Complexity - Revision history

FloraC: /* Complexity of a temperament */ - badness (moved to the dedicated article)

2025-11-13T10:06:51Z

Complexity of a temperament: - badness (moved to the dedicated article)

← Older revision		Revision as of 10:06, 13 November 2025
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	A commonly used temperament complexity measure is [[Tenney–Euclidean temperament measures #TE complexity\|Tenney–Euclidean complexity]], which works nicely for multirank temperaments and equal temperaments alike.		A commonly used temperament complexity measure is [[Tenney–Euclidean temperament measures #TE complexity\|Tenney–Euclidean complexity]], which works nicely for multirank temperaments and equal temperaments alike.

	For an [[equal temperament]], a simpler definition of the complexity is the number of notes per octave, which means that [[12edo\|12et]] has a complexity of 12, etc. For unusual mappings where 2 is mapped to a strange number of steps, that does not work. Norm-based complexities such as TE complexity are foolproof and equave-agnostic, however. For example, the TE complexity of 31et is 30.98, which is close to the edo number as expected for a patent val. But if one were to take the TE complexity of {{val\| 1 1900 2785 3370 }}, which is technically a tuning of 1et, they would get 1038.83, which matches the complexity of the tuning much better than the naive approach of simply taking 1 for the complexity, and means that that val is roughly equivalent to 1039et in complexity.		For an [[equal temperament]], a simpler definition of the complexity is the number of notes per octave, which means that [[12edo\|12et]] has a complexity of 12, etc. For unusual mappings where 2 is mapped to a strange number of steps, that does not work. Norm-based complexities such as TE complexity are foolproof and equave-agnostic, however. For example, the TE complexity of 31et is 30.98, which is close to the edo number as expected for a patent val. But if one were to take the TE complexity of {{val\| 1 1900 2785 3370 }}, which is technically a tuning of 1et, they would get 1038.83, which matches the complexity of the tuning much better than the naive approach of simply taking 1 for the complexity, and means that that val is roughly equivalent to 1039et in complexity.

	Complexity and [[error]] are both usually treated as undesirable characteristics, but there is a trade-off between them in that very low complexity temperaments (e.g. small [[edo\|edos]]) typically do not have low error, and very low error temperaments (e.g. [[microtemperament\|microtemperaments]]) typically do not have low complexity. [[Badness]] is a way to combine complexity and error such that a search for low-badness temperaments yields results with a particularly good trade-off between complexity and error.

	== Links ==		== Links ==

FloraC: Complexity of a just interval and complexity of a tempered interval should always go together

2025-11-13T10:02:05Z

Complexity of a just interval and complexity of a tempered interval should always go together

← Older revision		Revision as of 10:02, 13 November 2025
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	When a complexity measures is defined in terms of a vector space, it is usually called a '''norm'''.		When a complexity measures is defined in terms of a vector space, it is usually called a '''norm'''.

	== Complexity of just ~~intervals~~ ==		== Complexity of a just interval ==
	{{~~main~~\|~~height~~}}		{{Main\| Height }}

	The complexity of a just interval is often measured using height functions.		The complexity of a just interval is often measured using height functions.
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	It is usually true that simpler (i.e. less complex) JI intervals are more consonant, however the converse does not hold.		It is usually true that simpler (i.e. less complex) JI intervals are more consonant, however the converse does not hold.
	Examples of this are easy to find. Consider for example an interval such as 3001/2001, which is very complex but still sounds consonant due to its proximity to [[3/2]].		Examples of this are easy to find. Consider for example an interval such as 3001/2001, which is very complex but still sounds consonant due to its proximity to [[3/2]].

			== Complexity of an interval in a temperament ==
			Besides saying that an interval has a high or low complexity, we also speak of the complexity of an interval ''in a temperament''. If an interval has a low complexity in a certain temperament, that means it can be reached in only a few [[generator]]s, so it is likely to appear frequently in scales of that temperament. For example, in [[meantone]] temperament, the generator represents 3/2, so clearly 3/2 has a very low complexity, since it can be reached in only one generator. In contrast, 45/32 can only be reached in 6 generators so it has a higher complexity and will tend to appear much less frequently in meantone scales.

			An example of temperament interval complexity is the [[Tenney–Euclidean metrics #TE temperamental norm\|Tenney–Euclidean temperamental norm]].

			The ''complexity of a chord'' likewise refers to the number of generator steps required to generate all the pitches of the chord.
			For an example of this, see [[Graham complexity]].

			Note that the concept of complexity applies not only to [[rank-2 temperament]]s, but temperaments of any rank. For higher-rank temperaments, the lattice is a higher-dimensional space, so there could be different ways of measuring the area/volume/etc. that a chord takes up.

