Glossary - Revision history

Inthar: /* Mode of limited transposition */

2026-06-26T00:49:49Z

Mode of limited transposition

← Older revision		Revision as of 00:49, 26 June 2026
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	A scale is a '''mode of limited transposition''' or a '''multiperiod scale''' if at least two distinct [[ordinal]]s result in the same mode when the scale is transposed by the corresponding interval. This is equivalent to the scale having a [[period]] that is a smaller fraction of its equave.		A scale is a '''mode of limited transposition''' or a '''multiperiod scale''' if at least two distinct [[ordinal]]s result in the same mode when the scale is transposed by the corresponding interval. This is equivalent to the scale having a [[period]] that is a smaller fraction of its equave.

	The diatonic scale of 12edo is ''not'' a mode of of limited transposition, because each possible transposition changes at least one of the pitches, until you return to the scale's period. However, the ~~"diatonic" of 7edo '~~''is''' of limited transposition, because ~~multiple~~ transpositions ~~of n\7~~ will result in the same pitches.		The diatonic scale of 12edo is ''not'' a mode of of limited transposition, because each possible transposition changes at least one of the pitches, until you return to the scale's period. However, the Blackwood scale LsLsLsLsLs ''is'' of limited transposition, because transpositions by L + s will result in the same pitches.

	== Monzo ==		== Monzo ==

Inthar: /* JI group, subgroup */

2026-06-25T21:08:56Z

JI group, subgroup

← Older revision		Revision as of 21:08, 25 June 2026
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	== JI group, subgroup ==		== JI group, subgroup ==
	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. For historical reasons, "subgroup" is more commonly used in xen theory than "group" to refer to any group generated by stacking musical intervals, whether it is mathematically a subgroup of JI or not (e.g. groups generated by [[straddle primes]]).		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. For historical reasons, "subgroup" is more commonly used in less mathematically rigorous treatments of xen theory than "group" to refer to any group generated by stacking musical intervals, whether it is mathematically a subgroup of JI or not (e.g. groups generated by [[straddle primes]]).

	JI groups are denoted by generators, called ''basis elements'' (the standard mathematical term) 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'' (the standard mathematical term) 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.

Inthar: /* JI group, subgroup */

2026-06-25T21:06:53Z

JI group, subgroup

← Older revision		Revision as of 21:06, 25 June 2026
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	== JI group, subgroup ==		== JI group, subgroup ==
	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. For historical reasons, "subgroup" is more commonly used in xen theory than "group" to refer to any group generated by stacking musical intervals, whether it is mathematically a subgroup of JI or not (e.g. groups generated by [[straddle primes]]).

	JI groups are denoted by generators, called ''basis elements'' (the standard mathematical term) 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'' (the standard mathematical term) 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.
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	{{adv\|A subgroup ''generated by'' a subset of a group is the group formed by iterating the binary operation on elements in the subset. Equivalently, it is the smallest subgroup of the larger group containing that subset.}}		{{adv\|A subgroup ''generated by'' a subset of a group is the group formed by iterating the binary operation on elements in the subset. Equivalently, it is the smallest subgroup of the larger group containing that subset.}}

	{{Adv\|Groups in xen theory are typically a much more specific type of groups, namely [[wikipedia:Free abelian group\|free abelian groups]]. Note that "subgroup" is more commonly used in xen theory than "group" to refer to any group generated by stacking musical intervals, whether it is mathematically a subgroup of JI or not (e.g. groups generated by [[straddle primes]]).}}		{{Adv\|Groups in xen theory are typically a much more specific type of groups, namely [[wikipedia:Free abelian group\|free abelian groups]].}}

	<small>Categories: JI, RTT, Math terms</small>		<small>Categories: JI, RTT, Math terms</small>

Inthar: /* JI group */

2026-06-25T21:04:40Z

JI group

← Older revision		Revision as of 21:04, 25 June 2026
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	<small>Categories: Scales</small>		<small>Categories: Scales</small>

	== JI group ==		== JI group, subgroup ==
	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.

