Diamond function - Revision history

Sintel: typo

2025-04-25T15:57:08Z

typo

← Older revision		Revision as of 15:57, 25 April 2025
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	The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as [[Harry Partch]] did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[11-limit]] diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.		The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as [[Harry Partch]] did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[11-limit]] diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.

	The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth. These scales are known as [[~~crystall~~ ball]]s. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale products and scale powers\|scale powers]], provides a convenient means of obtaining ''p''-limit intervals, or intervals in a desired [[Just intonation subgroups\|JI subgroup]], in abundance.		The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth. These scales are known as [[crystal ball]]s. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale products and scale powers\|scale powers]], provides a convenient means of obtaining ''p''-limit intervals, or intervals in a desired [[Just intonation subgroups\|JI subgroup]], in abundance.

	[[Category:Diamond]]		[[Category:Diamond]]

Sintel: Remove sentece about 17/16, as it occurs in 17-odd limit so that doesn't make any sense. Also remove remark about multisets, which has no clear relevance or application. Add crystal ball link.

2025-04-25T15:56:56Z

Remove sentece about 17/16, as it occurs in 17-odd limit so that doesn't make any sense. Also remove remark about multisets, which has no clear relevance or application. Add crystal ball link.

← Older revision		Revision as of 15:56, 25 April 2025
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	== Definition ==		== Definition ==
	Given a collection of pitches S, the diamond of S, D = diamond(S), is found as the set of intervals between those pitches, taking the intervals in direct and inverted form, [[octave reduction\|reduced to an octave]], and from there D can be used either as those intervals (as is the case with targeted intervals of RTT tuning scheme) or interpreted as pitches themselves (as is the case with tonality diamonds). For instance, given the pitches {1, 3, 5}, diamond({1, 3, 5}) is {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with [[just intonation]], in which case S is a set of rational numbers, but it applies to any collection; for instance diamond({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd ''n'' is called the [[tonality diamond]], and is often taken as the set of theoretical consonances in the ''n'' [[odd limit]]. This can be justified on the grounds that these are ~~just~~ the intervals appearing in the ~~"chord of nature", or~~ harmonic series~~, hence objecting~~ to ~~17/16, for example, on the grounds it isn~~'~~t actually very consonant doesn~~'~~t take account of the fact that the integers up to 17~~, ~~a "chord of nature", contain this interval.~~		Given a collection of pitches S, the diamond of S, D = diamond(S), is found as the set of intervals between those pitches, taking the intervals in direct and inverted form, [[octave reduction\|reduced to an octave]], and from there D can be used either as those intervals (as is the case with targeted intervals of RTT tuning scheme) or interpreted as pitches themselves (as is the case with tonality diamonds). For instance, given the pitches {1, 3, 5}, diamond({1, 3, 5}) is {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with [[just intonation]], in which case S is a set of rational numbers, but it applies to any collection; for instance diamond({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd ''n'' is called the [[tonality diamond]], and is often taken as the set of theoretical consonances in the ''n'' [[odd limit]]. This can be justified on the grounds that these are all of the intervals appearing in the [[harmonic series]] up to ''n'', when accounting for [[octave equivalence]].

	~~The above definition is based on sets, but it is also possible to define diamonds in terms of~~ [[~~Wikipedia: Multiset\|multisets~~]], which can lead to different results. If S is a multiset, then diamult(S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.

	== Creating scales ==		== Creating scales ==
	The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as [[Harry Partch]] did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[11-limit]] diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.		The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as [[Harry Partch]] did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[11-limit]] diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.

	The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth~~, and the same is true of diamult~~. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale products and scale powers\|scale powers]], provides a convenient means of obtaining ''p''-limit intervals, or intervals in a desired [[Just intonation subgroups\|JI subgroup]], in abundance.		The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth. These scales are known as [[crystall ball]]s. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale products and scale powers\|scale powers]], provides a convenient means of obtaining ''p''-limit intervals, or intervals in a desired [[Just intonation subgroups\|JI subgroup]], in abundance.

	[[Category:Diamond]]		[[Category:Diamond]]

Fredg999: See also not needed (link already in body, including hints of its significance)

2023-02-26T07:49:53Z

See also not needed (link already in body, including hints of its significance)

← Older revision		Revision as of 07:49, 26 February 2023
Line 10:		Line 10:

	The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth, and the same is true of diamult. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale products and scale powers\|scale powers]], provides a convenient means of obtaining ''p''-limit intervals, or intervals in a desired [[Just intonation subgroups\|JI subgroup]], in abundance.		The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth, and the same is true of diamult. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale products and scale powers\|scale powers]], provides a convenient means of obtaining ''p''-limit intervals, or intervals in a desired [[Just intonation subgroups\|JI subgroup]], in abundance.

