music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...

, a tetrachord ( el, τετράχορδoν; lat, tetrachordum) is a series of four notes separated by three

intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval e ...

. In traditional music theory, a tetrachord always spanned the interval of a

perfect fourth A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...

, a 4:3 frequency proportion (approx. 498 cents)—but in modern use it means any four-note segment of a scale or

tone row In music, a tone row or note row (german: Reihe or '), also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets ar ...

, 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 greater and lesser perfect systems bounded by ''immovable'' notes ( ); the notes between these were ''movable'' ( ). It literally means ''four strings'', originally in reference to harp-like instruments such as the

lyre The lyre () is a stringed musical instrument that is classified by Hornbostel–Sachs as a member of the lute-family of instruments. In organology, a lyre is considered a yoke lute, since it is a lute in which the strings are attached to a yoke ...

or the kithara, with the implicit understanding that the four strings produced adjacent (i.e., conjunct) notes. Modern music theory uses the

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

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 In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...

Ancient Greek music theory

Ancient Greek music Music was almost universally present in ancient Greek society, from marriages, funerals, and religious ceremonies to theatre, folk music, and the ballad-like reciting of epic poetry. It thus played an integral role in the lives of ancient Greek ...

theory distinguishes three ''genera'' (singular: ''genus'') of tetrachords. These genera are characterized by the largest of the three intervals of the tetrachord: ;

Diatonic Diatonic and chromatic are terms in music theory that are most often used to characterize Scale (music), scales, and are also applied to musical instruments, Interval (music), intervals, Chord (music), chords, Musical note, notes, musical sty ...

: A diatonic tetrachord has a characteristic interval that is less than or equal to half the total interval of the tetrachord (or approximately 249

cent Cent may refer to: Currency * Cent (currency), a one-hundredth subdivision of several units of currency * Penny (Canadian coin), a Canadian coin removed from circulation in 2013 * 1 cent (Dutch coin), a Dutch coin minted between 1941 and 1944 * ...

s). This characteristic interval is usually slightly smaller (approximately 200 cents), becoming a

. Classically, the diatonic tetrachord consists of two intervals of a tone and one of a

semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...

, e.g. A–G–F–E. ;

Chromatic Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, ...

: 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 In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two com ...

(approximately 300 cents), and the two smaller intervals are equal semitones, e.g. A–G–F–E. ;

Enharmonic In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written n ...

: 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 In music, a ditone (, from , "of two tones") is the interval of a major third. The size of a ditone varies according to the sizes of the two tones of which it is compounded. The largest is the Pythagorean ditone, with a ratio of 81:64, also cal ...

or a

major third In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...

, and the two smaller intervals are

quarter tone A quarter tone is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide (aurally, or logarithmically) as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each, a ...

s, e.g. 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 tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...

s of the diatonic and chromatic tetrachords: Here is a representative Pythagorean tuning of the enharmonic genus attributed to

Archytas Archytas (; el, Ἀρχύτας; 435/410–360/350 BC) was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder ...

: 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 In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...

, 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 In mathematics, a superparticular ratio, also called a superparticular number or epimoric ratio, is the ratio of two consecutive integer numbers. More particularly, the ratio takes the form: :\frac = 1 + \frac where is a positive integer. Thu ...

divisions by Chalmers after Hofmann.

Variations

Romantic era

Tetrachords based upon

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...

tuning were used to explain common

heptatonic scale A heptatonic scale is a musical scale that has seven pitches, or tones, per octave. Examples include the major scale or minor scale; e.g., in C major: C D E F G A B C—and in the relative minor, A minor, natural minor: A B C D E F G A; the ...

s. 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 In Western culture, Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a interval (music), musical interval encompassing two adjacent staff positions ( ...

(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).

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. 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.

Atonal usage

Allen Forte Allen, Allen's or Allens may refer to: Buildings * Allen Arena, an indoor arena at Lipscomb University in Nashville, Tennessee * Allen Center, a skyscraper complex in downtown Houston, Texas * Allen Fieldhouse, an indoor sports arena on the Univer ...

occasionally uses the term ''tetrachord'' to mean what he elsewhere calls a ''

tetrad Tetrad ('group of 4') or tetrade may refer to: * Tetrad (area), an area 2 km x 2 km square * Tetrad (astronomy), four total lunar eclipses within two years * Tetrad (chromosomal formation) * Tetrad (general relativity), or frame field ** Tetrad fo ...

'' 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.

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

:Marcel Dupré, ''Cours Complet d'Improvisation a l'Orgue'', 2 vols., translated by John Fenstermaker (Paris: Alphonse Leduc, 1962): 2:35. ASIN: B0006CNH8E.

Indian-specific tetrachord system

''See also Carnatic rāga and

Hindustani classical music Hindustani classical music is the classical music of northern regions of the Indian subcontinent. It may also be called North Indian classical music or, in Hindustani, ''shastriya sangeet'' (). It is played in instruments like the violin, sita ...

.'' Tetrachords separated by a

halfstep A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...

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. These 36 combinations together with the 36 combinations described above produce the so-called "72 karnatic modes".

