Interval space - Revision history

Vector at 01:12, 8 February 2026

2026-02-08T01:12:36Z

← Older revision		Revision as of 01:12, 8 February 2026
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	[[File:5-lim.png\|thumb\|A segment of octave-reduced 5-limit interval space. The origin is 1/1, and the axes represent 3/2 and 5/4. Highlighted is the [[diatonic]] scale.]]		[[File:5-lim.png\|thumb\|A segment of octave-reduced 5-limit interval space, visualized as a lattice. The origin is 1/1, and the axes represent 3/2 and 5/4. Highlighted is the [[diatonic]] scale.With octaves, the 5-limit interval space is 3D.]]
	An '''interval space''' is a multi-dimensional space in which intervals in [[just intonation]] or a [[regular temperament]] exist. The ''dimensionality'' or ''rank'' of an interval space is the minimum number of different "step sizes", called '''generators''', that you need to [[stack]] to all the intervals in the space. An [[Equal temperament\|equal]] (rank-1) temperament has a 1D interval space, because any interval can be represented by one integer (the number of steps). A temperament such as [[Meantone]] has a 2D interval space (because it has both a generator and a period), so any interval must be represented with 2 integers (the number of periods and the number of generators), defining it as a rank-2 temperament. For [[just intonation]], the number of primes in a JI [[subgroup]] determines its rank. For example, the [[7-limit]] (subgroup 2.3.5.7) is rank-4, because intervals are composed from 4 primes: 2, 3, 5, and 7.		An '''interval space''' is a multi-dimensional space in which intervals in [[just intonation]] or a [[regular temperament]] exist. The ''dimensionality'' or ''rank'' of an interval space is the minimum number of different "step sizes", called '''generators''', that you need to [[stack]] to all the intervals in the space. An [[Equal temperament\|equal]] (rank-1) temperament has a 1D interval space, because any interval can be represented by one integer (the number of steps). A temperament such as [[Meantone]] has a 2D interval space (because it has both a generator and a period), so any interval must be represented with 2 integers (the number of periods and the number of generators), defining it as a rank-2 temperament. For [[just intonation]], the number of primes in a JI [[subgroup]] determines its rank. For example, the [[7-limit]] (subgroup 2.3.5.7) is rank-4, because intervals are composed from 4 primes: 2, 3, 5, and 7.

Vector at 01:12, 8 February 2026

2026-02-08T01:12:11Z

← Older revision		Revision as of 01:12, 8 February 2026
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	[[File:5-lim.png\|thumb\|A segment of 5-limit interval space. The origin is 1/1, and the axes represent 3/2 and 5/4. Highlighted is the [[diatonic]] scale.]]		[[File:5-lim.png\|thumb\|A segment of octave-reduced 5-limit interval space. The origin is 1/1, and the axes represent 3/2 and 5/4. Highlighted is the [[diatonic]] scale.]]
	An '''interval space''' is a multi-dimensional space in which intervals in [[just intonation]] or a [[regular temperament]] exist. The ''dimensionality'' or ''rank'' of an interval space is the minimum number of different "step sizes", called '''generators''', that you need to [[stack]] to all the intervals in the space. An [[Equal temperament\|equal]] (rank-1) temperament has a 1D interval space, because any interval can be represented by one integer (the number of steps). A temperament such as [[Meantone]] has a 2D interval space (because it has both a generator and a period), so any interval must be represented with 2 integers (the number of periods and the number of generators), defining it as a rank-2 temperament. For [[just intonation]], the number of primes in a JI [[subgroup]] determines its rank. For example, the [[7-limit]] (subgroup 2.3.5.7) is rank-4, because intervals are composed from 4 primes: 2, 3, 5, and 7.		An '''interval space''' is a multi-dimensional space in which intervals in [[just intonation]] or a [[regular temperament]] exist. The ''dimensionality'' or ''rank'' of an interval space is the minimum number of different "step sizes", called '''generators''', that you need to [[stack]] to all the intervals in the space. An [[Equal temperament\|equal]] (rank-1) temperament has a 1D interval space, because any interval can be represented by one integer (the number of steps). A temperament such as [[Meantone]] has a 2D interval space (because it has both a generator and a period), so any interval must be represented with 2 integers (the number of periods and the number of generators), defining it as a rank-2 temperament. For [[just intonation]], the number of primes in a JI [[subgroup]] determines its rank. For example, the [[7-limit]] (subgroup 2.3.5.7) is rank-4, because intervals are composed from 4 primes: 2, 3, 5, and 7.

