5-limit: Difference between revisions
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The '''5-limit''' consists of all [[just intonation]] intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called [http://en.wikipedia.org/wiki/Regular_number regular numbers]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]]. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, [[4/3]], [[3/2]], [[8/5]], [[5/3]], [[2/1]]. Approximating these ratios has been basic to Western common-practice music since the Renaissance. | The '''5-limit''' consists of all [[just intonation]] intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called [http://en.wikipedia.org/wiki/Regular_number regular numbers]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]]. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, [[4/3]], [[3/2]], [[8/5]], [[5/3]], [[2/1]]. Approximating these ratios has been basic to Western common-practice music since the Renaissance. | ||
The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a [http://en.wikipedia.org/wiki/Hexagonal_lattice hexagonal lattice] or as a [http://en.wikipedia.org/wiki/Square_lattice square lattice]; this can be done automatically by [http://www.huygens-fokker.org/scala/ Scala]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [http://en.wikipedia.org/wiki/Hexagonal_tiling hexagonal tiling]. | The octave equivalence classes of 5-limit or quinquimal intervals can usefully be depicted on a lattice diagram, either as a [http://en.wikipedia.org/wiki/Hexagonal_lattice hexagonal lattice] or as a [http://en.wikipedia.org/wiki/Square_lattice square lattice]; this can be done automatically by [http://www.huygens-fokker.org/scala/ Scala]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [http://en.wikipedia.org/wiki/Hexagonal_tiling hexagonal tiling]. | ||
[[EDO]]s which do relatively well in approximating the 5-limit (harmonics 3 and 5) are [[2edo|2]], [[3edo|3]], [[7edo|7]], [[9edo|9]], [[10edo|10]], [[12edo|12]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[34edo|34]], [[53edo|53]], [[118edo|118]], [[289edo|289]], [[323edo|323]], [[441edo|441]], [[494edo|494]], [[559edo|559]], [[612edo|612]], [[1171edo|1171]], [[1783edo|1783]], [[2513edo|2513]], [[3684edo|3684]], [[4296edo|4296]], ... | [[EDO]]s which do relatively well in approximating the 5-limit (harmonics 3 and 5) are [[2edo|2]], [[3edo|3]], [[7edo|7]], [[9edo|9]], [[10edo|10]], [[12edo|12]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[34edo|34]], [[53edo|53]], [[118edo|118]], [[289edo|289]], [[323edo|323]], [[441edo|441]], [[494edo|494]], [[559edo|559]], [[612edo|612]], [[1171edo|1171]], [[1783edo|1783]], [[2513edo|2513]], [[3684edo|3684]], [[4296edo|4296]], ... | ||