Tritone
Inverse | tritone |
---|---|
Name | |
Other names | augmented fourth, diminished fifth, the Devil’s interval (obscure) |
Abbreviation | TT, A4, d5 |
Size | |
Semitones | 6 |
Interval class | 6 |
Just interval | Pythagorean: 729:512, 1024:729 5-limit: 25:18, 36:25; 45:32, 64:45 7-limit: 7:5, 10:7 13-limit: 13:9, 18:13 |
Cents | |
12-Tone equal temperament | 600 |
Just intonation | Pythagorean: 612, 588 5-limit: 569, 631; 590, 610 7-limit: 583, 617 13-limit: 563, 637 |
In music theory, the tritone is defined as a musical interval spanning three adjacent whole tones (six semitones).[1] For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B.
Narrowly defined, each of these whole tones must be a step in the
In classical music, the tritone is a harmonic and melodic dissonance and is important in the study of musical harmony. The tritone can be used to avoid traditional tonality: "Any tendency for a tonality to emerge may be avoided by introducing a note three whole tones distant from the key note of that tonality."[4] The tritone found in the dominant seventh chord can also drive the piece of music towards resolution with its tonic. These various uses exhibit the flexibility, ubiquity, and distinctness of the tritone in music.
The condition of having tritones is called tritonia; that of having no tritones is atritonia. A musical scale or chord containing tritones is called tritonic; one without tritones is atritonic.
Augmented fourth and diminished fifth
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Since a chromatic scale is formed by 12 pitches (each a semitone apart from its neighbors), it contains 12 distinct tritones, each starting from a different pitch and spanning six semitones. According to a complex but widely used naming convention, six of them are classified as augmented fourths, and the other six as diminished fifths.
Under that convention, a fourth is an interval encompassing four
Any augmented fourth can be decomposed into three whole tones. For instance, the interval F–B is an augmented fourth and can be decomposed into the three adjacent whole tones F–G, G–A, and A–B.
It is not possible to decompose a diminished fifth into three adjacent whole tones. The reason is that a whole tone is a major second, and according to a rule explained elsewhere,[where?] the composition of three seconds is always a fourth (for instance, an A4). To obtain a fifth (for instance, a d5), it is necessary to add another second. For instance, using the notes of the C major scale, the diminished fifth B–F can be decomposed into the four adjacent intervals
- B–C (minor second), C–D (major second), D–E (major second), and E–F (minor second).
Using the notes of a chromatic scale, B–F may be also decomposed into the four adjacent intervals
- B–C♯ (major second), C♯–D♯ (major second), D♯–E♯ (major second), and E♯–F♮ (diminished second).
Notice that the last diminished second is formed by two
Definitions
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A tritone (abbreviation: TT) is traditionally defined as a musical interval composed of three whole tones. As the symbol for whole tone is T, this definition may also be written as follows:
- TT = T+T+T
Only if the three tones are of the same size (which is not the case for many
- TT = 3T
This definition, however, has two different interpretations (broad and strict).
Broad interpretation (chromatic scale)
In a chromatic scale, the interval between any note and the previous or next is a semitone. Using the notes of a chromatic scale, each tone can be divided into two semitones:
- T = S+S
For instance, the tone from C to D (in short, C–D) can be decomposed into the two semitones C–C♯ and C♯–D by using the note C♯, which in a chromatic scale lies between C and D. This means that, when a chromatic scale is used, a tritone can be also defined as any musical interval spanning six semitones:
- TT = T+T+T = S+S+S+S+S+S.
According to this definition, with the twelve notes of a chromatic scale it is possible to define twelve different tritones, each starting from a different note and ending six notes above it. Although all of them span six semitones, six of them are classified as augmented fourths, and the other six as diminished fifths.
Strict interpretation (diatonic scale)
Within a diatonic scale, whole tones are always formed by adjacent notes (such as C and D) and therefore they are regarded as incomposite intervals. In other words, they cannot be divided into smaller intervals. Consequently, in this context the above-mentioned "decomposition" of the tritone into six semitones is typically not allowed.
If a diatonic scale is used, with its 7 notes it is possible to form only one sequence of three adjacent whole tones (T+T+T). This interval is an A4. For instance, in the C major diatonic scale (C–D–E–F–G–A–B–...), the only tritone is from F to B. It is a tritone because F–G, G–A, and A–B are three adjacent whole tones. It is a fourth because the notes from F to B are four (F, G, A, B). It is augmented (i.e., widened) because it is wider than most of the fourths found in the scale (they are perfect fourths).
According to this interpretation, the d5 is not a tritone. Indeed, in a diatonic scale, there is only one d5, and this interval does not meet the strict definition of tritone, as it is formed by one semitone, two whole tones, and another semitone:
- d5 = S+T+T+S.
For instance, in the C major diatonic scale, the only d5 is from B to F. It is a fifth because the notes from B to F are five (B, C, D, E, F). It is diminished (i.e. narrowed) because it is smaller than most of the fifths found in the scale (they are perfect fifths).
