Tetrachord

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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 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:

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.

Contents

History

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.

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[citation needed]:

(The extents of the Greek system are from Chalmers, Divisions of the Tetrachord.[2])

Pythagorean tunings

Here are the traditional Pythagorean tunings of the diatonic and chromatic tetrachords:

Diatonic About this sound Play 
hypate   parhypate                lichanos                   mese
 4/3       81/64                    9/8                      1/1
  | 256/243  |          9/8          |          9/8           |
-498       -408                    -204                       0 cents
Chromatic About this sound Play 
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 About this sound Play 
hypate parhypate lichanos                                    mese
 4/3     9/7   5/4                                           1/1
  | 28/27 |36/35|                     5/4                     |
-498    -435  -386                                            0 cents

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. Originally, the lyre had only five to seven strings(see also the Kithara, a larger form), so only a single tetrachord was needed.[citation 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 [3] [4]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

This is a partial table of the superparticular divisions by Chalmers after Hofmann.[5]

[edit] Variations

[edit] Persian

Persian music divide the tetrachord differently than the Greek. For example, Farabi presented ten possible intervals used to divide the tetrachord [6]:

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 Persian tone system before the quarter tone scale. A more inclusive description (where Ottoman, Persian and Arabic overlap), of the scale divisions is that of 24 tones, 24 equal quarter tones, where a quarter tone equals half a semitone (50 cents) in a 12 tone equal-tempered scale (see also Arabian maqam). It should be mentioned that Al-Farabi's among other Islamic treatises also contained other division schemes and provided a gloss of the Greek system as Aristoxenian doctrines where often included.[7]

[edit] Uses

Polyphonic complex of three tetrachords from early sketch for Arnold Schoenberg's Suite for Piano, Op. 25 [8].

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

Allen Forte occasionally uses the term tetrachord to mean what other theorists[weasel words] call a tetrad, and what Forte himself also calls a "4-element set"—a set of any four pitches or pitch classes[9].

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