Quintile family: Difference between revisions

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m FloraC moved page Pental family to Quintile family: Implement the community-anticipated rename
+CTE tunings. +error maps. Spell out the period in cents. Note the badness metric being used. Misc. cleanup
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The '''quintile family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quintile comma]] ([[ratio]]: 847288609443/838860800000, {{monzo|legend=1| -28 25 -5 }}.
The '''quintile family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quintile comma]] ({{monzo|legend=1| -28 25 -5 }}, [[ratio]]: 847 288 609 443 / 838 860 800 000).  


== Quintile ==
== Quintile ==
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: mapping generators: ~59049/51200, ~3
: mapping generators: ~59049/51200, ~3


[[Optimal tuning]] ([[POTE]]): ~59049/51200 = 1\5, ~3/2 = 701.210 (~81/80 = 18.790)
[[Optimal tuning]]s:
* [[CTE]]: ~59049/51200 = 240.000, ~3/2 = 701.317 (~81/80 = 18.683)
: [[error map]]: {{val| 0.000 -0.638 +0.274 }}
* [[POTE]]: ~59049/51200 = 240.000, ~3/2 = 701.210 (~81/80 = 18.790)
: error map: {{val| 0.000 -0.745 -0.265 }}


{{Optimal ET sequence|legend=1| 5, 60, 65, 190, 255, 575, 830b, 1405b }}
{{Optimal ET sequence|legend=1| 5, 60, 65, 190, 255, 575, 830b, 1405b }}


[[Badness]]: 0.240050
[[Badness]] (Smith): 0.240050


== Pentacloud ==
== Pentacloud ==
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[[Comma list]]: 245/243, 16807/16384
[[Comma list]]: 245/243, 16807/16384


{{Mapping|legend=1| 5 0 -28 14 |0 1 5 0 }}
{{Mapping|legend=1| 5 0 -28 14 | 0 1 5 0 }}


[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~3/2 = 700.548 (~81/80 = 19.452)
[[Optimal tuning]]s:
* [[CTE]]: ~8/7 = 240.000, ~3/2 = 701.317 (~81/80 = 18.683)
: [[error map]]: {{val| 0.000 -0.638 +0.274 -8.826 }}
* [[POTE]]: ~8/7 = 240.000, ~3/2 = 700.548 (~81/80 = 19.452)
: error map: {{val| 0.000 -1.407 -3.574 -8.826 }}


{{Optimal ET sequence|legend=1| 5, 60, 65, 125d, 185cdd }}
{{Optimal ET sequence|legend=1| 5, 60, 65, 125d, 185cdd }}


[[Badness]]: 0.120942
[[Badness]] (Smith): 0.120942


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 5 0 -28 14 49 | 0 1 5 0 -4 }}
Mapping: {{mapping| 5 0 -28 14 49 | 0 1 5 0 -4 }}


Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 701.377 (~81/80 = 18.623)
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|legend=1| 5, 60, 65 }}
{{Optimal ET sequence|legend=0| 5, 60, 65 }}


Badness: 0.093248
Badness (Smith): 0.093248


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 5 0 -28 14 49 -29 | 0 1 5 0 -4 6 }}
Mapping: {{mapping| 5 0 -28 14 49 -29 | 0 1 5 0 -4 6 }}


Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 700.996 (~81/80 = 19.004)
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|legend=1| 5, 60, 65, 125de, 190ddef }}
{{Optimal ET sequence|legend=0| 5, 60, 65, 125de, 190ddef }}


Badness: 0.067549
Badness (Smith): 0.067549


== Hemiquintile ==
== Hemiquintile ==
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: mapping generators: ~147/128, ~140/81
: mapping generators: ~147/128, ~140/81


[[Optimal tuning]] ([[POTE]]): ~147/128 = 1\5, ~140/81 = 950.6536 (~1029/1024 = 9.3464)
[[Optimal tuning]]s:
* [[CTE]]: ~147/128 = 240.0000, ~140/81 = 950.6620 (~1029/1024 = 9.3380)
: [[error map]]: {{val| 0.000 -0.6311 +0.3059 +0.5121 }}
* [[POTE]]: ~147/128 = 240.0000, ~140/81 = 950.6536 (~1029/1024 = 9.3464)
: error map: {{val| 0.000 -0.6473 +0.2249 +0.5202 }}


{{Optimal ET sequence|legend=1| 125, 130, 255, 385 }}
{{Optimal ET sequence|legend=1| 125, 130, 255, 385 }}


[[Badness]]: 0.104163
[[Badness]] (Smith): 0.104163


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 5 0 -28 18 -54 | 0 2 10 -1 18 }}
Mapping: {{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 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|legend=1| 125, 130, 255, 385, 640 }}
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 640 }}


Badness: 0.047624
Badness (Smith): 0.047624


==== Hemiquint ====
==== Hemiquint ====
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Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 -13 }}
Mapping: {{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 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|legend=1| 125, 130, 255, 385, 515 }}
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 515 }}


Badness: 0.041043
Badness (Smith): 0.041043


==== Hemiquintilis ====
==== Hemiquintilis ====
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Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 13 }}
Mapping: {{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 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|legend=1| 125f, 130, 255f, 385f }}
{{Optimal ET sequence|legend=0| 125f, 130, 255f, 385f }}


Badness: 0.033542
Badness (Smith): 0.033542


== Decile ==
== Decile ==
Decile (formerly ''decal'') can be described as 130 & 190 and tempers out the varunisma, 321489/320000 in the 7-limit, as well as 235298/234375 (triwellisma, tribizo-asepgu), 2460375/2458624 ([[2460375/2458624|breeze comma]], laquadru-atriyo), and the [[linus comma]], {{monzo| 11 -10 -10 10 }}.
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]] {{monzo| 11 -10 -10 10 }}.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~15/14, ~3
: mapping generators: ~15/14, ~3


Wedgie: {{multival| -10 -50 -60 -56 -67 1 }}
{{Multival|legend=1| -10 -50 -60 -56 -67 1 }}


[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~3/2 = 701.303 (~81/80 = 18.697)
[[Optimal tuning]]s:
* [[CTE]]: ~15/14 = 120.000, ~3/2 = 701.390 (~81/80 = 18.610)
: [[error map]]: {{val| 0.000 -0.565 +0.639 -0.483 }}
* [[POTE]]: ~15/14 = 120.000, ~3/2 = 701.303 (~81/80 = 18.697)
: error map: {{val| 0.000 -0.652 +0.200 -1.009 }}


{{Optimal ET sequence|legend=1| 60, 130, 320, 450, 770d }}
{{Optimal ET sequence|legend=1| 60, 130, 320, 450, 770d }}


[[Badness]]: 0.104859
[[Badness]] (Smith): 0.104859


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 10 0 -56 -67 -108 | 0 1 5 6 9 }}
Mapping: {{mapping| 10 0 -56 -67 -108 | 0 1 5 6 9 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 701.240 (~99/98 = 18.760)
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|legend=1| 60e, 130, 190, 320 }}
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}


Badness: 0.040633
Badness (Smith): 0.040633


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 10 0 -56 -67 -108 37 | 0 1 5 6 9 0 }}
Mapping: {{mapping| 10 0 -56 -67 -108 37 | 0 1 5 6 9 0 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 701.252 (~91/90 = 18.748)
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|legend=1| 60e, 130, 190, 320 }}
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}


Badness: 0.023948
Badness (Smith): 0.023948


== Notes ==
== Notes ==

Revision as of 12:26, 19 February 2025

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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