Prime interval: Difference between revisions

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A '''prime interval''' or '''~~primary interval~~''' 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 [[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.

~~For example, the~~ [[~~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~~.

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.

The [[monzo]] notation of each prime interval ~~has only one <code>1</code>~~ entry: (2 = {{monzo|1}}, 3 = {{monzo|0 1}}, 5 = {{monzo|0 0 1}}, 7 = {{monzo|0 0 0 1}}, 11 = {{monzo|0 0 0 0 1}}, ...)

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.

The [[monzo]] notation of each prime interval consists of all-zeros except for a single unity entry: (2: {{monzo| 1 }}, 3: {{monzo| 0 1 }}, 5: {{monzo| 0 0 1 }}, 7: {{monzo| 0 0 0 1 }}, 11: {{monzo| 0 0 0 0 1 }}, …)

The opposite of a prime interval is a [[highly composite interval]].

== Individual pages ==

See [[:Category: Prime harmonics]].

== See also ==

* [[Patent val]]

* [[~~Consistency~~]]

* [[Harmonic limit]]

* [[Prime harmonic series]]

[[Category:~~Term~~]]

[[Category:Terms]]

[[Category:Prime]]

[[Category:Harmonic]]

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

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

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-A '''prime interval''' or '''primary interval''' 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 [[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.
-For example, the [[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.
+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.
-The [[monzo]] notation of each prime interval has only one <code>1</code> entry: (2 = {{monzo|1}}, 3 = {{monzo|0 1}}, 5 = {{monzo|0 0 1}}, 7 = {{monzo|0 0 0 1}}, 11 = {{monzo|0 0 0 0 1}}, ...)
+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.
+The [[monzo]] notation of each prime interval consists of all-zeros except for a single unity entry: (2: {{monzo| 1 }}, 3: {{monzo| 0 1 }}, 5: {{monzo| 0 0 1 }}, 7: {{monzo| 0 0 0 1 }}, 11: {{monzo| 0 0 0 0 1 }}, …)
+The opposite of a prime interval is a [[highly composite interval]].
+== Individual pages ==
+See [[:Category: Prime harmonics]].
 == See also ==
 * [[Patent val]]
-* [[Consistency]]
+* [[Harmonic limit]]
 * [[Prime harmonic series]]
-[[Category:Term]]
+[[Category:Terms]]
 [[Category:Prime]]
+[[Category:Harmonic]]
-[[Category:Todo:review]]
-[[Category:Todo:expand]]