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

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]

Middle Ages

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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

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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

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Sources

[edit]
  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. ^ Forte, Allen (1973). The Structure of Atonal Music. Yale University Press. ISBN 0-300-02120-8.[page needed]
  10. ^ Hanson, Howard (1960). Harmonic Materials of Modern Music: Resources of the Tempered Scale. New York: Appleton-Century-Crofts. ISBN 0891972072. {{cite book}}: ISBN / Date incompatibility (help)[page needed]
  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

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Hexachord

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A hexachord is a six-note diatonic scale segment in Western music theory, characterized by the intervallic pattern of two whole tones followed by a semitone and then two more whole tones, serving as the foundational unit for solmization and sight-singing in medieval and Renaissance music. Developed by the Benedictine monk Guido d'Arezzo in the early 11th century, with the hexachord introduced in his treatise Micrologus (c. 1025) and the solmization syllables in the Epistola ad Michahelem (c. 1032), the system revolutionized music pedagogy by enabling singers to read and perform unfamiliar chants directly from notation without prior memorization, using syllables derived from the hymn Ut queant laxis. This approach, which persisted as a core element of musical practice through the 16th century, emphasized overlapping hexachords to cover the full musical gamut from G to e''. Guido's hexachordal framework comprised three mutually overlapping types, each starting on a different pitch to accommodate the and occasional accidentals: the natural hexachord (C–D–E–F–G–A), the hard hexachord (G–A–B–C–D–E, incorporating B natural), and the soft hexachord (F–G–A–B♭–C–D, with B flat). Singers applied the syllables ut (later do), re, mi, fa, sol, and la to these notes, with mi and fa marking the critical to guide intonation and avoid dissonance. To traverse melodies spanning multiple hexachords, performers employed , a seamless shift between adjacent hexachords where a shared note retained its syllable, facilitating fluid performance across the . The hexachord system's influence extended beyond sight-singing to composition and theory, underpinning pedagogy and inspiring symbolic uses in , such as structures in motets. In modern , the term has been repurposed in atonal and twelve-tone contexts to denote any collection of six distinct pitch classes, as explored in serialist analyses like the hexachordal theorem, which relates complementary hexachords under transposition and inversion. Despite its obsolescence in everyday practice by the , the hexachord remains a cornerstone for understanding the evolution of Western and .

Fundamentals

Definition and Etymology

A hexachord is a six-note diatonic segment in Western music theory, serving primarily as a structural unit for solmization, the system of assigning syllables to notes for sight-singing. It typically comprises six consecutive scale degrees with whole steps between most intervals and a single semitone between the third and fourth notes, as in the pattern from C to A. This framework enabled musicians to navigate melodies within the diatonic collection without fixed absolute pitches. In modern music theory, particularly in atonal and serial contexts, the term hexachord has evolved to refer to any unordered collection of six distinct pitch classes, contrasting with its historical diatonic specificity and emphasizing combinatorial properties in set analysis. The word "hexachord" originates from the Greek "hexa" (six) and "chordē" (gut string or note), initially denoting a six-stringed instrument or the interval spanning six strings in ancient Greek music theory. This etymological root reflects an early association with the major sixth interval, later adapted to describe the six-note solmization unit. Historically, the hexachord functioned as a foundational pedagogical tool for sight-singing and melodic orientation in Gregorian chant and early polyphony, with syllables like ut-re-mi-fa-sol-la applied to its notes to internalize interval patterns.

Solmization System

The solmization system of the hexachord assigns the syllables ut, re, mi, fa, sol, and la to its six successive notes, establishing a consistent pattern where the only semitone occurs between mi and fa. This fixed interval structure ensures that performers associate the semitone with the same syllables regardless of the hexachord's position in the gamut. In later solmization traditions, the syllable ut evolved into do to improve vocal ease and pronunciation, particularly as the system expanded to include a seventh note (si). However, within the strict hexachordal framework, the original ut is retained to preserve the pedagogical integrity of the six-note unit and its interval relationships. As a movable do-like system, hexachordal emphasizes relative intervals over absolute pitches, enabling singers to recognize and reproduce melodic patterns through syllable associations alone. This approach facilitates sight-singing and transposition by applying the syllables uniformly across the three hexachord types—natural, hard, and soft—without reference to specific keys.

