16edo: Difference between revisions

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In general, 16edo tends to better approximate the differences between odd [[harmonic]]s than odd harmonics themselves, though it has a [[5/1|5th harmonic]] which is only 11 cents flat, and a [[7/1|7th harmonic]] which is only 6 cents sharp. Most low harmonics are tuned very flat, but some such as [[21/16|21]]:[[11/8|22]]:[[23/16|23]]:[[3/2|24]]:[[25/16|25]]:[[13/8|26]] are well in tune with each other.

The [[3/2|perfect fifth]] of 16edo~~, so to speak,~~ is 27 cents flat of just, flatter than that of [[7edo]] so that it generates an [[2L 5s|antidiatonic]] instead of [[5L 2s|diatonic]] scale. This befits a tuning of [[mavila]], the [[5-limit]] [[regular temperament|temperament]] that [[tempering out|tempers out]] [[135/128]], such that a stack of four fifths gives a [[6/5]] minor third instead of the familiar [[5/4]] major third as in [[meantone]].

The [[3/2|perfect fifth]] of 16edo is 27 cents flat of just, flatter than that of [[7edo]] so that it generates an [[2L 5s|antidiatonic]] instead of [[5L 2s|diatonic]] scale. This befits a tuning of [[mavila]], the [[5-limit]] [[regular temperament|temperament]] that [[tempering out|tempers out]] [[135/128]], such that a stack of four fifths gives a [[6/5]] minor third instead of the familiar [[5/4]] major third as in [[meantone]].

Four steps of 16edo gives the 300{{c}} minor third interval shared by [[12edo]] (and other multiples of [[4edo]]), tempering out 648/625, the [[diminished comma]], and thus the familiar [[diminished seventh chord]] may be built on any scale step with four unique tetrads up to [[octave equivalence]]. The minor third is of course not distinguished from the septimal subminor third, [[7/6]], so [[36/35]] and moreover [[50/49]] are tempered out, making 16edo a possible tuning for [[diminished (temperament)|septimal diminished]].

16edo shares several similarities with 15edo. They both share mappings of [[8/7]], [[5/4]], and [[3/2]] in terms of edosteps – in fact, they are both [[valentine]] tunings, and thus [[slendric]] tunings. 16edo and 15edo also both have three types of seconds and two types of thirds (not including arto/tendo thirds). However, 15edo's fifth is sharp while 16's is flat.

16edo works as a tuning for [[extraclassical tonality]], due to its ultramajor third of 450 cents.

=== Odd harmonics ===

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 In general, 16edo tends to better approximate the differences between odd [[harmonic]]s than odd harmonics themselves, though it has a [[5/1|5th harmonic]] which is only 11 cents flat, and a [[7/1|7th harmonic]] which is only 6 cents sharp. Most low harmonics are tuned very flat, but some such as [[21/16|21]]:[[11/8|22]]:[[23/16|23]]:[[3/2|24]]:[[25/16|25]]:[[13/8|26]] are well in tune with each other.
-The [[3/2|perfect fifth]] of 16edo, so to speak, is 27 cents flat of just, flatter than that of [[7edo]] so that it generates an [[2L 5s|antidiatonic]] instead of [[5L 2s|diatonic]] scale. This befits a tuning of [[mavila]], the [[5-limit]] [[regular temperament|temperament]] that [[tempering out|tempers out]] [[135/128]], such that a stack of four fifths gives a [[6/5]] minor third instead of the familiar [[5/4]] major third as in [[meantone]].
+The [[3/2|perfect fifth]] of 16edo is 27 cents flat of just, flatter than that of [[7edo]] so that it generates an [[2L 5s|antidiatonic]] instead of [[5L 2s|diatonic]] scale. This befits a tuning of [[mavila]], the [[5-limit]] [[regular temperament|temperament]] that [[tempering out|tempers out]] [[135/128]], such that a stack of four fifths gives a [[6/5]] minor third instead of the familiar [[5/4]] major third as in [[meantone]].
 Four steps of 16edo gives the 300{{c}} minor third interval shared by [[12edo]] (and other multiples of [[4edo]]), tempering out 648/625, the [[diminished comma]], and thus the familiar [[diminished seventh chord]] may be built on any scale step with four unique tetrads up to [[octave equivalence]]. The minor third is of course not distinguished from the septimal subminor third, [[7/6]], so [[36/35]] and moreover [[50/49]] are tempered out, making 16edo a possible tuning for [[diminished (temperament)|septimal diminished]].
+edo shares several similarities with 15edo. They both share mappings of [[8/7]], [[5/4]], and [[3/2]] in terms of edosteps – in fact, they are both [[valentine]] tunings, and thus [[slendric]] tunings. 16edo and 15edo also both have three types of seconds and two types of thirds (not including arto/tendo thirds). However, 15edo's fifth is sharp while 16's is flat.
+edo works as a tuning for [[extraclassical tonality]], due to its ultramajor third of 450 cents.
 === Odd harmonics ===