In music theory, traditionally, a tetrachord (Greek: τετράχορδoν, Latin: tetrachordum) is a series of three smaller intervals that span the interval of a perfect fourth, a 4:3 frequency proportion. In modern usage a tetrachord is any four-note segment of a scale or tone row, not necessarily related to a particular system of tuning. The term tetrachord derives from ancient Greek music theory, where it signified a segment of the Greater and Lesser Perfect Systems bounded by unmovable notes (Greek: ἑστῶτες). 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 must be contiguous.
Ancient Greek music theory distinguishes three genera (singular: genus) of tetrachords. These genera are characterised by the largest of the three intervals of the tetrachord:
- A diatonic tetrachord has a characteristic interval that is less than or equal to half the total interval of the tetrachord (or approximately 249 cents). This characteristic interval is usually slightly smaller (approximately 200 cents), becoming a whole tone. Classically, the diatonic tetrachord consists of two intervals of a tone and one of a semitone.
- 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.
- 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, and the two smaller intervals are quartertones.
As the three genera simply represent ranges of possible intervals within the tetrachord, various shades (chroai) of tetrachord with specific tunings were specified. Once the genus and shade of tetrachord are specified the three internal intervals could be arranged in six possible permutations.[contradiction]
Modern music theory makes use of the octave as the basic unit for determining tuning: ancient Greeks used the tetrachord for this purpose. Ancient Greek theorists recognised that the octave is a fundamental interval, but saw it as built from two tetrachords and a whole tone.
The three permutations of this shade of diatonic tetrachord are:
- Lydian mode
- A rising scale of two whole tones followed by a semitone, or C D E F. (same hypatē and mesē for the ancient Greeks)
- Dorian mode
- A rising scale of tone, semitone and tone, C D E♭ F, or D E F G (E to A for the ancient Greeks).
- Phrygian mode
- A rising scale of a semitone followed by two tones, C D♭ E♭ F, or E F G A (D to G for the ancient Greeks).
(The extents of the Greek system are from Chalmers, Divisions of the Tetrachord.)
Pythagorean tunings 
Here are the traditional Pythagorean tunings of the diatonic and chromatic tetrachords:
Diatonic Play (help·info) hypate parhypate lichanos mese 4/3 81/64 9/8 1/1 | 256/243 | 9/8 | 9/8 | -498 -408 -204 0 cents
Chromatic Play (help·info) hypate parhypate lichanos mese 4/3 81/64 32/27 1/1 | 256/243 | 2187/2048 | 32/27 | -498 -408 -294 0 cents
Since there is no reasonable Pythagorean tuning of the enharmonic genus, here is a representative tuning due to Archytas:
Enharmonic Play (help·info) hypate parhypate lichanos mese 4/3 9/7 5/4 1/1 | 28/27 |36/35| 5/4 | -498 -435 -386 0 cents
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. Originally, the lyre had only five to seven strings(see also the Kithara, a larger form), so only a single tetrachord was needed. 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, the latter being the color between the two other types of modes, which were seen as black and white. Scales are constructed from conjunct or disjunct tetrachords: the tetrachords of the chromatic genus contained a minor third on top and two semitones at the bottom, the diatonic contained a minor second at top with two major seconds at the bottom, and the enharmonic contained a major third on top with two quarter tones at the bottom, all filling in the perfect fourth[not in citation given][not in citation given] of the fixed outer strings. However, the closest term used by the Greeks to our modern usage of chromatic is pyknon, the density ("condensation") of chromatic or enharmonic genera.
|Didymos chromatic tetrachord||16:15, 25:24, 6:5|
|Eratosthenes chromatic tetrachord||20:19, 19:18, 6:5|
|Ptolemy soft chromatic||28:27, 15:14, 6:5|
|Ptolemy intense chromatic||22:21, 12:11, 7:6|
|Archytas enharmonic||28:27, 36:35, 5:4|
This is a partial table of the superparticular divisions by Chalmers after Hofmann.
