Editing Tetrachord

{{Short description|Series of four notes separated by three intervals}}
{{Use American English|date=July 2014}}
In [[music theory]], a '''tetrachord''' ({{langx|el|τετράχορδoν}}; {{langx|la|tetrachordum}}) is a series of four notes separated by three [[interval (music)|intervals]]. In traditional music theory, a tetrachord always spanned the interval of a [[perfect fourth]], a 4:3 frequency proportion (approx. 498 [[cent (music)|cents]])—but in modern use it means any four-note segment of a [[scale (music)|scale]] or [[tone row]], not necessarily related to a particular tuning system.

==History==
The name comes from ''tetra'' (from Greek—"four of something") and ''chord'' (from Greek ''chordon''—"string" or "note"). In ancient Greek music theory, ''tetrachord'' signified a segment of the [[Musical system of ancient Greece#Systema ametabolon, an overview of the tone system|greater and lesser perfect systems]] bounded by ''immovable'' notes ({{math|{{langx|el|ἑστῶτες}}}}); the notes between these were ''movable'' ({{math|{{langx|el|κινούμενοι}}}}). It literally means ''four strings'', originally in reference to harp-like instruments such as the [[lyre]] or the kithara, with the implicit understanding that the four strings produced adjacent (i.e., conjunct) notes.

Modern music theory uses the [[octave]] as the basic unit for determining tuning, where ancient Greeks used the tetrachord. Ancient Greek theorists recognized that the octave is a fundamental interval but saw it as built from two tetrachords and a [[major second|whole tone]].<ref>{{cite book |first=Thomas J. |last=Mathiesen |section=Greece&nbsp;§I: Ancient |title=The New Grove Dictionary of Music and Musicians |title-link=The New Grove Dictionary of Music and Musicians |edition=second |editor1-first=S. |editor1-last=Sadie |editor1-link=Stanley Sadie |editor2-link=John Tyrrell (musicologist) |editor2-first=J. |editor2-last=Tyrrell |place=London, UK |publisher=Macmillan |year=2001 |at=6&nbsp;Music Theory, (iii)&nbsp;Aristoxenian Tradition, (d)&nbsp;Scales}}</ref>

===Ancient Greek music theory===
{{Main|Genus (music)}}
[[Music of ancient Greece|Ancient Greek music]] theory distinguishes three ''genera'' (singular: ''genus'') of tetrachords. These genera are characterized by the largest of the three intervals of the tetrachord:

;[[Genus (music)#Diatonic|Diatonic]]
: A diatonic tetrachord has a characteristic interval that is less than or equal to half the total interval of the tetrachord (or approximately 249&nbsp;[[cent (music)|cent]]s). This characteristic interval is usually slightly smaller (approximately 200&nbsp;cents), becoming a [[major second|whole tone]]. Classically, the diatonic tetrachord consists of two intervals of a tone and one of a [[semitone]], e.g. A–G–F–E.
;[[Genus (music)#Chromatic|Chromatic]]
: A chromatic tetrachord has a characteristic interval that is greater than about half the total interval of the tetrachord, yet not as great as four-fifths of the interval (between about 249 and 398 cents). Classically, the characteristic interval is a [[minor third]] (approximately 300 cents), and the two smaller intervals are equal semitones, e.g. A–G{{Music|flat}}–F–E.
;[[Genus (music)#Enharmonic|Enharmonic]]
[[File:Greek Dorian enharmonic genus.png|thumb|Two Greek tetrachords in the enharmonic genus, forming an enharmonic Dorian scale]]
: An enharmonic tetrachord has a characteristic interval that is greater than about four-fifths the total tetrachord interval. Classically, the characteristic interval is a [[ditone]] or a [[major third]],{{sfn|Chalmers|1993|p=8}} and the two smaller intervals are variable, but ''approximately'' [[quarter tone]]s, e.g. {{nobr|A–G{{Music|double flat}}–F{{Music|half flat}}–E.}} 

When the composite of the two smaller intervals is less than the remaining ([[incomposite interval|incomposite]]) interval, the three-note group is called the ''[[pyknon|pyknón]]'' (from ''pyknós'', meaning "compressed"). This is the case for the chromatic and enharmonic tetrachords, but not the diatonic (meaning "stretched out") tetrachord.

