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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.
{{Interwiki
| en = Tetrachord
| ja = テトラコード
}}
{{Distinguish| Tetrad }}
{{Wikipedia}}


[[John Chalmers]], in [http://eamusic.dartmouth.edu/%7Elarry/published_articles/divisions_of_the_tetrachord/index.html Divisions of the Tetrachord], tells us:
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 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 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:


Related pages: [[22edo tetrachords]], [[17edo tetrachords]], [[Tricesimoprimal Tetrachordal Tesseract]], [[Armodue armonia#Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords|16edo tetrachords]], [[Gallery of Wakalixes#Divisions of the Tetrachord|Wakalix tetrachords]]
<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 ==
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{{c}}.
 
; Enharmonic genus: The CI approximates a major third, falling between 375–425{{c}}.
; hyperenharmonic genus: The CI is larger than 425 cents.
; Chromatic genus: The CI approximates a minor or neutral third, falling between 250–375{{c}}.
; enharmonic genus: The CI approximates a major third, falling between 425 cents and 375 cents.
; Diatonic genus: The CI (and the other intervals) approximates a "[[tone]]", measuring less than 250{{c}}.
; chromatic genus: The CI approximates a minor or neutral third, falling between 375 cents and 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.


{| class="wikitable"
{| class="wikitable"
|-
|+ style="font-size: 105%;" | Archytas's genera
|+ Archytas's Genera
|-
|-
| 28/27, 36/35, 5/4
| 28/27, 36/35, 5/4
| 63 + 49 + 386
| 63 + 49 + 386
| enharmonic
| Enharmonic
|-
|-
| 28/27, 243/224, 32/27
| 28/27, 243/224, 32/27
| 63 + 141 + 294
| 63 + 141 + 294
| chromatic
| Chromatic
|-
|-
| 28/27, 8/7, 9/8
| 28/27, 8/7, 9/8
| 63 + 231 + 204
| 63 + 231 + 204
| diatonic
| Diatonic
|}
|}


{| class="wikitable"
{| class="wikitable"
|-
|+ style="font-size: 105%;" | Eratosthenes's genera
|+ Eratosthenes's Genera
|-
|-
| 40/39, 39/38, 19/15
| 40/39, 39/38, 19/15
| 44 + 45 + 409
| 44 + 45 + 409
| enharmonic
| Enharmonic
|-
|-
| 20/19, 19/18, 6/5
| 20/19, 19/18, 6/5
| 89 + 94 + 316
| 89 + 94 + 316
| chromatic
| Chromatic
|-
|-
| 256/243, 9/8, 9/8
| 256/243, 9/8, 9/8
| 90 + 204 + 204
| 90 + 204 + 204
| diatonic
| Diatonic
|}
|}


{| class="wikitable"
{| class="wikitable"
|+ style="font-size: 105%;" | Didymos's genera
|-
|-
|+ Didymos's Genera
| 32/31, 31/30, 5/4
| 55 + 57 + 386
| Enharmonic
|-
|-
| | 32/31, 31/30, 5/4
| 16/15, 25/24, 6/5
| | 55 + 57 + 386
| 112 + 74 + 316
| | enharmonic
| Chromatic
|-
|-
| | 16/15, 25/24, 6/5
| 16/15, 10/9, 9/8
| | 112 + 74 + 316
| 112 + 182 + 204
| | chromatic
| Diatonic
|-
| | 16/15, 10/9, 9/8
| | 112 + 182 + 204
| | diatonic
|}
|}


{| class="wikitable"
{| class="wikitable"
|-
|+ style="font-size: 105%;" | Ptolemy's Tunings
|+ Ptolemy's Tunings
|-
|-
| 46/45, 24/23, 5/4
| 46/45, 24/23, 5/4
| 38 + 75 + 386
| 38 + 75 + 386
| enharmonic
| Enharmonic
|-
|-
| 28/27, 15/14, 6/5
| 28/27, 15/14, 6/5
| 63 + 119 + 316
| 63 + 119 + 316
| soft chromatic
| Soft chromatic
|-
|-
| 22/21, 12/11, 7/6
| 22/21, 12/11, 7/6
| 81 + 151 + 267
| 81 + 151 + 267
| intense chromatic
| Intense chromatic
|-
|-
| 21/20, 10/9, 8/7
| 21/20, 10/9, 8/7
| 85 + 182 + 231
| 85 + 182 + 231
| soft diatonic
| Soft diatonic
|-
|-
| 28/27, 8/7, 9/8
| 28/27, 8/7, 9/8
| 63 + 231 + 204
| 63 + 231 + 204
| diatonon toniaion
| Diatonon toniaion
|-
|-
| 256/243, 9/8, 9/8
| 256/243, 9/8, 9/8
| 90 + 204 + 204
| 90 + 204 + 204
| diatonon ditoniaion
| Diatonon ditoniaion
|-
|-
| 16/15, 9/8, 10/9
| 16/15, 9/8, 10/9
| 112 + 182 + 204
| 112 + 204 + 182
| intense diatonic
| Intense diatonic
|-
|-
| 12/11, 11/10, 10/9
| 12/11, 11/10, 10/9
| 151 + 165 + 182
| 151 + 165 + 182
| equable diatonic
| Equable diatonic
|}
|}


