Quintile family

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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. It 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[1]. It did not get named until 2012, when Petr Pařízek called it pental[2]. In 2024, the community has decided to rename it for fear of confusion.

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

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

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, ~140/81 = 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

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

Wedgie: ⟨⟨ -10 -50 -60 -56 -67 1 ]]

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

Notes

  1. ↑ Yahoo! Tuning Group | Some new 5-limit microtemperaments
  2. ↑ Yahoo! Tuning Group | 2D temperaments, part II -- new 5-limit temperaments
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