Whatever the tuning of the tetrachord, its four degrees are named, in ascending order, ''hypate'', ''parhypate'', ''lichanos'' (or ''hypermese''), and ''mese'' and, for the second tetrachord in the construction of the system, ''paramese'', ''trite'', ''paranete'', and ''nete''. The ''hypate'' and ''mese'', and the ''paramese'' and ''nete'' are fixed, and a perfect fourth apart, while the position of the ''parhypate'' and ''lichanos'', or ''trite'' and ''paranete'', are movable.

As the three genera simply represent ranges of possible intervals within the tetrachord, various ''shades'' (''chroai'') with specific tunings were specified. Once the genus and shade of tetrachord are specified, their arrangement can produce three main types of scales, depending on which note of the tetrachord is taken as the first note of the scale. The tetrachords themselves remain independent of the scales that they produce, and were never named after these scales by Greek theorists.{{sfn|Chalmers|1993|p=103}}

;Dorian scale : The first note of the tetrachord is also the first note of the scale.
:Diatonic: E–D–C–B | A–G–F–E
:Chromatic: E–D{{Music|flat}}–C–B | A–G{{Music|flat}}–F–E
:Enharmonic: E–D{{Music|double flat}}–C{{Music|half flat}}–B │ A–G{{Music|double flat}}–F{{Music|half flat}}–E

;Phrygian scale: The second note of the tetrachord (in descending order) is the first of the scale.

:Diatonic: D–C–B | A–G–F–E | D
:Chromatic: D{{Music|flat}}–C–B  | A–G{{Music|flat}}–F–E | D{{Music|flat}}
:Enharmonic: D{{Music|double flat}}–C{{Music|half flat}}–B | A–G{{Music|double flat}}–F{{Music|half flat}}–E | D{{Music|double flat}}

; Lydian scale: The third note of the tetrachord (in descending order) is the first of the scale.
:Diatonic: C–B | A–G–F–E | D–C
:Chromatic: C–B | A–G{{Music|flat}}–F–E | D{{Music|flat}}–C
:Enharmonic: C{{Music|half flat}}–B | A–G{{Music|double flat}}–F{{Music|half flat}}–E | D{{Music|double flat}}–C{{Music|half flat}}

In all cases, the extreme notes of the tetrachords, E – B, and A – E, remain fixed, while the notes in between are different depending on the genus.

===Pythagorean tunings===
Here are the traditional [[Pythagorean tuning]]s of the diatonic and chromatic tetrachords:

{| style="text-align:center; margin-left: 4em;"
 |-
 !colspan=7| Diatonic
 |-
 !  hypate || &emsp;  &emsp; || parhypate || &emsp; &emsp; &emsp; &emsp; &nbsp; || lichanos || &emsp; &emsp; &emsp; &emsp; &nbsp; || mese
 |- 
 | {{sfrac|4|3}} ||   || {{sfrac|81|64}} ||  || {{sfrac|9|8}} ||      || {{sfrac|1|1}}
 |- 
 |  │  || {{sfrac|256|243}} || │  || {{sfrac|9|8}}  || │ || {{sfrac|9|8}} ||  │
 |- 
 | −498[[cents (music)|¢]] ||  || −408[[cents (music)|¢]] ||  || −204[[cents (music)|¢]] ||  || 0[[cents (music)|¢]]
 |-
 |colspan=7| {{Listen|filename=Diatonic tetrachord pythagorean tuning.mid|title=Diatonic tetrachord pythagorean tuning}}
 |}

{| style="text-align:center; margin-left: 4em;"
 |-
 !colspan=7| Chromatic
 |-
 ! hypate  ||   &emsp;  &emsp; ||  parhypate ||  &emsp; &emsp; || lichanos || &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; || mese
 |-
 | {{sfrac|4|3}} || || {{sfrac|81|64}}   || || {{sfrac|32|27}} ||  || {{sfrac|1|1}}
 |-
 |   │  || {{sfrac|256|243}} ||  │ || {{sfrac|2187|2048}}  ||  │  || {{sfrac|32|27}} || │
 |-
 | −498[[cents (music)|¢]] ||  || −408[[cents (music)|¢]] ||  ||  −294[[cents (music)|¢]] ||         ||  0[[cents (music)|¢]]
 |-
 |colspan=7| {{Listen|help=no|filename=Chromatic tetrachord pythagorean tuning.mid|title=Chromatic tetrachord pythagorean tuning}}
 |}
Here is a representative Pythagorean tuning of the enharmonic genus attributed to [[Archytas]]:

{| style="text-align:center; margin-left: 4em;"
 |- 
 !colspan=7| Enharmonic
 |- 
 !  hypate ||       || parhypate ||       || lichanos || &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; &nbsp; || mese
 |- 
 | {{sfrac|4|3}} || || {{sfrac|9|7}} || || {{sfrac|5|4}} ||           || {{sfrac|1|1}}
 |- 
 | │ || {{sfrac|28|27}} || │ || {{sfrac|36|35}} || │ || {{sfrac|5|4}} ||  │
 |- 
 |  −498[[cents (music)|¢]] ||  || −435[[cents (music)|¢]] ||   || −386[[cents (music)|¢]]   ||    || 0[[cents (music)|¢]]
 |-
 |colspan=7| {{Listen|help=no|filename=Enharmonic tetrachord pythagorean tuning.mid|title=Enharmonic tetrachord pythagorean tuning}}
 |}

The number of strings on the classical lyre varied at different epochs, and possibly in different localities – four, seven and ten having been favorite numbers. Larger scales are constructed from conjunct or disjunct tetrachords. Conjunct tetrachords share a note, while disjunct tetrachords are separated by a ''disjunctive tone'' of 9/8 (a Pythagorean major second). Alternating conjunct and disjunct tetrachords form a scale that repeats in octaves (as in the familiar [[diatonic scale]], created in such a manner from the diatonic genus), but this was not the only arrangement.

The Greeks analyzed genera using various terms, including diatonic, enharmonic, and chromatic. Scales are constructed from conjunct or disjunct tetrachords.

{| class="wikitable" style="margin-left: 4em;"
 | Didymos’ chromatic tetrachord
 | 4:3 || (6:5) || 10:9 || (25:24) || 16:15 || (16:15) || 1:1
 | [[File:Didymos chromatic tetrachord.mid]]
 |-
 | Eratosthenes’ chromatic tetrachord
 | 4:3 || (6:5) || 10:9 || (19:18) || 20:19 || (20:19) || 1:1
 | [[File:Eratosthenes chromatic tetrachord.mid]]
 |-
 | Ptolemy’s soft chromatic
 | 4:3 || (6:5) || 10:9 || (15:14) || 28:27 || (28:27) || 1:1
 | [[File:Ptolemy soft chromatic tetrachord.mid]]
 |-
 | Ptolemy’s intense chromatic
 | 4:3 || (7:6) || 8:7 || (12:11) || 22:21 || (22:21) || 1:1
 | [[File:Ptolemy intense chromatic tetrachord.mid]]
 |-
 |Archytas’ enharmonic
 | 4:3 || (5:4) || 9:7 || (36:35) || 28:27 || (28:27) || 1:1
 | [[File:Enharmonic tetrachord pythagorean tuning.mid]]
 |}

This is a partial table of the [[superparticular interval|superparticular]] divisions by Chalmers after Hofmann.{{Who|date=July 2013}}<!-- The cited online publication says "I.E. Hofmann (Vogel 1975)", but fails to identify this mysterious source. -->{{sfn|Chalmers|1993|p=11}}

==Variations==

===Romantic era===
[[File:Locrian tetrachord.png|thumb|upright=1.3|[[Descending tetrachord]] in the modern [[locrian mode|B Locrian]] (also known as the upper minor tetrachord): <sub>{{music|scale|8}}</sub>–{{music|b}}<sub>{{music|scale|7}}</sub>–{{music|b}}<sub>{{music|scale|6}}</sub>–{{music|b}}<sub>{{music|scale|5}}</sub> (b–a–g–f). This tetrachord spans a tritone instead of a perfect fourth.[[File:Locrian tetrachord.mid]]]]

[[File:Phrygian half cadence in C.png|thumb|upright=1.3|The [[Phrygian mode|Phrygian]] [[chord progression|progression]] creates a descending tetrachord<ref>[http://classicalmusicblog.com/2007/08/phrygian-progression.html "Phrygian Progression"], ''Classical Music Blog''. {{Webarchive|url=https://web.archive.org/web/20111006081406/http://classicalmusicblog.com/2007/08/phrygian-progression.html |date=2011-10-06 }}</ref>{{Unreliable source?|reason=just a blog, describes itself as a "work in progress" (i.e., it is unstable), makes several dubious claims, such as that there were musicologists in ancient Greece|date=November 2015}} [[bassline]]: <sub>{{music|scale|8}}</sub>–{{music|b}}<sub>{{music|scale|7}}</sub>–{{music|b}}<sub>{{music|scale|6}}</sub>–<sub>{{music|scale|5}}</sub>. [[Phrygian half cadence]]: i–v6–iv6–V in C minor (bassline: c–b{{music|b}}–a{{music|b}}–g)[[File:Phrygian half cadence in C.mid]]]]

