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 [[wikipedia: 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 [[ratio|numerators and denominators]] are both products of the primes 2, 3, and 5; these are sometimes called [[wikipedia: Regular number|regular numbers]]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]]. The 5-limit is the third prime limit and is a superset of the [[3-limit]] and a subset of the [[7-limit]].

These things are contained by the 5-limit, but not the 3-limit:

* The [[5-odd-limit]];

* Mode 3 of the harmonic or subharmonic series.

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 [[wikipedia: Hexagonal lattice|hexagonal lattice]] or as a [[wikipedia: Square lattice|square lattice]]; this can be done automatically by [[Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [[wikipedia:Hexagonal tiling|hexagonal tiling]].

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== See also ==

* [[~~5-odd-limit]]~~

* [[Table of 5-limit commas]]

* [[Harmonic limit]]

* [[5-limit commas]]

== Notes ==

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 {{Prime limit navigation|5}}
 {{Wikipedia|Five-limit tuning}}
-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 [[wikipedia: 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 [[ratio|numerators and denominators]] are both products of the primes 2, 3, and 5; these are sometimes called [[wikipedia: Regular number|regular numbers]]. Some examples of 5-limit intervals are [[5/4]], [[6/5]], [[10/9]] and [[81/80]]. The 5-limit is the third prime limit and is a superset of the [[3-limit]] and a subset of the [[7-limit]].
+These things are contained by the 5-limit, but not the 3-limit:
+* The [[5-odd-limit]];
+* Mode 3 of the harmonic or subharmonic series.
+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 [[wikipedia: Hexagonal lattice|hexagonal lattice]] or as a [[wikipedia: Square lattice|square lattice]]; this can be done automatically by [[Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [[wikipedia:Hexagonal tiling|hexagonal tiling]].
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 == See also ==
-* [[5-odd-limit]]
+* [[Table of 5-limit commas]]
-* [[Harmonic limit]]
-* [[5-limit commas]]
 == Notes ==