	== Complexity of a temperament ==		== Complexity of a temperament ==
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	Complexity and [[error]] are both usually treated as undesirable characteristics, but there is a trade-off between them in that very low complexity temperaments (e.g. small [[edo\|edos]]) typically do not have low error, and very low error temperaments (e.g. [[microtemperament\|microtemperaments]]) typically do not have low complexity. [[Badness]] is a way to combine complexity and error such that a search for low-badness temperaments yields results with a particularly good trade-off between complexity and error.		Complexity and [[error]] are both usually treated as undesirable characteristics, but there is a trade-off between them in that very low complexity temperaments (e.g. small [[edo\|edos]]) typically do not have low error, and very low error temperaments (e.g. [[microtemperament\|microtemperaments]]) typically do not have low complexity. [[Badness]] is a way to combine complexity and error such that a search for low-badness temperaments yields results with a particularly good trade-off between complexity and error.

	== ~~Complexity of an interval in a temperament~~ ==		== Links ==
	Besides saying that a temperament has a high or low complexity, we also speak of the ''complexity of an interval'' in a temperament. If an interval has a low complexity in a certain temperament, that means it can be reached in only a few [~~[generator]]s, so it is likely to appear frequently in scales of that temperament. For example, in [[meantone]] temperament, the generator represents 3~~/~~2, so clearly 3~~/~~2 has a very low complexity, since it can be reached in only one generator~~. ~~In contrast, 45~~/~~32 can only be reached in 6 generators so it has a higher complexity and will tend to appear much less frequently in meantone scales~~.		* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19636.html Yahoo! Tuning Group \| ''Complexity terminology wars'']

	~~An example of temperament interval complexity is the [[Tenney-Euclidean_metrics #TE temperamental norm~~\|~~Tenney-Euclidean temperamental norm]].~~

	~~The~~ ''~~complexity of a chord~~'' ~~likewise refers to the number of generator steps required to generate all the pitches of the chord.~~
	~~For an example of this, see [[Graham complexity]].~~

	~~Note that the concept of complexity applies not only to [[rank-2 temperament~~]~~]s, but temperaments of any rank. For higher-rank temperaments, the lattice is a higher-dimensional space, so there could be different ways of measuring the area/volume/etc. that a chord takes up.~~

	[[Category:Complexity\| ]] <!-- Main article -->		[[Category:Complexity\| ]] <!-- Main article -->

	~~== Links ==~~
	* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19636.html Yahoo! Tuning Group \| ''Complexity terminology wars'']

FloraC: /* Complexity of a temperament */ move some of Vector's stuff here

2025-11-13T07:58:06Z

Complexity of a temperament: move some of Vector's stuff here

← Older revision		Revision as of 07:58, 13 November 2025
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	Being a characteristic of [[temperament]]s, complexity can be used to evaluate and compare them. Generally speaking, if a temperament has high complexity, that means that interesting pitches (e.g. ones [[consonant]] with each other) are many [[generator]]s apart, so useful scales tend to have many notes. If a temperament has low complexity, fewer generators are required, and scales with fewer notes are more likely to be useful.		Being a characteristic of [[temperament]]s, complexity can be used to evaluate and compare them. Generally speaking, if a temperament has high complexity, that means that interesting pitches (e.g. ones [[consonant]] with each other) are many [[generator]]s apart, so useful scales tend to have many notes. If a temperament has low complexity, fewer generators are required, and scales with fewer notes are more likely to be useful.

	For an [[equal temperament]], a ~~simple~~ definition of the complexity is the number of notes per octave~~. Which~~ means that [[12edo]] has a complexity of 12, etc. ~~This notion can be generalized~~ to ~~temperaments~~ of ~~higher rank~~.		A commonly used temperament complexity measure is [[Tenney–Euclidean temperament measures #TE complexity\|Tenney–Euclidean complexity]], which works nicely for multirank temperaments and equal temperaments alike.

			For an [[equal temperament]], a simpler definition of the complexity is the number of notes per octave, which means that [[12edo\|12et]] has a complexity of 12, etc. For unusual mappings where 2 is mapped to a strange number of steps, that does not work. Norm-based complexities such as TE complexity are foolproof and equave-agnostic, however. For example, the TE complexity of 31et is 30.98, which is close to the edo number as expected for a patent val. But if one were to take the TE complexity of {{val\| 1 1900 2785 3370 }}, which is technically a tuning of 1et, they would get 1038.83, which matches the complexity of the tuning much better than the naive approach of simply taking 1 for the complexity, and means that that val is roughly equivalent to 1039et in complexity.

