Tetrachord

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Polyphonic complex of three tetrachords from early sketch for Arnold Schoenberg's Suite for Piano, Op. 25.[1]

In music theory, traditionally, a tetrachord (Greek: τετράχορδoν, Latin: tetrachordum) is a series of three smaller intervals that span the interval of a perfect fourth, a 4:3 frequency proportion. In modern usage a tetrachord is any four-note segment of a scale or tone row, not necessarily related to a particular system of tuning.

History[edit]

The term tetrachord derives from ancient Greek music theory, where it signified a segment of the Greater and Lesser Perfect Systems bounded by unmovable notes (Greek: ἑστῶτες). It literally means four strings, originally in reference to harp-like instruments such as the lyre or the kithara, with the implicit understanding that the four strings must be contiguous.

Modern music theory makes use of the octave as the basic unit for determining tuning: ancient Greeks used the tetrachord for this purpose. Ancient Greek theorists recognised that the octave is a fundamental interval, but saw it as built from two tetrachords and a whole tone.[2]

Ancient Greek music theory[edit]

Ancient Greek music theory distinguishes three genera (singular: genus) of tetrachords. These genera are characterised by the largest of the three intervals of the tetrachord:

Diatonic
A diatonic tetrachord has a characteristic interval that is less than or equal to half the total interval of the tetrachord (or approximately 249 cents). This characteristic interval is usually slightly smaller (approximately 200 cents), becoming a whole tone. Classically, the diatonic tetrachord consists of two intervals of a tone and one of a semitone.
Chromatic
A chromatic tetrachord has a characteristic interval that is greater than about half the total interval of the tetrachord, yet not as great as four-fifths of the interval (between about 249 and 398 cents). Classically, the characteristic interval is a minor third (approximately 300 cents), and the two smaller intervals are equal semitones.
Enharmonic
An enharmonic tetrachord has a characteristic interval that is greater than about four-fifths the total tetrachord interval. Classically, the characteristic interval is a ditone or a major third,[3] and the two smaller intervals are quartertones.

As the three genera simply represent ranges of possible intervals within the tetrachord, various shades (chroai) of tetrachord with specific tunings were specified. Once the genus and shade of tetrachord are specified the three internal intervals could be arranged in three possible permutations.

Diatonic Lydian tetrachord: scale degree 4-scale degree 3-scale degree 2-scale degree 1 (F-E-D-C) About this sound Play .
Diatonic Dorian tetrachord: scale degree 4-scale degree 3- scale degree 2- scale degree 1 (g-f-e-d) About this sound Play .
Diatonic Phrygian tetrachord: scale degree 4-scale degree 3-scale degree 2- scale degree 1 (e-f-g-a) About this sound Play .

The three permutations of the diatonic tetrachord are:[4]

Lydian mode
A rising scale of two whole tones followed by a semitone, or C D E F. (same hypatē and mesē for the ancient Greeks)
Dorian mode
A rising scale of tone, semitone and tone, C D E♭ F, or D E F G (E to A for the ancient Greeks).
Phrygian mode
A rising scale of a semitone followed by two tones, C D♭ E♭ F, or E F G A (D to G for the ancient Greeks).

Pythagorean tunings[edit]

Here are the traditional Pythagorean tunings of the diatonic and chromatic tetrachords:

 Diatonic About this sound Play  hypate   parhypate                lichanos                   mese  4/3       81/64                    9/8                      1/1   | 256/243  |          9/8          |          9/8           | -498       -408                    -204                       0 cents 
 Chromatic About this sound Play  hypate   parhypate      lichanos                             mese  4/3       81/64         32/27                               1/1   | 256/243  |  2187/2048  |              32/27               | -498       -408          -294                                 0 cents 

Since there is no reasonable[clarification needed] Pythagorean tuning of the enharmonic genus, here is a representative tuning attributed to Archytas:

 Enharmonic About this sound Play  hypate parhypate lichanos                                    mese  4/3     9/7   5/4                                           1/1   | 28/27 |36/35|                     5/4                     | -498    -435  -386                                            0 cents 

The number of strings on the classical lyre varied at different epochs, and possibly in different localities – four, seven and ten having been favorite numbers. Originally, the lyre had only five to seven strings(see also the Kithara, a larger form), so only a single tetrachord was needed.[citation needed] Larger scales are constructed from conjunct or disjunct tetrachords. Conjunct tetrachords share a note, while disjunct tetrachords are separated by a disjunctive tone of 9/8 (a Pythagorean major second). Alternating conjunct and disjunct tetrachords form a scale that repeats in octaves (as in the familiar diatonic scale, created in such a manner from the diatonic genus), but this was not the only arrangement.