Persian

Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples ** Persian language, an Iranian language of the ...

music divides the interval of a fourth differently than the Greek. For example,

Al-Farabi Abu Nasr Muhammad Al-Farabi ( fa, ابونصر محمد فارابی), ( ar, أبو نصر محمد الفارابي), known in the Western world, West as Alpharabius; (c. 872 – between 14 December, 950 and 12 January, 951)PDF version was a reno ...

describes four genres of the division of the fourth: * 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, three quarter tones and three quarter tones, as G–A–B–C. * The third genre has a tone and a quarter, three quarter tones and a semitone, as G–A–B–C. * The fourth genre, corresponding to the Greek chromatic, has a tone and a half, a semitone and a semitone, as G–A–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". Later, he presents possible positions of the frets on the lute, producing ten intervals dividing the interval of a fourth between the strings: If one considers that the interval of a fourth between the strings of the lute (

Oud , image=File:oud2.jpg , image_capt=Syrian oud made by Abdo Nahat in 1921 , background= , classification= * String instruments *Necked bowl lutes , hornbostel_sachs=321.321-6 , hornbostel_sachs_desc=Composite chordophone sounded with a plectrum , ...

) corresponds to a tetrachord, and that there are two tetrachords and a

major tone In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...

in an octave, this would create a 25-tone scale. A more inclusive description (where Ottoman, Persian and Arabic overlap), of the scale divisions is that of 24 quarter tones (see also

Arabian maqam Arabic maqam ( ar, مقام, maqām, literally "rank"; ') is the system of melodic modes used in traditional Arabic music, which is mainly melodic. The word ''maqam'' in Arabic means place, location or position. The Arabic ''maqam'' is a melo ...

). 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.

Compositional forms

The tetrachord, a fundamentally incomplete fragment, is the basis of two compositional forms constructed upon repetition of that fragment: the

complaint In legal terminology, a complaint is any formal legal document that sets out the facts and legal reasons (see: cause of action) that the filing party or parties (the plaintiff(s)) believes are sufficient to support a claim against the party ...

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 Henry Purcell (, rare: September 1659 – 21 November 1695) was an English composer. Purcell's style of Baroque music was uniquely English, although it incorporated Italian and French elements. Generally considered among the greatest E ...

's ''Dido and Aeneas'', the ''Crucifixus'' in

Johann Sebastian Bach Johann Sebastian Bach (28 July 1750) was a German composer and musician of the late Baroque period. He is known for his orchestral music such as the '' Brandenburg Concertos''; instrumental compositions such as the Cello Suites; keyboard w ...

's Mass in B minor, BWV 232, or the ''Qui tollis'' in

Mozart Wolfgang Amadeus Mozart (27 January 17565 December 1791), baptised as Joannes Chrysostomus Wolfgangus Theophilus Mozart, was a prolific and influential composer of the Classical period (music), Classical period. Despite his short life, his ra ...

's Mass in C minor, KV 427, etc.Ellen Rosand, "The Descending Tetrachord: An Emblem of Lament", ''The Musical Quarterly'' 65, no. 3 (1979): 346–59. 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''. There exists a short, free musical form of the

Romantic Era Romanticism (also known as the Romantic movement or Romantic era) was an artistic, literary, musical, and intellectual movement that originated in Europe towards the end of the 18th century, and in most areas was at its peak in the approximate ...

, called ''complaint'' or ''complainte'' (Fr.) or

lament A lament or lamentation is a passionate expression of grief, often in music, poetry, or song form. The grief is most often born of regret, or mourning. Laments can also be expressed in a verbal manner in which participants lament about somethin ...

.Marcel Dupré, ''Cours complet d'improvisation a l'orgue: Exercices preparées'', 2 vols., translated by John Fenstermaker. Paris: Alphonse Leduc, 1937): 1:14. It is typically a set of harmonic

variations Variation or Variations may refer to: Science and mathematics * Variation (astronomy), any perturbation of the mean motion or orbit of a planet or satellite, particularly of the moon * Genetic variation, the difference in DNA among individuals ...

homophonic In music, homophony (;, Greek: ὁμόφωνος, ''homóphōnos'', from ὁμός, ''homós'', "same" and φωνή, ''phōnē'', "sound, tone") is a texture in which a primary part is supported by one or more additional strands that flesh ...

texture, wherein the bass descends through some tetrachord, possibly that of the previous paragraph, but usually one suggesting a

minor mode In music theory, the minor scale is three scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending) – rather than just two as with the major scale, which also ...

. This tetrachord, treated as a very short

ground bass In music, an ostinato (; derived from Italian word for ''stubborn'', compare English ''obstinate'') is a motif or phrase that persistently repeats in the same musical voice, frequently in the same pitch. Well-known ostinato-based pieces include ...

, 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).Marcel Dupré, (1962). ''Cours complet d'improvisation a l'orgue'', 2 vols., translated by John Fenstermaker (Paris: Alphonse Leduc, 1962): 2:110. It is also a set of harmonic

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 In music, a chorale prelude or chorale setting is a short liturgical composition for organ using a chorale tune as its basis. It was a predominant style of the German Baroque era and reached its culmination in the works of J.S. Bach, who wrote 46 ...

. Because of the extreme brevity of the theme and number of repetitions required, and free of the binding of

chord progression In a musical composition, a chord progression or harmonic progression (informally chord changes, used as a plural) is a succession of chords. Chord progressions are the foundation of harmony in Western musical tradition from the common practice ...

to tetrachord in the lament, the breadth of the harmonic excursion in litany is usually notable.

References

Sources * *