16edo: Difference between revisions

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== Theory ==

~~In general,~~ 16edo ~~tends~~ to ~~better approximate~~ the ~~differences between odd [[harmonic]]s than odd harmonics themselves~~, ~~though there are exceptions:~~ it ~~has~~ a [[~~5/1~~]] ~~which~~ is ~~only eleven cents flat,~~ and a [[~~7/1~~]] ~~which is only six 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~~.

16edo's fifth is too flat to be a reasonable approximation of 3/2, but still within the range where it can serve the role of a fifth, similarly to 9edo. If it is treated as a diatonic fifth, it generates a scale that is similar to diatonic, but where minor is larger than major, which is called an [[antidiatonic]] scale (with 5 small steps and 2 large steps). It is thus often useful to switch "minor" and "major" from their standard diatonic meanings in interval names (but see below in [[#Intervals]]). If treated as [[3/2]], this fifth produces the [[mavila]] temperament, as stacking it four times produces a [[6/5]] minor third as opposed to the standard [[5/4]] major third found in meantone temperaments. (This tempers out [[135/128]], as that is the difference between 6/5 and 81/64, the interval generated by a stack of 4 untempered fifths. Note that 5/4 is tempered to 32/27.)

~~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]]~~.

This minor third is 300 cents (supporting the [[diminished]] temperament), however in this tuning its inaccuracy comes more from the error on the fifth than on the 5/4 major third, which is much closer to (and flat of) just intonation at 375 cents. However, 16edo's best low prime is 7, which is mapped with only 6 cents of error. 13 is also mapped decently accurately (if 12edo's error on 5/4 is acceptable). This contributes to a good tuning of certain [[interseptimal]] intervals, specifically 13/10 and 20/13. As such, 16edo can most reasonably be considered a 2.5.7.13 temperament (2.3.5.7.13, if the mavila fifth is acceptable as a 3/2).

~~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]]~~.

7/5 is mapped to the tritone of 600 cents, and 14/13 to the neutral second of 150 cents. 35/32 is mapped to the same interval, and 28/25 is mapped to 225 cents.

16edo can be seen as an approach to tuning that takes advantage of the idea that simpler ratios can be functionally approximated with greater error: by choosing a tuning with greater error on lower primes as opposed to higher ones, one can create a much more consistent feeling than if the highest errors are on higher primes. In essence, 16edo's 3/1, 5/1, and 7/1 are backwards from 12edo's, with 7 being nearly perfect, 5 being decent, and 3 being distinctly out-of-tune.

12edo's symmetrical diminished seventh chord may be constructed in 16edo by stacking minor thirds.

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

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 temperaments, and thus slendric temperaments. 16edo and 15edo also both have 3 types of seconds and 2 types of thirds (excluding arto/tendo thirds).

However, 15edo's fifth is sharp while 16's is flat.

=== Odd harmonics ===

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 == Theory ==
-In general, 16edo tends to better approximate the differences between odd [[harmonic]]s than odd harmonics themselves, though there are exceptions: it has a [[5/1]] which is only eleven cents flat, and a [[7/1]] which is only six 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.
+edo's fifth is too flat to be a reasonable approximation of 3/2, but still within the range where it can serve the role of a fifth, similarly to 9edo. If it is treated as a diatonic fifth, it generates a scale that is similar to diatonic, but where minor is larger than major, which is called an [[antidiatonic]] scale (with 5 small steps and 2 large steps). It is thus often useful to switch "minor" and "major" from their standard diatonic meanings in interval names (but see below in [[#Intervals]]). If treated as [[3/2]], this fifth produces the [[mavila]] temperament, as stacking it four times produces a [[6/5]] minor third as opposed to the standard [[5/4]] major third found in meantone temperaments. (This tempers out [[135/128]], as that is the difference between 6/5 and 81/64, the interval generated by a stack of 4 untempered fifths. Note that 5/4 is tempered to 32/27.)
-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]].
+This minor third is 300 cents (supporting the [[diminished]] temperament), however in this tuning its inaccuracy comes more from the error on the fifth than on the 5/4 major third, which is much closer to (and flat of) just intonation at 375 cents. However, 16edo's best low prime is 7, which is mapped with only 6 cents of error. 13 is also mapped decently accurately (if 12edo's error on 5/4 is acceptable). This contributes to a good tuning of certain [[interseptimal]] intervals, specifically 13/10 and 20/13. As such, 16edo can most reasonably be considered a 2.5.7.13 temperament (2.3.5.7.13, if the mavila fifth is acceptable as a 3/2).
-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]].
+/5 is mapped to the tritone of 600 cents, and 14/13 to the neutral second of 150 cents. 35/32 is mapped to the same interval, and 28/25 is mapped to 225 cents.
+edo can be seen as an approach to tuning that takes advantage of the idea that simpler ratios can be functionally approximated with greater error: by choosing a tuning with greater error on lower primes as opposed to higher ones, one can create a much more consistent feeling than if the highest errors are on higher primes. In essence, 16edo's 3/1, 5/1, and 7/1 are backwards from 12edo's, with 7 being nearly perfect, 5 being decent, and 3 being distinctly out-of-tune.
+edo's symmetrical diminished seventh chord may be constructed in 16edo by stacking minor thirds.
+edo works as a tuning for [[extraclassical tonality]], due to its ultramajor third of 450 cents.
+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 temperaments, and thus slendric temperaments. 16edo and 15edo also both have 3 types of seconds and 2 types of thirds (excluding arto/tendo thirds).
+However, 15edo's fifth is sharp while 16's is flat.
 === Odd harmonics ===