Quintile family: Difference between revisions

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== Quintile ==
== Quintile ==
Quintile reaches the interval class of 5 by five [[3/2|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 [[4/3|perfect fourth]] minus a [[81/80|syntonic comma]]. Quintile is a member of the [[syntonic–diatonic equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}, so it equates a [[256/243|Pythagorean limma]] with a stack of five [[81/80|syntonic commas]].  
Quintile reaches the interval class of 5 by five [[3/2|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 [[4/3|perfect fourth]] minus a [[81/80|syntonic comma]]. Quintile is a member of the [[syntonic–limmic equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}, so it equates a [[256/243|Pythagorean limma]] with a stack of five [[81/80|syntonic commas]].  


The temperament was first introduced by [[Mike Battaglia]] in 2011 along with other temperaments in the continuum mentioned above<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_99315.html#99323 Yahoo! Tuning Group | ''Some new 5-limit microtemperaments'']</ref>. It did not get named until 2012, when [[Petr Pařízek]] called it ''pental''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104270.html Yahoo! Tuning Group | ''2D temperaments, part II -- new 5-limit temperaments'']</ref>. 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]].  
The temperament was first introduced by [[Mike Battaglia]] in 2011 along with other temperaments in the continuum mentioned above<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_99315.html#99323 Yahoo! Tuning Group | ''Some new 5-limit microtemperaments'']</ref>. It did not get named until 2012, when [[Petr Pařízek]] called it ''pental''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104270.html Yahoo! Tuning Group | ''2D temperaments, part II -- new 5-limit temperaments'']</ref>. 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]].  
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{{Mapping|legend=1| 5 0 -28 | 0 1 5 }}
{{Mapping|legend=1| 5 0 -28 | 0 1 5 }}
: mapping generators: ~59049/51200, ~3
: mapping generators: ~59049/51200, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~59049/51200 = 240.000, ~3/2 = 701.317 (~81/80 = 18.683)
* [[WE]]: ~59049/51200 = 240.0389{{c}}, ~3/2 = 701.3235{{c}} (~81/80 = 18.7932{{c}})
: [[error map]]: {{val| 0.000 -0.638 +0.274 }}
: [[error map]]: {{val| +0.195 -0.437 +0.187 }}
* [[POTE]]: ~59049/51200 = 240.000, ~3/2 = 701.210 (~81/80 = 18.790)
* [[CWE]]: ~59049/51200 = 240.0000{{c}}, ~3/2 = 701.2536{{c}} (~81/80 = 18.7464{{c}})
: error map: {{val| 0.000 -0.745 -0.265 }}
: error map: {{val| 0.000 -0.701 -0.046 }}


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


[[Badness]] (Smith): 0.240050
[[Badness]] (Sintel): 5.63


== Pentacloud ==
== 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 [[245/243|sensamagic comma]] 245/243 in the 7-limit.
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
[[Subgroup]]: 2.3.5.7
Line 35: Line 34:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~8/7 = 240.000, ~3/2 = 701.317 (~81/80 = 18.683)
* [[WE]]: ~8/7 = 240.2778{{c}}, ~3/2 = 701.3587{{c}} (~81/80 = 19.4746{{c}})
: [[error map]]: {{val| 0.000 -0.638 +0.274 -8.826 }}
: [[error map]]: {{val| +1.389 +0.793 -0.353 -4.937 }}
* [[POTE]]: ~8/7 = 240.000, ~3/2 = 700.548 (~81/80 = 19.452)
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 700.7696{{c}} (~81/80 = 19.2304{{c}})
: error map: {{val| 0.000 -1.407 -3.574 -8.826 }}
: error map: {{val| 0.000 -1.185 -2.466 -8.826 }}


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


[[Badness]] (Smith): 0.120942
[[Badness]] (Sintel): 3.06


=== 11-limit ===
=== 11-limit ===
Line 49: Line 48:
Comma list: 245/243, 385/384, 3087/3025
Comma list: 245/243, 385/384, 3087/3025


