In music theory, a tetrachord (Greek: τετράχορδoν; Latin: tetrachordum) is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion (approx. 498 cents)—but in modern use it means any four-note segment of a scale or tone row, not necessarily related to a particular tuning system. Three modal patterns are possible:
and the tritone:
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 greater and lesser perfect systems bounded by immovable notes (Greek: ἑστῶτες); the notes between these were movable (Greek: 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 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:
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 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, e.g., A–G–F–E. 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♭–F–E. Enharmonic
An enharmonic tetrachord has a characteristic interval that is greater than about four-fifths of the total tetrachord interval. Classically, the characteristic interval is a ditone or a major third, and the two smaller intervals are variable, but approximately quarter tones, e.g. A–G–F–E. When the composite of the two smaller intervals is less than the remaining (incomposite) interval, the three-note group is called the 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.
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♭–C–B | A–G♭–F–E Enharmonic: E–D–C–B │ A–G–F–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♭–C–B | A–G♭–F–E | D♭ Enharmonic: D–C–B | A–G–F–E | D 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♭–F–E | D♭–C Enharmonic: C–B | A–G–F–E | D–C 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 tunings of the diatonic and chromatic tetrachords:
Here is a representative Pythagorean tuning of the enharmonic genus attributed to Archytas:
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.
This is a partial table of the superparticular divisions by Chalmers after Hofmann.
Variations
Romantic era
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):
The following scales could be derived by joining two tetrachords with a whole step (2) between:
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♯–G, the C minor one C–D–E♭–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).
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