	Complexity and [[error]] are both usually treated as undesirable characteristics, but there is a trade-off between them in that very low complexity temperaments (e.g. small [[edo\|edos]]) typically do not have low error, and very low error temperaments (e.g. [[microtemperament\|microtemperaments]]) typically do not have low complexity. [[Badness]] is a way to combine complexity and error such that a search for low-badness temperaments yields results with a particularly good trade-off between complexity and error.		Complexity and [[error]] are both usually treated as undesirable characteristics, but there is a trade-off between them in that very low complexity temperaments (e.g. small [[edo\|edos]]) typically do not have low error, and very low error temperaments (e.g. [[microtemperament\|microtemperaments]]) typically do not have low complexity. [[Badness]] is a way to combine complexity and error such that a search for low-badness temperaments yields results with a particularly good trade-off between complexity and error.

	~~A commonly used definition of temperament complexity is [[Tenney-Euclidean temperament_measures #TE complexity\|Tenney-Euclidean complexity]].~~

	== Complexity of an interval in a temperament ==		== Complexity of an interval in a temperament ==

Sintel: /* Complexity of just intervals */ Rewording relation to height functions. + add example

2025-04-26T13:23:35Z

Complexity of just intervals: Rewording relation to height functions. + add example

← Older revision		Revision as of 13:23, 26 April 2025
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	== Complexity of just intervals ==		== Complexity of just intervals ==
	~~:''Main article: [[~~height~~]]''~~		{{main\|height}}

	The complexity of a just interval is often ~~called the '''~~height~~'''~~.		The complexity of a just interval is often measured using height functions.
	~~There are various measures of complexity for rational intervals.~~ Generally these can be thought of as measuring the size of the numerator and denominator when expressed in lowest terms.		Generally these can be thought of as measuring the size of the numerator and denominator when expressed in lowest terms.
			A simple example of a height function is the [[Weil height]] (or [[integer limit]]), which is simply the maximum of the numerator and denominator of the ratio.

	~~Specific examples~~ of ~~these~~ are [[Benedetti height]], [[Tenney height]], [[Wilson height]] and the [[Tenney-Euclidean metrics#TE_norm\|Tenney-Euclidean norm]].		There are various measures of complexity for rational intervals.
			Commonly used are [[Benedetti height]], [[Tenney height]], [[Wilson height]] and the [[Tenney-Euclidean metrics#TE_norm\|Tenney-Euclidean norm]].

	=== Relationship to consonance ===		=== Relationship to consonance ===

Sintel: /* Complexity of just intervals */ typo

2025-04-24T10:28:06Z

Complexity of just intervals: typo

← Older revision		Revision as of 10:28, 24 April 2025
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	The complexity of a just interval is often called the '''height'''.		The complexity of a just interval is often called the '''height'''.
	There are various measures of complexity for rational intervals. Generally these can be ~~tought~~ of as measuring the size of the numerator and denominator when expressed in lowest terms.		There are various measures of complexity for rational intervals. Generally these can be thought of as measuring the size of the numerator and denominator when expressed in lowest terms.

	Specific examples of these are [[Benedetti height]], [[Tenney height]], [[Wilson height]] and the [[Tenney-Euclidean metrics#TE_norm\|Tenney-Euclidean norm]].		Specific examples of these are [[Benedetti height]], [[Tenney height]], [[Wilson height]] and the [[Tenney-Euclidean metrics#TE_norm\|Tenney-Euclidean norm]].

Sintel: typo

2025-04-19T17:14:03Z

typo

← Older revision		Revision as of 17:14, 19 April 2025
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	When a complexity measures is defined in terms of a vector space, it is usually called a '''norm'''.		When a complexity measures is defined in terms of a vector space, it is usually called a '''norm'''.

	== ~~Comexity~~ of just intervals ==		== Complexity of just intervals ==
	:''Main article: [[height]]''		:''Main article: [[height]]''

Sintel: Some cleanup and extra definitions/terminology

2025-04-19T14:48:13Z

Some cleanup and extra definitions/terminology

← Older revision		Revision as of 14:48, 19 April 2025
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	In tuning, '''complexity''' can be said with respect to individual [[interval]]s, [[chord]]s, [[scale]]s as well as the entire [[tuning system]]. While mathematically rigorous measurements of complexity are not available for all contexts and purposes, some of them have been extensively studied, including those of [[regular temperament]]s and of just or tempered [[interval]]s.		In tuning, '''complexity''' can be said with respect to individual [[interval]]s, [[chord]]s, [[scale]]s as well as the entire [[tuning system]]. While mathematically rigorous measurements of complexity are not available for all contexts and purposes, some of them have been extensively studied, including those of [[regular temperament]]s and of just or tempered [[interval]]s.

