12edo: Difference between revisions

Line 12:

== Theory ==

12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.

It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament ~~(specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone)~~.

It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s ~~each unless [[stretched and compressed tuning|octave stretching or compression]] is employed~~. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is ~~flat of just by~~ even ~~more~~, 15.6 cents.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.

@@ Line 12: / Line 12: @@
 == Theory ==
 edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
-It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).
+It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament.
-It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.
+It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.
 Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.