Historical Development

Origins in the Middle Ages

The hexachord system emerged in medieval through the innovations of Guido d'Arezzo, an Italian Benedictine monk and music theorist active in the early . Guido outlined aspects of the system in his earlier Micrologus (c. 1025) before detailing it around 1030 in his treatise Epistola de ignoto cantu, a letter addressed to his colleague Michael of Pomposa, as a pedagogical tool to reform the notation and teaching of . This system divided the into overlapping six-note segments, facilitating easier memorization and sight-singing of melodies by assigning specific syllables to each note, known as . In the context of 11th-century monastic music education, Guido's hexachord addressed longstanding challenges in training singers for . Prior to his reforms, instruction relied heavily on rote and imprecise notation—early symbols indicating melodic contour without fixed pitches—which often led to errors and inefficiency, as monks could take months to learn a single . Guido, frustrated by these limitations during his time at monasteries like Pomposa and Arezzo, developed the hexachord alongside a four-line staff to provide a more systematic, visual method for identifying intervals and pitches, enabling faster and more accurate performance of . Guido's innovations gained papal endorsement when summoned him to around 1028–1033 to demonstrate the system to the , leading to its rapid adoption in schools across . By the mid-12th century, the hexachord had become integral to music curricula in institutions such as those at . This widespread integration transformed from an oral tradition into a more accessible and consistent practice, laying foundational principles for later theory.

The Guidonian Hexachords

The Guidonian hexachords comprise three distinct six-note segments that underpin the solmization system developed by Guido d'Arezzo (c. 995–1050) in his Epistola ad Michahelem (c. 1032), enabling singers to associate fixed syllables with specific intervals across a defined pitch range. These hexachords—naturale, molle, and durum—are positioned within the gamut, the complete medieval scale beginning at gamma (the lowest G, notated as Γ and serving as the foundational ut), and they overlap on common pitches to span the full extent from Γ to high e. The hexachordum naturale (natural hexachord) consists of the pitches C–D–E–F–G–A, containing no accidentals and representing the diatonic core without alteration. It is the central hexachord, often starting on middle C, and embodies the "natural" property (proprietas per naturam) in Guido's framework. The hexachordum molle (soft hexachord) encompasses F–G–A–B♭–C–D, distinguished by the inclusion of B♭ (the "round b" or b rotundum), which softens the third degree relative to the natural hexachord. This configuration, known as the "soft" property (proprietas per b rotundum), allows coverage of pitches below and overlapping with the naturale on notes like C and D. The hexachordum durum (hard hexachord) includes G–A–B–C–D–E, featuring B natural (the "square b" or b quadratum), which hardens the third scale degree. Referred to as the "hard" (proprietas per b quadratum), it starts from G (including the initial gamma ut) and overlaps with the naturale on A, C, D, and E to extend the upward. These overlapping hexachords form a modular structure within the gamut (Γ to ee, spanning roughly two octaves plus a sixth), where shared notes such as A (la in the durum and sol in the naturale) and D (la in the and re in the durum) provide continuity for assigning the solmization syllables ut, re, mi, fa, sol, la to all pitches in the system.

Mutations and Musica Ficta

In theory, refers to the process of shifting from one to another by reassigning syllables to a note that belongs to both, allowing singers to navigate the without introducing new syllables. For instance, the note A could serve as la in the on C (hexachordum naturale) and then mutate to mi in the on F (hexachordum molle), facilitating a smooth transition while incorporating B-flat as fa. This technique, essential for extending the across the three primary —naturalis on C, mollis on F, and durus on G—enabled composers to compose melodies spanning more than a single without disrupting the framework. Musica ficta emerged as a complementary practice in polyphonic music, involving the unwritten alteration of notes (such as raising E to E♯ or lowering B to B♭) to conform to hexachordal rules, particularly to avoid the forbidden mi contra fa interval—the —between mi and fa degrees. These alterations were applied during performance to resolve dissonances, prevent parallel fifths or octaves, and ensure euphony in contrapuntal lines, often guided by rules like una nota super la (one note above la), which required flattening a B above A to avoid the tritone. In the context of , ficta notes facilitated shifts into temporary hexachords outside the standard , such as introducing F♯ via a mutation on G to the hexachord on D. Examples of these techniques abound in 13th- to 15th-century repertory, where mutations and ficta expanded the expressive range of polyphony. In the early 14th-century motet Garrit gallus / In nova fert by Philippe de Vitry, an indirect mutation occurs around measures 54–60, shifting hexachords on an overlapping note to accommodate chromatic inflections and maintain solmization continuity amid dense counterpoint. Similarly, organa from the Notre Dame school, such as those in the Magnus liber organi, employ mutations on notes like G to transition between hexachords, often with ficta sharps on F to resolve dissonances in the upper voices and avoid parallel motion. By the 15th century, these practices were integral to motets by composers like Jean Mouton, where ficta mutations on leading tones enhanced cadential resolutions without notational explicitness.