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~8/7 = 240.000, ~3/2 = 701.496 (~81/80 = 18.304)
* WE: ~8/7 = 240.0919{{c}}, ~3/2 = 701.6455{{c}} (~81/80 = 18.6301{{c}})
* POTE: ~8/7 = 240.000, ~3/2 = 701.377 (~81/80 = 18.623)
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 701.3981{{c}} (~81/80 = 18.6019{{c}})


{{Optimal ET sequence|legend=0| 5, 60, 65 }}
{{Optimal ET sequence|legend=0| 60, 65 }}


Badness (Smith): 0.093248
Badness (Sintel): 3.08


=== 13-limit ===
=== 13-limit ===
Line 64: Line 63:
Comma list: 105/104, 144/143, 245/243, 3087/3025
Comma list: 105/104, 144/143, 245/243, 3087/3025


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~8/7 = 240.000, ~3/2 = 701.085 (~81/80 = 18.915)
* WE: ~8/7 = 240.0470{{c}}, ~3/2 = 701.1333{{c}} (~81/80 = 19.0078{{c}})
* POTE: ~8/7 = 240.000, ~3/2 = 700.996 (~81/80 = 19.004)
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 701.0105{{c}} (~81/80 = 18.9895{{c}})


{{Optimal ET sequence|legend=0| 5, 60, 65, 125de, 190ddef }}
{{Optimal ET sequence|legend=0| 60, 65 }}


Badness (Smith): 0.067549
Badness (Sintel): 2.79


== Hemiquintile ==
== 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 temperaments|hewuermera]], satribiru-agu) and 5250987/5242880 ([[mitonismic temperaments|mitonisma]], laquadzo-agu).
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 comma]]) and 5250987/5242880 ([[mitonisma]]).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 5 0 -28 18 | 0 2 10 -1 }}
{{Mapping|legend=1| 5 0 -28 18 | 0 2 10 -1 }}
: mapping generators: ~147/128, ~140/81
: mapping generators: ~147/128, ~140/81


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~147/128 = 240.0000, ~140/81 = 950.6620 (~1029/1024 = 9.3380)
* [[WE]]: ~147/128 = 240.0063{{c}}, ~140/81 = 950.6787{{c}} (~1029/1024 = 9.3464{{c}})
: [[error map]]: {{val| 0.000 -0.6311 +0.3059 +0.5121 }}
: [[error map]]: {{val| +0.031 -0.598 +0.298 +0.608 }}
* [[POTE]]: ~147/128 = 240.0000, ~140/81 = 950.6536 (~1029/1024 = 9.3464)
* [[CWE]]: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6561{{c}} (~1029/1024 = 9.3439{{c}})
: error map: {{val| 0.000 -0.6473 +0.2249 +0.5202 }}
: error map: {{val| 0.000 -0.643 +0.247 +0.518 }}


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


[[Badness]] (Smith): 0.104163
[[Badness]] (Sintel): 2.64


=== 11-limit ===
=== 11-limit ===
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Comma list: 540/539, 8019/8000, 180224/180075
Comma list: 540/539, 8019/8000, 180224/180075


Mapping: {{mapping| 5 0 -28 18 -54 | 0 2 10 -1 18 }}
{{Mapping|legend=0| 5 0 -28 18 -54 | 0 2 10 -1 18 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~147/128 = 240.0000, ~140/81 = 950.6430 (~176/175 = 9.3570)
* WE: ~147/128 = 240.0080{{c}}, ~140/81 = 950.6658{{c}} (~176/175 = 9.3662{{c}})
* POTE: ~147/128 = 240.0000, ~140/81 = 950.6341 (~176/175 = 9.3659)
* CWE: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6365{{c}} (~176/175 = 9.3635{{c}})


{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 640 }}
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 640 }}


Badness (Smith): 0.047624
Badness (Sintel): 1.57


==== Hemiquintilis ====
==== Hemiquintilis ====
Line 115: Line 113:
Comma list: 351/350, 540/539, 676/675, 124215/123904
Comma list: 351/350, 540/539, 676/675, 124215/123904


Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 13 }}
{{Mapping|legend=0| 5 0 -28 18 -54 34 | 0 2 10 -1 18 13 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~147/128 = 240.0000, ~26/15 = 950.6775 (~176/175 = 9.3225)
* WE: ~147/128 = 240.0155{{c}}, ~26/15 = 950.7207{{c}} (~176/175 = 9.3413{{c}})
* POTE: ~147/128 = 240.0000, ~26/15 = 950.6593 (~176/175 = 9.3407)
* CWE: ~147/128 = 240.0000{{c}}, ~26/15 = 950.6641{{c}} (~176/175 = 9.3359{{c}})


{{Optimal ET sequence|legend=0| 125f, 130, 255f, 385f }}
{{Optimal ET sequence|legend=0| 125f, 130, 255f, 385f }}


Badness (Smith): 0.033542
Badness (Sintel): 1.39


==== Hemiquint ====
==== Hemiquint ====
Line 130: Line 128:
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000


Mapping: {{mapping| 5 0 -28 18 -54 34 | 0 2 10 -1 18 -13 }}
{{Mapping|legend=0| 5 0 -28 18 -54 34 | 0 2 10 -1 18 -13 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~147/128 = 240.0000, ~140/81 = 950.6607 (~144/143 = 9.3393)
* WE: ~147/128 = 239.9879{{c}}, ~140/81 = 950.6196{{c}} (~176/175 = 9.3318{{c}})
* POTE: ~147/128 = 240.0000, ~140/81 = 950.6677 (~144/143 = 9.3323)
* CWE: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6666{{c}} (~176/175 = 9.3334{{c}})


{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 515 }}
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 515 }}


Badness (Smith): 0.041043
Badness (Sintel): 1.70


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 148: Line 146:


{{Mapping|legend=1| 10 0 -56 -67 | 0 1 5 6 }}
{{Mapping|legend=1| 10 0 -56 -67 | 0 1 5 6 }}
: mapping generators: ~15/14, ~3
: mapping generators: ~15/14, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~15/14 = 120.000, ~3/2 = 701.390 (~81/80 = 18.610)
* [[WE]]: ~15/14 = 120.0171{{c}}, ~3/2 = 701.4027{{c}} (~81/80 = 18.6999{{c}})
: [[error map]]: {{val| 0.000 -0.565 +0.639 -0.483 }}
: [[error map]]: {{val| +0.171 -0.381 +0.597 -0.529 }}
* [[POTE]]: ~15/14 = 120.000, ~3/2 = 701.303 (~81/80 = 18.697)
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 701.3381{{c}} (~81/80 = 18.6619{{c}})
: error map: {{val| 0.000 -0.652 +0.200 -1.009 }}
: error map: {{val| 0.000 -0.617 +0.377 -0.797 }}


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


[[Badness]] (Smith): 0.104859
[[Badness]] (Sintel): 2.65


=== 11-limit ===
=== 11-limit ===
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Comma list: 441/440, 8019/8000, 234375/234256
Comma list: 441/440, 8019/8000, 234375/234256


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~15/14 = 120.000, ~3/2 = 701.336 (~99/98 = 18.664)
* WE: ~15/14 = 120.0208{{c}}, ~3/2 = 701.3614{{c}} (~99/98 = 18.7632{{c}})
* POTE: ~15/14 = 120.000, ~3/2 = 701.240 (~99/98 = 18.760)
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 701.2776{{c}} (~99/98 = 18.7224{{c}})


{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}


Badness (Smith): 0.040633
Badness (Sintel): 1.34


=== 13-limit ===
=== 13-limit ===
Line 181: Line 178:
Comma list: 441/440, 729/728, 1001/1000, 4225/4224
Comma list: 441/440, 729/728, 1001/1000, 4225/4224


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~15/14 = 120.000, ~3/2 = 701.336 (~91/90 = 18.664)
* WE: ~15/14 = 120.0182{{c}}, ~3/2 = 701.3582{{c}} (~91/90 = 18.7510{{c}})
* POTE: ~15/14 = 120.000, ~3/2 = 701.252 (~91/90 = 18.748)
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 701.2754{{c}} (~91/90 = 18.7246{{c}})


{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }}


Badness (Smith): 0.023948
Badness (Sintel): 0.990


== Notes ==
== References ==


[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Quintile family| ]] <!-- main article -->
[[Category:Quintile family| ]] <!-- main article -->
[[Category:Quintile| ]] <!-- key article -->
[[Category:Quintile| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 12:33, 24 September 2026

