Glossary - Revision history

Inthar: /* Extension */

2026-04-05T01:41:46Z

Extension

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	For instance, [[Meantone]] introduces [[5-limit]] interpretations of intervals on a [[Chain of fifths\|chain]] of tempered fifths by making the equivalence ([[3/2]])<sup>4</sup> = [[Octave\|2]]<sup>2</sup> × 5/4 (tempering out the comma [[81/80]] and finding 5 at 4 fifths up). But if the chain of fifths is continued further, [[7-limit]] harmonies can be introduced: (3/2)<sup>2</sup> × (5/4)<sup>2</sup> = 2 × [[7/4]], which can be worked out to place 7 at 10 fifths up, a mapping of 7 known as ''septimal Meantone'', which is a strong extension of 5-limit Meantone.		For instance, [[Meantone]] introduces [[5-limit]] interpretations of intervals on a [[Chain of fifths\|chain]] of tempered fifths by making the equivalence ([[3/2]])<sup>4</sup> = [[Octave\|2]]<sup>2</sup> × 5/4 (tempering out the comma [[81/80]] and finding 5 at 4 fifths up). But if the chain of fifths is continued further, [[7-limit]] harmonies can be introduced: (3/2)<sup>2</sup> × (5/4)<sup>2</sup> = 2 × [[7/4]], which can be worked out to place 7 at 10 fifths up, a mapping of 7 known as ''septimal Meantone'', which is a strong extension of 5-limit Meantone.

	Weak extensions are ~~often~~ created by dividing the original period or (a choice of) generator into equal parts and then interpreting the split parts. As an example, Mothra is a temperament where the 3/2 Meantone generator is split into 3 parts, and then (3/2)^(1/3) is interpreted as [[8/7]]. It is a weak extension of pental Meantone, as Meantone natively doesn't have something that is one-third of a 3/2, to the 7-limit. Sometimes a weak extension may split the period instead of the generator; for example, Pajara (2.3.5.7[10 & 22]) is a weak extension of Archy (2.3.7[5 & 22]) that splits 2/1 into two 7/5's. {{Adv\|If you don't interpret the new intervals of a weak extension, the result is called ''contorsion''.}}		Weak extensions are created by dividing the original period or (a choice of) generator into equal parts and then interpreting the split parts. As an example, Mothra is a temperament where the 3/2 Meantone generator is split into 3 parts, and then (3/2)^(1/3) is interpreted as [[8/7]]. It is a weak extension of pental Meantone, as Meantone natively doesn't have something that is one-third of a 3/2, to the 7-limit. Sometimes a weak extension may split the period instead of the generator; for example, Pajara (2.3.5.7[10 & 22]) is a weak extension of Archy (2.3.7[5 & 22]) that splits 2/1 into two 7/5's. {{Adv\|If you don't interpret the new intervals of a weak extension, the result is called ''contorsion''.}}

	Note that temperaments of different ranks are ''not'' considered extensions or restrictions of one another.		Note that temperaments of different ranks are ''not'' considered extensions or restrictions of one another.

Inthar: /* Regular temperament */

2026-03-15T00:44:10Z

Regular temperament

← Older revision		Revision as of 00:44, 15 March 2026
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	== Regular temperament ==		== Regular temperament ==
	''Main article: [[Regular temperament]]''		:''Main article: [[Regular temperament]]''

	A '''regular temperament''' (often just '''temperament''') is a way of assigning JI interpretations (from a chosen JI group) to intervals in a non-JI tuning. We assign the interpretations so that the stack of two JI ratios gets assigned to the stack of the corresponding tempered versions of the two ratios. This is what makes a regular temperament "regular".		A '''regular temperament''' (often just '''temperament''') is a way of assigning JI interpretations (from a chosen JI group) to intervals in a non-JI tuning. We assign the interpretations so that the stack of two JI ratios gets assigned to the stack of the corresponding tempered versions of the two ratios. This is what makes a regular temperament "regular".

Inthar: /* JI group */

2026-03-09T02:41:35Z

JI group

← Older revision		Revision as of 02:41, 9 March 2026
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	JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group.		JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group.

	Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to. 2.b/a.c/a.d/a may be written {{nowrap\|2.(a:b:c:d)}} for brevity, for example {{nowrap\|2.(5:7:11:13)}} = 2.7/5.11/5.13/5 (note that this is the JI group generated by 2/1 and the intervals of 5:7:11:13).		Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to. On XR, 2.b/a.c/a.d/a may be written {{nowrap\|2.(a:b:c:d)}} for brevity, for example {{nowrap\|2.(5:7:11:13)}} = 2.7/5.11/5.13/5 (note that this is the JI group generated by 2/1 and the intervals of 5:7:11:13).

	Groups can be generalized to non-JI generators, for example 2.√6 (representing a chain of perfect hemififths), or 2.φ.		Groups can be generalized to non-JI generators, for example 2.√6 (representing a chain of perfect hemififths), or 2.φ.

Inthar: /* JI group */

2026-03-09T02:38:43Z

JI group

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	A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.		A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

	JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written {{nowrap\|2.(a:b:c:d)}} for brevity, for example {{nowrap\|2.(5:7:11:13)}} = 2.7/5.11/5.13/5 (note that this is the JI group generated by 2/1 and the intervals of 5:7:11:13).		JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group.

	Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.		Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to. 2.b/a.c/a.d/a may be written {{nowrap\|2.(a:b:c:d)}} for brevity, for example {{nowrap\|2.(5:7:11:13)}} = 2.7/5.11/5.13/5 (note that this is the JI group generated by 2/1 and the intervals of 5:7:11:13).