Tetrachords based upon [[equal temperament]] tuning were used to explain common [[heptatonic scale]]s. Given the following vocabulary of tetrachords (the digits give the number of semitones in consecutive intervals of the tetrachord, adding to five):
{| class="wikitable" style="margin-left: 4em;"
|-
! Tetrachord !! Halfstep String
|-
| Major || 2 2 1
|-
| Minor || 2 1 2
|-
| Harmonic || 1 3 1
|-
| Upper Minor || 1 2 2
|}
the following scales could be derived by joining two tetrachords with a [[whole step]] (2) between:{{sfn|Dupré|1962|loc=2:35}}<ref>[[Joseph Schillinger]], ''The Schillinger System of Musical Composition'', 2 vols. (New York: Carl Fischer, 1941), 1:112–114. {{ISBN|978-0306775215}}.</ref>
{| class="wikitable" style="margin-left: 4em;"
|-
! Component tetrachords !! Halfstep string !! Resulting scale
|-
| Major + major || 2 2 1 : 2 : 2 2 1 || Diatonic major
|-
| Minor + upper minor || 2 1 2 : 2 : 1 2 2 || Natural minor
|-
| Major + harmonic || 2 2 1 : 2 : 1 3 1 || Harmonic major
|-
| Minor + harmonic || 2 1 2 : 2 : 1 3 1 || Harmonic minor
|-
| Harmonic + harmonic || 1 3 1 : 2 : 1 3 1 ||[[Double harmonic scale]]<ref>Joshua Craig Podolsky, ''Advanced Lead Guitar Concepts'' (Pacific, Missouri: Mel Bay, 2010): 111. {{ISBN|978-0-7866-8236-2}}.</ref><ref>{{cite web|url=http://www.docs.solfege.org/3.21/C/scales/dha.html|title=Double harmonic scale and its modes|website=docs.solfege.org|access-date=2015-04-12|url-status=dead|archive-url=https://web.archive.org/web/20150618200916/http://www.docs.solfege.org/3.21/C/scales/dha.html|archive-date=2015-06-18}}</ref> or Gypsy major<ref>[[Jonathan Bellman]], ''The "Style hongrois" in the Music of Western Europe'' (Boston: Northeastern University Press, 1993): 120. {{ISBN|1-55553-169-5}}.</ref>
|-
| Major + upper minor || 2 2 1 : 2 : 1 2 2 || Melodic major
|-
| Minor + major || 2 1 2 : 2 : 2 2 1 || Melodic minor
|-
| Upper minor + harmonic || 1 2 2 : 2 : 1 3 1 || Neapolitan minor
|}
All these scales are formed by two complete disjunct tetrachords: contrarily to Greek and Medieval theory, the tetrachords change here from scale to scale (i.e., the C major tetrachord would be C–D–E–F, the D major one D–E–F{{music|sharp}}–G, the C minor one C–D–E{{music|flat}}–F, etc.). The 19th-century theorists of ancient Greek music believed that this had also been the case in Antiquity, and imagined that there had existed Dorian, Phrygian or Lydian tetrachords. This misconception was denounced in Otto Gombosi's thesis (1939).<ref>Otto Johannes Gombosi, ''Tonarten und Stimmungen der Antiken Musik'', Kopenhagen, Ejnar Munksgaard, 1939.</ref>