Theoretical Aspects

Interval Structure

The hexachord in d'Arezzo's system consists of six diatonic pitches arranged in a specific interval pattern of two whole tones, followed by a , and then two more whole tones (TTSTTT). This structure, equivalent to four whole tones and one semitone, positions the semitone between the third and fourth degrees, corresponding to the syllables mi and fa. The syllables facilitate identification of this semitone, enabling singers to orient themselves within the hexachord during . A key structural feature is the perfect fourth spanning the first four notes (from ut to fa), which encompasses the initial three intervals (two whole tones and a semitone) and serves as an anchor for the hexachord's diatonic framework. This interval provides a consonant foundation, reflecting the medieval emphasis on perfect consonances like the fourth and fifth, and helps delineate the position of the internal semitone. The overall span from the first to the sixth note (ut to la) forms a major sixth, but the embedded perfect fourth reinforces the hexachord's role as a self-contained unit for interval recognition. Although each hexachord contains only six notes, the system's design achieves diatonic completeness across the natural scale through overlaps among multiple hexachords. For instance, the natural hexachord on C (C-D-E-F-G-A) overlaps with the hard hexachord on G (G-A-B-C-D-E), sharing notes like G, A, C, D, and E to collectively encompass all seven diatonic pitches (A through G) without gaps. This overlapping mechanism ensures coverage of the full diatonic collection via positioned units, each limited to six notes, while maintaining the consistent TTSTTT pattern for pedagogical consistency.

Relation to Modes and Scales

The hexachord system, as formulated by d'Arezzo, integrated seamlessly with the eight church modes by mapping specific hexachords onto modal frameworks, enabling consistent across modal structures. For example, the naturale hexachord aligns with the on D, allowing its pitches to be sung using the syllables ut-re-mi-fa-sol-la while preserving the mode's characteristic intervals. Similarly, the durum hexachord corresponds to the on G, and the molle hexachord to the on F, with analogous alignments for the plagal modes through transposition. This mapping ensured that each mode could be navigated using the hexachord's fixed interval pattern, bridging with modal theory. Central to this integration was the hexachord's role in constructing the full , the complete range of pitches from Gamma-ut to E-la, as a composite of overlapping hexachords. By arranging the three primary hexachords—durum, naturale, and —in sequence with partial overlaps, the system encompassed approximately two octaves plus a sixth, sufficient for the demands of and . This composite structure avoided the limitations of a single extended scale by relying on the hexachord's repeatable , thus organizing the entire modal repertoire within a unified framework. In terms of scale , hexachords influenced teaching practices by segmenting longer modal scales into manageable six-note units, which simplified the learning of interval relationships and for singers and composers. This approach transformed the complex modal system into accessible building blocks, where students could master one hexachord before extending to others via overlaps, fostering greater accuracy in sight-singing and modal recognition. As Guido's innovations spread, the hexachord emerged as the primary organizing agent of the modal system from the onward, shaping for centuries.