The Greeks analyzed genera using various terms, including diatonic, enharmonic, and chromatic, the latter being the color between the two other types of modes, which were seen as black and white. Scales are constructed from conjunct or disjunct tetrachords: the tetrachords of the chromatic genus contained a minor third on top and two semitones at the bottom, the diatonic contained a minor second at top with two major seconds at the bottom, and the enharmonic contained a major third on top with two quarter tones at the bottom, all filling in the perfect fourth[5][not in citation given][6][not in citation given] of the fixed outer strings. However, the closest term used by the Greeks to our modern usage of chromatic is pyknon, the density ("condensation") of chromatic or enharmonic genera.

Didymos chromatic tetrachord16:15, 25:24, 6:5About this sound Play 
Eratosthenes chromatic tetrachord20:19, 19:18, 6:5About this sound Play 
Ptolemy soft chromatic28:27, 15:14, 6:5About this sound Play 
Ptolemy intense chromatic22:21, 12:11, 7:6About this sound Play 
Archytas enharmonic28:27, 36:35, 5:4About this sound Play 

This is a partial table of the superparticular divisions by Chalmers after Hofmann.[who?][7]

Variations[edit]

Romantic era[edit]

Descending tetrachord in the modern B Locrian (also known as the Upper Minor Tetrachord): scale degree 8-scale degree 7-scale degree 6-scale degree 5 (b-a-g-f) About this sound Play .
The Phrygian progression creates a descending tetrachord[8] bassline: scale degree 8-scale degree 7-scale degree 6- scale degree 5. Phrygian half cadence: i-v6-iv6-V in c minor (bassline: c -b-a-g) About this sound Play .

Tetrachords based upon Equal temperament tuning were used to explain common Heptatonic scales. Given the following vocabulary of tetrachords:

TetrachordHalfstep String
Major221
Minor212
Harmonic131
Upper Minor122

the following scales could be derived by joining 2 tetrachords with a whole step (2) between:[9][10]

Component TetrachordsHalfstep StringResulting Scale
Major + Major221 2 221Diatonic Major
Minor + Upper Minor212 2 122Natural Minor
Major + Harmonic221 2 131Harmonic Major
Minor + Harmonic212 2 131Harmonic Minor
Harmonic + Harmonic131 2 131Double Harmonic
Major + Upper Minor221 2 122Melodic Major
Minor + Major212 2 221Melodic Minor
Upper Minor + Harmonic122 2 131Neapolitan Minor

20th-century analysis[edit]

Theorists of the later 20th century often use the term "tetrachord" to describe any four-note set when analysing music of a variety of styles and historical periods.[11] The expression "chromatic tetrachord" may be used in two different senses: to describe the special case consisting of a four-note segment of the chromatic scale,[12] or, in a more historically oriented context, to refer to the six chromatic notes used to fill the interval of a perfect fourth, usually found in descending bass lines.[13] It may also be used to describes sets of fewer than four notes, when used in scale-like fashion to span the interval of a perfect fourth.[14]

Atonal usage[edit]

Allen Forte occasionally uses the term tetrachord to mean what he elsewhere calls a tetrad or simply a "4-element set"—a set of any four pitches or pitch classes.[15] In twelve-tone theory, the term may have the special sense of any consecutive four notes of a twelve-tone row.[16]

Non-Western scales[edit]

Tetrachords based upon Equal temperament tuning were also used to approximate common Heptatonic scales in use in Indian, Hungarian, Arabian and Greek musics. The following elements produce 36 combinations when joined by whole step:[17]

Lower tetrachordsUpper tetrachords
311311
221221
131131
212212
122122
113113

Indian-specific tetrachord system[edit]

Tetrachords separated by a halfstep also appear particularly in Indian music. The following elements produce 36 combinations when joined by halfstep:[17]

Lower tetrachordsUpper tetrachords
321311
312221
222131
132212
213122
123113

Persian[edit]

Persian music divides the tetrachord differently than the Greek. For example, Farabi presented ten possible intervals used to divide the tetrachord:[18]

Ratio:1/1256/24318/17162/14954/499/832/2781/6827/2281/644/3
Note name:CCCCthree quarter sharpCthree quarter sharpDEEEhalf flatEF
Cents:09098145168204294303355408498

Since there are two tetrachords and a major tone in an octave, this creates a 25 tone scale as used in the Persian tone system before the quarter tone scale. A more inclusive description (where Ottoman, Persian and Arabic overlap), of the scale divisions is that of 24 tones, 24 equal quarter tones, where a quarter tone equals half a semitone (50 cents) in a 12 tone equal-tempered scale (see also Arabian maqam). It should be mentioned that Al-Farabi's, among other Islamic treatises, also contained additional division schemes as well as providing a gloss of the Greek system as Aristoxenian doctrines were often included.[19]

See also[edit]

Sources[edit]

