Tetrachord - Revision history

ArrowHead294 at 13:30, 30 January 2026

2026-01-30T13:30:19Z

Overthink: /* Dividing the perfect fourth into more parts */ mention tetrachords in section

2026-01-15T23:26:06Z

Dividing the perfect fourth into more parts: mention tetrachords in section

← Older revision		Revision as of 23:26, 15 January 2026
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	== Dividing the perfect fourth into more parts ==		== Dividing the perfect fourth into more parts ==
	A scale segment spanning a perfect fourth ~~can also contain~~ more than 4 notes~~; for example, a~~ segment with 5 notes is called a '''pentachord''', a segment with 6 notes a '''hexachord''', etc. ~~These are useful in larger scales, such as the decatonic scales of [[pajara]] temperament. In particular~~, the ''pentachordal'' decatonic scale contains two equal pentachords and a 9/8 interval split into two semitones.		While tetrachords are useful for classifying heptatonic scales, in larger scales, such as the decatonic scales of [[pajara]] temperament, a scale segment spanning a perfect fourth often contains more than 4 notes. A segment with 5 notes is called a '''pentachord''', a segment with 6 notes a '''hexachord''', etc.

			For example, the ''pentachordal'' pajara decatonic scale contains two equal pentachords and a 9/8 interval split into two semitones.

	== Dividing other-than-perfect fourths ==		== Dividing other-than-perfect fourths ==

Overthink: Add section on larger divisions

2026-01-15T23:21:25Z

Add section on larger divisions

← Older revision		Revision as of 23:21, 15 January 2026
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	If you'd like to wrestle some tetrachords out of an equal temperament, please consider making a page for them and linking to that page from here!		If you'd like to wrestle some tetrachords out of an equal temperament, please consider making a page for them and linking to that page from here!

			== Dividing the perfect fourth into more parts ==
			A scale segment spanning a perfect fourth can also contain more than 4 notes; for example, a segment with 5 notes is called a '''pentachord''', a segment with 6 notes a '''hexachord''', etc. These are useful in larger scales, such as the decatonic scales of [[pajara]] temperament. In particular, the ''pentachordal'' decatonic scale contains two equal pentachords and a 9/8 interval split into two semitones.

	== Dividing other-than-perfect fourths ==		== Dividing other-than-perfect fourths ==

Dummy index: /* Ptolemy's catalog */ fix cent value order

2025-11-07T16:34:36Z

Ptolemy's catalog: fix cent value order

← Older revision		Revision as of 16:34, 7 November 2025
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	\|-		\|-
	\| 16/15, 9/8, 10/9		\| 16/15, 9/8, 10/9
	\| 112 + 182 ~~+ 204~~		\| 112 + 204 + 182
	\| intense diatonic		\| intense diatonic
	\|-		\|-

FloraC: +interwiki

2024-12-18T08:21:25Z

+interwiki

← Older revision		Revision as of 08:21, 18 December 2024
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			{{Interwiki
			\| en = Tetrachord
			\| ja = テトラコード
			}}
	{{Distinguish\| Tetrad }}		{{Distinguish\| Tetrad }}
	{{Wikipedia}}		{{Wikipedia}}

FloraC: Rework the intro; + category

2024-11-15T10:00:57Z

Rework the intro; + category

← Older revision		Revision as of 10:00, 15 November 2024
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	{{Wikipedia}}		{{Wikipedia}}

	~~The word~~ '''tetrachord''' usually ~~refers to~~ the interval of a [[perfect fourth]] (~~just or not~~) ~~divided~~ into three subintervals by the interposition of two additional notes.		A '''tetrachord''' is a four-note segment of a [[scale]] or tone row, usually spanning the interval of a [[perfect fourth]] (possibly tempered). It can be formed by dividing the perfect fourth into three subintervals by the interposition of two additional notes.

	[[John Chalmers]], in [~~http~~://eamusic.dartmouth.edu/~~%7Elarry~~/published_articles/divisions_of_the_tetrachord/index.html Divisions of the Tetrachord], tells us:		Tetrachords are fundamental to many musical traditions around the world. [[John Chalmers]], in [https://eamusic.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/index.html ''Divisions of the Tetrachord''], tells us:

	''Tetrachords are modules from which more complex scalar and harmonic structures may be built. These structures range from the simple heptatonic scales known to the classical civilizations of the eastern Mediterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the world's music, including that of Europe, the [[Arabic, Turkish, Persian\|Near East]], the Catholic and Orthodox churches, Iran and India, are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the world's music.''		<blockquote>
			Tetrachords are modules from which more complex scalar and harmonic structures may be built. These structures range from the simple heptatonic scales known to the classical civilizations of the eastern Mediterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the world's music, including that of Europe, the [[Arabic, Turkish, Persian\|Near East]], the Catholic and Orthodox churches, Iran and India, are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the world's music.
			</blockquote>

