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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.



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. [1]

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, e.g. A–G–F–E.
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.
Two Greek tetrachords in the enharmonic genus, forming an enharmonic Dorian scale Greek Dorian enharmonic genus.png
Two Greek tetrachords in the enharmonic genus, forming an enharmonic Dorian scale
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, [2] and the two smaller intervals are quarter tones, e.g. A–G Doubleflat.svg –F Llpd- 1/2 .svg –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. [3]

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 Doubleflat.svg –C Llpd- 1/2 .svg –B │ A–G Doubleflat.svg –F Llpd- 1/2 .svg –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 Doubleflat.svg –C Llpd- 1/2 .svg –B | A–G Doubleflat.svg –F Llpd- 1/2 .svg –E | D Doubleflat.svg
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 Llpd- 1/2 .svg –B | A–G Doubleflat.svg –F Llpd- 1/2 .svg –E | D Doubleflat.svg –C Llpd- 1/2 .svg

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:

hypateparhypate lichanos mese
−498¢ −408¢ −204¢ 0¢
−498¢ −408¢ −294¢ 0¢

Here is a representative Pythagorean tuning of the enharmonic genus attributed to Archytas:

hypateparhypatelichanos mese
−498¢ −435¢ −386¢ 0¢

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 4:3(6:5)10:9(25:24)16:15(16:15)1:1
Eratosthenes’ chromatic tetrachord 4:3(6:5)10:9(19:18)20:19(20:19)1:1
Ptolemy’s soft chromatic 4:3(6:5)10:9(15:14)28:27(28:27)1:1
Ptolemy’s intense chromatic 4:3(7:6)8:7(12:11)22:21(22:21)1:1
Archytas’ enharmonic 4:3(5:4)9:7(36:35)28:27(28:27)1:1

This is a partial table of the superparticular divisions by Chalmers after Hofmann.[ who? ] [4]


Romantic era

Descending tetrachord in the modern B Locrian (also known as the upper minor tetrachord): --- (b-a-g-f). This tetrachord spans a tritone instead of a perfect fourth. Locrian tetrachord.png
Descending tetrachord in the modern B Locrian (also known as the upper minor tetrachord): Scale deg 8.svg Scale deg 7.svg Scale deg 6.svg Scale deg 5.svg (b–a–g–f). This tetrachord spans a tritone instead of a perfect fourth.
The Phrygian progression creates a descending tetrachord
bassline: ---. Phrygian half cadence: i-v6-iv6-V in C minor (bassline: c-b-a-g) Phrygian half cadence in C.png
The Phrygian progression creates a descending tetrachord bassline: Scale deg 8.svg Scale deg 7.svg Scale deg 6.svg Scale deg 5.svg . Phrygian half cadence: i–v6–iv6–V in C minor (bassline: c–b–a–g)

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):

TetrachordHalfstep String
Major2 2 1
Minor2 1 2
Harmonic1 3 1
Upper Minor1 2 2

the following scales could be derived by joining two tetrachords with a whole step (2) between: [6] [7]

Component tetrachordsHalfstep stringResulting scale
Major + major2 2 1 : 2 : 2 2 1Diatonic major
Minor + upper minor2 1 2 : 2 : 1 2 2Natural minor
Major + harmonic2 2 1 : 2 : 1 3 1Harmonic major
Minor + harmonic2 1 2 : 2 : 1 3 1Harmonic minor
Harmonic + harmonic1 3 1 : 2 : 1 3 1 Double harmonic scale [8] [9] or Gypsy major [10]
Major + upper minor2 2 1 : 2 : 1 2 2Melodic major
Minor + major2 1 2 : 2 : 2 2 1Melodic minor
Upper minor + harmonic1 2 2 : 2 : 1 3 1Neapolitan minor

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). [11]

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. [12] 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, [13] 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. [14] 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. [15]

Atonal usage

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. [16] In twelve-tone theory, the term may have the special sense of any consecutive four notes of a twelve-tone row. [17]

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 whole step: [18]

Lower tetrachordsUpper tetrachords
3 1 13 1 1
2 2 12 2 1
1 3 11 3 1
2 1 22 1 2
1 2 21 2 2
1 1 31 1 3

Indian-specific tetrachord system

Tetrachords separated by a halfstep 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. [18] These 36 combinations together with the 36 combinations described above produce the so-called "72 karnatic modes". [19]

Lower tetrachordsUpper tetrachords
3 2 13 1 1
3 1 22 2 1
2 2 21 3 1
1 3 22 1 2
2 1 31 2 2
1 2 31 1 3


Persian music divides the interval of a fourth differently than the Greek. For example, Al-Farabi describes four genres of the division of the fourth: [20]

He continues with four other possible genres "dividing the tone in quarters, eighths, thirds, half thirds, quarter thirds, and combining them in diverse manners". [21] Later, he presents possible positions of the frets on the lute, producing ten intervals dividing the interval of a fourth between the strings: [22]

Note name:CCCC Llpd+1 1/2 .svg C Llpd+1 1/2 .svg DEEE Llpd- 1/2 .svg EF

If one considers that the interval of a fourth between the strings of the lute (Oud) corresponds to a tetrachord, and that there are two tetrachords and a major tone 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). 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. [23]

Compositional forms

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. [24] 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. [25] 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). [26] 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.

See also

Related Research Articles

In music theory, a diatonic scale is any heptatonic scale that includes five whole steps 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, depending on their position in the scale. This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other.

<span class="mw-page-title-main">Major scale</span> Musical scale made of seven notes

The major scale is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at double its frequency so that it is called a higher octave of the same note.

<span class="mw-page-title-main">Pythagorean tuning</span> Method of tuning a musical instrument

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈702 cents wide.