			When a complexity measures is defined in terms of a vector space, it is usually called a '''norm'''.

			== Comexity of just intervals ==
			:''Main article: [[height]]''

			The complexity of a just interval is often called the '''height'''.
			There are various measures of complexity for rational intervals. Generally these can be tought of as measuring the size of the numerator and denominator when expressed in lowest terms.

			Specific examples of these are [[Benedetti height]], [[Tenney height]], [[Wilson height]] and the [[Tenney-Euclidean metrics#TE_norm\|Tenney-Euclidean norm]].

			=== Relationship to consonance ===
			It is usually true that simpler (i.e. less complex) JI intervals are more consonant, however the converse does not hold.
			Examples of this are easy to find. Consider for example an interval such as 3001/2001, which is very complex but still sounds consonant due to its proximity to [[3/2]].

	== Complexity of a temperament ==		== Complexity of a temperament ==
	Being a characteristic of [[temperament]]s, complexity can be used to evaluate and compare them. Generally speaking, if a temperament has high complexity, that means that interesting pitches (e.g. ones [[consonant]] with each other) are many [[generator]]s apart, so useful scales tend to have many notes. If a temperament has low complexity, fewer generators are required, and scales with fewer notes are more likely to be useful.		Being a characteristic of [[temperament]]s, complexity can be used to evaluate and compare them. Generally speaking, if a temperament has high complexity, that means that interesting pitches (e.g. ones [[consonant]] with each other) are many [[generator]]s apart, so useful scales tend to have many notes. If a temperament has low complexity, fewer generators are required, and scales with fewer notes are more likely to be useful.

	Complexity and [[error]] are both usually treated as undesirable characteristics, but there is a trade-off between them in that very low complexity temperaments (e.g. small [[edo]]s) typically do not have low error, and very low error temperaments (e.g. [[microtemperament]]~~s or [[JI]] itself~~) typically do not have low complexity. [[Badness]] is a way to combine complexity and error such that a search for low-badness temperaments yields results with a particularly good trade-off between complexity and error.		For an [[equal temperament]], a simple definition of the complexity is the number of notes per octave. Which means that [[12edo]] has a complexity of 12, etc. This notion can be generalized to temperaments of higher rank.

			Complexity and [[error]] are both usually treated as undesirable characteristics, but there is a trade-off between them in that very low complexity temperaments (e.g. small [[edo\|edos]]) typically do not have low error, and very low error temperaments (e.g. [[microtemperament\|microtemperaments]]) typically do not have low complexity. [[Badness]] is a way to combine complexity and error such that a search for low-badness temperaments yields results with a particularly good trade-off between complexity and error.

			A commonly used definition of temperament complexity is [[Tenney-Euclidean temperament_measures #TE complexity\|Tenney-Euclidean complexity]].

	== Complexity of an interval in a temperament ==		== Complexity of an interval in a temperament ==
	Besides saying that a temperament has a high or low complexity, we also speak of the ''complexity of an interval'' in a temperament. If an interval has a low complexity in a certain temperament, that means it can be reached in only a few [[generator]]s, so it is likely to appear frequently in scales of that temperament. For example, in [[meantone]] temperament, the generator represents 3/2, so clearly 3/2 has a very low complexity, since it can be reached in only one generator. In contrast, 45/32 can only be reached in 6 generators so it has a higher complexity and will tend to appear much less frequently in meantone scales.		Besides saying that a temperament has a high or low complexity, we also speak of the ''complexity of an interval'' in a temperament. If an interval has a low complexity in a certain temperament, that means it can be reached in only a few [[generator]]s, so it is likely to appear frequently in scales of that temperament. For example, in [[meantone]] temperament, the generator represents 3/2, so clearly 3/2 has a very low complexity, since it can be reached in only one generator. In contrast, 45/32 can only be reached in 6 generators so it has a higher complexity and will tend to appear much less frequently in meantone scales.

			An example of temperament interval complexity is the [[Tenney-Euclidean_metrics #TE temperamental norm\|Tenney-Euclidean temperamental norm]].

	The ''complexity of a chord'' likewise refers to the number of generator steps required to generate all the pitches of the chord.		The ''complexity of a chord'' likewise refers to the number of generator steps required to generate all the pitches of the chord.
			For an example of this, see [[Graham complexity]].

