Harmonic: Difference between revisions

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== Prime harmonic ==

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.

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. It is an element of the [[prime harmonic series]].

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.

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The [[monzo]] notation of each prime interval consists of all-zeros except for a single entry equal to 1: (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 }}, …)

Prime harmonics are often used to define [[harmonic limit]]s, also known as prime limits.

== Highly composite harmonic ==

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 == Prime harmonic ==
 {{See also|Category:Prime harmonics}}
-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.
+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. It is an element of the [[prime harmonic series]].
 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.
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 The [[monzo]] notation of each prime interval consists of all-zeros except for a single entry equal to 1: (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 }}, …)
+Prime harmonics are often used to define [[harmonic limit]]s, also known as prime limits.
 == Highly composite harmonic ==