5th-octave temperaments: Difference between revisions

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[[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]] ~~- tempers~~ 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 [[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.

Considered below is a temperament called quint, which uses exactly the same 5-limit as 5et, but the harmonic 7 is mapped to an independent generator. In what way is this useful is unexplained.

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

* [[Pental family|Pental 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.

* [[Quintosec family|Quintosec temperaments]]

* [[Trisedodge family|Trisedodge temperaments]]

== Quint ==

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~9/8, ~7

[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\5, ~7/4 = 1017.903

[[Badness]]: 0.048312

== Pentonismic (rank-5) ==

Subgroup: 2.3.5.7.11.13

@@ Line 2: / Line 2: @@
 [[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]] - tempers 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 [[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.
+Considered below is a temperament called quint, which uses exactly the same 5-limit as 5et, but the harmonic 7 is mapped to an independent generator. In what way is this useful is unexplained.
 Other families of 5-limit 5th-octave commas are:
 * [[Pental family|Pental 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.
 * [[Quintosec family|Quintosec temperaments]]
 * [[Trisedodge family|Trisedodge temperaments]]
+== Quint ==
+[[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
+{{Multival|legend=1| 0 0 5 0 8 12 }}
+[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\5, ~7/4 = 1017.903
+{{Optimal ET sequence|legend=1| 5, 15ccd }}
+[[Badness]]: 0.048312
 == Pentonismic (rank-5) ==
-{{Main|Pentonisma}}
+{{Main| Pentonisma }}
 Subgroup: 2.3.5.7.11.13