Tetrachord

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 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 cents). This characteristic interval is usually slightly smaller (approximating to 200 cents), becoming a whole tone. Classically, the diatonic tetrachord consists of two intervals of a tone and one of a semitone.

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 (approximately 300 cents), and the two smaller intervals are equal semitones.

Enharmonic

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 (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) 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 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. 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 and perfect fifth) (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 and India, but in Europe they were maintained only in certain types of folk music. 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

A rising scale of two whole tones followed by a semitone, or C D E F.

Phrygian mode

A rising scale of tone, semitone and tone, C D E F, or D E F G.

Dorian 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 tunings of the diatonic and chromatic tetrachords:

Diatonic

hypate   parhypate                lichanos                   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:

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 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, 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 on top and two semitones 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 on top with two quarter tones at the bottom, all filling in the perfect fourth (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 ("condensation") of chromatic or enharmonic genera.

Didymos chromatic tetrachord   16:15, 25:24, 6:5

Eratosthenes chromatic tetrachord         20:19, 19:18, 6:5

Ptolemy soft chromatic 28:27, 15:14, 6:5

Ptolemy intense chromatic          22:21, 12:11, 7:6

Archytas enharmonic     28:27, 36:35, 5:4

Variations

Arabic and Indian

Arabic and Indian music divide the tetrachord differently than the Greek. For example, al-Farabi presented ten possible intervals used to divide the tetrachord (Touma 1996, p.19):

Ratio:    1/1         256/243                18/17     162/149                54/49     9/8         32/27     81/68     27/22     81/64     4/3

Note name:        c                                                                              d                                                             e             f

Cents:   0              90           98           145         168         204         294         303         355         408         498

Since there are two tetrachords and a major tone in an octave, this creates a 25 tone scale as used in the Arab tone system 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, a set of four pitches or pitch classes, rather than a series of four contiguous pitches within a scale or tone row.

Source

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

 
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