Hexachord
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In music, a hexachord 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.
Middle Ages
| 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 | ||||||
The basis of most if not all medieval European music theory is the hexachordal system. This system attempted to explain the pitch relationships in the music of the time by grouping pitches into sets of six, or hexachords. In each hexachord, all adjacent pitches are a whole tone apart, except for the middle two, which are separated by a half tone. In each hexachord, these pitches are called ut, re, mi, fa, sol, and la, so the half tone is always between mi and fa. Note that most pitches have several syllables associated with them, one for each hexachord they belong to. These syllables were often used to disambiguate octaves: thus, C was "C fa ut" (or "Cefaut"), c was "C sol fa ut", and cc was "C sol fa". Since the lowest pitch was designated by the Greek letter Γ (gamma), it was known as "Gamma ut" or "Gamut". This term came to designate the whole system, and later extended its meaning outside music to denote an entire range of anything.
Since hexachords could start on G, C, or F, the hexachordal system had to account for both B♭ (fa from F) and B♮ (mi from G). The former was called "B molle" (soft B), and the latter was "B durum" (hard B). 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". Since the two varieties of B were called soft and hard, it was natural to depict them with rounded and squared-off letters b respectively. These eventually developed into the modern flat and natural signs.
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.