Inthar: /* JI group */

2026-06-25T21:02:36Z

JI group

← Older revision		Revision as of 21:02, 25 June 2026
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	{{adv\|A subgroup ''generated by'' a subset of a group is the group formed by iterating the binary operation on elements in the subset. Equivalently, it is the smallest subgroup of the larger group containing that subset.}}		{{adv\|A subgroup ''generated by'' a subset of a group is the group formed by iterating the binary operation on elements in the subset. Equivalently, it is the smallest subgroup of the larger group containing that subset.}}

	{{Adv\|Groups in xen theory are typically a much more specific type of groups, namely [[wikipedia:Free abelian group\|free abelian groups]].}}		{{Adv\|Groups in xen theory are typically a much more specific type of groups, namely [[wikipedia:Free abelian group\|free abelian groups]]. Note that "subgroup" is more commonly used in xen theory than "group" to refer to any group generated by stacking musical intervals, whether it is mathematically a subgroup of JI or not (e.g. groups generated by [[straddle primes]]).}}

	<small>Categories: JI, RTT, Math terms</small>		<small>Categories: JI, RTT, Math terms</small>

Inthar: /* Mode of limited transposition */

2026-06-16T02:26:14Z

Mode of limited transposition

← Older revision		Revision as of 02:26, 16 June 2026
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	A scale is a '''mode of limited transposition''' or a '''multiperiod scale''' if at least two distinct [[ordinal]]s result in the same mode when the scale is transposed by the corresponding interval. This is equivalent to the scale having a [[period]] that is a smaller fraction of its equave.		A scale is a '''mode of limited transposition''' or a '''multiperiod scale''' if at least two distinct [[ordinal]]s result in the same mode when the scale is transposed by the corresponding interval. This is equivalent to the scale having a [[period]] that is a smaller fraction of its equave.

	The diatonic scale of 12edo is '''not''' a mode of of limited transposition, because each possible transposition changes at least one of the pitches, until you return to the scale's period. However, the "diatonic" of 7edo '''is''' of limited transposition, because multiple transpositions of n\7 will result in the same pitches.		The diatonic scale of 12edo is ''not'' a mode of of limited transposition, because each possible transposition changes at least one of the pitches, until you return to the scale's period. However, the "diatonic" of 7edo '''is''' of limited transposition, because multiple transpositions of n\7 will result in the same pitches.

	== Monzo ==		== Monzo ==

Inthar: /* Consistency */

2026-06-15T20:18:44Z

Consistency

← Older revision		Revision as of 20:18, 15 June 2026
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	== Consistency ==		== Consistency ==
	An approximation of an interval in an EDO (or ~~otherwise an~~ equal-step tuning) is '''consistent''' when it is both the closest direct approximation of the just interval available in the tuning, and the approximation regularly dictated by the [[#Val\|val]] being used (whether [[#Patent val\|patent]] or otherwise). Approximations where these two deviate from each other are correspondingly '''inconsistent'''.		An approximation of an interval in an EDO (or another equal-step tuning) is '''consistent''' when it is both the closest direct approximation of the just interval available in the tuning, and the approximation regularly dictated by the [[#Val\|val]] being used (whether [[#Patent val\|patent]] or otherwise). Approximations where these two deviate from each other are correspondingly '''inconsistent'''.

	For example, the interval [[7/6]] is inconsistent in [[34edo\|34et]], since while [[7/4]] is defined as 27 steps and [[3/2]] as 20 steps (implying 7/6 to be 7 steps), 7/6 itself is slightly closer to 8 steps than to 7 steps of 34edo. {{Adv\|In the alternative val 34d, where 7/4 is mapped to 28 steps instead, 7/6 becomes consistent but 7/4 itself is now inconsistent.}}		For example, the interval [[7/6]] is inconsistent in [[34edo\|34et]], since while [[7/4]] is defined as 27 steps and [[3/2]] as 20 steps (implying 7/6 to be 7 steps), 7/6 itself is slightly closer to 8 steps than to 7 steps of 34edo. {{Adv\|In the alternative val 34d, where 7/4 is mapped to 28 steps instead, 7/6 becomes consistent but 7/4 itself is now inconsistent.}}

