Quintile family: Difference between revisions
m Notes -> references; recategorize |
m Units & misc. cleanup |
||
| Line 3: | Line 3: | ||
== 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 [[ | 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]]. | ||
| Line 12: | Line 12: | ||
{{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) | * [[CTE]]: ~59049/51200 = 240.000{{c}}, ~3/2 = 701.317{{c}} (~81/80 = 18.683{{c}}) | ||
: [[error map]]: {{val| 0.000 -0.638 +0.274 }} | : [[error map]]: {{val| 0.000 -0.638 +0.274 }} | ||
* [[POTE]]: ~59049/51200 = 240.000, ~3/2 = 701.210 (~81/80 = 18.790) | * [[POTE]]: ~59049/51200 = 240.000{{c}}, ~3/2 = 701.210{{c}} (~81/80 = 18.790{{c}}) | ||
: error map: {{val| 0.000 -0.745 -0.265 }} | : error map: {{val| 0.000 -0.745 -0.265 }} | ||
| Line 26: | Line 25: | ||
== 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 [[ | 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) | * [[CTE]]: ~8/7 = 240.000{{c}}, ~3/2 = 701.317{{c}} (~81/80 = 18.683{{c}}) | ||
: [[error map]]: {{val| 0.000 -0.638 +0.274 -8.826 }} | : [[error map]]: {{val| 0.000 -0.638 +0.274 -8.826 }} | ||
* [[POTE]]: ~8/7 = 240.000, ~3/2 = 700.548 (~81/80 = 19.452) | * [[POTE]]: ~8/7 = 240.000{{c}}, ~3/2 = 700.548{{c}} (~81/80 = 19.452{{c}}) | ||
: error map: {{val| 0.000 -1.407 -3.574 -8.826 }} | : error map: {{val| 0.000 -1.407 -3.574 -8.826 }} | ||
| Line 52: | Line 51: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~8/7 = 240.000, ~3/2 = 701.496 (~81/80 = 18.304) | * CTE: ~8/7 = 240.000{{c}}, ~3/2 = 701.496{{c}} (~81/80 = 18.304{{c}}) | ||
* POTE: ~8/7 = 240.000, ~3/2 = 701.377 (~81/80 = 18.623) | * POTE: ~8/7 = 240.000{{c}}, ~3/2 = 701.377{{c}} (~81/80 = 18.623{{c}}) | ||
{{Optimal ET sequence|legend=0| 5, 60, 65 }} | {{Optimal ET sequence|legend=0| 5, 60, 65 }} | ||
| Line 67: | Line 66: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~8/7 = 240.000, ~3/2 = 701.085 (~81/80 = 18.915) | * CTE: ~8/7 = 240.000{{c}}, ~3/2 = 701.085{{c}} (~81/80 = 18.915{{c}}) | ||
* POTE: ~8/7 = 240.000, ~3/2 = 700.996 (~81/80 = 19.004) | * POTE: ~8/7 = 240.000{{c}}, ~3/2 = 700.996{{c}} (~81/80 = 19.004{{c}}) | ||
{{Optimal ET sequence|legend=0| 5, 60, 65, 125de, 190ddef }} | {{Optimal ET sequence|legend=0| 5, 60, 65, 125de, 190ddef }} | ||
| Line 75: | Line 74: | ||
== 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 | 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 | ||
| Line 82: | Line 81: | ||
{{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) | * [[CTE]]: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6620{{c}} (~1029/1024 = 9.3380{{c}}) | ||
: [[error map]]: {{val| 0.000 -0.6311 +0.3059 +0.5121 }} | : [[error map]]: {{val| 0.000 -0.6311 +0.3059 +0.5121 }} | ||
* [[POTE]]: ~147/128 = 240.0000, ~140/81 = 950.6536 (~1029/1024 = 9.3464) | * [[POTE]]: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6536{{c}} (~1029/1024 = 9.3464{{c}}) | ||
: error map: {{val| 0.000 -0.6473 +0.2249 +0.5202 }} | : error map: {{val| 0.000 -0.6473 +0.2249 +0.5202 }} | ||
