5-limit: Difference between revisions

Line 352:

== Rank-2 temperaments ==

=== Meantone ===

[[Meantone]] is the rank-2 temperament tempering out 81/80 in the 5-limit. It is generated by a flat perfect fifth, around ~~696 or 697~~ cents in ideal tunings. It equates complex pythagorean intervals with simpler 5-limit ones, such as [[32/27]] with [[6/5]], and [[81/64]] with [[5/4]]. It is the most historically prevalent regular temperament, and it forms much of the basis of modern harmony, notably being supported by [[12edo]]. There are, however, better tunings for meantone than 12edo, such as [[19edo]] and [[31edo]]. Meantone has an obvious extension to the 7-limit known as septimal meantone, which maps [[7/4]] to the augmented sixth, and adds [[126/125]] and [[225/224]] to the commas.

[[Meantone]] is the rank-2 temperament tempering out 81/80 in the 5-limit. It is generated by a flat perfect fifth, around 696–697 cents in ideal tunings. It equates complex pythagorean intervals with simpler 5-limit ones, such as [[32/27]] with [[6/5]], and [[81/64]] with [[5/4]]. It is the most historically prevalent regular temperament, and it forms much of the basis of modern harmony, notably being supported by [[12edo]]. There are, however, better tunings for meantone than 12edo, such as [[19edo]] and [[31edo]]. Meantone has an obvious extension to the 7-limit known as septimal meantone, which maps [[7/4]] to the augmented sixth, and adds [[126/125]] and [[225/224]] to the commas.

=== Magic ===

[[Magic]] is the rank-2 temperament tempering out [[3125/3072]], the magic comma. It is generated by a flat major third of around 380 cents, and equates five of them with a [[3/1|perfect twelfth]]. It is one of the simplest 5-limit temperaments with decent accuracy to not temper out the syntonic comma, 81/80. It has an obvious extension to the 7-limit, tempering out [[225/224]] and [[245/243]], and mapping 7/4 to +12 generators. Edos supporting magic include [[19edo]], [[22edo]], and [[41edo]].

=== Diaschismic ===

[[Diaschismic]] tempers out [[2048/2025]], the diaschisma, which equates [[45/32]] with its octave complement [[64/45]]. This creates a half-octave period. This temperament is generated by a semitone of around 103–105 cents in optimal tunings, representing [[16/15]]. A half-octave plus a semitone reaches the perfect fifth, and a half-octave minus two semitones reaches the major third. This temperament has an obvious extension to prime [[17/1|17]] by equating the semitone to [[17/16]] and [[18/17]], known as [[srutal archagall]]. This temperament remarkably contains ''two'' relatively simple and accurate extensions to the full [[17-limit]]; one is simply called diaschismic, and the other is known as [[srutal]]. Another notable extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]].

=== Kleismic ===

@@ Line 352: / Line 352: @@
 == Rank-2 temperaments ==
 === Meantone ===
-[[Meantone]] is the rank-2 temperament tempering out 81/80 in the 5-limit. It is generated by a flat perfect fifth, around 696 or 697 cents in ideal tunings. It equates complex pythagorean intervals with simpler 5-limit ones, such as [[32/27]] with [[6/5]], and [[81/64]] with [[5/4]]. It is the most historically prevalent regular temperament, and it forms much of the basis of modern harmony, notably being supported by [[12edo]]. There are, however, better tunings for meantone than 12edo, such as [[19edo]] and [[31edo]]. Meantone has an obvious extension to the 7-limit known as septimal meantone, which maps [[7/4]] to the augmented sixth, and adds [[126/125]] and [[225/224]] to the commas.
+[[Meantone]] is the rank-2 temperament tempering out 81/80 in the 5-limit. It is generated by a flat perfect fifth, around 696–697 cents in ideal tunings. It equates complex pythagorean intervals with simpler 5-limit ones, such as [[32/27]] with [[6/5]], and [[81/64]] with [[5/4]]. It is the most historically prevalent regular temperament, and it forms much of the basis of modern harmony, notably being supported by [[12edo]]. There are, however, better tunings for meantone than 12edo, such as [[19edo]] and [[31edo]]. Meantone has an obvious extension to the 7-limit known as septimal meantone, which maps [[7/4]] to the augmented sixth, and adds [[126/125]] and [[225/224]] to the commas.
 === Magic ===
 [[Magic]] is the rank-2 temperament tempering out [[3125/3072]], the magic comma. It is generated by a flat major third of around 380 cents, and equates five of them with a [[3/1|perfect twelfth]]. It is one of the simplest 5-limit temperaments with decent accuracy to not temper out the syntonic comma, 81/80. It has an obvious extension to the 7-limit, tempering out [[225/224]] and [[245/243]], and mapping 7/4 to +12 generators. Edos supporting magic include [[19edo]], [[22edo]], and [[41edo]].
+=== Diaschismic ===
+[[Diaschismic]] tempers out [[2048/2025]], the diaschisma, which equates [[45/32]] with its octave complement [[64/45]]. This creates a half-octave period. This temperament is generated by a semitone of around 103–105 cents in optimal tunings, representing [[16/15]]. A half-octave plus a semitone reaches the perfect fifth, and a half-octave minus two semitones reaches the major third. This temperament has an obvious extension to prime [[17/1|17]] by equating the semitone to [[17/16]] and [[18/17]], known as [[srutal archagall]]. This temperament remarkably contains ''two'' relatively simple and accurate extensions to the full [[17-limit]]; one is simply called diaschismic, and the other is known as [[srutal]]. Another notable extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]].
 === Kleismic ===

5-limit: Difference between revisions

Overthink (talk | contribs)