Vector at 01:11, 8 February 2026

2026-02-08T01:11:39Z

← Older revision		Revision as of 01:11, 8 February 2026
Line 1:		Line 1:
			[[File:5-lim.png\|thumb\|A segment of 5-limit interval space. The origin is 1/1, and the axes represent 3/2 and 5/4. Highlighted is the [[diatonic]] scale.]]
	An '''interval space''' is a multi-dimensional space in which intervals in [[just intonation]] or a [[regular temperament]] exist. The ''dimensionality'' or ''rank'' of an interval space is the minimum number of different "step sizes", called '''generators''', that you need to [[stack]] to all the intervals in the space. An [[Equal temperament\|equal]] (rank-1) temperament has a 1D interval space, because any interval can be represented by one integer (the number of steps). A temperament such as [[Meantone]] has a 2D interval space (because it has both a generator and a period), so any interval must be represented with 2 integers (the number of periods and the number of generators), defining it as a rank-2 temperament. For [[just intonation]], the number of primes in a JI [[subgroup]] determines its rank. For example, the [[7-limit]] (subgroup 2.3.5.7) is rank-4, because intervals are composed from 4 primes: 2, 3, 5, and 7.		An '''interval space''' is a multi-dimensional space in which intervals in [[just intonation]] or a [[regular temperament]] exist. The ''dimensionality'' or ''rank'' of an interval space is the minimum number of different "step sizes", called '''generators''', that you need to [[stack]] to all the intervals in the space. An [[Equal temperament\|equal]] (rank-1) temperament has a 1D interval space, because any interval can be represented by one integer (the number of steps). A temperament such as [[Meantone]] has a 2D interval space (because it has both a generator and a period), so any interval must be represented with 2 integers (the number of periods and the number of generators), defining it as a rank-2 temperament. For [[just intonation]], the number of primes in a JI [[subgroup]] determines its rank. For example, the [[7-limit]] (subgroup 2.3.5.7) is rank-4, because intervals are composed from 4 primes: 2, 3, 5, and 7.

Lériendil at 07:19, 15 December 2025

2025-12-15T07:19:31Z

← Older revision		Revision as of 07:19, 15 December 2025
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	An '''interval space''' is a multi-dimensional space in which intervals in [[just intonation]] or a [[regular temperament]] exist. The ''dimensionality'' or ''rank'' of an interval space is the minimum number of different "step sizes" you need to stack to all the intervals in the space. An [[Equal temperament\|equal]] (rank-1) temperament has a 1D interval space, because any interval can be represented by one integer (the number of steps). A temperament such as [[~~meantone~~]] has a 2D interval space (because it has both a generator and a period), so any interval must be represented with 2 integers (the number of periods and the number of generators)~~. This means~~ it's a rank-2 temperament. For just intonation, the number of primes in a JI subgroup determines its rank. For example, the subgroup 2...7 is rank-4, because ~~there~~ are 4 primes: 2, 3, 5, and 7.		An '''interval space''' is a multi-dimensional space in which intervals in [[just intonation]] or a [[regular temperament]] exist. The ''dimensionality'' or ''rank'' of an interval space is the minimum number of different "step sizes", called '''generators''', that you need to [[stack]] to all the intervals in the space. An [[Equal temperament\|equal]] (rank-1) temperament has a 1D interval space, because any interval can be represented by one integer (the number of steps). A temperament such as [[Meantone]] has a 2D interval space (because it has both a generator and a period), so any interval must be represented with 2 integers (the number of periods and the number of generators), defining it as a rank-2 temperament. For [[just intonation]], the number of primes in a JI [[subgroup]] determines its rank. For example, the [[7-limit]] (subgroup 2.3.5.7) is rank-4, because intervals are composed from 4 primes: 2, 3, 5, and 7.

	== Monzos ==		== Monzos ==
	An interval's location in interval space is notated using a ''monzo''. which functions as a list of coordinates. In just intonation, a monzo encodes the prime factorization of an interval's frequency ratio as a list of exponents. For example, 125/96 is 2^-5 * 3^-1 * 5^3, so the monzo is {{Monzo\|-5 -1 3\|\|}}. In general, {{Monzo\|n<sub>1</sub> n<sub>2</sub> n<sub>3</sub> ... n<sub>k</sub>\|\|}} means 2^n<sub>1</sub> * 3^n<sub>2</sub> * 5^n<sub>3</sub> * ... * p^n<sub>k</sub> for k-th prime p.		An interval's location in interval space is notated using a ''monzo''. which functions as a list of coordinates. In just intonation, a monzo encodes the prime factorization of an interval's frequency ratio as a list of exponents. For example, 125/96 is 2^-5 * 3^-1 * 5^3, so the monzo is {{Monzo\|-5 -1 3\|\|}}. In general, {{Monzo\|n<sub>1</sub> n<sub>2</sub> n<sub>3</sub> ... n<sub>k</sub>\|\|}} means 2^n<sub>1</sub> * 3^n<sub>2</sub> * 5^n<sub>3</sub> * ... * p^n<sub>k</sub> for k-th prime p.

