudy comma]], 16807/16384 and the [[245/243|sensamagic comma]], 245/243 in the 7-limit.
The pental temperament can be described as 5&60 temperament (also known as ''pentacloud''), tempering out the [[cloudy comma]], 16807/16384 and the [[245/243|sensamagic comma]], 245/243 in the 7-limit.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 16807/16384
[[Comma list]]: 245/243, 16807/16384


[[Mapping]]: [{{val|5 0 -28 14}}, {{val|0 1 5 0}}]
{{Mapping|legend=1| 5 0 -28 14 |0 1 5 0 }}


[[POTE generator]]: ~3/2 = 700.548
[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~3/2 = 700.548


{{Optimal ET sequence|legend=1| 5, 60, 65, 125d, 185cdd }}
{{Optimal ET sequence|legend=1| 5, 60, 65, 125d, 185cdd }}
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Comma list: 245/243, 385/384, 3087/3025
Comma list: 245/243, 385/384, 3087/3025


Mapping: [{{val|5 0 -28 14 49}}, {{val|0 1 5 0 -4}}]
Mapping: {{mapping| 5 0 -28 14 49 | 0 1 5 0 -4 }}


POTE generator: ~3/2 = 701.377
Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 701.377


{{Optimal ET sequence|legend=1| 5, 60, 65 }}
{{Optimal ET sequence|legend=1| 5, 60, 65 }}
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Comma list: 105/104, 144/143, 245/243, 3087/3025
Comma list: 105/104, 144/143, 245/243, 3087/3025


Mapping: [{{val|5 0 -28 14 49 -29}}, {{val|0 1 5 0 -4 6}}]
Mapping: {{mapping| 5 0 -28 14 49 -29 | 0 1 5 0 -4 6 }}


POTE generator: ~3/2 = 700.996
Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 700.996


{{Optimal ET sequence|legend=1| 5, 60, 65, 125de, 190ddef }}
{{Optimal ET sequence|legend=1| 5, 60, 65, 125de, 190ddef }}
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== Hemipental ==
== Hemipental ==
The ''hemipental'' temperament (125&130) tempers out the cataharry comma, 19683/19600 in the 7-limit, as well as 589824/588245 ([[Hewuermera temperaments|hewuermera]], satribiru-agu) and 5250987/5242880 ([[Mitonismic temperaments|mitonisma]], laquadzo-agu).
The ''hemipental'' temperament (125 & 130) tempers out the cataharry comma, 19683/19600 in the 7-limit, as well as 589824/588245 ([[Hewuermera temperaments|hewuermera]], satribiru-agu) and 5250987/5242880 ([[Mitonismic temperaments|mitonisma]], laquadzo-agu).


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 19683/19600, 589824/588245
[[Comma list]]: 19683/19600, 589824/588245


[[Mapping]]: [{{val| 5 0 -28 18 }}, {{val| 0 2 10 -1 }}]
{{Mapping|legend=1| 5 0 -28 18 | 0 2 10 -1 }}


Mapping generators: ~147/128, ~140/81
: mapping generators: ~147/128, ~140/81


[[POTE generator]]: ~140/81 = 950.6536 (~1029/1024 = 9.3464)
[[Optimal tuning]] ([[POTE]]): ~147/128 = 1\5, ~140/81 = 950.6536 (~1029/1024 = 9.3464)


{{Optimal ET sequence|legend=1| 125, 130, 255, 385 }}
{{Optimal ET sequence|legend=1| 125, 130, 255, 385 }}
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Comma list: 540/539, 8019/8000, 180224/180075
Comma list: 540/539, 8019/8000, 180224/180075


Mapping: [{{val| 5 0 -28 18 -54 }}, {{val| 0 2 10 -1 18 }}]
Mapping: {{mapping| 5 0 -28 18 -54 | 0 2 10 -1 18 }}


POTE generator: ~140/81 = 950.6341 (~176/175 = 9.3659)
Optimal tuning (POTE): ~147/128 = 1\5, ~140/81 = 950.6341 (~176/175 = 9.3659)