===20th-century analysis===
Theorists of the later 20th century often use the term "tetrachord" to describe any four-note set when analysing music of a variety of styles and historical periods.<ref>See the following:
* {{cite journal
 |first=Benedict |last=Taylor
 |date=Spring 2010
 |title=Modal four-note pitch collections in the music of [[Antonín Dvořák|Dvořák]]'s American period
 |journal=[[Music Theory Spectrum]]
 |volume=32 |issue=1 |pages=44–59
}}
* {{cite journal
 |first1=Steven |last1=Block  |author1-link=Steven Block
 |first2=Jack   |last2=Douthett
 |date=Spring 1994
 |title=[[Vector product]]s and intervallic weighting
 |journal=[[Journal of Music Theory]]
 |volume=38 |issue=1 |pages=21–41
}}
* {{cite journal
 |first=Ian |last=Quinn
 |date=Summer 2001
 |title=Listening to similarity relations
 |journal=[[Perspectives of New Music]]
 |volume=39 |issue=2 |pages=108–158
}}
* {{cite journal
 |first=Joseph N. |last=Straus
 |date=Spring 1999
 |title=[[Igor Stravinsky|Stravinsky]]'s ''Construction of twelve verticals'': An aspect of harmony in the serial music
 |journal=[[Music Theory Spectrum]]
 |volume=21 |issue=1 |pages=43–73
}}
* {{cite journal
 |first=Tuire |last=Kuusi
 |date=September 2007
 |title=Subset-class relation, common pitches, and common interval structure guiding estimations of similarity
 |journal=[[Music Perception]]
 |volume=25 |issue=1 |pages=1–11
}}
* {{cite journal
 |first=J.B. |last=Mailman |author-link=Joshua Banks Mailman
 |date=July–October 2009
 |title=An imagined drama of competitive opposition in [[Elliott Carter|Carter]]'s ''Scrivo in Vento'', with notes on narrative, symmetry, quantitative flux, and [[Heraclitus]]
 |journal=[[Music Analysis (journal)|Music Analysis]]
 |volume=28 |issue=2-3 |pages=373–422
}}
* {{cite journal
 |first1=J. |last1=Harbison |author1-link=John Harbison
 |first2=E. |last2=Cory     |author2-link=Eleanor Cory
 |date=Spring–Summer 1973
 |title=[[Martin Boykan]]: ''String Quartet'' (1967): Two views
 |journal=[[Perspectives of New Music]] |volume=11 |issue=2 |pages=204–209 }}
* {{cite journal
 |first=M. |last=Babbitt |author-link=Milton Babbitt
 |date=Spring–Summer 1973
 |title=[[Edgard Varèse]]: A few observations of his music
 |journal=[[Perspectives of New Music]]
 |volume=4 |issue=2 |pages=14–22
}}
* {{cite journal
 |first=Annie K. |last=Yih
 |date=July 2000
 |title=Analysing [[Claude Debussy|Debussy]]: Tonality, motivic sets and the referential pitch-class specific collection
 |journal=[[Music Analysis (journal)|Music Analysis]]
 |volume=19 |issue=2 |pages=203–229
}}
* {{cite journal
 |first=J.K. |last=Randall |author-link=James K. Randall
 |date=Autumn–Winter 1963
 |title=[[Godfrey Winham]]'s ''Composition for Orchestra'' &hairsp;
 |journal=[[Perspectives of New Music]]
 |volume=2 |issue=1 |pages=102–113
}}</ref> 
The expression "chromatic tetrachord" may be used in two different senses: to describe the special case consisting of a four-note segment of the chromatic scale,<ref>
{{cite journal
 |first=Brent |last=Auerbach
 |date=Fall 2008
 |title=Tiered polyphony and its determinative role in the piano music of [[Johannes Brahms]]
 |journal=[[Journal of Music Theory]]
 |volume=52 |issue=2 |pages=273–320
}}
</ref>
or, in a more historically oriented context, to refer to the six chromatic notes used to fill the interval of a perfect fourth, usually found in descending bass lines.<ref>
{{cite journal
 |first=R. |last=Gauldin |author-link=Robert Gauldin
 |year=1991
 |title=[[Beethoven]]'s interrupted tetrachord and the [[Symphony No. 7 (Beethoven)|''Seventh Symphony'']] &hairsp;
 |journal=[[Intégral (journal)|Intégral]]
 |volume=5 |pages=77–100
}}
</ref>
It may also be used to describes sets of fewer than four notes, when used in scale-like fashion to span the interval of a perfect fourth.<ref>
{{cite journal
 |first=Nors S. |last=Josephson
 |year=2004
 |title=On some apparent sketches for [[Jean Sibelius|Sibelius's]] &hairsp; [[Symphony No. 8 (Sibelius)|''Eighth Symphony'']] &hairsp;
 |journal=[[Archiv für Musikwissenschaft]]
 |volume=61 |issue=1 |pages=54–67
}}
</ref>

