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


[[Mapping]]: [{{val|5 0 -28}}, {{val|0 1 5}}]
[[Mapping]]: [{{val| 5 0 -28 }}, {{val| 0 1 5 }}]
 
Mapping generators: ~59049/51200, ~3


[[POTE generator]]: ~3/2 = 701.210
[[POTE generator]]: ~3/2 = 701.210
Line 14: Line 16:
[[Badness]]: 0.240050
[[Badness]]: 0.240050


=== 7-limit ===
== 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 [[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.


Line 62: Line 64:
[[Comma list]]: 19683/19600, 589824/588245
[[Comma list]]: 19683/19600, 589824/588245


[[Mapping]]: [{{val|5 8 12 14}}, {{val|0 -2 -10 1}}]
[[Mapping]]: [{{val| 5 0 -28 18 }}, {{val| 0 2 10 -1 }}]
 
Mapping generators: ~147/128, ~140/81


[[POTE generator]]: ~8/7 = 230.654
[[POTE generator]]: ~8/7 = 230.654
Line 75: Line 79:
Comma list: 540/539, 8019/8000, 180224/180075
Comma list: 540/539, 8019/8000, 180224/180075


Mapping: [{{val|5 8 12 14 18}}, {{val|0 -2 -10 1 -18}}]
Mapping: [{{val| 5 0 -28 18 -54 }}, {{val| 0 2 10 -1 18 }}]


POTE generator: ~8/7 = 230.634
POTE generator: ~8/7 = 230.634
Line 88: Line 92:
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000


Mapping: [{{val|5 8 12 14 18 18}}, {{val|0 -2 -10 1 -18 13}}]
Mapping: [{{val| 5 0 -28 18 -54 34 }}, {{val| 0 2 10 -1 18 -13 }}]


POTE generator: ~8/7 = 230.668
POTE generator: ~8/7 = 230.668
Line 103: Line 107:
[[Comma list]]: 235298/234375, 321489/320000
[[Comma list]]: 235298/234375, 321489/320000


[[Mapping]]: [{{val|10 0 -56 -67}}, {{val|0 1 5 6}}]
[[Mapping]]: [{{val| 10 0 -56 -67 }}, {{val| 0 1 5 6 }}]
 
Mapping generators: ~15/14, ~3


[[POTE generator]]: ~3/2 = 701.303
[[POTE generator]]: ~3/2 = 701.303
Line 116: Line 122:
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: [{{val| 10 0 -56 -67 -108 }}, {{val| 0 1 5 6 9 }}]


POTE generator: ~3/2 = 701.240
POTE generator: ~3/2 = 701.240
Line 129: Line 135:
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: [{{val| 10 0 -56 -67 -108 37 }}, {{val| 0 1 5 6 9 0 }}]


POTE generator: ~3/2 = 701.252
POTE generator: ~3/2 = 701.252
Line 137: Line 143:
Badness: 0.023948
Badness: 0.023948


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Pental]]
[[Category:Pental family| ]] <!-- main article -->
[[Category:Rank 2]]

Revision as of 06:37, 28 July 2021

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

POTE generator: ~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]]

POTE generator: ~3/2 = 700.548

Template:Val list

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]]

POTE generator: ~3/2 = 701.377

Vals: Template:Val list

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]]

POTE generator: ~3/2 = 700.996

Vals: Template:Val list

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 (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

POTE generator: ~8/7 = 230.654

Template:Val list

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]]

POTE generator: ~8/7 = 230.634

Vals: Template:Val list

Badness: 0.047624

13-limit

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]]

POTE generator: ~8/7 = 230.668

Vals: Template:Val list

Badness: 0.041043

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

POTE generator: ~3/2 = 701.303

Template:Val list

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]]

POTE generator: ~3/2 = 701.240

Vals: Template:Val list

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]]

POTE generator: ~3/2 = 701.252

Vals: Template:Val list

Badness: 0.023948

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