Quintile family
The pental family tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5⟩.
Pental
Comma: 847288609443/838860800000
Mapping: [⟨5 0 -28], ⟨0 1 5]]
POTE generator: ~3/2 = 701.210
Optimal ET sequence: 5, 60, 65, 190, 255, 575, 830b, 1405b
Badness: 0.240050
7-limit
The pental temperament can be described as 5&60 temperament (pentacloud, named by Xenllium), tempering out the cloudy comma, 16807/16384 and the sensamagic comma, 245/243 in the 7-limit.
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
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
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, named by Xenllium) tempers out the cataharry comma, 19683/19600 in the 7-limit, as well as 589824/588245 (hewuermera, satribiru-agu) and 5250987/5242880 (laquadzo-agu).
Comma list: 19683/19600, 589824/588245
Mapping: [⟨5 8 12 14], ⟨0 -2 -10 1]]
POTE generator: ~8/7 = 230.654
Badness: 0.104163
11-limit
Comma list: 540/539, 8019/8000, 180224/180075
Mapping: [⟨5 8 12 14 18], ⟨0 -2 -10 1 -18]]
POTE generator: ~8/7 = 230.634
Vals: Template:Val list
Badness: 0.047624
13-limit
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000
Mapping: [⟨5 8 12 14 18 18], ⟨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⟩.
Comma list: 235298/234375, 321489/320000
Mapping: [⟨10 0 -56 -67], ⟨0 1 5 6]]
POTE generator: ~3/2 = 701.303
Badness: 0.104859
11-limit
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
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