Prime interval - Revision history

VectorGraphics: don't expect people to know what "unity" means.

2025-11-07T03:04:17Z

don't expect people to know what "unity" means.

← Older revision		Revision as of 03:04, 7 November 2025
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	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.		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 [[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 }}, …)

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

FloraC: +link to highly composite interval

2025-04-01T15:54:59Z

+link to highly composite interval

← Older revision		Revision as of 15:54, 1 April 2025
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	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 [[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 ==		== Individual pages ==
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	* [[Harmonic limit]]		* [[Harmonic limit]]
	* [[Prime harmonic series]]		* [[Prime harmonic series]]

	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Prime]]		[[Category:Prime]]
	[[Category:Harmonic]]		[[Category:Harmonic]]

FloraC: Rework: link to harmonic, be specific about "JI" instead of "p-limit", and note its property of no redundancy and therefore forms a basis

2025-04-01T15:13:57Z

Rework: link to harmonic, be specific about "JI" instead of "p-limit", and note its property of no redundancy and therefore forms a basis

← Older revision		Revision as of 15:13, 1 April 2025
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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.		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:Prime]]
	[[Category:Harmonic]]		[[Category:Harmonic]]

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

FloraC: +link to individual pages

2025-04-01T14:26:18Z

+link to individual pages

← Older revision		Revision as of 14:26, 1 April 2025
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	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 [[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 }}, …)

			== Individual pages ==
			See [[:Category: Prime harmonics]].

	== See also ==		== See also ==

FloraC: Improve formatting and recategorize

2025-04-01T14:25:39Z

Improve formatting and recategorize

← Older revision		Revision as of 14:25, 1 April 2025
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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 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''.

	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.		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 [[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 }}, …)

	== See also ==		== See also ==
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	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Prime]]		[[Category:Prime]]
	[[Category:~~Ratio~~]]		[[Category:Harmonic]]

	[[Category:Todo:review]]		[[Category:Todo:review]]
	[[Category:Todo:expand]]		[[Category:Todo:expand]]

FloraC: +link

2022-12-29T16:59:14Z

+link

← Older revision		Revision as of 16:59, 29 December 2022
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	== See also ==		== See also ==
	* [[Patent val]]		* [[Patent val]]
			* [[Harmonic limit]]
	* [[Prime harmonic series]]		* [[Prime harmonic series]]

FloraC: +"prime harmonic"; improve links and categories

2022-12-29T16:48:36Z

+"prime harmonic"; improve links and categories

← Older revision		Revision as of 16:48, 29 December 2022
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	A '''prime 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 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''.

	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.		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.

	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}}, ...)		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}}, ...)

	== See also ==		== See also ==

	* [[Patent val]]		* [[Patent val]]
	* [[Consistency]]
	* [[Prime harmonic series]]		* [[Prime harmonic series]]

	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Prime]]		[[Category:Prime]]
			[[Category:Ratio]]

	[[Category:Todo:review]]		[[Category:Todo:review]]
	[[Category:Todo:expand]]		[[Category:Todo:expand]]

Inthar: "primary" is not the adjective corresponding to "prime number" in English

2021-03-17T11:05:13Z

"primary" is not the adjective corresponding to "prime number" in English

← Older revision		Revision as of 11:05, 17 March 2021
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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''' 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.

	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.		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.

Keenan Pepper category edits: Moving from Category:Term to Category:Terms using Cat-a-lot

2020-12-04T22:44:58Z

Moving from Category:Term to Category:Terms using Cat-a-lot

← Older revision		Revision as of 22:44, 4 December 2020
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	* [[Prime harmonic series]]		* [[Prime harmonic series]]

	[[Category:~~Term~~]]		[[Category:Terms]]
	[[Category:Prime]]		[[Category:Prime]]

	[[Category:Todo:review]]		[[Category:Todo:review]]
	[[Category:Todo:expand]]		[[Category:Todo:expand]]

Xenwolf: absolutely makes sense, but should be further elaborated

2020-11-20T11:27:30Z

absolutely makes sense, but should be further elaborated

← Older revision		Revision as of 11:27, 20 November 2020
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	:''~~<tt>"Proposal", see [[#Open questions]]</tt>~~''		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''' is a musical interval which as a ratio of frequencies is a [[prime number]]; that is, a number such as 2, 3, 5, 7, ... 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.

	~~The~~ prime intervals ~~are~~ [[2/1]], [[1/2]], [[3/1]], [[1/3]], [[5/1]], [[1/5]], [[7/1]], etc.		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.
	~~The interval [[5/3]], for example, is not considered a prime interval~~.

	== ~~Open questions~~ ==		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}}, ...)

	* What about [[1/1~~]]?~~
	* What about [[octave equivalence]]?

	== See also ==		== See also ==

	* [[~~Octave~~]]		* [[Patent val]]
			* [[Consistency]]
	* [[Prime harmonic series]]		* [[Prime harmonic series]]

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	[[Category:Todo:review]]		[[Category:Todo:review]]
			[[Category:Todo:expand]]