Inthar: /* Period */

2026-06-15T20:16:15Z

Period

← Older revision		Revision as of 20:16, 15 June 2026
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	== Period ==		== Period ==
	'''Period''' has the following related but different senses:		'''Period''' has the following related but different senses:
	* The smallest unit at which a given scale repeats — a fraction of the equave but not necessarily the equave itself.		* The smallest unit at which a given scale repeats — a fraction of the equave but not necessarily the equave itself. If the period is smaller than the equave, the scale is called a ''multiperiod scale'' or a ''[[#Mode of limited transposition\|mode of limited transposition]]''.
	** Example: Pentawood (5L5s, LsLsLsLsLs) has period 1\5 (240c).		** Example: Pentawood (5L5s, LsLsLsLsLs) has period 1\5 (240c).
	* One of the generators of a regular temperament, specifically chosen to be a fraction of the equave (usually 2/1). (We make this choice for musical reasons, though a group mathematically doesn't have a distinguished element called the "period".)		* One of the generators of a regular temperament, specifically chosen to be a fraction of the equave (usually 2/1). (We make this choice for musical reasons, though a group mathematically doesn't have a distinguished element called the "period".)

Inthar: /* Mode of limited transposition */

2026-06-15T20:15:08Z

Mode of limited transposition

← Older revision		Revision as of 20:15, 15 June 2026
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	== Mode of limited transposition ==		== Mode of limited transposition ==
	A scale is a '''mode of limited transposition''' or a '''multiperiod scale''' if at least two distinct [[ordinal]]s result in the same mode when the scale is transposed by the corresponding interval. This is equivalent to the scale having a period that is a smaller fraction of its equave.		A scale is a '''mode of limited transposition''' or a '''multiperiod scale''' if at least two distinct [[ordinal]]s result in the same mode when the scale is transposed by the corresponding interval. This is equivalent to the scale having a [[period]] that is a smaller fraction of its equave.

	The diatonic scale of 12edo is '''not''' a mode of of limited transposition, because each possible transposition changes at least one of the pitches, until you return to the scale's period. However, the "diatonic" of 7edo '''is''' of limited transposition, because multiple transpositions of n\7 will result in the same pitches.		The diatonic scale of 12edo is '''not''' a mode of of limited transposition, because each possible transposition changes at least one of the pitches, until you return to the scale's period. However, the "diatonic" of 7edo '''is''' of limited transposition, because multiple transpositions of n\7 will result in the same pitches.

Inthar: /* Mode of limited transposition */

2026-06-15T20:14:40Z

Mode of limited transposition

← Older revision		Revision as of 20:14, 15 June 2026
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	== Mode of limited transposition ==		== Mode of limited transposition ==
	A scale is ~~considered to be~~ a '''mode of limited transposition''' if at least two distinct [[ordinal]]s result in the same mode when the scale is transposed by the corresponding interval.		A scale is a '''mode of limited transposition''' or a '''multiperiod scale''' if at least two distinct [[ordinal]]s result in the same mode when the scale is transposed by the corresponding interval. This is equivalent to the scale having a period that is a smaller fraction of its equave.

	The diatonic scale of 12edo is '''not''' a mode of of limited transposition, because each possible transposition changes at least one of the pitches, until you return to the scale's period. However, the "diatonic" of 7edo '''is''' of limited transposition, because multiple transpositions of n\7 will result in the same pitches.		The diatonic scale of 12edo is '''not''' a mode of of limited transposition, because each possible transposition changes at least one of the pitches, until you return to the scale's period. However, the "diatonic" of 7edo '''is''' of limited transposition, because multiple transpositions of n\7 will result in the same pitches.