5th-octave temperaments: Difference between revisions

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{{Infobox fractional-octave|5}}[[5edo]] is the smallest xenharmonic system, as 1edo, 2edo, 3edo and 4edo are all subsets of [[12edo]].

The most notable 5th-octave family is [[~~limmic temperaments~~]] – [[tempering out]] [[256/243]] and associates 3\5 to [[3/2]] as well as 1\5 to [[9/8]], producing temperaments like [[blackwood]]. Equally notable among small equal divisions are the [[Cloudy ~~clan~~|cloudy temperaments]] – identifying [[8/7]] with one step of 5edo.

The most notable 5th-octave family is [[blackwood family]] – [[tempering out]] [[256/243]] and associates 3\5 to [[3/2]] as well as 1\5 to [[9/8]], producing temperaments like [[blackwood]]. Equally notable among small equal divisions are the [[Cloudy comma #Temperaments|cloudy temperaments]] – identifying [[8/7]] with one step of 5edo.

Other families of 5-limit 5th-octave commas are:

* [[Quintile family~~|Quintile temperaments~~]] - tempers out the {{monzo|-28 25 -5}} comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.

* [[Quintile family]] – tempers out the {{monzo|-28 25 -5}} comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.

* [[Quintosec family~~|Quintosec temperaments~~]]

* [[Quintosec family]]

* [[Trisedodge family~~|Trisedodge temperaments~~]]

* [[Trisedodge family]]

== Slendroschismic ==

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Supporting ETs: {{Optimal ET sequence|10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585}}

~~== Quint ==~~

Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 16/15, 27/25~~

~~{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}~~

~~: Mapping generators: ~9/8, ~7~~

~~[[Optimal tuning]]s:~~

* [[POTE]]: ~9/8 = 1\5, ~7/4 = 1017.903 (~21/20 = 57.903)

* [[CTE]]: ~9/8 = 1\5, ~7/4 = 968.8259 (~63/64 = 8.8259)

~~{{Optimal ET sequence|legend=1| 5, 15ccd }}~~

~~[[Badness]]: 0.048312~~

== Obscenity ==

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 {{Infobox fractional-octave|5}}[[5edo]] is the smallest xenharmonic system, as 1edo, 2edo, 3edo and 4edo are all subsets of [[12edo]].
-The most notable 5th-octave family is [[limmic temperaments]] – [[tempering out]] [[256/243]] and associates 3\5 to [[3/2]] as well as 1\5 to [[9/8]], producing temperaments like [[blackwood]]. Equally notable among small equal divisions are the [[Cloudy clan|cloudy temperaments]] – identifying [[8/7]] with one step of 5edo.
+The most notable 5th-octave family is [[blackwood family]] – [[tempering out]] [[256/243]] and associates 3\5 to [[3/2]] as well as 1\5 to [[9/8]], producing temperaments like [[blackwood]]. Equally notable among small equal divisions are the [[Cloudy comma #Temperaments|cloudy temperaments]] – identifying [[8/7]] with one step of 5edo.
 Other families of 5-limit 5th-octave commas are:
-* [[Quintile family|Quintile temperaments]] - tempers out the {{monzo|-28 25 -5}} comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.
+* [[Quintile family]] – tempers out the {{monzo|-28 25 -5}} comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.
-* [[Quintosec family|Quintosec temperaments]]
+* [[Quintosec family]]
-* [[Trisedodge family|Trisedodge temperaments]]
+* [[Trisedodge family]]
 == Slendroschismic ==
@@ Line 212: / Line 212: @@
 Supporting ETs: {{Optimal ET sequence|10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585}}
-== Quint ==
-Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 16/15, 27/25
-{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
-: Mapping generators: ~9/8, ~7
-[[Optimal tuning]]s:
-* [[POTE]]: ~9/8 = 1\5, ~7/4 = 1017.903 (~21/20 = 57.903)
-* [[CTE]]: ~9/8 = 1\5, ~7/4 = 968.8259 (~63/64 = 8.8259)
-{{Optimal ET sequence|legend=1| 5, 15ccd }}
-[[Badness]]: 0.048312
 == Obscenity ==
@@ Line 246: / Line 227: @@
 {{Navbox fractional-octave}}
+{{Todo| review | cleanup }}