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–limmic 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 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:

  • WE: ~59049/51200 = 240.0389 ¢, ~3/2 = 701.3235 ¢ (~81/80 = 18.7932 ¢)
error map: ⟨+0.195 -0.437 +0.187]
  • CWE: ~59049/51200 = 240.0000 ¢, ~3/2 = 701.2536 ¢ (~81/80 = 18.7464 ¢)
error map: ⟨0.000 -0.701 -0.046]

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

Badness (Sintel): 5.63

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:

  • WE: ~8/7 = 240.2778 ¢, ~3/2 = 701.3587 ¢ (~81/80 = 19.4746 ¢)
error map: ⟨+1.389 +0.793 -0.353 -4.937]
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 700.7696 ¢ (~81/80 = 19.2304 ¢)
error map: ⟨0.000 -1.185 -2.466 -8.826]

Optimal ET sequence: 60, 125d, 185cdd

Badness (Sintel): 3.06

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:

  • WE: ~8/7 = 240.0919 ¢, ~3/2 = 701.6455 ¢ (~81/80 = 18.6301 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 701.3981 ¢ (~81/80 = 18.6019 ¢)

Optimal ET sequence: 60, 65

Badness (Sintel): 3.08

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:

  • WE: ~8/7 = 240.0470 ¢, ~3/2 = 701.1333 ¢ (~81/80 = 19.0078 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~3/2 = 701.0105 ¢ (~81/80 = 18.9895 ¢)

Optimal ET sequence: 60, 65

Badness (Sintel): 2.79

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 comma) and 5250987/5242880 (mitonisma).

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:

  • WE: ~147/128 = 240.0063 ¢, ~140/81 = 950.6787 ¢ (~1029/1024 = 9.3464 ¢)
error map: ⟨+0.031 -0.598 +0.298 +0.608]
  • CWE: ~147/128 = 240.0000 ¢, ~140/81 = 950.6561 ¢ (~1029/1024 = 9.3439 ¢)
error map: ⟨0.000 -0.643 +0.247 +0.518]

Optimal ET sequence: 125, 130, 255, 385

Badness (Sintel): 2.64

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:

  • WE: ~147/128 = 240.0080 ¢, ~140/81 = 950.6658 ¢ (~176/175 = 9.3662 ¢)
  • CWE: ~147/128 = 240.0000 ¢, ~140/81 = 950.6365 ¢ (~176/175 = 9.3635 ¢)

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

Badness (Sintel): 1.57

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:

  • WE: ~147/128 = 240.0155 ¢, ~26/15 = 950.7207 ¢ (~176/175 = 9.3413 ¢)
  • CWE: ~147/128 = 240.0000 ¢, ~26/15 = 950.6641 ¢ (~176/175 = 9.3359 ¢)

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

Badness (Sintel): 1.39

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:

  • WE: ~147/128 = 239.9879 ¢, ~140/81 = 950.6196 ¢ (~176/175 = 9.3318 ¢)
  • CWE: ~147/128 = 240.0000 ¢, ~140/81 = 950.6666 ¢ (~176/175 = 9.3334 ¢)

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

Badness (Sintel): 1.70

Decile

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

  • WE: ~15/14 = 120.0171 ¢, ~3/2 = 701.4027 ¢ (~81/80 = 18.6999 ¢)
error map: ⟨+0.171 -0.381 +0.597 -0.529]
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 701.3381 ¢ (~81/80 = 18.6619 ¢)
error map: ⟨0.000 -0.617 +0.377 -0.797]

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

Badness (Sintel): 2.65

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:

  • WE: ~15/14 = 120.0208 ¢, ~3/2 = 701.3614 ¢ (~99/98 = 18.7632 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 701.2776 ¢ (~99/98 = 18.7224 ¢)

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

Badness (Sintel): 1.34

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:

  • WE: ~15/14 = 120.0182 ¢, ~3/2 = 701.3582 ¢ (~91/90 = 18.7510 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 701.2754 ¢ (~91/90 = 18.7246 ¢)

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

Badness (Sintel): 0.990

References

  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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