Tetrachord

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The tetrachord is a concept of music theory borrowed from ancient Greece. The name means four strings, and refers to the Greek lyre. Traditionally, the lowest and highest strings of the lyre were tuned at the interval of a perfect fourth, or a just ratio of 3:4. The two middle strings were tuned to various divisions of this set interval to create different musical modes. The series of these four tones defines the tetrachord. Greek music theory distinguished between 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 equal to, or less than 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 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.

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.

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, mediaeval and some modern music theory uses these names for different modes than those for which they were originally intended.

Arab 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.

In musical set theory, a tetrachord is a collection of four pitch classes, often one of the three ordered tetrachords in a tone row or set form. Tetrachords may be used to create derived rows and invariance.

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Source

Chalmers, John. Divisions of the tetrachord. Frog Peak Publications.
Habib Hassan Touma (1996). The Music of the Arabs, trans. Laurie Schwartz. Portland, Oregon: Amadeus Press. ISBN 0931340888.

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