| Line 103: | Line 101: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~147/128 = 240.0000, ~140/81 = 950.6430 (~176/175 = 9.3570) | * CTE: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6430{{c}} (~176/175 = 9.3570{{c}}) | ||
* POTE: ~147/128 = 240.0000, ~140/81 = 950.6341 (~176/175 = 9.3659) | * POTE: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6341{{c}} (~176/175 = 9.3659{{c}}) | ||
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 640 }} | {{Optimal ET sequence|legend=0| 125, 130, 255, 385, 640 }} | ||
| Line 118: | Line 116: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~147/128 = 240.0000, ~26/15 = 950.6775 (~176/175 = 9.3225) | * CTE: ~147/128 = 240.0000{{c}}, ~26/15 = 950.6775{{c}} (~176/175 = 9.3225{{c}}) | ||
* POTE: ~147/128 = 240.0000, ~26/15 = 950.6593 (~176/175 = 9.3407) | * POTE: ~147/128 = 240.0000{{c}}, ~26/15 = 950.6593{{c}} (~176/175 = 9.3407{{c}}) | ||
{{Optimal ET sequence|legend=0| 125f, 130, 255f, 385f }} | {{Optimal ET sequence|legend=0| 125f, 130, 255f, 385f }} | ||
| Line 133: | Line 131: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~147/128 = 240.0000, ~140/81 = 950.6607 (~144/143 = 9.3393) | * CTE: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6607{{c}} (~144/143 = 9.3393{{c}}) | ||
* POTE: ~147/128 = 240.0000, ~140/81 = 950.6677 (~144/143 = 9.3323) | * POTE: ~147/128 = 240.0000{{c}}, ~140/81 = 950.6677{{c}} (~144/143 = 9.3323{{c}}) | ||
{{Optimal ET sequence|legend=0| 125, 130, 255, 385, 515 }} | {{Optimal ET sequence|legend=0| 125, 130, 255, 385, 515 }} | ||
| Line 141: | Line 139: | ||
== Decile == | == Decile == | ||
Decile (formerly ''decal'') can be described as 130 & 190 and tempers out the [[ | 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) | * [[CTE]]: ~15/14 = 120.000{{c}}, ~3/2 = 701.390{{c}} (~81/80 = 18.610{{c}}) | ||
: [[error map]]: {{val| 0.000 -0.565 +0.639 -0.483 }} | : [[error map]]: {{val| 0.000 -0.565 +0.639 -0.483 }} | ||
* [[POTE]]: ~15/14 = 120.000, ~3/2 = 701.303 (~81/80 = 18.697) | * [[POTE]]: ~15/14 = 120.000{{c}}, ~3/2 = 701.303{{c}} (~81/80 = 18.697{{c}}) | ||
: error map: {{val| 0.000 -0.652 +0.200 -1.009 }} | : error map: {{val| 0.000 -0.652 +0.200 -1.009 }} | ||
| Line 169: | Line 166: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~15/14 = 120.000, ~3/2 = 701.336 (~99/98 = 18.664) | * CTE: ~15/14 = 120.000{{c}}, ~3/2 = 701.336{{c}} (~99/98 = 18.664{{c}}) | ||
* POTE: ~15/14 = 120.000, ~3/2 = 701.240 (~99/98 = 18.760) | * POTE: ~15/14 = 120.000{{c}}, ~3/2 = 701.240{{c}} (~99/98 = 18.760{{c}}) | ||
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }} | {{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }} | ||
| Line 184: | Line 181: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~15/14 = 120.000, ~3/2 = 701.336 (~91/90 = 18.664) | * CTE: ~15/14 = 120.000{{c}}, ~3/2 = 701.336{{c}} (~91/90 = 18.664{{c}}) | ||
* POTE: ~15/14 = 120.000, ~3/2 = 701.252 (~91/90 = 18.748) | * POTE: ~15/14 = 120.000{{c}}, ~3/2 = 701.252{{c}} (~91/90 = 18.748{{c}}) | ||
{{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }} | {{Optimal ET sequence|legend=0| 60e, 130, 190, 320 }} | ||