	Note that the concept of complexity applies not only to [[rank-2 temperament]]s, but temperaments of any rank. For higher-rank temperaments, the lattice is a higher-dimensional space, so there could be different ways of measuring the area/volume/etc. that a chord takes up.		Note that the concept of complexity applies not only to [[rank-2 temperament]]s, but temperaments of any rank. For higher-rank temperaments, the lattice is a higher-dimensional space, so there could be different ways of measuring the area/volume/etc. that a chord takes up.

	~~== Complexity measures ==~~
	~~Some specific, mathematically rigorous definitions of "complexity" are…~~

	* [[Graham complexity]] (chord complexity)
	* [[Generator complexity]] (temperament complexity)
	* [[Tenney-Euclidean temperament_measures #TE complexity\|TE complexity]] (temperament complexity)
	* Various [[height]] functions (interval complexity in JI)
	* [[Tenney-Euclidean_metrics #TE temperamental norm\|Interval temperamental complexity]] (interval complexity in temperaments)

	[[Category:Complexity\| ]] <!-- Main article -->		[[Category:Complexity\| ]] <!-- Main article -->

Sintel: irrelevant wikipedia page

2025-04-19T14:07:31Z

irrelevant wikipedia page

← Older revision		Revision as of 14:07, 19 April 2025
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	~~{{Wikipedia\| Complexity }}~~
	In tuning, '''complexity''' can be said with respect to individual [[interval]]s, [[chord]]s, [[scale]]s as well as the entire [[tuning system]]. While mathematically rigorous measurements of complexity are not available for all contexts and purposes, some of them have been extensively studied, including those of [[regular temperament]]s and of just or tempered [[interval]]s.		In tuning, '''complexity''' can be said with respect to individual [[interval]]s, [[chord]]s, [[scale]]s as well as the entire [[tuning system]]. While mathematically rigorous measurements of complexity are not available for all contexts and purposes, some of them have been extensively studied, including those of [[regular temperament]]s and of just or tempered [[interval]]s.

Sintel: Remove quote from top of article

2025-04-15T13:39:26Z

Remove quote from top of article

← Older revision		Revision as of 13:39, 15 April 2025
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	~~<blockquote>~~		{{Wikipedia\| Complexity }}
	~~In [[amity]], all of the lower-complexity intervals are more complex.<ref group="note">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19636.html Yahoo! Tuning Group~~ \| ''Complexity ~~terminology wars'']</ref>~~
	~~</blockquote>~~

	In tuning, '''complexity''' can be said with respect to individual [[interval]]s, [[chord]]s, [[scale]]s as well as the entire [[tuning system]]. While mathematically rigorous measurements of complexity are not available for all contexts and purposes, some of them have been extensively studied, including those of [[regular temperament]]s and of just or tempered [[interval]]s.		In tuning, '''complexity''' can be said with respect to individual [[interval]]s, [[chord]]s, [[scale]]s as well as the entire [[tuning system]]. While mathematically rigorous measurements of complexity are not available for all contexts and purposes, some of them have been extensively studied, including those of [[regular temperament]]s and of just or tempered [[interval]]s.

	~~{{Wikipedia\| Complexity }}~~

	== Complexity of a temperament ==		== Complexity of a temperament ==
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	* [[Tenney-Euclidean_metrics #TE temperamental norm\|Interval temperamental complexity]] (interval complexity in temperaments)		* [[Tenney-Euclidean_metrics #TE temperamental norm\|Interval temperamental complexity]] (interval complexity in temperaments)

	~~== Notes ==~~		[[Category:Complexity\| ]] <!-- Main article -->
	<~~references group="note" /~~>

	[~~[Category~~:Complexity\| ]~~] <!-- Main article -->~~		== Links ==
			* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19636.html Yahoo! Tuning Group \| ''Complexity terminology wars'']

Roeesi: /* Complexity measures */ Corrected a link to a non-existing paragraph

2024-12-03T18:42:47Z

Complexity measures: Corrected a link to a non-existing paragraph

← Older revision		Revision as of 18:42, 3 December 2024
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	* [[Tenney-Euclidean temperament_measures #TE complexity\|TE complexity]] (temperament complexity)		* [[Tenney-Euclidean temperament_measures #TE complexity\|TE complexity]] (temperament complexity)
	* Various [[height]] functions (interval complexity in JI)		* Various [[height]] functions (interval complexity in JI)
	* [[Tenney-Euclidean_metrics #~~Temperamental complexity~~\|Interval temperamental complexity]] (interval complexity in temperaments)		* [[Tenney-Euclidean_metrics #TE temperamental norm\|Interval temperamental complexity]] (interval complexity in temperaments)

	== Notes ==		== Notes ==