	== Ancient Greek genera ==		== Ancient Greek genera ==
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	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Terms]]		[[Category:Terms]]
			[[Category:Ancient Greek music]]
	[[Category:Arabic music]]		[[Category:Arabic music]]
	[[Category:Historical]]		[[Category:Historical]]

BudjarnLambeth at 03:15, 8 November 2024

2024-11-08T03:15:35Z

← Older revision		Revision as of 03:15, 8 November 2024
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	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Arabic]]		[[Category:Arabic music]]
	[[Category:Historical]]		[[Category:Historical]]

BudjarnLambeth: Categories

2024-11-08T03:15:09Z

BudjarnLambeth at 23:41, 18 September 2024

2024-09-18T23:41:17Z

← Older revision		Revision as of 23:41, 18 September 2024
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	== Ancient Greek genera ==		== Ancient Greek genera ==
	The ancient Greeks distinguished between three primary genera depending on the size of the largest interval, the ''characteristic interval'' (CI): the enharmonic, chromatic, and diatonic. Modern theorists have added a fourth genera, called hyperenharmonic.		The ancient Greeks distinguished between three primary [[genera]] depending on the size of the largest interval, the ''characteristic interval'' (CI): the enharmonic, chromatic, and diatonic. Modern theorists have added a fourth genera, called hyperenharmonic.

	; hyperenharmonic genus: The CI is larger than 425 cents.		; hyperenharmonic genus: The CI is larger than 425 cents.

Fredg999: "Jins" now has a dedicated page, sentence case in headings, misc. edits

2023-12-21T02:07:50Z

"Jins" now has a dedicated page, sentence case in headings, misc. edits

← Older revision		Revision as of 02:07, 21 December 2023
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	{{Distinguish\| Tetrad }}		{{Distinguish\| Tetrad }}
	{{Wikipedia~~\| Tetrachord~~ }}		{{Wikipedia}}

	The word '''tetrachord''' usually refers to the interval of a [[perfect fourth]] (just or not) divided into three subintervals by the interposition of two additional notes.		The word '''tetrachord''' usually refers to the interval of a [[perfect fourth]] (just or not) divided into three subintervals by the interposition of two additional notes.
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	''Tetrachords are modules from which more complex scalar and harmonic structures may be built. These structures range from the simple heptatonic scales known to the classical civilizations of the eastern Mediterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the world's music, including that of Europe, the [[Arabic, Turkish, Persian\|Near East]], the Catholic and Orthodox churches, Iran and India, are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the world's music.''		''Tetrachords are modules from which more complex scalar and harmonic structures may be built. These structures range from the simple heptatonic scales known to the classical civilizations of the eastern Mediterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the world's music, including that of Europe, the [[Arabic, Turkish, Persian\|Near East]], the Catholic and Orthodox churches, Iran and India, are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the world's music.''

	== Ancient Greek ~~Genera~~ ==		== Ancient Greek genera ==

	The ancient Greeks distinguished between three primary genera depending on the size of the largest interval, the ''characteristic interval'' (CI): the enharmonic, chromatic, and diatonic. Modern theorists have added a fourth genera, called hyperenharmonic.		The ancient Greeks distinguished between three primary genera depending on the size of the largest interval, the ''characteristic interval'' (CI): the enharmonic, chromatic, and diatonic. Modern theorists have added a fourth genera, called hyperenharmonic.

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	; diatonic genus: The CI (and the other intervals) approximates a "[[tone]]", measuring less than 250 cents.		; diatonic genus: The CI (and the other intervals) approximates a "[[tone]]", measuring less than 250 cents.

	=== Ptolemy's ~~Catalog~~ ===		=== Ptolemy's catalog ===

	In the ''Harmonics'', Ptolemy catalogs several historical tetrachords and attributes them to particular theorists.		In the ''Harmonics'', Ptolemy catalogs several historical tetrachords and attributes them to particular theorists.

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	\|}		\|}

	=== Superparticular ~~Intervals~~ ===		=== Superparticular intervals ===

	In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are [[superparticular]].		In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are [[superparticular]].

	== ~~Ajnas~~ (tetrachords in middle-eastern music) ==		== Jins/ajnas (tetrachords in middle-eastern music) ==
	{{~~Wikipedia~~\|Jins}}		{{Main\|Jins}}
	~~The~~ concept of the tetrachord ~~is extensively used~~ in [[Arabic, Turkish, Persian\|~~middle eastern~~]] ~~music theory. The Arabic word for tetrachord is "~~jins" (~~singular form~~) or ~~"ajnas" (plural form).~~		A concept similar to the tetrachord exists in [[Arabic, Turkish, Persian music\|Arabic music theory]]: a [[jins]] (pl. ajnas) is a set of three, four or five stepwise pitches used to build an Arabic [[maqam]].

