Quintile family: Difference between revisions
m Removing from Category:Regular temperament theory using Cat-a-lot |
→Hemipental: +an alternative 13-limit extension since I'm not convinced by the original one. Also normalize generators |
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| Line 68: | Line 68: | ||
Mapping generators: ~147/128, ~140/81 | Mapping generators: ~147/128, ~140/81 | ||
[[POTE generator]]: ~ | [[POTE generator]]: ~140/81 = 950.6536 (~1029/1024 = 9.3464) | ||
{{Val list|legend=1| | {{Val list|legend=1| 125, 130, 255, 385 }} | ||
[[Badness]]: 0.104163 | [[Badness]]: 0.104163 | ||
| Line 81: | Line 81: | ||
Mapping: [{{val| 5 0 -28 18 -54 }}, {{val| 0 2 10 -1 18 }}] | Mapping: [{{val| 5 0 -28 18 -54 }}, {{val| 0 2 10 -1 18 }}] | ||
POTE generator: ~ | POTE generator: ~140/81 = 950.6341 (~176/175 = 9.3659) | ||
Optimal GPV sequence: {{Val list| | Optimal GPV sequence: {{Val list| 125, 130, 255, 385, 640 }} | ||
Badness: 0.047624 | Badness: 0.047624 | ||
=== | ==== Hemipent ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 94: | Line 94: | ||
Mapping: [{{val| 5 0 -28 18 -54 34 }}, {{val| 0 2 10 -1 18 -13 }}] | Mapping: [{{val| 5 0 -28 18 -54 34 }}, {{val| 0 2 10 -1 18 -13 }}] | ||
POTE generator: ~ | POTE generator: ~140/81 = 950.6677 (~144/143 = 9.3323) | ||
Optimal GPV sequence: {{Val list| | Optimal GPV sequence: {{Val list| 125, 130, 255, 385, 515 }} | ||
Badness: 0.041043 | Badness: 0.041043 | ||
==== Hemipentalis ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
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 }}] | |||
POTE generator: ~26/15 = 950.6593 (~176/175 = 9.3407) | |||
Optimal GPV sequence: {{Val list| 125f, 130, 255f, 385f }} | |||
Badness: 0.033542 | |||
== Decal == | == Decal == | ||
Revision as of 21:35, 2 April 2022
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
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
Optimal GPV sequence: 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
Optimal GPV sequence: 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 (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
POTE generator: ~140/81 = 950.6536 (~1029/1024 = 9.3464)
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: ~140/81 = 950.6341 (~176/175 = 9.3659)
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~140/81 = 950.6677 (~144/143 = 9.3323)
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~26/15 = 950.6593 (~176/175 = 9.3407)
Optimal GPV sequence: Template:Val list
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
POTE generator: ~3/2 = 701.303
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
Optimal GPV sequence: 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
Optimal GPV sequence: Template:Val list
Badness: 0.023948