Tetrachord: Difference between revisions
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{{Interwiki | |||
| en = Tetrachord | |||
| ja = テトラコード | |||
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{{Distinguish| Tetrad }} | |||
{{Wikipedia}} | |||
[[ | 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. | ||
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: | |||
<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 | == 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. | ||
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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 | === 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 | === 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]]. | ||
== | == Jins/ajnas (tetrachords in middle-eastern music) == | ||
{{Main|Jins}} | |||
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]]. | |||
== 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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Note that these tetrachords are all rotations of each other, and in terms of Greek genera, they are all "diatonic" (because the largest interval, or characteristic interval, at 240 cents, is less than 250 cents). | Note that these tetrachords are all rotations of each other, and in terms of Greek genera, they are all "diatonic" (because the largest interval, or characteristic interval, at 240 cents, is less than 250 cents). | ||
=== Tetrachords in other equal temperaments === | |||
* [[Armodue armonia#Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords|16edo tetrachords]] | |||
* [[17edo tetrachords]] | |||
* [[22edo tetrachords]] | |||
* [[Tricesimoprimal Tetrachordal Tesseract]] (tetrachords of [[31edo]]) | |||
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 == | |||
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 | == 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]) | ||
[[Category:Tetrachords| ]] | == See also == | ||
<!-- main article --> | * [[Gallery of Wakalixes#Divisions of the Tetrachord|Wakalix tetrachords]] | ||
[[Category:Tetrachords| ]] <!-- main article --> | |||
[[Category:Scale]] | [[Category:Scale]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Ancient Greek music]] | |||
[[Category:Arabic music]] | |||
[[Category:Historical]] | |||