{{Optimal ET sequence|legend=1| 125, 130, 255, 385, 640 }}
{{Optimal ET sequence|legend=1| 125, 130, 255, 385, 640 }}
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Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000


Mapping: [{{val| 5 0 -28 18 -54 34 }}, {{val| 0 2 10 -1 18 -13 }}]
Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 -13 }}


POTE generator: ~140/81 = 950.6677 (~144/143 = 9.3323)
Optimal tuning (POTE): ~147/128 = 1\5, ~140/81 = 950.6677 (~144/143 = 9.3323)


{{Optimal ET sequence|legend=1| 125, 130, 255, 385, 515 }}
{{Optimal ET sequence|legend=1| 125, 130, 255, 385, 515 }}
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Comma list: 351/350, 540/539, 676/675, 124215/123904
Comma list: 351/350, 540/539, 676/675, 124215/123904


Mapping: [{{val| 5 0 -28 18 -54 34 }}, {{val| 0 2 10 -1 18 13 }}]
Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 13 }}


POTE generator: ~26/15 = 950.6593 (~176/175 = 9.3407)
Optimal tuning (POTE): ~147/128 = 1\5, ~26/15 = 950.6593 (~176/175 = 9.3407)


{{Optimal ET sequence|legend=1| 125f, 130, 255f, 385f }}
{{Optimal ET sequence|legend=1| 125f, 130, 255f, 385f }}
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== Decal ==
== Decal ==
The decal temperament (130&190) tempers out the varunisma, 321489/320000 in the 7-limit, as well as 235298/234375 (triwellisma, tribizo-asepgu), 2460375/2458624 (sesquiquartisma, laquadru-atriyo), and the [[15/14ths equal temperament|linus comma]], {{Monzo|11 -10 -10 10}}.
The decal temperament (130 & 190) tempers out the varunisma, 321489/320000 in the 7-limit, as well as 235298/234375 (triwellisma, tribizo-asepgu), 2460375/2458624 (sesquiquartisma, laquadru-atriyo), and the [[15/14 equal-step tuning|linus comma]], {{monzo| 11 -10 -10 10 }}.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 235298/234375, 321489/320000
[[Comma list]]: 235298/234375, 321489/320000


[[Mapping]]: [{{val| 10 0 -56 -67 }}, {{val| 0 1 5 6 }}]
{{Mapping|legend=1| 10 0 -56 -67 | 0 1 5 6 }}


Mapping generators: ~15/14, ~3
: mapping generators: ~15/14, ~3


[[POTE generator]]: ~3/2 = 701.303
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~3/2 = 701.303


{{Optimal ET sequence|legend=1| 60, 130, 320, 450, 770d }}
{{Optimal ET sequence|legend=1| 60, 130, 320, 450, 770d }}
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Comma list: 441/440, 8019/8000, 234375/234256
Comma list: 441/440, 8019/8000, 234375/234256


Mapping: [{{val| 10 0 -56 -67 -108 }}, {{val| 0 1 5 6 9 }}]
Mapping: {{mapping| 10 0 -56 -67 -108 | 0 1 5 6 9 }}


POTE generator: ~3/2 = 701.240
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 701.240


{{Optimal ET sequence|legend=1| 60e, 130, 190, 320 }}
{{Optimal ET sequence|legend=1| 60e, 130, 190, 320 }}
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Comma list: 441/440, 729/728, 1001/1000, 4225/4224
Comma list: 441/440, 729/728, 1001/1000, 4225/4224


Mapping: [{{val| 10 0 -56 -67 -108 37 }}, {{val| 0 1 5 6 9 0 }}]
Mapping: {{mapping| 10 0 -56 -67 -108 37 | 0 1 5 6 9 0 }}


POTE generator: ~3/2 = 701.252
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 701.252


{{Optimal ET sequence|legend=1| 60e, 130, 190, 320 }}
{{Optimal ET sequence|legend=1| 60e, 130, 190, 320 }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pental family| ]] <!-- main article -->
[[Category:Pental family| ]] <!-- main article -->
[[Category:Pental| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 09:13, 3 September 2023

The pental family tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5⟩.

