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.
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 recognized that the octave is a fundamental interval, but saw it as built from two tetrachords and a whole tone.
Ancient Greek music theory
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:
- 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 three possible permutations.
Although the Dorian, Phrygian, and Lydian modes have distinctive tetrachords at their base, these tetrachords were never named after their parent modes by Greek theorists:
- Dorian mode
- A rising scale of quartertone, quartertone, ditone (enharmonic): E–F–F–A; semitone, semitone, minor third (chromatic): E–F–F♯–A; or semitone, tone and tone (diatonic): E–F–G–A
- Phrygian mode
- A rotation of the Dorian scale beginning with a tetrachord of a ditone, quartertone, quartertone (enharmonic): C–E–F–F; minor third followed by two semitones (chromatic): D♭–E–F–G♭; or whole tone, semitone, whole tone (diatonic): D–E–F–G.
- Lydian mode
- A rotation of the Dorian scale beginning with a tetrachord of quartertone, ditone, quartertone (enharmonic): C–C–E–F; semitone, minor third, semitone (chromatic): C–D♭–E–F; and two whole tones and a semitone (diatonic): C–D–E–F (same hypatē and mesē for the ancient Greeks)[clarification needed]
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
Here is a representative Pythagorean tuning of the enharmonic genus attributed 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. 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.
|Didymos chromatic tetrachord||16:15, 25:24, 6:5||Play (help·info)|
|Eratosthenes chromatic tetrachord||20:19, 19:18, 6:5||Play (help·info)|
|Ptolemy soft chromatic||28:27, 15:14, 6:5||Play (help·info)|
|Ptolemy intense chromatic||22:21, 12:11, 7:6||Play (help·info)|
|Archytas enharmonic||28:27, 36:35, 5:4||Play (help·info)|
Tetrachords based upon Equal temperament tuning were used to explain common Heptatonic scales. Given the following vocabulary of tetrachords (the digits give the number of semitones in consecutive intervals of the tetrachord, adding to five):
|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 Scale or Gypsy Major|
|Major + Upper Minor||221 2 122||Melodic Major|
|Minor + Major||212 2 221||Melodic Minor|
|Upper Minor + Harmonic||122 2 131||Neapolitan Minor|
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. 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, 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. 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.
Allen Forte occasionally uses the term tetrachord to mean what he elsewhere calls a tetrad or simply a "4-element set" – a set of any four pitches or pitch classes. In twelve-tone theory, the term may have the special sense of any consecutive four notes of a twelve-tone row.
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|
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.
The tetrachord, a fundamentally incomplete fragment, is the basis of two compositional forms constructed upon repetition of that fragment: the 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. This tetrachord, known as lamento ("complaint", "lamentation"), has been used until today. A variant form, the full chromatic descent (e.g. A–G♯–G–F♯–F–E in A minor), has been known as Passus duriusculus in the Baroque Figurenlehre.[full citation needed]
There exists a short, free musical form of the Romantic Era, called complaint or complainte (Fr.) or lament. It is typically a set of harmonic variations in homophonic texture, wherein the bass descends through some tetrachord, possibly that of the previous paragraph, but usually one suggesting a 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). It is also a set of harmonic variations in 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 harmonic excursion in litany is usually notable.
- 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.
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