Pental family
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 (breeze comma, 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