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 octave equivalence classes of 5-limit, ''~~classical~~'', ''pental'', ''quinquimal'' or ''ptolemaic''<ref>[https://marsbat.space/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref>~~, 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]]~~.

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]].

== Terminology ==

Due to its historical significance, 5-limit intervals go by various names, including '''classic(al)'''<ref>[https://dkeenan.com/Music/IntervalNaming.htm ''A note on the naming of musical intervals''] by [[Dave Keenan]]</ref>, '''pental'''{{Citation needed}}, or '''quinquimal'''{{Citation needed}}. Although the corresponding Latin numerals are used to refer to higher prime limits such as ''septimal'' for the 7-limit and ''undecimal'' for the 11-limit, ''quintal'' is ''never'' used to refer to the 5-limit because it is the adjective associated with the fifth [[5L 2s|diatonic]] degree. (Quintal harmony does ''not'' mean 5-limit harmony, but harmony with chords stacked by fifths – cf. secundal harmony, tertian harmony, quartal harmony.)

A finite set of 5-limit intervals are labeled ''just'', especially when the interval in question is the simplest in the [[Interval category|category]]. For example, 5/4 is known as the ''just major third''<ref>[https://marsbat.space/pdfs/HEJI2_legend+series.pdf ''The Helmholtz-Ellis JI Pitch Notation (HEJI)''] by [[Marc Sabat]] and [[Thomas Nicholson]] from Plainsound Music Edition</ref>. Indeed, ''just intonation'' traditionally meant specifically the 5-limit version thereof. Even so, justness is not to be generalized to all 5-limit intervals, nor can we assume all just intervals 5-limit in contemporary usage.

The term ''ptolemaic'' could also refer to the 5-limit<ref>[https://marsbat.space/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref>. On this wiki it is part of the [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic interval naming system]], and refers specifically to intervals that contains a single factor of harmonic 5. We distinguish multi-order 5-limit intervals by ''diptolemaic'', ''triptolemaic'', and so on.

== Edo approximation ==

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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 octave equivalence classes of 5-limit, ''classical'', ''pental'', ''quinquimal'' or ''ptolemaic''<ref>[https://marsbat.space/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref>, 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]].
+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]].
+== Terminology ==
+Due to its historical significance, 5-limit intervals go by various names, including '''classic(al)'''<ref>[https://dkeenan.com/Music/IntervalNaming.htm ''A note on the naming of musical intervals''] by [[Dave Keenan]]</ref>, '''pental'''{{Citation needed}}, or '''quinquimal'''{{Citation needed}}. Although the corresponding Latin numerals are used to refer to higher prime limits such as ''septimal'' for the 7-limit and ''undecimal'' for the 11-limit, ''quintal'' is ''never'' used to refer to the 5-limit because it is the adjective associated with the fifth [[5L 2s|diatonic]] degree. (Quintal harmony does ''not'' mean 5-limit harmony, but harmony with chords stacked by fifths – cf. secundal harmony, tertian harmony, quartal harmony.)
+A finite set of 5-limit intervals are labeled ''just'', especially when the interval in question is the simplest in the [[Interval category|category]]. For example, 5/4 is known as the ''just major third''<ref>[https://marsbat.space/pdfs/HEJI2_legend+series.pdf ''The Helmholtz-Ellis JI Pitch Notation (HEJI)''] by [[Marc Sabat]] and [[Thomas Nicholson]] from Plainsound Music Edition</ref>. Indeed, ''just intonation'' traditionally meant specifically the 5-limit version thereof. Even so, justness is not to be generalized to all 5-limit intervals, nor can we assume all just intervals 5-limit in contemporary usage.
+The term ''ptolemaic'' could also refer to the 5-limit<ref>[https://marsbat.space/pdfs/JI.pdf ''Fundamental Principles of Just Intonation and Microtonal Composition''] by Thomas Nicholson and Marc Sabat —"'Ptolemaic' refers to intervals combining only the primes 2, 3, and 5."</ref>. On this wiki it is part of the [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic interval naming system]], and refers specifically to intervals that contains a single factor of harmonic 5. We distinguish multi-order 5-limit intervals by ''diptolemaic'', ''triptolemaic'', and so on.
 == Edo approximation ==