	~~== See also ==~~
	* [[Tonality diamond]] – for musical applications and scale examples
	* [[Wikipedia: Tonality diamond]]

	[[Category:Diamond]]		[[Category:Diamond]]
	[[Category:Math]]		[[Category:Math]]
	~~[[Category:Theory]]~~

	{{todo\|improve synopsis}}		{{todo\|improve synopsis}}

Cmloegcmluin: eliminate self-contradiction in article (in first paragraph D is "notes", probably meaning "pitches", but then in the second paragraph it is "intervals", and in practice D is sometimes one and sometimes the other); including use case of target tunings; otherwise improving clarity and consistency; simplifying formatting

2022-08-25T00:44:49Z

eliminate self-contradiction in article (in first paragraph D is "notes", probably meaning "pitches", but then in the second paragraph it is "intervals", and in practice D is sometimes one and sometimes the other); including use case of target tunings; otherwise improving clarity and consistency; simplifying formatting

← Older revision		Revision as of 00:44, 25 August 2022
Line 1:		Line 1:
	The '''diamond function''' creates a set of ~~notes~~ D from a given set of ~~notes~~ S. It can be applied to generate scales or [[tonality diamond]]s.		The '''diamond function''' creates a set of intervals or pitches D from a given set of pitches S. It can be applied, for example, to generate scales, [[tonality diamond]]s, or [[Target tuning\|targeted intervals of an RTT tuning scheme]].

	== Definition ==		== Definition ==
	Given a collection of ~~notes~~ S, the ''diamond'' of S, ~~'''~~diamond~~'''~~ (S), is the set of intervals between those ~~notes~~, taking the intervals in direct and inverted form, [[octave reduction\|reduced to an octave]]. For instance, given the ~~notes~~ {1, 3, 5}, diamond ({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with [[just intonation]], in which case S is a set of rational numbers, but it applies to any collection; for instance diamond ({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd ''n'' is called the [[tonality diamond]], and is often taken as the set of theoretical consonances in the ''n'' [[odd limit]]. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or harmonic series, hence objecting to 17/16, for example, on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.		Given a collection of pitches S, the diamond of S, D = diamond(S), is found as the set of intervals between those pitches, taking the intervals in direct and inverted form, [[octave reduction\|reduced to an octave]], and from there D can be used either as those intervals (as is the case with targeted intervals of RTT tuning scheme) or interpreted as pitches themselves (as is the case with tonality diamonds). For instance, given the pitches {1, 3, 5}, diamond({1, 3, 5}) is {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with [[just intonation]], in which case S is a set of rational numbers, but it applies to any collection; for instance diamond({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd ''n'' is called the [[tonality diamond]], and is often taken as the set of theoretical consonances in the ''n'' [[odd limit]]. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or harmonic series, hence objecting to 17/16, for example, on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.

	The above definition is based on sets, but it is also possible to define diamonds in terms of [[Wikipedia: Multiset\|multisets]], which can lead to different results. If S is a multiset, then ~~'''~~diamult~~'''~~ (S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.		The above definition is based on sets, but it is also possible to define diamonds in terms of [[Wikipedia: Multiset\|multisets]], which can lead to different results. If S is a multiset, then diamult(S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.

	== Creating scales ==		== Creating scales ==
	The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as [[Harry Partch]] did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[11-limit]] diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.		The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as [[Harry Partch]] did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[11-limit]] diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.

	The diamond construction can be iterated, giving diamond (diamond (S)), diamond (diamond (diamond (S))), and so forth, and the same is true of diamult. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale products and scale powers\|scale powers]], provides a convenient means of obtaining ''p''-limit intervals, or intervals in a desired [[Just intonation subgroups\|JI subgroup]], in abundance.		The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth, and the same is true of diamult. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale products and scale powers\|scale powers]], provides a convenient means of obtaining ''p''-limit intervals, or intervals in a desired [[Just intonation subgroups\|JI subgroup]], in abundance.

	== See also ==		== See also ==

Wolftune: fixed "overtone series" vs "harmonic series"

2022-01-22T20:42:11Z

fixed "overtone series" vs "harmonic series"

← Older revision		Revision as of 20:42, 22 January 2022
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	== Definition ==		== Definition ==
	Given a collection of notes S, the ''diamond'' of S, '''diamond''' (S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave reduction\|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond ({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with [[just intonation]], in which case S is a set of rational numbers, but it applies to any collection; for instance diamond ({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd ''n'' is called the [[tonality diamond]], and is often taken as the set of theoretical consonances in the ''n'' [[odd limit]]. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or ~~overtone~~ series, hence objecting to 17/16, for example, on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.		Given a collection of notes S, the ''diamond'' of S, '''diamond''' (S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave reduction\|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond ({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with [[just intonation]], in which case S is a set of rational numbers, but it applies to any collection; for instance diamond ({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd ''n'' is called the [[tonality diamond]], and is often taken as the set of theoretical consonances in the ''n'' [[odd limit]]. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or harmonic series, hence objecting to 17/16, for example, on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.

