View source for Hexachord

{{short description|Six-note series in musical notation}}

[[Image:Schoenberg - Die Jakobsleiter opening hexachord.png|thumb|right|400px|Hexachord [[ostinato]], in cello, which opens ''[[Die Jakobsleiter]]'' by [[Arnold Schoenberg]], notable for its compositional use of hexachords<ref>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).</ref> {{audio|Schoenberg - Die Jakobsleiter opening hexachord.mid|Play}}]]

In [[music]], a '''hexachord''' (also '''hexachordon''') is a six-[[Musical note|note]] series, as exhibited in a [[musical scale|scale]] ([[hexatonic 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 [[serialism|serial theory]]. The word is taken from the {{langx|el|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]]).<ref>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.</ref><ref>[[Ephraim Chambers]], ''[[Cyclopaedia, or an Universal Dictionary of Arts and Sciences|Cyclopædia: or, an Universal Dictionary of Arts and Sciences]]'', 2 vols. (London: Printed for J. and J. Knapton [and 18 others], 1728): 1, part&nbsp;2:247.</ref>

==Middle Ages==
{{Main|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''.<ref>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}}.</ref> 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|''Ut'' queant laxis ''re''sonare fibris / ''Mi''ra gestorum ''fa''muli tuorum]], etc.<ref>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.</ref> 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{{Music|flat}}. A melody moving a semitone higher than ''la'' (namely, from A to the B{{Music|flat}} above) required changing the ''la'' to ''mi'', so that the required B{{Music|flat}} becomes ''fa''. Because B{{Music|flat}} 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{{Music|natural}} and C was called the ''hexachordum durum'' (hard hexachord), because the B{{Music|natural}} 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.<ref name="Jehoash Hirshberg 2001">Jehoash Hirshberg, "Hexachord", ''The New Grove Dictionary of Music and Musicians'', second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (professor of music)|John Tyrrell]] (London: Macmillan Publishers, 2001).</ref>

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{{Music|natural}}, but also below its newly created variant, this entailing, as a result of the "original sin" committed by the well-meant innovation B{{Music|flat}}, the introduction of the still newer respective notes F{{Music|sharp}} and E{{Music|flat}}, with as consequences of these last C{{Music|sharp}} and A{{Music|flat}}, and so on. The new notes, being outside the [[Gamut (music)|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''.<ref>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 (professor of music)|John Tyrrell]] (London: Macmillan Publishers, 2001): §4, "Expansion of the Hexachord System".</ref>

==20th century==
[[Image:Hauer trope.png|thumb|Example of [[Josef Matthias Hauer|Hauer]]'s [[trope (music)#In 20th-century music|tropes]].<ref>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}}.</ref> {{audio|Hauer trope.mid|Play}}]]

[[Allen Forte]] in ''The Structure of Atonal Music''<ref>{{Cite book|first=Allen|last=Forte|title=The Structure of Atonal Music|publisher=Yale University Press|year=1973|isbn=0-300-02120-8}}{{Page needed|date=December 2013}}</ref> redefines the term ''hexachord'' to mean what other theorists (notably [[Howard Hanson]] in his ''Harmonic Materials of Modern Music: Resources of the Tempered Scale''<ref>{{cite book|first=Howard|last=Hanson|title=Harmonic Materials of Modern Music: Resources of the Tempered Scale|location=New York|publisher=Appleton-Century-Crofts|year=1960|isbn=0891972072}}{{Page needed|date=October 2009}}</ref>) 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.<ref>David Lewin, "Re: Intervallic Relations Between Two Collections of Notes", ''Journal of Music Theory'' 3, no. 2 (November 1959): 298–301, citation on 300.</ref> [[Carlton Gamer]] uses both terms interchangeably.<ref>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.</ref>

==See also==
*[[Hexatonic scale]]
*[[Musica ficta]]
*[[Guidonian hand]]
*[[Combinatoriality]]
*[[Complement (music)|Hexachordal complementation]]
*[["Ode-to-Napoleon" hexachord|6-20]], [[Mystic chord|6-34]], [[All-trichord hexachord|6-Z43]], and [[Schoenberg hexachord|6-Z44]]

==Sources==
{{Reflist}}

==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 (professor of music)|John Tyrrell]]. London: Macmillan Publishers, 2001.

==External links==
* [http://www.medieval.org/emfaq/harmony/hex.html Hexachords, solmization, and musica ficta]

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{{Twelve-tone technique}}
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[[Category:Hexachords| ]]