	~~See ''~~[~~http://www.maqamworld.com/en/jins.php Arabic Jins~~]~~'' on maqamworld.com for details~~.

	~~== Tetrachords Generalized ==~~

			== Generalized tetrachords ==
	All tetrachords share the interval of a perfect fourth, but they vary in the other two intervals. Assuming a just fourth, we can name the two variable intervals ''a'' & ''b'', & then write our generalized tetrachord like this:		All tetrachords share the interval of a perfect fourth, but they vary in the other two intervals. Assuming a just fourth, we can name the two variable intervals ''a'' & ''b'', & then write our generalized tetrachord like this:

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	=== Modes of a [tetrachord], 9/8, [tetrachord] scale ===		=== Modes of a [tetrachord], 9/8, [tetrachord] scale ===

	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
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	=== Tetrachord rotations ===		=== Tetrachord rotations ===

	If you think of a tetrachord as three steps which total to a perfect fourth, then it makes sense that we can put those steps in any order. If we have a tetrachord with three step sizes, s, M, and L, then we have six rotations:		If you think of a tetrachord as three steps which total to a perfect fourth, then it makes sense that we can put those steps in any order. If we have a tetrachord with three step sizes, s, M, and L, then we have six rotations:

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	== Tetrachords in equal temperaments ==		== Tetrachords in equal temperaments ==

	Naturally, any equally divided scale which contains an approximation of 4/3 will have its own family of tetrachords, starting with [[7edo]], which has one tetrachord:		Naturally, any equally divided scale which contains an approximation of 4/3 will have its own family of tetrachords, starting with [[7edo]], which has one tetrachord:

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	=== Tetrachords of [[10edo]] ===		=== Tetrachords of [[10edo]] ===

	Another example: 10edo has an interval that can function as a perfect fourth at 4 degrees, measuring 480 cents. It can thus be divided into any arrangement of two 1-degree steps and one 2-degree step:		Another example: 10edo has an interval that can function as a perfect fourth at 4 degrees, measuring 480 cents. It can thus be divided into any arrangement of two 1-degree steps and one 2-degree step:

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	=== Tetrachords in other equal temperaments ===		=== Tetrachords in other equal temperaments ===

	* [[Armodue armonia#Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords\|16edo tetrachords]]		* [[Armodue armonia#Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords\|16edo tetrachords]]
	* [[17edo tetrachords]]		* [[17edo tetrachords]]
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	If you'd like to wrestle some tetrachords out of an equal temperament, please consider making a page for them and linking to that page from here!		If you'd like to wrestle some tetrachords out of an equal temperament, please consider making a page for them and linking to that page from here!

	== Dividing ~~Other~~-~~Than~~-~~Perfect Fourths~~ ==		== Dividing other-than-perfect fourths ==

	A composer may choose to treat other-than-perfect fourths as material for constructing tetrachords. Some of the low-number equal temperaments contain diminished or augmented fourths, but nothing resembling a perfect fourth: [[6edo]], [[8edo]], [[9edo]], [[11edo]], [[13edo]], [[16edo]]. Also, one may divide a just other-than-perfect fourth, such as 21/16, 43/32, 26/19, 11/8. At what point does the concept of "tetrachord" stop being useful?		A composer may choose to treat other-than-perfect fourths as material for constructing tetrachords. Some of the low-number equal temperaments contain diminished or augmented fourths, but nothing resembling a perfect fourth: [[6edo]], [[8edo]], [[9edo]], [[11edo]], [[13edo]], [[16edo]]. Also, one may divide a just other-than-perfect fourth, such as 21/16, 43/32, 26/19, 11/8. At what point does the concept of "tetrachord" stop being useful?

	== Tetrachords ~~And Non-Octave Scales~~ ==		== Tetrachords and nonoctave scales ==

	Dividing a tenth into three equal parts generates a cycle of three fourths, they resembling perfect fourths when the division is done on a minor tenth.		Dividing a tenth into three equal parts generates a cycle of three fourths, they resembling perfect fourths when the division is done on a minor tenth.

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	* [http://www.seraph.it/dep/det/GloriousGuitars.mp3 Glorious Guitars] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/e8a36018d6b782c8ff7bc2416fa7ea5b-47.html blog entry])		* [http://www.seraph.it/dep/det/GloriousGuitars.mp3 Glorious Guitars] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/e8a36018d6b782c8ff7bc2416fa7ea5b-47.html blog entry])

	== ~~Related pages~~ ==		== See also ==

	* [[Gallery of Wakalixes#Divisions of the Tetrachord\|Wakalix tetrachords]]		* [[Gallery of Wakalixes#Divisions of the Tetrachord\|Wakalix tetrachords]]