Quintile family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The quintile family of temperaments tempers out the quintile comma (monzo: [-28 25 -5⟩, ratio: 847 288 609 443 / 838 860 800 000).

Quintile

Quintile reaches the interval class of 5 by five perfect fifths (i.e. a major seventh) plus two periods of 1/5-octave; this two-period interval represents a grave fourth of 320/243, that is, a perfect fourth minus a syntonic comma. Quintile is a member of the syntonic–diatonic equivalence continuum with n = 5, so it equates a Pythagorean limma with a stack of five syntonic commas.

The temperament was first introduced by Mike Battaglia in 2011 along with other temperaments in the continuum mentioned above. It did not get named until 2012, when Petr Pařízek called it pental. In 2024, the community has decided to rename it for fear of confusion with the more common usage of the term pental to refer to the 5-limit.

Subgroup: 2.3.5

Comma list: 847288609443/838860800000

Mapping: [⟨5 0 -28], ⟨0 1 5]]

mapping generators: ~59049/51200, ~3

Optimal tunings:

• CTE: ~59049/51200 = 240.000, ~3/2 = 701.317 (~81/80 = 18.683)

error map: ⟨0.000 -0.638 +0.274]

• POTE: ~59049/51200 = 240.000, ~3/2 = 701.210 (~81/80 = 18.790)

error map: ⟨0.000 -0.745 -0.265]

Optimal ET sequence: 5, 60, 65, 190, 255, 575, 830b, 1405b

Badness (Smith): 0.240050

Pentacloud

Pentacloud can be described as the 5 & 60 temperament. It identifies the period as ~8/7, 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 tunings:

• CTE: ~8/7 = 240.000, ~3/2 = 701.317 (~81/80 = 18.683)

error map: ⟨0.000 -0.638 +0.274 -8.826]

• POTE: ~8/7 = 240.000, ~3/2 = 700.548 (~81/80 = 19.452)

error map: ⟨0.000 -1.407 -3.574 -8.826]

Optimal ET sequence: 5, 60, 65, 125d, 185cdd

Badness (Smith): 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 tunings:

• CTE: ~8/7 = 240.000, ~3/2 = 701.496 (~81/80 = 18.304)
• POTE: ~8/7 = 240.000, ~3/2 = 701.377 (~81/80 = 18.623)

Optimal ET sequence: 5, 60, 65

Badness (Smith): 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 tunings:

• CTE: ~8/7 = 240.000, ~3/2 = 701.085 (~81/80 = 18.915)
• POTE: ~8/7 = 240.000, ~3/2 = 700.996 (~81/80 = 19.004)

Optimal ET sequence: 5, 60, 65, 125de, 190ddef

Badness (Smith): 0.067549

Hemiquintile

Hemiquintile (formerly hemipental) can be described as 125 & 130 and 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 tunings:

• CTE: ~147/128 = 240.0000, ~140/81 = 950.6620 (~1029/1024 = 9.3380)

error map: ⟨0.000 -0.6311 +0.3059 +0.5121]

• POTE: ~147/128 = 240.0000, ~140/81 = 950.6536 (~1029/1024 = 9.3464)

error map: ⟨0.000 -0.6473 +0.2249 +0.5202]

Optimal ET sequence: 125, 130, 255, 385

Badness (Smith): 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 tunings:

• CTE: ~147/128 = 240.0000, ~140/81 = 950.6430 (~176/175 = 9.3570)
• POTE: ~147/128 = 240.0000, ~140/81 = 950.6341 (~176/175 = 9.3659)

Optimal ET sequence: 125, 130, 255, 385, 640

Badness (Smith): 0.047624

Hemiquintilis

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 tunings:

• CTE: ~147/128 = 240.0000, ~26/15 = 950.6775 (~176/175 = 9.3225)
• POTE: ~147/128 = 240.0000, ~26/15 = 950.6593 (~176/175 = 9.3407)

Optimal ET sequence: 125f, 130, 255f, 385f

Badness (Smith): 0.033542

Hemiquint

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 tunings:

• CTE: ~147/128 = 240.0000, ~140/81 = 950.6607 (~144/143 = 9.3393)
• POTE: ~147/128 = 240.0000, ~140/81 = 950.6677 (~144/143 = 9.3323)

Optimal ET sequence: 125, 130, 255, 385, 515

Badness (Smith): 0.041043

Decile

Decile (formerly decal) can be described as 130 & 190 and tempers out the varunisma 321489/320000 in the 7-limit, as well as the triwellisma 235298/234375, the breeze comma 2460375/2458624, 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 tunings:

• CTE: ~15/14 = 120.000, ~3/2 = 701.390 (~81/80 = 18.610)

error map: ⟨0.000 -0.565 +0.639 -0.483]

• POTE: ~15/14 = 120.000, ~3/2 = 701.303 (~81/80 = 18.697)

error map: ⟨0.000 -0.652 +0.200 -1.009]

Optimal ET sequence: 60, 130, 320, 450, 770d

Badness (Smith): 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 tunings:

• CTE: ~15/14 = 120.000, ~3/2 = 701.336 (~99/98 = 18.664)
• POTE: ~15/14 = 120.000, ~3/2 = 701.240 (~99/98 = 18.760)

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

Badness (Smith): 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 tunings:

• CTE: ~15/14 = 120.000, ~3/2 = 701.336 (~91/90 = 18.664)
• POTE: ~15/14 = 120.000, ~3/2 = 701.252 (~91/90 = 18.748)

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

Badness (Smith): 0.023948