	The above definition is based on sets, but it is also possible to define diamonds in terms of [[Wikipedia: Multiset\|multisets]], which can lead to different results. If S is a multiset, then '''diamult''' (S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.		The above definition is based on sets, but it is also possible to define diamonds in terms of [[Wikipedia: Multiset\|multisets]], which can lead to different results. If S is a multiset, then '''diamult''' (S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.

FloraC: Cleanup

2021-01-23T05:05:35Z

Cleanup

← Older revision		Revision as of 05:05, 23 January 2021
Line 1:		Line 1:
	The '''diamond function''' creates a set of notes D from a given set of notes S. It can be applied to generate scales or [[tonality diamond]]s.		The '''diamond function''' creates a set of notes D from a given set of notes S. It can be applied to generate scales or [[tonality diamond]]s.

	~~__TOC__~~

	== Definition ==		== Definition ==
	Given a collection of notes S, the ''diamond'' of S, '''diamond'''(S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave~~-reduce~~\|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers, but it applies to any collection; for instance diamond({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd n is called the [[~~Tonality_diamond\|~~tonality diamond]], and is often taken as the set of theoretical consonances in the n odd limit. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or overtone series, hence objecting to 17/16 on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.		Given a collection of notes S, the ''diamond'' of S, '''diamond''' (S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave reduction\|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond ({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with [[just intonation]], in which case S is a set of rational numbers, but it applies to any collection; for instance diamond ({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd ''n'' is called the [[tonality diamond]], and is often taken as the set of theoretical consonances in the ''n'' [[odd limit]]. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or overtone series, hence objecting to 17/16, for example, on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.

	The above definition is based on sets, but it is also possible to define diamonds in terms of [~~http~~:~~//en.wikipedia.org/wiki/~~Multiset multisets], which can lead to different results. If S is a multiset, then '''diamult'''(S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.		The above definition is based on sets, but it is also possible to define diamonds in terms of [[Wikipedia: Multiset\|multisets]], which can lead to different results. If S is a multiset, then '''diamult''' (S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.

	== Creating scales ==		== Creating scales ==
	The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as [[Harry Partch]] did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[11-limit]] diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.		The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as [[Harry Partch]] did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[11-limit]] diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.

	The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth, and the same is true of diamult. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[~~Scale_products_and_scale_powers~~\|scale powers]], provides a convenient means of obtaining p-limit		The diamond construction can be iterated, giving diamond (diamond (S)), diamond (diamond (diamond (S))), and so forth, and the same is true of diamult. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale products and scale powers\|scale powers]], provides a convenient means of obtaining ''p''-limit intervals, or intervals in a desired [[Just intonation subgroups\|JI subgroup]], in abundance.

	intervals, or intervals in a desired [[~~Just_intonation_subgroups~~\|JI subgroup]], in abundance.

	== See also ==		== See also ==
	* [[Tonality diamond]] - for musical applications and scale examples		* [[Tonality diamond]] – for musical applications and scale examples
	* [~~http~~:~~//en.wikipedia.org/wiki/Tonality_diamond~~ Tonality diamond ~~- Wikipedia~~]		* [[Wikipedia: Tonality diamond]]

	[[Category:Diamond]]		[[Category:Diamond]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Theory]]		[[Category:Theory]]
	~~[[Category:~~todo:improve synopsis]]
			{{todo\|improve synopsis}}

Spt3125 at 23:57, 1 December 2018

2018-12-01T23:57:12Z

← Older revision		Revision as of 23:57, 1 December 2018
Line 1:		Line 1:
	The '''diamond function''' creates a set of ~~fractions~~ D from a given set of ~~fractions~~ S. It can be applied to generate scales or [[tonality diamond]]s.		The '''diamond function''' creates a set of notes D from a given set of notes S. It can be applied to generate scales or [[tonality diamond]]s.

	__TOC__		__TOC__

	== Definition ==		== Definition ==
	Given a collection of notes S, the ''diamond'' of S, diamond(S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave-reduce\|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers, but it applies to any collection; for instance diamond({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd n is called the [[Tonality_diamond\|tonality diamond]], and is often taken as the set of theoretical consonances in the n odd limit. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or overtone series, hence objecting to 17/16 on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.		Given a collection of notes S, the ''diamond'' of S, '''diamond'''(S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave-reduce\|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers, but it applies to any collection; for instance diamond({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd n is called the [[Tonality_diamond\|tonality diamond]], and is often taken as the set of theoretical consonances in the n odd limit. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or overtone series, hence objecting to 17/16 on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.

