5-limit: Difference between revisions

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== 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 696–697 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|rank-2]] [[regular temperament|temperament]] [[tempering out]] [[81/80]] in the 5-limit. It is generated by a flat [[3/2|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 [[support]]ed 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 ===

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=== Kleismic ===

[[Kleismic]]~~/Hanson~~ tempers out [[15625/15552]], the kleisma. It is generated by a slightly sharp minor third of around 317 cents, with the perfect twelfth being equated to six of them. It has much better accuracy than meantone or magic with not much more complexity, but it doesn't extend as well to the 7-limit. It does, however, have an obvious extension to prime [[13/1|13]] called [[cata]]. Edos supporting kleismic include {{edos| 15, 19, 34, and 53.}}

[[Kleismic]] (a.k.a. hanson) tempers out [[15625/15552]], the kleisma. It is generated by a slightly sharp minor third of around 317 cents, with the perfect twelfth being equated to six of them. It has much better accuracy than meantone or magic with not much more complexity, but it doesn't extend as well to the 7-limit. It does, however, have an obvious extension to prime [[13/1|13]] called [[cata]]. Edos supporting kleismic include {{edos| 15, 19, 34, and 53}}.

=== Schismic ===

[[Schismic]]~~/Schismatic/Helmholtz~~ tempers out [[32805/32768]], the schisma. It is generated by a very slightly flat perfect fifth of around 701.73 cents in an ideal tuning. It equates the major third 5/4 with the Pythagorean diminished fourth 8192/6561, or 8 fifths down. It is a [[microtemperament]] in the 5-limit, with errors well under a cent. A notable extension of schismic to the 7-limit is [[~~Garibaldi~~]], which maps 7/4 to the ~~doubly~~-diminished octave (-14 fifths), tempering out the [[~~Garischisma~~]]. Other notable commas it tempers out include [[225/224]] and [[5120/5103]] (the difference between [[64/63]] and 81/80). While it is still quite accurate, ~~[[Garibaldi]]~~ is no longer a microtemperament unlike schismic. ~~EDOs~~ supporting schismic include {{edos| 12, 41, 53, 65, 118, and 171.}}

[[Schismic]] (a.k.a. schismatic and helmholtz) tempers out [[32805/32768]], the schisma. It is generated by a very slightly flat perfect fifth of around 701.73 cents in an ideal tuning. It equates the major third 5/4 with the Pythagorean diminished fourth 8192/6561, or 8 fifths down. It is a [[microtemperament]] in the 5-limit, with errors well under a cent. A notable extension of schismic to the 7-limit is [[garibaldi]], which maps 7/4 to the double-diminished octave (-14 fifths), tempering out the [[garischisma]]. Other notable commas it tempers out include [[225/224]] and [[5120/5103]] (the difference between [[64/63]] and 81/80). While it is still quite accurate, garibaldi is no longer a microtemperament unlike schismic. Edos supporting schismic include {{edos| 12, 41, 53, 65, 118, and 171.}}

== Music ==

@@ 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 696–697 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|rank-2]] [[regular temperament|temperament]] [[tempering out]] [[81/80]] in the 5-limit. It is generated by a flat [[3/2|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 [[support]]ed 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 ===
@@ Line 361: / Line 361: @@
 === Kleismic ===
-[[Kleismic]]/Hanson tempers out [[15625/15552]], the kleisma. It is generated by a slightly sharp minor third of around 317 cents, with the perfect twelfth being equated to six of them. It has much better accuracy than meantone or magic with not much more complexity, but it doesn't extend as well to the 7-limit. It does, however, have an obvious extension to prime [[13/1|13]] called [[cata]]. Edos supporting kleismic include {{edos| 15, 19, 34, and 53.}}
+[[Kleismic]] (a.k.a. hanson) tempers out [[15625/15552]], the kleisma. It is generated by a slightly sharp minor third of around 317 cents, with the perfect twelfth being equated to six of them. It has much better accuracy than meantone or magic with not much more complexity, but it doesn't extend as well to the 7-limit. It does, however, have an obvious extension to prime [[13/1|13]] called [[cata]]. Edos supporting kleismic include {{edos| 15, 19, 34, and 53}}.
 === Schismic ===
-[[Schismic]]/Schismatic/Helmholtz tempers out [[32805/32768]], the schisma. It is generated by a very slightly flat perfect fifth of around 701.73 cents in an ideal tuning. It equates the major third 5/4 with the Pythagorean diminished fourth 8192/6561, or 8 fifths down. It is a [[microtemperament]] in the 5-limit, with errors well under a cent. A notable extension of schismic to the 7-limit is [[Garibaldi]], which maps 7/4 to the doubly-diminished octave (-14 fifths), tempering out the [[Garischisma]]. Other notable commas it tempers out include [[225/224]] and [[5120/5103]] (the difference between [[64/63]] and 81/80). While it is still quite accurate, [[Garibaldi]] is no longer a microtemperament unlike schismic. EDOs supporting schismic include {{edos| 12, 41, 53, 65, 118, and 171.}}
+[[Schismic]] (a.k.a. schismatic and helmholtz) tempers out [[32805/32768]], the schisma. It is generated by a very slightly flat perfect fifth of around 701.73 cents in an ideal tuning. It equates the major third 5/4 with the Pythagorean diminished fourth 8192/6561, or 8 fifths down. It is a [[microtemperament]] in the 5-limit, with errors well under a cent. A notable extension of schismic to the 7-limit is [[garibaldi]], which maps 7/4 to the double-diminished octave (-14 fifths), tempering out the [[garischisma]]. Other notable commas it tempers out include [[225/224]] and [[5120/5103]] (the difference between [[64/63]] and 81/80). While it is still quite accurate, garibaldi is no longer a microtemperament unlike schismic. Edos supporting schismic include {{edos| 12, 41, 53, 65, 118, and 171.}}
 == Music ==