In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

<span class="mw-page-title-main">Chromatic scale</span> Musical scale set of twelve pitches

The chromatic scale is a set of twelve pitches used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the chromatic scale, while other instruments capable of continuously variable pitch, such as the trombone and violin, can also produce microtones, or notes between those available on a piano.

<span class="mw-page-title-main">Enharmonic</span>

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 note, interval, or chord is an alternative way to write that note, interval, or chord. The term is derived from Latin enharmonicus, from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος (enarmónios), from ἐν (en) and ἁρμονία (harmonía).

A jazz scale is any musical scale used in jazz. Many "jazz scales" are common scales drawn from Western European classical music, including the diatonic, whole-tone, octatonic, and the modes of the ascending melodic minor. All of these scales were commonly used by late nineteenth and early twentieth-century composers such as Rimsky-Korsakov, Debussy, Ravel and Stravinsky, often in ways that directly anticipate jazz practice. Some jazz scales, such as the bebop scales, add additional chromatic passing tones to the familiar diatonic scales.

<span class="mw-page-title-main">Semitone</span> Musical interval

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 notes in a 12-tone scale. For example, C is adjacent to C; the interval between them is a semitone.

Mixolydian mode may refer to one of three things: the name applied to one of the ancient Greek harmoniai or tonoi, based on a particular octave species or scale; one of the medieval church modes; or a modern musical mode or diatonic scale, related to the medieval mode.

<span class="mw-page-title-main">Quarter tone</span> Musical interval

A quarter tone is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each, and have 24 different pitches.

In the musical system of ancient Greece, genus is a term used to describe certain classes of intonations of the two movable notes within a tetrachord. The tetrachordal system was inherited by the Latin medieval theory of scales and by the modal theory of Byzantine music; it may have been one source of the later theory of the jins of Arabic music. In addition, Aristoxenus calls some patterns of rhythm "genera".

<span class="mw-page-title-main">Comma (music)</span> Very small interval arising from discrepancies in tuning

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning system, and another F tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B and A are both approximated by the same interval although they are a septimal kleisma apart.

<span class="mw-page-title-main">Heptatonic scale</span> Musical scale with seven pitches

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 melodic minor scale, A B C D E FGA ascending, A G F E D C B A descending; the harmonic minor scale, A B C D E F GA; and a scale variously known as the Byzantine, and Hungarian, scale, C D E F G A B C. Indian classical theory postulates seventy-two seven-tone scale types, collectively called melakarta or thaat, whereas others postulate twelve or ten seven-tone scale types.

<span class="mw-page-title-main">Nenano</span> Type of chant in medieval Christian churches

Phthora nenano is the name of one of the two "extra" modes in the Byzantine Octoechos—an eight mode system, which was proclaimed by a synod of 792. The phthorai nenano and nana were favoured by composers at the Monastery Agios Sabas, near Jerusalem, while hymnographers at the Stoudiou-Monastery obviously preferred the diatonic mele.

<span class="mw-page-title-main">Diminished second</span> Musical interval

In modern Western tonal music theory, a diminished second is the interval produced by narrowing a minor second by one chromatic semitone. It is enharmonically equivalent to a perfect unison. Thus, it is the interval between notes on two adjacent staff positions, or having adjacent note letters, altered in such a way that they have no pitch difference in twelve-tone equal temperament. An example is the interval from a B to the C immediately above; another is the interval from a B to the C immediately above.

In the musical system of ancient Greece, an octave species is a specific sequence of intervals within an octave. In Elementa harmonica, Aristoxenus classifies the species as three different genera, distinguished from each other by the largest intervals in each sequence: the diatonic, chromatic, and enharmonic genera, whose largest intervals are, respectively, a whole tone, a minor third, and a ditone; quarter tones and semitones complete the tetrachords.

<span class="mw-page-title-main">Diatonic and chromatic</span> Terms in music theory to characterize scales

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, especially when applied to contrasting features of the common practice music of the period 1600–1900.

The musical system of ancient Greece evolved over a period of more than 500 years from simple scales of tetrachords, or divisions of the perfect fourth, into several complex systems encompassing tetrachords and octaves, as well as octave scales divided into seven to thirteen intervals.

<span class="mw-page-title-main">Incomposite interval</span>

An incomposite interval is a concept in the Ancient Greek theory of music concerning melodic musical intervals between neighbouring notes in a tetrachord or scale which, for that reason, do not encompass smaller intervals. Aristoxenus defines melodically incomposite intervals in the following context:

Let us assume that given a systēma, whether pyknon or non-pyknon, no interval less than the remainder of the first concord can be placed next above it, and no interval less than a tone next below it. Let us also assume that each of the notes which are melodically successive in each genus will either form with the fourth note in order from it the concord of a fourth, or will form with the fifth note from it in order the concord of a fifth, or both, and that any note of which none of these things is true is unmelodic relative to those with which it forms no concord. Let us further assume that given that there are four intervals in the fifth, of which two are usually equal and two unequal, the unequal ones are placed next to the equal ones in the opposite order above and below. Let us assume that notes standing at the same concordant interval from successive notes are in succession with one another. Let us assume that in each genus an interval is melodically incomposite if the voice, in singing a melody, cannot divide it into intervals.

Pyknon, sometimes also transliterated as pycnon in the music theory of Antiquity is a structural property of any tetrachord in which a composite of two smaller intervals is less than the remaining (incomposite) interval. The makeup of the pyknon serves to identify the melodic genus and the octave species made by compounding two such tetrachords, and the rules governing the ways in which such compounds may be made centre on the relationships of the two pykna involved.


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Further reading