Pental

Subgroup: 2.3.5

Comma list: 847288609443/838860800000

Mapping: [⟨5 0 -28], ⟨0 1 5]]

mapping generators: ~59049/51200, ~3

Optimal tuning (POTE): ~59049/51200 = 1\5, ~3/2 = 701.210

Optimal ET sequence: 5, 60, 65, 190, 255, 575, 830b, 1405b

Badness: 0.240050

Septimal pental

The pental temperament can be described as 5&60 temperament (also known as pentacloud), tempering out the cloudy comma, 16807/16384 and the sensamagic comma, 245/243 in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 245/243, 16807/16384

Mapping: [⟨5 0 -28 14], ⟨0 1 5 0]]

Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 700.548

Optimal ET sequence: 5, 60, 65, 125d, 185cdd

Badness: 0.120942

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 3087/3025

Mapping: [⟨5 0 -28 14 49], ⟨0 1 5 0 -4]]

Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 701.377

Optimal ET sequence: 5, 60, 65

Badness: 0.093248

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 245/243, 3087/3025

Mapping: [⟨5 0 -28 14 49 -29], ⟨0 1 5 0 -4 6]]

Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 700.996

Optimal ET sequence: 5, 60, 65, 125de, 190ddef

Badness: 0.067549

Hemipental

The hemipental temperament (125 & 130) tempers out the cataharry comma, 19683/19600 in the 7-limit, as well as 589824/588245 (hewuermera, satribiru-agu) and 5250987/5242880 (mitonisma, laquadzo-agu).

Subgroup: 2.3.5.7

Comma list: 19683/19600, 589824/588245

Mapping: [⟨5 0 -28 18], ⟨0 2 10 -1]]

mapping generators: ~147/128, ~140/81

Optimal tuning (POTE): ~147/128 = 1\5, ~140/81 = 950.6536 (~1029/1024 = 9.3464)

Optimal ET sequence: 125, 130, 255, 385

Badness: 0.104163

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 8019/8000, 180224/180075

Mapping: [⟨5 0 -28 18 -54], ⟨0 2 10 -1 18]]

Optimal tuning (POTE): ~147/128 = 1\5, ~140/81 = 950.6341 (~176/175 = 9.3659)

Optimal ET sequence: 125, 130, 255, 385, 640

Badness: 0.047624

Hemipent

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000

Mapping: [⟨5 0 -28 18 -54 34], ⟨0 2 10 -1 18 -13]]

Optimal tuning (POTE): ~147/128 = 1\5, ~140/81 = 950.6677 (~144/143 = 9.3323)

Optimal ET sequence: 125, 130, 255, 385, 515

Badness: 0.041043

Hemipentalis

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 124215/123904

Mapping: [⟨5 0 -28 18 -54 34], ⟨0 2 10 -1 18 13]]

Optimal tuning (POTE): ~147/128 = 1\5, ~26/15 = 950.6593 (~176/175 = 9.3407)

Optimal ET sequence: 125f, 130, 255f, 385f

Badness: 0.033542

Decal

The decal temperament (130 & 190) tempers out the varunisma, 321489/320000 in the 7-limit, as well as 235298/234375 (triwellisma, tribizo-asepgu), 2460375/2458624 (sesquiquartisma, laquadru-atriyo), and the linus comma, [11 -10 -10 10⟩.

Subgroup: 2.3.5.7

Comma list: 235298/234375, 321489/320000

Mapping: [⟨10 0 -56 -67], ⟨0 1 5 6]]

mapping generators: ~15/14, ~3

Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 701.303

Optimal ET sequence: 60, 130, 320, 450, 770d

Badness: 0.104859

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 234375/234256

Mapping: [⟨10 0 -56 -67 -108], ⟨0 1 5 6 9]]

Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 701.240

Optimal ET sequence: 60e, 130, 190, 320

Badness: 0.040633

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 729/728, 1001/1000, 4225/4224

Mapping: [⟨10 0 -56 -67 -108 37], ⟨0 1 5 6 9 0]]

Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 701.252

Optimal ET sequence: 60e, 130, 190, 320

Badness: 0.023948

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