===Atonal usage===
[[Allen Forte]] occasionally uses the term ''tetrachord'' to mean what he elsewhere calls a ''[[Tetrad (music)|tetrad]]'' or simply a "4-element set" – a set of any four pitches or ''pitch classes''.<ref>[[Allen Forte]] (1973). ''The Structure of Atonal Music'', pp. 1, 18, 68, 70, 73, 87, 88, 21, 119, 123, 124, 125, 138, 143, 171, 174, and 223. New Haven and London: Yale University Press. {{ISBN|0-300-01610-7}} (cloth) {{ISBN|0-300-02120-8}} (pbk). Allen Forte (1985). "Pitch-Class Set Analysis Today". ''[[Music Analysis (journal)|Music Analysis]]'' 4, nos. 1 & 2 (March–July: Special Issue: [[King's College London]] ''Music Analysis'' Conference 1984): 29–58, citations on 48–51, 53.</ref> In [[Twelve-tone technique|twelve-tone theory]], the term may have the special sense of any consecutive four notes of a twelve-tone row.<ref>Reynold Simpson, "New Sketches, Old Fragments, and Schoenberg's Third String Quartet, Op. 30",  ''Theory and Practice'' 17, In Celebration of Arnold Schoenberg (1) (1992): 85–101.</ref>

==Non-Western scales==

Tetrachords based upon equal-tempered tuning were also used to approximate common heptatonic scales in use in Indian, Hungarian, Arabian and Greek musics.  Western theorists of the 19th and 20th centuries, convinced that any scale should consist of two tetrachords and a tone, described various combinations supposed to correspond to a variety of exotic scales. For instance, the following diatonic intervals of one, two or three semitones, always totaling five semitones, produce 36 combinations when joined by [[whole step]]:{{sfn|Dupré|1962|loc=2:35}}

{| class="wikitable" style="margin-left: 4em;"
|-
! Lower tetrachords !! Upper tetrachords  
|-
| 3 1 1 || 3 1 1 
|-
| 2 2 1 || 2 2 1
|-
| 1 3 1 || 1 3 1 
|-
| 2 1 2 || 2 1 2
|-
| 1 2 2 || 1 2 2
|-
| 1 1 3 || 1 1 3 
|}

===India-specific tetrachord system===
{{See also|Carnatic rāga|Hindustani classical music}}

Tetrachords separated by a [[halfstep]] are said to also appear particularly in Indian music. In this case, the lower "tetrachord" totals six semitones (a tritone). The following elements produce 36 combinations when joined by halfstep.{{sfn|Dupré|1962|loc=2:35}} These 36 combinations together with the 36 combinations described above produce the so-called "72 karnatic modes".<ref>Joanny Grosset, "Inde. Histoire de la musique depuis l'origine jusqu'à nos jours", ''Encyclopédie de la musique et Dictionnaire du Conservatoire'', vol. 1, Paris, Delagrave, 1914, p. 325.</ref>

{| class="wikitable" style="margin-left: 4em;"
|-
! Lower tetrachords !! Upper tetrachords  
|-
| 3 2 1 || 3 1 1 
|-
| 3 1 2 || 2 2 1
|-
| 2 2 2 || 1 3 1 
|-
| 1 3 2 || 2 1 2
|-
| 2 1 3 || 1 2 2
|-
| 1 2 3 || 1 1 3 
|}

