5-limit: Difference between revisions

The '''5-limit''' (a.k.a. ''ya'' in [[color notation]]) consists of all [[just intonation]] intervals whose [[ratio|numerators and denominators]] are both products of the primes 2, 3, and 5. The 5-limit is the third prime limit and is a superset of the [[3-limit]] and a subset of the [[7-limit]]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]].  

[[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.

[[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]] (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.}}