Pental family

From Xenharmonic Wiki
(Redirected from Pental)
Jump to navigation Jump to search

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 GPV sequence5, 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

Optimal GPV sequence5, 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]]

POTE generator: ~3/2 = 701.377

Optimal GPV 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]]

POTE generator: ~3/2 = 700.996

Optimal GPV 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

POTE generator: ~140/81 = 950.6536 (~1029/1024 = 9.3464)

Optimal GPV sequence125, 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]]

POTE generator: ~140/81 = 950.6341 (~176/175 = 9.3659)

Optimal GPV 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]]

POTE generator: ~140/81 = 950.6677 (~144/143 = 9.3323)

Optimal GPV 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]]

POTE generator: ~26/15 = 950.6593 (~176/175 = 9.3407)

Optimal GPV 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 (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

Optimal GPV sequence60, 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]]

POTE generator: ~3/2 = 701.240

Optimal GPV 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]]

POTE generator: ~3/2 = 701.252

Optimal GPV sequence: 60e, 130, 190, 320

Badness: 0.023948