Prime interval: Difference between revisions

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A '''prime interval''' or '''prime harmonic''' is a ~~musical interval~~ which as a [[ratio]] of frequencies is a [[prime number]]; that is, a number such as 2, 3, 5, 7, 11, … which is divisible only by itself and 1. Any ~~musical~~ interval ~~in the~~ [[~~Harmonic limit~~|~~''p''-limit~~]] can be expressed in terms of a product of prime ~~numbers less than or equal~~ to ~~''p''~~.

A '''prime interval''' or '''prime harmonic''' is a [[harmonic]] which as a [[ratio]] of frequencies is a [[prime number]]; that is, a number such as 2, 3, 5, 7, 11, … which is divisible only by itself and 1.

Any interval of [[just intonation|just intonation (JI)]] can be expressed in terms of a product of prime intervals, allowing us to decompose a complex JI interval into simpler parts. A prime interval itself cannot be expressed by other prime intervals, so no prime intervals are redundant for reconstructing the entirety of JI. For those reasons and for the fact that prime intervals occur in [[harmonic series]], they form a very important [[basis]] (literally and mathematically) for JI.

For example, the [[2/1|octave]] is a prime interval whereas the intervals [[5/3]] or even [[1/1]] are not. In traditional ratio notation, the prime intervals are [[2/1]], [[3/1]], [[5/1]], [[7/1]], [[11/1]] etc.

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[[Category:Prime]]

[[Category:Harmonic]]

~~[[Category:Todo:review]]~~

~~[[Category:Todo:expand]]~~

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-A '''prime interval''' or '''prime harmonic''' is a musical interval which as a [[ratio]] of frequencies is a [[prime number]]; that is, a number such as 2, 3, 5, 7, 11, … which is divisible only by itself and 1. Any musical interval in the [[Harmonic limit|''p''-limit]] can be expressed in terms of a product of prime numbers less than or equal to ''p''.
+A '''prime interval''' or '''prime harmonic''' is a [[harmonic]] which as a [[ratio]] of frequencies is a [[prime number]]; that is, a number such as 2, 3, 5, 7, 11, … which is divisible only by itself and 1.
+Any interval of [[just intonation|just intonation (JI)]] can be expressed in terms of a product of prime intervals, allowing us to decompose a complex JI interval into simpler parts. A prime interval itself cannot be expressed by other prime intervals, so no prime intervals are redundant for reconstructing the entirety of JI. For those reasons and for the fact that prime intervals occur in [[harmonic series]], they form a very important [[basis]] (literally and mathematically) for JI.
 For example, the [[2/1|octave]] is a prime interval whereas the intervals [[5/3]] or even [[1/1]] are not. In traditional ratio notation, the prime intervals are [[2/1]], [[3/1]], [[5/1]], [[7/1]], [[11/1]] etc.
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 [[Category:Prime]]
 [[Category:Harmonic]]
-[[Category:Todo:review]]
-[[Category:Todo:expand]]