Hexachord

In music, a hexachord (also hexachordon) is a six-note series, as exhibited in a scale (hexatonic or hexad) or tone row. The term was adopted in this sense during the Middle Ages and adapted in the 20th century in Milton Babbitt's serial theory. The word is taken from the Greek: ἑξάχορδος, compounded from ἕξ (hex, six) and χορδή (chordē, string [of the lyre], whence "note"), and was also the term used in music theory up to the 18th century for the interval of a sixth ("hexachord major" being the major sixth and "hexachord minor" the minor sixth).[2][3]

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

Middle Ages
Main article: Guidonian hand § The hexachord in the Middle Ages

The hexachord as a mnemonic device was first described by Guido of Arezzo, in his Epistola de ignoto cantu.[4] 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-verse of the first stanza of the 8th-century Vesper hymn Ut queant laxis resonare fibris / Mira gestorum famuli tuorum, etc.[5] Melodies with a range wider than a major sixth required the device of mutation to a new hexachord. For example, the hexachord beginning on C and rising to A, named hexachordum naturale, has its only semitone between the notes E and F, and stops short of the note B or B♭. A melody moving a semitone higher than la (namely, from A to the B♭ above) required changing the la to mi, so that the required B♭ becomes fa. Because B♭ was named by the "soft" or rounded letter B, the hexachord with this note in it was called the hexachordum molle (soft hexachord). Similarly, the hexachord with mi and fa expressed by the notes B♮ and C was called the hexachordum durum (hard hexachord), because the B♮ was represented by a squared-off, or "hard" B. Starting in the 14th century, these three hexachords were extended in order to accommodate the increasing use of signed accidentals on other notes.[6]

The introduction of these new notes was principally a product of polyphony, which required the placing of a perfect fifth not only above the old note B♮, but also below its newly created variant, this entailing, as a result of the "original sin" committed by the well-meant innovation B♭, the introduction of the still newer respective notes F♯ and E♭, with as consequences of these last C♯ and A♭, and so on. The new notes, being outside the gamut of those ordinarily available, had to be "imagined", or "feigned" (it was long forbidden to write them), and for this reason music containing them was called musica ficta or musica falsa.[7]

20th century
Example of Hauer's tropes.[8] Playⓘ

Allen Forte in The Structure of Atonal Music[9] redefines the term hexachord to mean what other theorists (notably Howard Hanson in his Harmonic Materials of Modern Music: Resources of the Tempered Scale[10]) mean by the term hexad, a six-note pitch collection which is not necessarily a contiguous segment of a scale or a tone row. David Lewin used the term in this sense as early as 1959.[11] Carlton Gamer uses both terms interchangeably.[12]

See also

Sources
  1. Arnold Whittall, The Cambridge Introduction to Serialism, Cambridge Introductions to Music (New York: Cambridge University Press, 2008): 23. ISBN 978-0-521-86341-4 (hardback) ISBN 978-0-521-68200-8 (pbk).
  2. William Holder, A Treatise of the Natural Grounds and Principles of Harmony (London: Printed by J. Heptinstall, for John Carr, at the Middle-Temple-Gate, in Fleet-Street, 1694): 192. Facsimile reprint, New York: Broude Brothers, 1967.
  3. Ephraim Chambers, Cyclopædia: or, an Universal Dictionary of Arts and Sciences, 2 vols. (London: Printed for J. and J. Knapton [and 18 others], 1728): 1, part 2:247.
  4. Guido d'Arezzo, "Epistola de ignotu cantu [ca. 1030]", abridged translation by Oliver Strink in Source Readings in Music History, selected and annotated by Oliver Strunk, 5 vols. (New York: W. W. Norton, 1965): 1:121–25. Latin test in Martin Gerbert, Scriptores ecclesistici de musica sacra potissimum, 3 vols. (St. Blasien, 1784), 2:43–46, 50. See also Clause V. Palisca, "Introduction" to Guido's Micrologus, in Hucbald, Guido, and John on Music: Three Medieval Treatises, translated by Warren Babb, edited, with introductions by Claude V. Palisca, index of chants by Alejandro Enrique Planchart, 49–56, Music Theory Translation Series 3 (New Haven and London: Yale University Press, 1978): esp. 49–50. ISBN 0-300-02040-6.
  5. Guido d'Arezzo, "Epistola de ignotu cantu [ca. 1030]", abridged translation by Oliver Strink in Source Readings in Music History, selected and annotated by Oliver Strunk, 5 vols. (New York: W. W. Norton, 1965): 1:121–25. Citation on p. 124.
  6. Jehoash Hirshberg, "Hexachord", The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell (London: Macmillan Publishers, 2001).
  7. Andrew Hughes and Edith Gerson-Kiwi, "Solmization [solfatio, solmifatio]", The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell (London: Macmillan Publishers, 2001): §4, "Expansion of the Hexachord System".
  8. George Perle, Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, sixth edition, revised (Berkeley: University of California Press, 1991): 145. ISBN 978-0-520-07430-9.
  9. Allen Forte, The Structure of Atonal Music (New Haven and London: Yale University Press, 1973). ISBN 0-300-01610-7 (cloth) ISBN 0-300-02120-8 (pbk).
  10. Howard Hanson, Harmonic Materials of Modern Music: Resources of the Tempered Scale (New York: Appleton-Century-Crofts, 1960): .
  11. David Lewin, "Re: Intervallic Relations Between Two Collections of Notes", Journal of Music Theory 3, no. 2 (November 1959): 298–301, citation on 300.
  12. Carlton Gamer, "Some Combinational Resources of Equal-Tempered Systems", Journal of Music Theory 11, no. 1 (Spring 1967): 32–59. The term "hexad" appears just once, in a table on p. 37; the word "hexachord" also occurs once, on p. 41.

Further reading
  • Rahn, John. 1980. Basic Atonal Theory. Longman Music Series. New York and London: Longman Inc. ISBN 0-582-28117-2.
  • Roeder, John. "Set (ii)". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers, 2001.
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