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Traditionally, a tetrachord is a series of four tones filling in 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. The term tetrachord derives from ancient Greek music theory. It literally means four strings, originally in reference to harp-like instruments such as the lyre
Lyre

The lyre is a string instrument well known for its use in classical antiquity and later. The recitations of the Ancient Greece were accompanied by lyre playing....
 or the kithara, with the implicit understanding that the four strings must be contiguous.






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Traditionally, a tetrachord is a series of four tones filling in 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. The term tetrachord derives from ancient Greek music theory. It literally means four strings, originally in reference to harp-like instruments such as the lyre
Lyre

The lyre is a string instrument well known for its use in classical antiquity and later. The recitations of the Ancient Greece were accompanied by lyre playing....
 or the kithara, with the implicit understanding that the four strings must be contiguous. Ancient Greek music theory distinguishes three genera of tetrachords. These genera are characterised 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 249 cent
Cent (music)

The cent is a logarithmic scale unit of measure used for musical interval . Typically cents are used to measure extremely small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is much too small to be heard between successive notes....
s). This characteristic interval is usually slightly smaller (approximating to 200 cents), becoming a whole tone
Major second

A major second , also called a whole step or a whole tone,One source says step is "chiefly US."The preferred usage has been argued since the 19th century:...
. Classically, the diatonic tetrachord consists of two intervals of a tone and one of a semitone
Semitone

A semitone, also called a half step or a half tone,Aaron Copland, Leonard Bernstein, and others use "half tone".One source says that step is "chiefly US", and that half-tone is "chiefly N....
.
Chromatic
A chromatic tetrachord has a characteristic interval that is greater than half the total interval of the tetrachord, yet not as great as four-fifths of the interval (between 249 and 398 cents). Classically, the characteristic interval is a minor third
Minor third

A minor third is a Interval of three semitones. It is the smaller of two commonly occurring musical intervals compounded of two steps of the diatonic scale....
 (approximately 300 cents), and the two smaller intervals are equal semitones.
Enharmonic
Enharmonic

In modern music and musical notation, an enharmonic equivalent is a note , interval , or key signature which is equivalence to some other note, interval, or key signature, but "spelled", or named, differently....
An enharmonic tetrachord has a characteristic interval that is greater than four-fifths the total tetrachord interval (greater than 398 cents). Classically, the characteristic interval is a major third
Major third

A major third is one of two commonly occurring musical intervals that span three diatonic scale degrees, the other being the minor third. It is denoted 'major' because it is the larger of the two: the major third is a leap of four semitones, the minor third three....
 (otherwise known as a ditone), and the two smaller intervals are quartertones.
As the three genera simply represent ranges of possible intervals within the tetrachord, various shades (chroai
Chroa

Chroa is a genus of alga. Chroa produces tetraspores.References ...
) 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.

About

Modern music theory makes use of the octave
Octave

In music, an octave The octave is occasionally referred to as a diapason.The octave above an indicated note is sometimes abbreviated 8va, and the octave below 8vb....
 as the basic unit for determining tuning: ancient Greeks used the tetrachord for this purpose. The octave was recognised by ancient Greece as a fundamental interval, but it was seen as being built from two tetrachords and a whole tone
Major second

A major second , also called a whole step or a whole tone,One source says step is "chiefly US."The preferred usage has been argued since the 19th century:...
. Ancient Greek music always seems to have used two identical tetrachords to build the octave. The single tone could be placed between the two tetrachords (between perfect fourth
Perfect fourth

The perfect fourth is a musical interval which spans four diatonic scale scale degree. It consists of the note and the note five semitones above it on the musical scale....
 and perfect fifth
Perfect fifth

The perfect fifth is the musical interval between a note and the note seven semitones above it on the musical scale. For example, the note G lies a perfect fifth above C; D is a perfect fifth above G, C is a perfect fifth above F, and so on....
) (termed disjunctive), or it could be placed at either end of the scale (termed conjunctive).

