Odd-limit - Revision history

Lériendil at 19:52, 20 May 2026

2026-05-20T19:52:19Z

← Older revision		Revision as of 19:52, 20 May 2026
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	An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.		An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-odd-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

	It may also be useful to restrict an odd-limit further to within a specific [[prime limit]] or [[subgroup]], and these restrictions are constructed from fractions amongst those odd numbers that are both below the odd-limit and composed of primes in the subgroup: e.g. the [[7-limit\|7-prime-limited]] 21-odd-limit comprises intervals of odds 3, 5, 7, 9, 15, and 21; and the [[2.3.5.11 subgroup]] 15-odd-limit comprises intervals of odds 3, 5, 9, 11, and 15. The set of intervals constructed as fractions among an arbitrary set of odds, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond''', and tonality diamonds generalize the idea of odd-limits.		It may also be useful to restrict an odd-limit further to within a specific [[prime limit]] or [[subgroup]], and these restrictions are constructed from fractions amongst those odd numbers that are both below the odd-limit and composed of primes in the subgroup: e.g. the [[7-limit\|7-prime-limited]] 21-odd-limit comprises intervals of odds 3, 5, 7, 9, 15, and 21; and the [[2.3.5.11 subgroup]] 15-odd-limit comprises intervals of odds 3, 5, 9, 11, and 15. The set of intervals constructed as fractions among an arbitrary set of odds, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond''', and tonality diamonds generalize the idea of odd-limits.

Lériendil at 19:45, 20 May 2026

2026-05-20T19:45:37Z

← Older revision		Revision as of 19:45, 20 May 2026
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	An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.		An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

	It may also be useful to restrict an odd-limit further to within a specific [[prime limit]] or [[subgroup]], and these restrictions are constructed from fractions amongst those odd numbers that are both below the odd-limit and composed of primes in the subgroup: e.g. the [[7-limit\|7-limited]] 21-odd-limit comprises intervals of odds 3, 5, 7, 9, 15, and 21; and the [[2.3.5.11 subgroup]] 15-odd-limit comprises intervals of odds 3, 5, 9, 11, and 15. The set of intervals constructed as fractions among an arbitrary set of odds, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond''', and tonality diamonds generalize the idea of odd-limits.		It may also be useful to restrict an odd-limit further to within a specific [[prime limit]] or [[subgroup]], and these restrictions are constructed from fractions amongst those odd numbers that are both below the odd-limit and composed of primes in the subgroup: e.g. the [[7-limit\|7-prime-limited]] 21-odd-limit comprises intervals of odds 3, 5, 7, 9, 15, and 21; and the [[2.3.5.11 subgroup]] 15-odd-limit comprises intervals of odds 3, 5, 9, 11, and 15. The set of intervals constructed as fractions among an arbitrary set of odds, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond''', and tonality diamonds generalize the idea of odd-limits.

	'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.		'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

Lériendil: restricted odd-limits

2026-05-20T19:45:26Z

restricted odd-limits

← Older revision		Revision as of 19:45, 20 May 2026
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	An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7. The set of intervals ~~comprising~~ an ~~odd-limit~~, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond'''.		An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

			It may also be useful to restrict an odd-limit further to within a specific [[prime limit]] or [[subgroup]], and these restrictions are constructed from fractions amongst those odd numbers that are both below the odd-limit and composed of primes in the subgroup: e.g. the [[7-limit\|7-limited]] 21-odd-limit comprises intervals of odds 3, 5, 7, 9, 15, and 21; and the [[2.3.5.11 subgroup]] 15-odd-limit comprises intervals of odds 3, 5, 9, 11, and 15. The set of intervals constructed as fractions among an arbitrary set of odds, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond''', and tonality diamonds generalize the idea of odd-limits.

	'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.		'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

Lériendil at 19:32, 20 May 2026

2026-05-20T19:32:21Z

← Older revision		Revision as of 19:32, 20 May 2026
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	An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.		An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7. The set of intervals comprising an odd-limit, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond'''.

	'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.		'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

Lériendil: Reverted edit by Lériendil (talk) to last revision by Inthar

2026-05-20T19:32:10Z

Reverted edit by Lériendil (talk) to last revision by Inthar

← Older revision		Revision as of 19:32, 20 May 2026
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	An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7~~. The set of intervals comprising an odd-limit, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond'''~~.		An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.

	'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.		'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.
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	* [[9-odd-limit]]		* [[9-odd-limit]]

	The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9 and 9/8 consonant. These may be considered "secondary" consonances.		The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9, 9/8, and 9/7 consonant. These may be considered "secondary" consonances.

	* [[11-odd-limit]]		* [[11-odd-limit]]

	More info is on each individual odd-limit page.		More info is on each individual odd-limit page.

Lériendil: diamond...

2026-05-20T19:31:40Z

diamond...

← Older revision		Revision as of 19:31, 20 May 2026
Line 1:		Line 1:
	An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7.		An '''odd-limit''' is a set of intervals defined by a maximum allowable number in the numerator or denominator once all factors of two are removed. It may be considered as a set of consonant intervals, generalizing the 12edo concept of consonance and dissonance. The 5-limit, for instance, includes 8/5, 6/5, and 3/2, but not 7/5 as that contains 7 (an odd number higher than 5); it does not contain 14/5 either because 14 reduces to 7. The set of intervals comprising an odd-limit, or more specifically that set itself treated as a [[scale]], is also known as a '''tonality diamond'''.