=== 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) ==
 
{{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/ajnas.html 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 {{nowrap|''a'' &amp; ''b''}}, and 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'' &amp; ''b'', &amp; then write our generalized tetrachord like this:


1/1, a, b, 4/3
1/1, a, b, 4/3
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=== Modes of a [tetrachord], 9/8, [tetrachord] scale ===
=== Modes of a [tetrachord], 9/8, [tetrachord] scale ===
{| class="wikitable"
{| class="wikitable"
|-
|-
! mode 1
! Mode 1
| 1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1
| 1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1
|-
|-
! mode 2
! Mode 2
| 1/1, b/a, 4/3a, 3/2a, 3/2, 3b/2a, 2/a, 2/1
| 1/1, b/a, 4/3a, 3/2a, 3/2, 3b/2a, 2/a, 2/1
|-
|-
! mode 3
! Mode 3
| 1/1, 4b/3, 3b/2, 3a/2b, 3/2, 2/b, 2/b, 2/1
| 1/1, 4b/3, 3b/2, 3a/2b, 3/2, 2/b, 2/b, 2/1
|-
|-
! mode 4
! Mode 4
| 1/1, 9/8, 9a/8, 9b/8, 3/2, 3a, 3b, 2/1
| 1/1, 9/8, 9a/8, 9b/8, 3/2, 3a, 3b, 2/1
|-
|-
! mode 5
! Mode 5
| 1/1, a, b, 4/3, 4a/3, 4b/3, 16/9, 2/1
| 1/1, a, b, 4/3, 4a/3, 4b/3, 16/9, 2/1
|-
|-
! mode 6
! Mode 6
| 1/1, b/a, 4/3a, 4/3, 4b/3a, 16/9a, 2/1
| 1/1, b/a, 4/3a, 4/3, 4b/3a, 16/9a, 2/1
|-
|-
! mode 7
! Mode 7
| 1/1, 4/3b, 4a/3b, 4/3, 16/9b, 2/b, 2a/b, 2/1
| 1/1, 4/3b, 4a/3b, 4/3, 16/9b, 2/b, 2a/b, 2/1
|}
|}
Line 181: Line 177:


=== 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:


Line 190: Line 185:
ssL, sLs, Lss
ssL, sLs, Lss


And, if you have only one step size (as is the case in [[Porcupine]] temperament, for instance), you have an evenly-divided fourth, and only one possible rotation. (A tetrachord of this type can be found in [[22edo]] - see [[22edo tetrachords]].)
And, if you have only one step size (as is the case in [[Porcupine]] temperament, for instance), you have an evenly-divided fourth, and only one possible rotation. (A tetrachord of this type can be found in [[22edo]]—see [[22edo tetrachords]].)


== 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:


Line 202: Line 196:
{| class="wikitable"
{| class="wikitable"
|-
|-
! tetrachord notation
! Tetrachord notation
! cents between steps
! Cents between steps
! cents from 0
! Cents from 0
|-
|-
| 1-1-1
| 1-1-1
Line 212: Line 206:


=== 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:


{| class="wikitable"
{| class="wikitable"
|-
|-
! tetrachord notation
! Tetrachord notation
! cents between
! Cents between
! cents from 0
! Cents from 0
|-
|-
| 1-1-2
| 1-1-2
Line 236: Line 229:
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).


See also: [[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!
=== 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]])


== Dividing Other-Than-Perfect Fourths ==
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 ==
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 ==
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])


[[Category:Theory]]
== See also ==
[[Category:Term]]
* [[Gallery of Wakalixes#Divisions of the Tetrachord|Wakalix tetrachords]]
 
[[Category:Tetrachords| ]] <!-- main article -->
[[Category:Scale]]
[[Category:Scale]]
[[Category:Tetrachord| ]] <!-- main article -->
[[Category:Terms]]
[[Category:Ancient Greek music]]
[[Category:Arabic music]]
[[Category:Historical]]
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