Scales built on chromatic and enharmonic tetrachords continued to be used in the classical music of the Middle East
Middle East

File:GreaterMiddleEast1.pngThe Middle East is a region that spans southwestern Asia, western Asia, and northeastern Africa. It has no clear boundaries, often used as a synonym to Near East, in opposition to Far East....
 and India
India

India, officially the Republic of India , is a country in South Asia. It is the List of countries and outlying territories by total area country by geographical area, the List of countries by population country, and the most populous liberal democracy in the world....
, but in Europe
Europe

Europe is, conventionally, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally divided from Asia to its east by the water divide of the Ural Mountains, the Ural , the Caspian Sea, and by the Caucasus Mountains to the southeast....
 they were maintained only in certain types of folk music
Folk music

Folk music can have a number of different meanings, including:* Traditional music: The original meaning of the term "folk music" was synonymous with the term "Traditional music", also often including World Music and Roots music; the term "Traditional music" was given its more specific meaning to distinguish it from the other definition...
. The diatonic tetrachord, however, and particularly the shade built around two tones and a semitone, became the dominant tuning in European music.

Permutations

The three permutations of this shade of diatonic tetrachord are: Lydian mode
Lydian mode

Due to historical confusion, Lydian mode can refer to two very different musical modes or diatonic scales....
A rising scale of two whole tones followed by a semitone, or C D E F.
Dorian mode
Dorian mode

Due to historical confusion, Dorian mode or Doric mode can refer to two very different musical modes or diatonic scales....
A rising scale of tone, semitone and tone, C D E? F, or D E F G.
Phrygian mode
Phrygian mode

Modes are early forms of scales used in music. The Phrygian mode can refer to two different musical modes or diatonic scales: the ancient Greek Phrygian mode and the Medieval Phrygian mode....
A rising scale of a semitone followed by two tones, C D? E? F, or E F G A.
Medieval music scholars misinterpreted Greek texts, and, therefore, medieval and some modern music theory uses these names for different modes than those for which they were originally intended.

Pythagorean tunings

Here are the traditional Pythagorean tuning
Pythagorean tuning

Pythagorean tuning is a system of musical tuning in which the frequency relationships of all interval are based on the ratio sesquialterum. Its name comes from medieval texts which attribute its discovery to Pythagoras, but its use has been documented as long ago as 3500 B.C....
s of the diatonic and chromatic tetrachords:

Diatonic hypate parhypatelichanos mese 4/3 81/64 9/8 1/1 | 256/243 | 9/8 | 9/8 | -498 -408 -204 0 cents

Chromatic 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
Archytas

Archytas was an Ancient Greece philosopher, mathematician, astronomer, statesman, and military strategy. He was a scientist of the Pythagorean school and famous for being the reputed founder of mathematical mechanics, as well as a good friend of Plato....
:

Enharmonic hypate parhypate lichanos mese 4/3 9/7 5/4 1/1 | 28/27 |36/35| 5/4 | -498 -435 -386 0 cents

Originally, the lyre
Lyre

The lyre is a string instrument well known for its use in classical antiquity and later. The recitations of the Ancient Greece were accompanied by lyre playing....
 had only four strings, 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
Diatonic scale

In music theory, a diatonic scale is a seven note musical scale comprising five whole steps and two half steps, in which the half steps are maximally separated....
, 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 being black and white. Scales are constructed from conjunct or disjunct tetrachords: the tetrachords of the chromatic genus contained a minor third
Minor third

A minor third is a Interval of three semitones. It is the smaller of two commonly occurring musical intervals compounded of two steps of the diatonic scale....
 on top and two semitone
Semitone

A semitone, also called a half step or a half tone,Aaron Copland, Leonard Bernstein, and others use "half tone".One source says that step is "chiefly US", and that half-tone is "chiefly N....
s 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
Major third

A major third is one of two commonly occurring musical intervals that span three diatonic scale degrees, the other being the minor third. It is denoted 'major' because it is the larger of the two: the major third is a leap of four semitones, the minor third three....
 on top with two quarter tone
Quarter tone

A quarter tone is an interval about half as wide as a semitone, which is half a whole tone.Many composers are known for having written music including quarter tones or the quarter tone scale, first proposed by 19th-century music theorist Mikha'il Mishaqah , including: Pierre Boulez, Juli?n Carrillo, Mildred Couper, Alberto Ginas...
s at the bottom, all filling in the perfect fourth
Perfect fourth

The perfect fourth is a musical interval which spans four diatonic scale scale degree. It consists of the note and the note five semitones above it on the musical scale....
 (Miller and Lieberman, 1998) of the fixed outer strings. However, the closest term used by the Greeks to our modern usage of chromatic is pyknon or the density
Density

The density of a material is defined as its mass per unit volume. The symbol of density is ....
 ("condensation
Condensation

Condensation is the change of the physical state of aggregation of matter from gaseous phase into liquid phase. When the transition happens from the gaseous phase into the solid phase directly, bypassing the liquid phase the change is called Deposition , which is the opposite of sublimation....
") of chromatic or enharmonic genera.