	Groups can be generalized to non-JI generators, for example 2.√6 (representing a chain of perfect hemififths), or 2.φ.		Groups can be generalized to non-JI generators, for example 2.√6 (representing a chain of perfect hemififths), or 2.φ.

Inthar: /* JI group */

2026-03-09T02:36:18Z

JI group

← Older revision		Revision as of 02:36, 9 March 2026
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	A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.		A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

	JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written {{nowrap\|2.(a:b:c:d)}} for brevity, for example {{nowrap\|2.(5:7:11:13)}} = 2.7/5.11/5.13/5.		JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written {{nowrap\|2.(a:b:c:d)}} for brevity, for example {{nowrap\|2.(5:7:11:13)}} = 2.7/5.11/5.13/5 (note that this is the JI group generated by 2/1 and the intervals of 5:7:11:13).

	Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.		Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.

Inthar: /* JI group */

2026-03-09T02:35:14Z

JI group

← Older revision		Revision as of 02:35, 9 March 2026
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	A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.		A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

	JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written 2.(a:b:c:d) for brevity, for example 2.(5:7:11:13) = 2.7/5.11/5.13/5.		JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written {{nowrap\|2.(a:b:c:d)}} for brevity, for example {{nowrap\|2.(5:7:11:13)}} = 2.7/5.11/5.13/5.

	Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.		Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.

Inthar: /* JI group */

2026-03-09T02:34:39Z

JI group

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	A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.		A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

	JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written 2.(a:b:c:d) for brevity.		JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written 2.(a:b:c:d) for brevity, for example 2.(5:7:11:13) = 2.7/5.11/5.13/5.

	Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.		Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.

Inthar: /* JI group */

2026-03-09T02:33:20Z

JI group

← Older revision		Revision as of 02:33, 9 March 2026
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	A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.		A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

	JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group.		JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written 2.(a:b:c:d) for brevity.

	Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.		Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.

Inthar: /* Square-superparticular */

2026-03-06T16:14:24Z

Square-superparticular

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	<math>\frac{k^2}{k^2-1} = \frac{k}{k-1}\frac{k}{k+1} = \frac{\frac{k}{k-1}}{\frac{k+1}{k}},</math>		<math>\frac{k^2}{k^2-1} = \frac{k}{k-1}\frac{k}{k+1} = \frac{\frac{k}{k-1}}{\frac{k+1}{k}},</math>

	denoted S''k'' in xen math.		denoted S''k'' or S(''k'') in xen math.

	A square-superparticular is the difference between consecutive suparparticulars. When a square-superparticular S''k'' is tempered out, it makes harmonics {{nowrap\|''k'' - 1}}, ''k'', and {{nowrap\|''k'' + 1}} equally spaced. For example, tempering out S9 = 81/80 makes harmonics 8, 9, and 10 equally spaced. Factoring a comma into a product of square-particulars, called an '''S-expression''', is often helpful for understanding it.		A square-superparticular is the difference between consecutive suparparticulars. When a square-superparticular S''k'' is tempered out, it makes harmonics {{nowrap\|''k'' - 1}}, ''k'', and {{nowrap\|''k'' + 1}} equally spaced. For example, tempering out S9 = 81/80 makes harmonics 8, 9, and 10 equally spaced. Factoring a comma into a product of square-particulars, called an '''S-expression''', is often helpful for understanding it.

Inthar: /* Binary */

2026-03-02T23:55:44Z

Binary

← Older revision		Revision as of 23:55, 2 March 2026
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	== Binary ==		== Binary ==
	A '''binary''' scale is a scale with exactly two step sizes (usually denoted L and s). MOS scales are binary, but binary scales need not be MOS scales (e.g. melodic minor, LsLLLLs).		A '''binary''' scale is a scale with exactly two step sizes (usually denoted L and s). [[MOS]] scales are binary, but binary scales need not be MOS scales (e.g. melodic minor, LsLLLLs, is binary but not a MOS).

	<small>Categories: Scales</small>		<small>Categories: Scales</small>

← Older revision		Revision as of 02:36, 9 March 2026
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	A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.		A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

	JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written {{nowrap\|2.(a:b:c:d)}} for brevity, for example {{nowrap\|2.(5:7:11:13)}} = 2.7/5.11/5.13/5.		JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written {{nowrap\|2.(a:b:c:d)}} for brevity, for example {{nowrap\|2.(5:7:11:13)}} = 2.7/5.11/5.13/5 (note that this is the JI group generated by 2/1 and the intervals of 5:7:11:13).

	Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.		Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.

← Older revision		Revision as of 02:35, 9 March 2026
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	A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.		A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

	JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written 2.(a:b:c:d) for brevity, for example 2.(5:7:11:13) = 2.7/5.11/5.13/5.		JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written {{nowrap\|2.(a:b:c:d)}} for brevity, for example {{nowrap\|2.(5:7:11:13)}} = 2.7/5.11/5.13/5.

	Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.		Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.

← Older revision		Revision as of 02:34, 9 March 2026
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	A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.		A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

	JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written 2.(a:b:c:d) for brevity.		JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written 2.(a:b:c:d) for brevity, for example 2.(5:7:11:13) = 2.7/5.11/5.13/5.

	Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.		Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.

← Older revision		Revision as of 02:33, 9 March 2026
Line 264:		Line 264:
	A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.		A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit\|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

	JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group.		JI groups are denoted by generators (called ''basis elements'' or ''formal primes'' in this context) separated by full stops: for example, 2.3.5.7 denotes the [[7-limit\|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1\|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group. 2.b/a.c/a.d/a may be written 2.(a:b:c:d) for brevity.

	Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.		Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup\|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to.