Hexachord

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In music, a hexachord, or Gamut, is a collection of six tones. The term was originally used in the Middle Ages, but gained new currency in the 20th century with the development of musical set theory.

[edit] Middle Ages

The medieval hexachordal system (c = middle C)
Note	Syllable
ee							la
dd						la	sol
cc						sol	fa
bb♮							mi
bb♭						fa
aa					la	mi	re
g					sol	re	ut
f					fa	ut
e				la	mi
d			la	sol	re
c			sol	fa	ut
b♮				mi
b♭			fa
a		la	mi	re
G		sol	re	ut
F		fa	ut
E	la	mi
D	sol	re
C	fa	ut
B	mi
A	re
Γ	ut

A hexachord is a group of six notes at specified intervals from each other, which is used as the basis of most, if not all, musical theory of the European middle ages. 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. The six two-letter names for each note being 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.

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.

[edit] Set theory

In musical set theory, a hexachord is a collection of six pitch classes, often one of two (complementary) ordered hexachords in a tone row or set form. Hexachords may be used to create invariance and combinatoriality, and often a melody and its accompaniment will be drawn from different hexachords. Josef Matthias Hauer's twelve tone technique uses unordered hexachords he referred to as tropes.