===Persian===
[[Persian traditional music|Persian]] music divides the interval of a fourth differently than the Greek. For example, [[Al-Farabi]] describes four genres of the division of the fourth:{{sfn|Al-Farabi|2001|pp=56–57}}
* The first genre, corresponding to the Greek diatonic, is composed of a tone, a tone, and a semitone, as G–A–B–C.
* The second genre is composed of a tone, a three-quarter tone, and a three-quarter tone, as G–A–B{{music|half flat}}–C.
* The third genre has a tone and a quarter, a three-quarter tone, and a semitone, as G–A{{music|half sharp}}–B–C.
* The fourth genre, corresponding to the Greek chromatic, has a tone and a half, a semitone, and a semitone, as G–A{{music|sharp}}–B–C.
He continues with four other possible genres "dividing the tone in quarters, eighths, thirds, half thirds, quarter thirds, and combining them in diverse manners".{{sfn|Al-Farabi|2001|p=58}} Later, he presents possible positions of the frets on the lute, producing ten intervals dividing the interval of a fourth between the strings:<ref>
{{harvnb|Al-Farabi|2001|pp=165–179}}
:
{{cite book
 |first=Liberty |last=Manik
 |year=1969
 |title=Das Arabische Tonsystem im Mittelalter
 |place=Leiden, NL
 |publisher=E.J. Brill
 |page=42
}}
:
{{cite book
 |first=H.H. |last=Touma |author-link=Habib Hassan Touma
 |year=1996
 |title=The Music of the Arabs
 |translator-first=Laurie |translator-last=Schwartz
 |place=Portland, OR
 |publisher=Amadeus Press
 |ISBN=0-931340-88-8
 |page=19
}}</ref>
{| style="text-align: center; margin-left: 4em;"
|style="text-align: left; width: 6em;"| Frequency ratio:
|style="width: 3em;"| {{math|{{sfrac| 1 | 1 }} }}
|style="width: 3em;"| {{math|{{sfrac| 256 | 243 }} }}
|style="width: 3em;"| {{math|{{sfrac| 18 | 17 }} }}
|style="width: 3em;"| {{math|{{sfrac| 162 | 149 }} }}
|style="width: 3em;"| {{math|{{sfrac| 54 | 49 }} }}
|style="width: 3em;"| {{math|{{sfrac| 9 | 8 }} }}
|style="width: 3em;"| {{math|{{sfrac| 32 | 27 }} }}
|style="width: 3em;"| {{math|{{sfrac| 81 | 68 }} }}
|style="width: 3em;"| {{math|{{sfrac| 27 | 22 }} }}
|style="width: 3em;"| {{math|{{sfrac| 81 | 64 }} }}
|style="width: 3em;"| {{math|{{sfrac| 4 | 3 }} }}
|-
|style="text-align: left;"| Note name:
| C
| C{{sup|{{music|#}}{{music|-}} }}
| C{{sup|{{music|#}}{{music|17}} }}
| C{{sup|{{music|#t}}}}
| C{{sup|{{music|#t}}{{music|7}} }}
| D
| E{{sup|{{music|b}}{{music|-}} }}
| E{{sup|{{music|b}}{{music|+}} }}
| E{{sup|{{music|d}} }}
| E{{sup|{{music|-}} }}
| F
|-
|style="text-align: left;"| [[cent (music)|Cent]]s:
|   0
|  90
|  99
| 145
| 168
| 204
| 294
| 303
| 355
| 408
| 498
|}

If one considers that the interval of a fourth between the strings of the lute ([[Oud]]) corresponds to a tetrachord, and that there are two tetrachords and a [[major second|major tone]] in an octave, this would create a 25&nbsp;tone scale. A more inclusive description (where [[Turkish makam|Ottoman]], [[Persian maqam|Persian]], and [[Arabic maqam|Arabic]] overlap), of the scale divisions is that of 24&nbsp;quarter tones (see also [[Arabian maqam]]). It should be mentioned that Al-Farabi's, among other [[maqam|Islamic musical treatises]], also contained additional division schemes as well as providing a gloss of the Greek system, as Aristoxenian doctrines were often included.{{sfn|Chalmers|1993|p=20}}

==Compositional forms==
The tetrachord, a fundamentally incomplete fragment, is the basis of two compositional forms constructed upon repetition of that fragment: the [[Lament bass|complaint]] and the litany.

The descending tetrachord from tonic to dominant, typically in minor (e.g. A–G–F–E in A minor), had been used since the Renaissance to denote a lamentation. Well-known cases include the ostinato bass of Dido's aria ''When I am laid in earth'' in [[Henry Purcell]]'s ''Dido and Aeneas'', the ''Crucifixus'' in [[Johann Sebastian Bach]]'s Mass in B minor, BWV 232, or the ''Qui tollis'' in [[Mozart]]'s Mass in C minor, KV 427, etc.<ref>Ellen Rosand, "The Descending Tetrachord: An Emblem of Lament", ''The Musical Quarterly'' 65, no.&nbsp;3 (1979): 346–59.</ref> This tetrachord, known as ''lamento'' ("complaint", "lamentation"), has been used until today. A variant form, the full chromatic descent (e.g. A–G{{Music|#}}–G–F{{Music|#}}–F–E in A minor), has been known as ''[[Chromatic fourth|Passus duriusculus]]'' in the Baroque ''Figurenlehre''.{{Full citation needed|date=January 2014}}<!--Author, year of publication, location, and publisher needed.-->

There exists a short, free musical form of the [[Romantic music|Romantic Era]], called ''complaint'' or ''complainte'' (Fr.) or [[Lament bass|lament]].<ref name="Dupre, Marcel 1937 p. 14">Marcel Dupré, ''Cours complet d'improvisation a l'orgue: Exercices preparées'', 2 vols., translated by John Fenstermaker. Paris: Alphonse Leduc, 1937): 1:14.</ref>  It is typically a set of harmonic [[Variation (music)|variations]] in [[Homophony|homophonic]] texture, wherein the bass descends through some tetrachord, possibly that of the previous paragraph, but usually one suggesting a [[Minor scale|minor mode]].  This tetrachord, treated as a very short [[ground bass]], is repeated again and again over the length of the composition.