Since there are two tetrachords and a major tone in an octave, this creates a 25 tone scale as used in the Persian tone system before the quarter tone scale. A more inclusive description (where Ottoman, Persian and Arabic overlap), of the scale divisions is that of 24 tones, 24 equal quarter tones, where a quarter tone equals half a semitone (50 cents) in a 12 tone equal-tempered scale (see also Arabian maqam). It should be mentioned that Al-Farabi's, among other Islamic treatises, also contained additional division schemes as well as providing a gloss of the Greek system as Aristoxenian doctrines were often included.
Romantic Era 
|Component Tetrachords||Halfstep String||Resulting Scale|
|Major + Major||221 2 221||Diatonic Major|
|Minor + Upper Minor||212 2 122||Natural Minor|
|Major + Harmonic||221 2 131||Harmonic Major|
|Minor + Harmonic||212 2 131||Harmonic Minor|
|Harmonic + Harmonic||131 2 131||Double Harmonic|
|Major + Upper Minor||221 2 122||Melodic Major|
|Minor + Major||212 2 221||Melodic Minor|
|Upper Minor + Harmonic||122 2 131||Neapolitan Minor|
Non-Western scales 
Tetrachords based upon Equal temperament tuning were also used to approximate common Heptatonic scales in use in Indian, Hungarian, Arabian and Greek musics. The following elements produce 36 combinations when joined by whole step:
|Lower tetrachords||Upper tetrachords|
Indian-specific tetrachord system 
|Lower tetrachords||Upper tetrachords|
Allen Forte occasionally uses the term tetrachord to mean what other theorists[weasel words] call a tetrad, and what Forte himself also calls a "4-element set"—a set of any four pitches or pitch classes.
See also 
- "Phrygian Progression", Classical Music Blog.
- Chalmers, John H. Jr. (1993). Divisions of the Tetrachord. Hanover, NH: Frog Peak Music. ISBN 0-945996-04-7 Chapter 2, Page 8
- Thomas J. Mathiesen, "Greece §I: Ancient”, The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell (London: Macmillan Publishers, 2001): 6. Music Theory, (iii) Aristoxenian Tradition, (d) Scales.
- Chalmers, John H. Jr. (1993). Divisions of the Tetrachord. Hanover, NH: Frog Peak Music. ISBN 0-945996-04-7 Chapter 6, Page 103
- Miller, Leta E. and Lieberman, Frederic (1998). Lou Harrison: Composing a World. Oxford University Press. ISBN 0-19-511022-6.
- Chalmers (1993). Chapter 1, Page 4
- Chalmers (1993). Chapter 2, Page 11
- Touma, Habib Hassan (1996). The Music of the Arabs, p. 19, trans. Laurie Schwartz. Portland, Oregon: Amadeus Press. ISBN 0-931340-88-8.
- Chalmers (1993). Chapter 3, Page 20
- Dupre, Marcel (1962). Cours Complet d'Improvisation a l'Orgue, v.2, p. 35, trans. John Fenstermaker. Paris: Alphonse Leduc. ASIN: B0006CNH8E.
- Schillinger, Joseph (1941). The Schillinger System of Musical Composition, v.1, p.112-114. New York: Carl Fischer. ISBN 978-0306775215.
- Dupre, Marcel (1962). Cours Complet d'Improvisation a l'Orgue, v. 2, p. 35, trans. John Fenstermaker. Paris: Alphonse Leduc. ASIN: B0006CNH8E.
- Whittall, Arnold (2008). The Cambridge Introduction to Serialism. Cambridge Introductions to Music, p. 34. New York: Cambridge University Press. ISBN 978-0-521-68200-8 (pbk).
- Forte, Allen (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).
Further reading 
- Anonymous. 2001. "Tetrachord". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
- Rahn, John. 1980. Basic Atonal Theory. Longman Music Series. New York and London: Longman Inc.. ISBN 0-582-28117-2.
- Roeder, John. 2001. "Set (ii)". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.