Quintile family: Difference between revisions
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== Septimal pental == | == 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. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
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POTE generator: ~3/2 = 701.377 | POTE generator: ~3/2 = 701.377 | ||
Optimal GPV sequence: {{Val list| 5, 60, 65 }} | |||
Badness: 0.093248 | Badness: 0.093248 | ||
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POTE generator: ~3/2 = 700.996 | POTE generator: ~3/2 = 700.996 | ||
Optimal GPV sequence: {{Val list| 5, 60, 65, 125de, 190ddef }} | |||
Badness: 0.067549 | Badness: 0.067549 | ||
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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 | ||
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POTE generator: ~8/7 = 230.634 | POTE generator: ~8/7 = 230.634 | ||
Optimal GPV sequence: {{Val list| 5e, 125, 130, 255, 385, 640, 1025b }} | |||
Badness: 0.047624 | Badness: 0.047624 | ||
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POTE generator: ~8/7 = 230.668 | POTE generator: ~8/7 = 230.668 | ||
Optimal GPV sequence: {{Val list| 5ef, 125, 130, 255, 385, 515, 900ef }} | |||
Badness: 0.041043 | Badness: 0.041043 | ||
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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 [[15/14ths equal temperament|linus comma]], {{Monzo|11 -10 -10 10}}. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
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POTE generator: ~3/2 = 701.240 | POTE generator: ~3/2 = 701.240 | ||
Optimal GPV sequence: {{Val list| 60e, 130, 190, 320 }} | |||
Badness: 0.040633 | Badness: 0.040633 | ||
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POTE generator: ~3/2 = 701.252 | POTE generator: ~3/2 = 701.252 | ||
Optimal GPV sequence: {{Val list| 60e, 130, 190, 320 }} | |||
Badness: 0.023948 | Badness: 0.023948 | ||
Revision as of 00:50, 31 December 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
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: ~8/7 = 230.654
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
Optimal GPV sequence: 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
Optimal GPV sequence: 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
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