Another musical form, of the same time period, is the ''litany'' or ''litanie'' (Fr.), or ''lytanie'' (OE spur).{{sfn|Dupré|1962|loc=2:110}} It is also a set of harmonic [[Variation (music)|variations]] in [[Homophony|homophonic]] texture, but in contrast to the lament, here the tetrachordal fragment – ascending or descending and possibly reordered – is set in the upper voice in the manner of a [[chorale prelude]].  Because of the extreme brevity of the theme and number of repetitions required, and free of the binding of [[chord progression]] to tetrachord in the lament, the breadth of the [[harmony (music)|harmonic excursion]] in litany is usually notable.

==See also==
*[[All-interval tetrachord]]
*[[Diatonic and chromatic]]
*[[Jins]]
*[[Lament bass]]
*[[Tetrad (music)|Tetrad]]
*[[Tetratonic scale]]

==References==
{{reflist}}

==Sources==
{{refbegin|small=yes}}
* {{cite book
 |author=[[Al-Farabi]]
 |year=2001  |orig-year=1930
 |title=Kitābu l-mūsīqī al-kabīr |lang=fr
 |translator-first=Rodolphe  |translator-last=d'Erlanger
 |trans-title=La musique arabe |edition=reprint
 |location=Paris, FR
 |publisher=Geuthner
}}
* {{cite book
 |last=Chalmers |first=John H., Jr.
 |year=1993
 |title=Divisions of the Tetrachord: A prolegomenon {{grey|[introduction]}} to the construction of musical scales
 |editor1-first=Larry  |editor1-last=Polansky |editor1-link=Larry Polansky
 |editor2-first=Carter |editor2-last=Scholz
 |others=[[Lou Harrison|Harrison, Lou]] (foreword)
 |location=Hanover, NH |publisher=Frog Peak Music
 |isbn=0-945996-04-7
 |url=http://eamusic.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/
 |via=eamusic.dartmouth.edu
}}
* {{cite book
 |last=Dupré |first=Marcel |author-link=Marcel Dupré
 |year=1962
 |orig-date=1925
 |title=Cours complet d'improvisation à l'orgue
 |trans-title=Complete Course in Organ Improvisation
 |language=French
 |translator-last=Fenstermaker |translator-first=John
 |location=Paris
 |publisher=[[Éditions Alphonse Leduc|Alphonse Leduc]]
 |asin=B0006CNH8E
}} (2 vols.)
{{refend}}

==Further reading==
{{refbegin|small=yes}}
* {{cite book
 |author-link=John Rahn |last=Rahn |first=John
 |year=1980
 |title=Basic Atonal Theory
 |series=Longman Music Series
 |place=New York, NY / London, UK
 |publisher=Longman Inc.
 |ISBN=0-582-28117-2
}}
* {{cite encyclopedia
 |first=John  |last=Roeder
 |year=2001
 |title=Set ({{mvar|ii}})
 |editor1-first=Stanley |editor1-last=Sadie   |editor1-link=Stanley Sadie
 |editor2-first=John    |editor2-last=Tyrrell |editor2-link=John Tyrrell (musicologist)
 |encyclopedia=[[The New Grove Dictionary of Music and Musicians]]
 |edition=2nd
 |place=London, UK 
 |publisher=Macmillan
}}
* {{cite encyclopedia
 |editor1-first=Stanley |editor1-last=Sadie   |editor1-link=Stanley Sadie
 |editor2-first=John    |editor2-last=Tyrrell |editor2-link=John Tyrrell (musicologist)
 |year=2001
 |title=Tetrachord
 |encyclopedia=[[The New Grove Dictionary of Music and Musicians]]
 |edition=2nd
 |place=London, UK 
 |publisher=Macmillan
}}
{{refend}}

{{Pitch segments}}

[[Category:Ancient Greek music theory]]
[[Category:Music of Greece]]
[[Category:Musical scales]]