Revival and Modern Use

20th-Century Serialism

In the early , the hexachord was revived as a structural element in atonal and serial composition, departing from its medieval roots to serve as a foundational unit in the twelve-tone chromatic space. employed a hexachordal in the cello introduction to his oratorio Die Jakobsleiter (1917–1922), where a six-note pattern (pitch classes 0,1,4,3,7,6) recurs invariantly under transposition and inversion, providing motivic unity amid the work's free and foreshadowing twelve-tone techniques. Independently, Josef Matthias Hauer developed the concept of tropes—pairs of complementary hexachords that partition the full chromatic aggregate—enumerating 44 such tropes by 1921 to generate unordered pitch collections for composition, as detailed in his Zwölftontechnik (1926). These tropes emphasized hexachordal content over linear order, influencing serial practice by prioritizing aggregate complementarity and melodic derivation from hexachordal fields. Allen Forte's seminal framework in The Structure of Atonal Music (1973) formalized the hexachord as a pitch-class set of six notes within the twelve-tone universe, cataloging all 50 unique hexachordal set classes (e.g., 6-1 through 6-35, with Z-pairs like 6-Z13) using prime forms and interval vectors to analyze similarity and inclusion relations. Forte's system highlighted combinatoriality, where certain hexachords (e.g., all-combinatorial types 6-1, 6-3, 6-5, 6-Z19) aggregate invariantly under the four basic row operations (prime, , retrograde-inversion) of twelve-tone rows, enabling periodic structures in serial works without pitch repetition. This approach provided a rigorous for dissecting atonal textures, such as the hexachordal rotations in Schoenberg's later pieces. Building on these foundations, Milton Babbitt and David Lewin advanced analyses of hexachordal invariance in serial music during the mid-20th century. Babbitt's essay "Twelve-Tone Invariants as Compositional Determinants" (1960) explored how invariant subsets, particularly hexachords, govern row derivations and cyclic formations, as in Webern's use of semi-combinatorial hexachords to link row forms through shared pitch content under transposition. Complementing this, Lewin's early work, including "The Intervallic Content of the Twelve-Tone Row" (1959), examined subset relations and invariance properties, demonstrating how hexachordal complements in rows like those of Schoenberg's String Trio (Op. 45) preserve intervallic structures across transformations, thus unifying surface details with deep serial organization. These contributions underscored the hexachord's role in extending twelve-tone syntax toward multidimensional invariance, influencing composers like Babbitt in works such as Composition for Twelve Instruments (1948).

Contemporary Applications

In contemporary music education, the hexachordal solmization system originally developed by d'Arezzo has been adapted into modern pedagogical methods for and sight-singing, particularly through the Kodály approach, which employs movable-do solfege syllables to emphasize tonal relationships and in diatonic music. This adaptation extends the original hexachord's focus on semitonal placement by incorporating hand signs and sequential exercises to build skills, widely implemented in conservatories and elementary programs to facilitate choral performance and melodic internalization. For instance, institutions like Oberlin Conservatory integrate hexachordal principles into theory curricula to enhance students' ability to navigate tonal progressions without fixed pitch associations. Extensions of the hexachord appear in spectralism and microtonal composition, where composers explore beyond twelve-tone to create expanded scale structures. In , figures like Horatiu Radulescu employ asymmetrical hexachords derived from harmonic spectra to generate dense, inharmonic textures that prioritize timbral evolution over traditional pitch organization, as seen in his Second Piano Sonata, where hexachords enrich fifteenth partial functions within compressed intervals. Similarly, in microtonal contexts, Carlton Gamer's work on equal-tempered systems introduces hexatonic scales that systematically combine intervals across divisions like 19- or 31-tone temperaments, enabling novel combinatorial resources for contemporary harmonic exploration. Analytical tools in software further apply hexachordal segmentation to post-tonal music, building on twentieth-century foundations to dissect complex structures. HexaChord, a Java-based environment developed at , facilitates topological representations of pitch spaces, allowing users to segment scores into hexachordal complexes for visualizing relational invariances in atonal works, thus supporting computational analysis of spectral and microtonal compositions. This tool addresses gaps in manual post-tonal segmentation by integrating geometric models, such as generalized lattices, to reveal hexachordal symmetries in modern repertoires.

References

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  17. https://www.areditions.com/blog/post/brief-guide-to-musica-ficta
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  25. https://music.arts.uci.edu/abauer/4.3/readings/Forte_Pitch-Class_Set_Analysis_Today.pdf
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  27. https://hdl.loc.gov/loc.music/eadmus.mu015002.3
  28. https://www.jstor.org/stable/832873
  29. https://music.arts.uci.edu/abauer/spectral/readings/TEM_Gilmore_Radulescu_Piano.pdf
  30. https://www.jstor.org/stable/842948
  31. http://repmus.ircam.fr/_media/moreno/BigoAndreatta_Computational_Musicology.pdf
  32. https://louisbigo.com/hexachord
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