	'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.		'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.
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	* [[9-odd-limit]]		* [[9-odd-limit]]

	The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9, 9/8~~, and 9/7~~ consonant. These may be considered "secondary" consonances.		The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9 and 9/8 consonant. These may be considered "secondary" consonances.

	* [[11-odd-limit]]		* [[11-odd-limit]]

	More info is on each individual odd-limit page.		More info is on each individual odd-limit page.

Inthar: /* Individual odd-limits */

2026-05-20T19:31:00Z

Individual odd-limits

← Older revision		Revision as of 19:31, 20 May 2026
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	* [[9-odd-limit]]		* [[9-odd-limit]]

	The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9 and 9/8 consonant. These may be considered "secondary" consonances.		The 7- and 9-odd limit contain more "exotic" [[septimal]] consonances (including a tritone, 7/5); expanding to the 9-odd-limit also implies considering 10/9, 9/8, and 9/7 consonant. These may be considered "secondary" consonances.

	* [[11-odd-limit]]		* [[11-odd-limit]]

	More info is on each individual odd-limit page.		More info is on each individual odd-limit page.

Lériendil: /* Distinction and monotonicity */

2026-05-20T19:29:51Z

Distinction and monotonicity

← Older revision		Revision as of 19:29, 20 May 2026
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	Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''<sup>2</sup>/(''n''<sup>2</sup> - 1) (though certain odd-limits can have a gap of size (''n''<sup>2</sup> + 1)/''n''<sup>2</sup> instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.		Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''<sup>2</sup>/(''n''<sup>2</sup> - 1) (though certain odd-limits can have a gap of size (''n''<sup>2</sup> + 1)/''n''<sup>2</sup> instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

	Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is ~~monotonic~~ in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly ~~monotonic~~''' in the ''n''-odd-limit.		Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotone in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotone''' in the ''n''-odd-limit.

	== Individual odd-limits ==		== Individual odd-limits ==

Lériendil: /* Distinction and monotonicity */

2026-05-20T19:26:04Z

Distinction and monotonicity

← Older revision		Revision as of 19:26, 20 May 2026
Line 6:		Line 6:
	Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''<sup>2</sup>/(''n''<sup>2</sup> - 1) (though certain odd-limits can have a gap of size (''n''<sup>2</sup> + 1)/''n''<sup>2</sup> instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.		Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''<sup>2</sup>/(''n''<sup>2</sup> - 1) (though certain odd-limits can have a gap of size (''n''<sup>2</sup> + 1)/''n''<sup>2</sup> instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

	Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotonic in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, ~~for~~ intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it ~~would~~ still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotonic''' in the ''n''-odd-limit.		Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotonic in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, if we expect intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it may still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotonic''' in the ''n''-odd-limit.

	== Individual odd-limits ==		== Individual odd-limits ==

Lériendil: added words about distinction, monotonicity, etc.

2026-05-20T19:25:18Z

added words about distinction, monotonicity, etc.

← Older revision		Revision as of 19:25, 20 May 2026
Line 2:		Line 2:

	'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.		'''Proper odd-limit''' is a stricter classification. The proper ''n''-odd-limit consists of all intervals in the ''n''-odd-limit that are in no lower odd-limits.

			== Distinction and monotonicity ==
			Distinguishing (having different representations of) intervals in a given odd-limit is an important way in which the resolution of a tuning system is specified. The smallest gap between two intervals of the ''n''-odd-limit is most commonly the ''n''th [[square superparticular]], S(''n'') = ''n''<sup>2</sup>/(''n''<sup>2</sup> - 1) (though certain odd-limits can have a gap of size (''n''<sup>2</sup> + 1)/''n''<sup>2</sup> instead: notably 7-odd, with [[50/49]] between [[7/5]] and [[10/7]]; or 21-odd, with [[442/441]] between [[21/17]] and [[26/21]]). Therefore, for a tuning system such as an [[EDO]] to distinguish the ''n''-odd-limit, it must be able to [[emancipate]] [[comma]]s the size of S(''n''): either by having a step size close to that size, or exaggerating intervals smaller than its step size.

			Another important property having to do with representations of odd-limits is '''monotonicity''': i.e. a tuning is monotonic in the ''n''-odd-limit if, for every two intervals within the odd-limit such that one is larger than the other, the tuning's representation of the larger interval is never smaller than its representation of the smaller interval. It is generally considered a minimal requirement, for intervals as tuned in a tuning to be registered as [[dyad]]s, for them to be tuned monotonically in relation to each other (though it would still be possible for an interval like [[10/9]] to exhibit its "structural" role as ([[4/3]])/([[6/5]]) even if it is tuned sharp of [[9/8]] or flat of [[11/10]], for example). Odd-limit monotonicity, in the context of [[equal-step tunings]], is also a necessary (but not sufficient) requirement for [[consistency]] in that odd-limit. If a tuning is monotonic in the ''n''-odd-limit and also distinguishes all intervals within that set, we can say it is '''strictly monotonic''' in the ''n''-odd-limit.

	== Individual odd-limits ==		== Individual odd-limits ==