Abridged from the article in the New Grove Dictionary of Music and Musicians
Contributor: William Drabkin
From Gk hex: 'six', chorde: string
An ascending series of six notes that proceeds by whole tones except between the third and fourth notes, which are a semitone aprt. It was first described by GUIDO OF AREZZO in the early 11th century and became the foundation of the earliest standard notational system for polyphonic music, being used as an aid to singing as well as a basis for composition.
The hexachord was not dependent on a particular starting point. The six degrees, or steps, were always called ut, re, mi, fa, sol and la regardless of which was the lowest note; it was thus potentially the basis of a totally relative pitch system. In the system described by Guido, the lowest hexachord was based on G, or 'gamma' as it was called, and so this note was named 'gamma ut' (whence the word 'gamut'). The next note A was named 'a re', the third B, 'b mi'. The second hexachord was based on the fourth note of the first, c; thus it overlapped with the first hexachord for the next three notes abd, accordingly, chared in the naming of these. 'c fa ut', 'd sol re', 'e la mi'. The next hexachord was based on f (f fa ut), the next three on g (g sol re ut), c' (cc sol fa ut) and f' (ff fa ut). Some notes, therefore, belonged to as many as three hexachords. These, and those belonging to just two were useful as pivot notes in moving from one hexachord to another, a procedure called mutation; by this a performer could sing (and a composer could write) melodic lines of greater compass than a 6th.
Perhaps the most striking feature of the hexachordal system was the, though each hexachord was strictly diatonic ( ie part of the seven-note species of of octave with a prescribed ordering of tones and semitones) both b natural and b flat belonged to the system. That is, there was one 'extra' note which, even in a strictly diatonic organzation permitted a modulation from one tonal (or modal) centre to another that used a slightly different collection of seven notes. In fact the hexacords were named according to whether they had B or Bb, or neither. The G hexachord was called the hexachordum durum, (or cantus durus) because it contained the 'hard' or square shaped B (ie B natural) on it's third step, 'b mi'. The F hexachord was called the hexachordum molle (cantus mollis) because it's fourth step was the 'soft' or rounded B (ie Bb), 'b fa'. The C hexachord, called the hexachordum naturale (cantus naturalis), included neither Bb or B.
The opposition of Bb and B went beyond there being in different hexachords: they could not be sung together or even consecutively either in the same part or in different parts. The segregation of 'b mi' from 'b fa' was extended in polyphonic music to the rule of 'mi contra fa', whereby the tritone, the medieval 'diabolo in musica' was a forbidden melodic and vertical interval and had to be corrected to a perfect fourth or fifth.
Hexachords could be transposed, usually up a fourth or down a major second (ie twice up a fourth and down an octave). This enabled composers to write for a particular vocal range as well as t make more distant modulations; the beginnings of chromatic writing in the early 16th century were, in fact, based on an extension of the hexachordal system to include hexachord on Bb and Eb. But by the end of te Renaissance the idea of absolute pitch which had taken a firm hold, and the hexachord, which was always a relative pitch scale, became obsolete. The expressions cantus durus and cantus mollis now stood for the entire pitch system without and with a one-flat signature.