	In a temperament, a monzo ~~exists~~ with respect to a set of generators (since the same temperament may have multiple valid sets of generators). For example, the monzo for ~5/4 in [[~~meantone~~]] with generators ~2, ~3/2 is {{Monzo\|-2 4\|\|}}, but with generators ~2, ~3 it is {{Monzo\|-6 4\|\|}}.		In a temperament, a monzo can be defined with respect to a set of generators (since the same temperament may have multiple valid sets of generators). For example, the monzo for ~[[5/4]] in [[Meantone]] with generators ~[[2/1\|2]] and ~[[3/2]] is {{Monzo\|-2 4\|\|}}, but with generators ~2, ~[[3/1\|3]] it is {{Monzo\|-6 4\|\|}}.

	When stacking intervals, simply add the corresponding entries of their monzos. For example, 125/96 * 81/64 = {{Monzo\|-5 -1 3\|\|}} + {{Monzo\|-6 4 0\|\|}} = {{Monzo\|-11 3 3\|\|}} = 3375/2048.		When stacking intervals, simply add the corresponding entries of their monzos. For example, 125/96 * 81/64 = {{Monzo\|-5 -1 3\|\|}} + {{Monzo\|-6 4 0\|\|}} = {{Monzo\|-11 3 3\|\|}} = 3375/2048.

Vector: Created page with "An '''interval space''' is a multi-dimensional space in which intervals in just intonation or a regular temperament exist. The ''dimensionality'' or ''rank'' of an interval space is the minimum number of different "step sizes" you need to stack to all the intervals in the space. An equal (rank-1) temperament has a 1D interval space, because any interval can be represented by one integer (the number of steps). A temperament such as meantone..."

2025-12-12T07:37:20Z

Created page with "An '''interval space''' is a multi-dimensional space in which intervals in just intonation or a regular temperament exist. The ''dimensionality'' or ''rank'' of an interval space is the minimum number of different "step sizes" you need to stack to all the intervals in the space. An equal (rank-1) temperament has a 1D interval space, because any interval can be represented by one integer (the number of steps). A temperament such as meantone..."

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An '''interval space''' is a multi-dimensional space in which intervals in [[just intonation]] or a [[regular temperament]] exist. The ''dimensionality'' or ''rank'' of an interval space is the minimum number of different "step sizes" you need to stack to all the intervals in the space. An [[Equal temperament|equal]] (rank-1) temperament has a 1D interval space, because any interval can be represented by one integer (the number of steps). A temperament such as [[meantone]] has a 2D interval space (because it has both a generator and a period), so any interval must be represented with 2 integers (the number of periods and the number of generators). This means it's a rank-2 temperament. For just intonation, the number of primes in a JI subgroup determines its rank. For example, the subgroup 2...7 is rank-4, because there are 4 primes: 2, 3, 5, and 7.

== Monzos ==
An interval's location in interval space is notated using a ''monzo''. which functions as a list of coordinates. In just intonation, a monzo encodes the prime factorization of an interval's frequency ratio as a list of exponents. For example, 125/96 is 2^-5 * 3^-1 * 5^3, so the monzo is {{Monzo|-5 -1 3||}}. In general, {{Monzo|n1 n2 n3 ... nk||}} means 2^n1 * 3^n2 * 5^n3 * ... * p^nk for k-th prime p.

In a temperament, a monzo exists with respect to a set of generators (since the same temperament may have multiple valid sets of generators). For example, the monzo for ~5/4 in [[meantone]] with generators ~2, ~3/2 is {{Monzo|-2 4||}}, but with generators ~2, ~3 it is {{Monzo|-6 4||}}.

When stacking intervals, simply add the corresponding entries of their monzos. For example, 125/96 * 81/64 = {{Monzo|-5 -1 3||}} + {{Monzo|-6 4 0||}} = {{Monzo|-11 3 3||}} = 3375/2048.