  1. ^ Whittall, Arnold (2008). The Cambridge Introduction to Serialism. Cambridge Introductions to Music, p. 34. New York: Cambridge University Press. ISBN 978-0-521-68200-8 (pbk).
  2. ^ Thomas J. Mathiesen, "Greece §I: Ancient”, The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell (London: Macmillan Publishers, 2001): 6. Music Theory, (iii) Aristoxenian Tradition, (d) Scales.
  3. ^ Chalmers, John H. Jr. (1993). Divisions of the Tetrachord. Hanover, NH: Frog Peak Music. ISBN 0-945996-04-7 Chapter 2, Page 8
  4. ^ Chalmers, John H. Jr. (1993). Divisions of the Tetrachord. Hanover, NH: Frog Peak Music. ISBN 0-945996-04-7 Chapter 6, Page 103
  5. ^ Miller, Leta E. and Lieberman, Frederic (1998). Lou Harrison: Composing a World. Oxford University Press. ISBN 0-19-511022-6.
  6. ^ Chalmers (1993). Chapter 1, Page 4
  7. ^ Chalmers (1993). Chapter 2, Page 11
  8. ^ "Phrygian Progression", Classical Music Blog.
  9. ^ Dupre, Marcel (1962). Cours Complet d'Improvisation a l'Orgue, v.2, p. 35, trans. John Fenstermaker. Paris: Alphonse Leduc. ASIN: B0006CNH8E.
  10. ^ Schillinger, Joseph (1941). The Schillinger System of Musical Composition, v.1, p.112-114. New York: Carl Fischer. ISBN 978-0306775215.
  11. ^ Benedict Taylor, "Modal Four-Note Pitch Collections in the Music of Dvořák's American Period", Music Theory Spectrum 32, no. 1 (Spring 2010): 44–59; Steven Block and Jack Douthett, "Vector Products and Intervallic Weighting", Journal of Music Theory 38, no. 1 (Spring 1994): 21–41; Ian Quinn, "Listening to Similarity Relations", Perspectives of New Music 39, no. 2 (Summer 2001): 108–58; Joseph N. Straus, "Stravinsky's 'Construction of Twelve Verticals': An Aspect of Harmony in the Serial Music", Music Theory Spectrum 21, no. 1 (Spring 1999): 43–73; Tuire Kuusi, "Subset-Class Relation, Common Pitches, and Common Interval Structure Guiding Estimations of Similarity", Music Perception: An Interdisciplinary Journal 25, no. 1 (September 2007): 1–11; Joshua B. Mailman, "An Imagined Drama of Competitive Opposition in Carter's Scrivo in Vento, With Notes on Narrative, Symmetry, Quantitative Flux and Heraclitus", Music Analysis 28, no. 2/3 (July–October 2009): 373–422; John Harbison and Eleanor Cory, "Martin Boykan: String Quartet (1967): Two Views", Perspectives of New Music 11, No. 2 (Spring–Summer 1973): 204–209; Milton Babbitt, "Edgard Varèse: A Few Observations of His Music", Perspectives of New Music 4, no. 2 (Spring–Summer 1966): 14–22; Annie K. Yih, "Analysing Debussy: Tonality, Motivic Sets and the Referential Pitch-Class Specific Collection", Music Analysis 19, no. 2 (July 2000): 203–29; J. K. Randall, "Godfrey Winham's Composition for Orchestra", Perspectives of New Music 2, no. 1 (Autumn–Winter 1963): 102–13.
  12. ^ Brent Auerbach, "Tiered Polyphony and Its Determinative Role in the Piano Music of Johannes Brahms", Journal of Music Theory 52, No. 2 (Fall 2008): 273–320.
  13. ^ Robert Gauldin, "Beethoven's Interrupted Tetrachord and the Seventh Symphony" Intégral 5 (1991): 77–100.
  14. ^ Nors S. Josephson, "On Some Apparent Sketches for Sibelius's Eighth Symphony", Archiv für Musikwissenschaft 61, no. 1 (2004): 54–67.
  15. ^ Forte, Allen (1973). The Structure of Atonal Music, pp. 1, 18, 68, 70, 73, 87, 88, 21, 119, 123, 124, 125, 138, 143, 171, 174, and 223. New Haven and London: Yale University Press. ISBN 0-300-01610-7 (cloth) ISBN 0-300-02120-8 (pbk). Allen Forte (1985). "Pitch-Class Set Analysis Today". Music Analysis 4, nos. 1 & 2 (March–July: Special Issue: King's College London Music Analysis Conference 1984): 29–58, citations on 48–51, 53.
  16. ^ Reynold Simpson, "New Sketches, Old Fragments, and Schoenberg's Third String Quartet, Op. 30", Theory and Practice 17, In Celebration of Arnold Schoenberg (1) (1992): 85–101.
  17. ^ a b Dupre, Marcel (1962). Cours Complet d'Improvisation a l'Orgue, v. 2, p. 35, trans. John Fenstermaker. Paris: Alphonse Leduc. ASIN: B0006CNH8E.
  18. ^ Touma, Habib Hassan (1996). The Music of the Arabs, p. 19, trans. Laurie Schwartz. Portland, Oregon: Amadeus Press. ISBN 0-931340-88-8.
  19. ^ Chalmers (1993). Chapter 3, Page 20

Further reading[edit]