hexachord

Hexachord ostinato, in cello, which opens Die Jakobsleiter by Arnold Schoenberg, notable for its compositional use of hexachords^[1]

In music, a hexachord is a collection of six pitch classes^[2] including six-note segments of a scale or tone row. The term was adopted in the Middle Ages and adapted in the twentieth-century in Milton Babbitt's serial theory.

Middle Ages

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The Guidonian hexachord (named after its inventor, Guido of Arezzo) was the most basic pedagogical tool for learning new music in the European Middle Ages, and was often referenced in contemporary musical theory. In each hexachord, all adjacent pitches are a whole tone apart, except for the middle two, which are separated by a semitone. These six pitches are named ut, re, mi, fa, sol, and la, with the semitone between mi and fa. These six names are derived from the first syllable of each half-line of the 8th century hymn Ut queant laxis.

Each hexachord could start on G, C or F and the table to the right, reading upwards from the bottom, shows the notes in each hexachord for each of three octaves. Reading from left to right could, within certain limits, permit notes within different octaves to be distinguished from each other. Thus, C (modern c) was "C fa ut" (or "Cefaut"), c (modern c') was "C sol fa ut", and cc (modern c") was "C sol fa". Since the lowest pitch was designated by the Greek letter Γ (gamma, for 'g'), the pitch was known as "Gamma ut" or " Gamut", a term which came to designate the range of notes available, and later, a complete range of anything.

The hexachordal system also distinguished between B♭ (fa in the F hexachord, and known as "B molle" for 'soft B') and B♮ (mi in the G hexachord, and known as "B durum" for 'hard B'). Over time, the soft and hard variants of 'b' were depicted as a rounded '♭' and a squared-off '♮' which gradually developed into the modern flat and natural signs.

Since a single hexachord did not cover every possible note in the range of the gamut (only C-A, F-D excluding B natural, or G-E excluding B flat), singers would have to "mutate" between hexachords if the range of a sixth was exceeded or if there was an alternation between B natural and B flat. In this way the "Guidonian" system of multiple hexachords was different from modern solfege, where a single set of syllables in one location suffices to name all possible pitches (including, often, chromatic pitches).

Because it included B durum, the G hexachord was called hexachordum durum; likewise, the F hexachord was called hexachordum molle. The C hexachord, containing neither B, was called hexachordum naturale.

Later changes to this system added hexachords that would accommodate chromatic pitches and a larger range.

20th century

Milton Babbitt's serial theory extends the term hexachord to refer to a six-note segment of a twelve-tone row. Allen Forte in his The Structure of Atonal Music redefines the term hexachord to mean what other theorists (notably including Howard Hanson in his Harmonic Materials of Modern Music: Resources of the Tempered Scale and Carlton Gamer in his "Some Combinational Resources of Equal-Tempered Systems") mean by the term hexad, a six-note pitch collection which is not necessarily a contiguous segment of a scale or a tone row.

An all-trichord hexachord is a hexachord from which all twelve possible trichords can be derived^[3].

Sacher hexachord^[4]: Eb (Es) A C B (H) E D (Re)

The Sacher hexachord is notable for its use in multiple compositions including Messagequisse by Pierre Boulez.

See also

• Hexatonic scale
• Musica ficta
• Guidonian hand
• Combinatoriality
• Hexachordal complementation
• Mystic chord

Source

Hexachords, solmization, and musica ficta