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{{Interwiki
| en = Tetrachord
| ja = テトラコード
}}
{{Distinguish| Tetrad }}
{{Distinguish| Tetrad }}
{{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 ==
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{{c}}.
; enharmonic genus: The CI approximates a major third, falling between 425 cents and 375 cents.
; Enharmonic genus: The CI approximates a major third, falling between 375–425{{c}}.
; chromatic genus: The CI approximates a minor or neutral third, falling between 375 cents and 250 cents.
; Chromatic genus: The CI approximates a minor or neutral third, falling between 250–375{{c}}.
; 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{{c}}.


=== Ptolemy's catalog ===
=== Ptolemy's catalog ===
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{| 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
|}
|}


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== Generalized tetrachords ==
== 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'' &amp; ''b'', &amp; 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 {{nowrap|''a'' &amp; ''b''}}, and then write our generalized tetrachord like this:


1/1, a, b, 4/3
1/1, a, b, 4/3
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{| 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
|}
|}
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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 ==
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{| 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
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{| 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 234: Line 236:


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 ==
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.
There are two ways to name such a scale segment. One way is to name it after the number of notes it contains. 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.
Another way is to name the scale segment after the interval it spans. The logic is that the "tetra" in tetrachord refers both to 4 notes and to a 4th, since these are identical in a heptatonic context. With this usage, the pajara decatonic scale contains two 5-note tetrachords and a 3-note '''dichord'''. For precedent for this usage, and for a discussion of its advantages, see [[Kite's thoughts on generalizing the term tetrachord]].


== Dividing other-than-perfect fourths ==
== Dividing other-than-perfect fourths ==
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== See also ==
== See also ==
* [[Gallery of Wakalixes#Divisions of the Tetrachord|Wakalix tetrachords]]
* [[Gallery of Wakalixes#Divisions of the Tetrachord|Wakalix tetrachords]]
* [[Kite's thoughts on generalizing the term tetrachord]]


[[Category:Tetrachords| ]] <!-- main article -->
[[Category:Tetrachords| ]] <!-- main article -->
[[Category:Scale]]
[[Category:Scale]]
[[Category:Terms]]
[[Category:Terms]]
[[Category:Ancient Greek music]]
[[Category:Arabic music]]
[[Category:Historical]]
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