Didymos chromatic tetrachord16:15, 25:24, 6:5
Eratosthenes chromatic tetrachord20:19, 19:18, 6:5
Ptolemy soft chromatic28:27, 15:14, 6:5
Ptolemy intense chromatic22:21, 12:11, 7:6
Archytas enharmonic28:27, 36:35, 5:4


Variations


Arabic and Indian

Arabic and Indian
Indian music

Indian music may refer to:*Music of India or other music of South Asia, the Indian subcontinent, also music of immigrant communities in the United States and Indo-Caribbean music...
 music divide the tetrachord differently than the Greek. For example, al-Farabi
Al-Farabi

Abu Nasr al-Farabi , known in the Western world as Alpharabius , was a Muslim polymath and one of the greatest Islamic sciences and Early Islamic philosophys of History of Iran and the Islamic Golden Age in his time....
 presented ten possible intervals used to divide the tetrachord (Touma 1996, p.19):
Ratio:1/1256/24318/17162/14954/499/832/2781/6827/2281/644/3
Note name:c    d   ef
Cents:09098145168204294303355408498


Since there are two tetrachords and a major tone
Major second

A major second , also called a whole step or a whole tone,One source says step is "chiefly US."The preferred usage has been argued since the 19th century:...
 in an octave, this creates a 25 tone scale as used in the Arab tone system
Arab tone system

The modern Arab tone system, or system of musical tuning, is based upon the theoretical division of the octave into twenty-four equal divisions or 24-tone equal temperament , the distance between each successive note being a quarter tone ....
 before the quarter tone scale.

Uses

Milton Babbitt's serial theory extends the term tetrachord to mean a four-note segment of a twelve-tone row.

Allen Forte in his The Structure of Atonal Music redefines the term tetrachord to mean what other theorists call a tetrad
Tetrad (music)

In music theory, a tetrad is a set of four notes. When these four notes form a tertian chord they are more particularly and more commonly referred to as a seventh chord, after the diatonic interval from the root of the chord to its fourth note ....
, a set of four pitches or pitch classes, rather than a series of four contiguous pitches within a scale or tone row.

See also

  • Tetrad
    Tetrad (music)

    In music theory, a tetrad is a set of four notes. When these four notes form a tertian chord they are more particularly and more commonly referred to as a seventh chord, after the diatonic interval from the root of the chord to its fourth note ....
  • All-interval tetrachord
    All-interval tetrachord

    An all-interval tetrachord is a tetrachord from which any interval class can be extracted by means of various Inversion . There are only two possible all-interval tetrachords....
  • Diatonic and chromatic
    Diatonic and chromatic

    Diatonic and chromatic are terms in music theory that are most often used to characterise Scale , and are also applied to Interval , Chord , notes, musical styles, and kinds of harmony....
  • Jins
    Jins

    A jins is a set of three, four, or five stepwise pitches used to build an Arabic maqam, or melodic mode. They correspond to the English terms trichord, tetrachord, and pentachord....
  • by John Chalmers, text available free online.


Source

  • Chalmers, John H. Jr. Divisions of the Tetrachord. Frog Peak Music
    Frog Peak Music

    Frog Peak Music is a composer's collective that produces and distributes experimental works, and functions as a home for its artists. It was co-founded in 1984 by Jody Diamond and Larry Polansky....
    , 1993. ISBN 0-945996-04-7
  • Habib Hassan Touma (1996). The Music of the Arabs, trans. Laurie Schwartz. Portland, Oregon: Amadeus Press. ISBN 0-931340-88-8.
  • Miller, Leta E. and Lieberman, Frederic (1998). Lou Harrison: Composing a World. Oxford University Press. ISBN 0-19-511022-6.