	The above definition is based on sets, but it is also possible to define diamonds in terms of [http://en.wikipedia.org/wiki/Multiset multisets], which can lead to different results. If S is a multiset, then diamult(S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.		The above definition is based on sets, but it is also possible to define diamonds in terms of [http://en.wikipedia.org/wiki/Multiset multisets], which can lead to different results. If S is a multiset, then '''diamult'''(S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.

	== Creating scales ==		== Creating scales ==
	The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as Harry Partch did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[~~11-limit\|~~11-limit]] diamond, ~~[[Harry_Partch\|~~Partch]] was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.		The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as [[Harry Partch]] did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the [[11-limit]] diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.

	The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth, and the same is true of diamult. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale_products_and_scale_powers\|scale powers]], provides a convenient means of obtaining p-limit		The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth, and the same is true of diamult. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its [[Scale_products_and_scale_powers\|scale powers]], provides a convenient means of obtaining p-limit
Line 14:		Line 16:

	== See also ==		== See also ==
	* [[Tonality diamond]] - for musical applications ~~ans~~ scale examples		* [[Tonality diamond]] - for musical applications and scale examples
	* [http://en.wikipedia.org/wiki/Tonality_diamond Tonality diamond -- Wikipedia]		* [http://en.wikipedia.org/wiki/Tonality_diamond Tonality diamond - Wikipedia]

	[[Category:Diamond]]		[[Category:Diamond]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Theory]]		[[Category:Theory]]

	[[Category:todo:improve synopsis]]		[[Category:todo:improve synopsis]]

Xenwolf: +toc

2018-10-25T11:54:41Z

+toc

← Older revision		Revision as of 11:54, 25 October 2018
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	The '''diamond function''' creates a set of fractions D from a given set of fractions S. It can be applied to generate scales or [[tonality diamond]]s.		The '''diamond function''' creates a set of fractions D from a given set of fractions S. It can be applied to generate scales or [[tonality diamond]]s.
			__TOC__
	== Definition ==		== Definition ==
	Given a collection of notes S, the ''diamond'' of S, diamond(S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave-reduce\|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers, but it applies to any collection; for instance diamond({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd n is called the [[Tonality_diamond\|tonality diamond]], and is often taken as the set of theoretical consonances in the n odd limit. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or overtone series, hence objecting to 17/16 on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.		Given a collection of notes S, the ''diamond'' of S, diamond(S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave-reduce\|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers, but it applies to any collection; for instance diamond({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd n is called the [[Tonality_diamond\|tonality diamond]], and is often taken as the set of theoretical consonances in the n odd limit. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or overtone series, hence objecting to 17/16 on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.

Xenwolf: exported parts to Tonality diamond; tried to improve the synopsis

2018-10-25T11:53:47Z

exported parts to Tonality diamond; tried to improve the synopsis

← Older revision		Revision as of 11:53, 25 October 2018
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	~~A Diamond is~~ a ~~tonal system that is generated in~~ a ~~specific way from some~~ given ~~notes~~.		The '''diamond function''' creates a set of fractions D from a given set of fractions S. It can be applied to generate scales or [[tonality diamond]]s.

	== Definition ==		== Definition ==
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	intervals, or intervals in a desired [[Just_intonation_subgroups\|JI subgroup]], in abundance.		intervals, or intervals in a desired [[Just_intonation_subgroups\|JI subgroup]], in abundance.

	~~== Examples of scales ==~~
	* [[diamond5]]
	* [[diamond7]]
	* [[diamond9]]
	* [[diamond11]]
	* [[diamond13]]
	* [[diamond15]]
	* [[diamond9plus-marvel]]

	~~== Music ==~~
	* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Oldani/GWS%20Scale%20Study-ModernJazzAtTheCrystalBall%20.mp3 Modern Jazz at the Crystal Ball] by Norbert Oldani in the [[7-limit diamond]].

	== See also ==		== See also ==
	* [[Tonality diamond]] - ~~related, todo~~		* [[Tonality diamond]] - for musical applications ans scale examples
	* [http://en.wikipedia.org/wiki/Tonality_diamond Tonality diamond -- Wikipedia]		* [http://en.wikipedia.org/wiki/Tonality_diamond Tonality diamond -- Wikipedia]

Xenwolf: Xenwolf moved page Diamonds to Diamond function: better lemma for separating math from music

2018-10-25T11:44:05Z

Xenwolf moved page Diamonds to Diamond function: better lemma for separating math from music

← Older revision	Revision as of 11:44, 25 October 2018
(No difference)