Size in different tuning systems
In twelve-tone equal temperament, the A4 is exactly half an
The half-octave or equal tempered A4 and d5 are unique in being equal to their own inverse (each to the other). In other meantone tuning systems, besides 12-tone equal temperament, A4 and d5 are distinct intervals because neither is exactly half an octave. In any meantone tuning near to 2⁄9-comma meantone the A4 is near to the ratio 7:5 (582.51) and the d5 to 10:7 (617.49), which is what these intervals are in septimal meantone temperament. In 31 equal temperament, for example, the A4 is 580.65 cents, whereas the d5 is 619.35 cents. This is perceptually indistinguishable from septimal meantone temperament.
Since they are the inverse of each other, by definition A4 and d5 always add up to exactly one perfect octave:
- A4 + d5 = P8.
On the other hand, two A4 add up to six whole tones. In equal temperament, this is equal to exactly one perfect octave:
- A4 + A4 = P8.
In quarter-comma meantone temperament, this is a diesis (128:125) less than a perfect octave:
- A4 + A4 = P8 − diesis.
In
These ratios are not in all contexts regarded as
Eleventh harmonic
The ratio of the eleventh harmonic, 11:8 (551.318 cents; approximated as F
Dissonance and expressiveness
Ján Haluska wrote:
The unstable character of the tritone sets it apart, as discussed in [Paul Hindemith. The Craft of Musical Composition, Book I. Associated Music Publishers, New York, 1945]. It can be expressed as a ratio by compounding suitable superparticular ratios. Whether it is assigned the ratio 64/45 or 45/32, depending on the musical context, or indeed some other ratio, it is not superparticular, which is in keeping with its unique role in music.[17]
Harry Partch has written:
Although this ratio [45/32] is composed of numbers which are multiples of 5 or under, they are excessively large for a 5-limit scale, and are sufficient justification, either in this form or as the tempered "tritone", for the epithet "diabolic", which has been used to characterize the interval. This is a case where, because of the largeness of the numbers, none but a temperament-perverted ear could possibly prefer 45/32 to a small-number interval of about the same width.
In the Pythagorean ratio 81/64 both numbers are multiples of 3 or under, yet because of their excessive largeness the ear certainly prefers 5/4 for this approximate degree, even though it involves a prime number higher than 3. In the case of the 45/32 "tritone" our theorists have gone around their elbows to reach their thumbs, which could have been reached simply and directly and non-"diabolically" via the number 7....[18]
Common uses
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Occurrences in diatonic scales
The augmented fourth (A4) occurs naturally between the fourth and seventh scale degrees of the major scale (for example, from F to B in the key of C major). It is also present in the natural minor scale as the interval formed between the second and sixth scale degrees (for example, from D to A♭ in the key of C minor). The melodic minor scale, having two forms, presents a tritone in different locations when ascending and descending (when the scale ascends, the tritone appears between the third and sixth scale degrees and the fourth and seventh scale degrees, and when the scale descends, the tritone appears between the second and sixth scale degrees). Supertonic chords using the notes from the natural minor mode thus contain a tritone, regardless of inversion. Containing tritones, these scales are tritonic.
Occurrences in chords
The dominant seventh chord in root position contains a diminished fifth (tritone) within its pitch construction: it occurs between the third and seventh above the root. In addition, augmented sixth chords, some of which are enharmonic to dominant seventh chords, contain tritones spelled as augmented fourths (for example, the German sixth, from A to D♯ in the key of A minor); the French sixth chord can be viewed as a superposition of two tritones a major second apart.
The diminished triad also contains a tritone in its construction, deriving its name from the diminished-fifth interval (i.e. a tritone). The half-diminished seventh chord contains the same tritone, while the fully diminished seventh chord is made up of two superposed tritones a minor third apart.
Other chords built on these, such as ninth chords, often include tritones (as diminished fifths).
Resolution
In all of the sonorities mentioned above, used in functional harmonic analysis, the tritone pushes towards resolution, generally resolving by
The augmented fourth resolves outward to a minor or major sixth (the first measure below). The inversion of this, a diminished fifth, resolves inward to a major or minor third (the second measure below). The diminished fifth is often called a tritone in modern
Other uses
The tritone is also one of the defining features of the Locrian mode, being featured between the and fifth scale degrees.
The half-octave tritone interval is used in the musical/auditory illusion known as the tritone paradox.
Historical uses
The tritone is a restless interval, classed as a
From then until the end of the Renaissance the tritone was regarded as an unstable interval and rejected as a consonance by most theorists.[21]
The name diabolus in musica (
It seems first to have been designated as a "dangerous" interval when Guido of Arezzo developed his system of hexachords and with the introduction of B flat as a diatonic note, at much the same time acquiring its nickname of "Diabolus in Musica" ("the devil in music").[27]
That original symbolic association with the devil and its avoidance led to Western cultural convention seeing the tritone as suggesting "evil" in music. However, stories that singers were
Later, with the rise of the Baroque and Classical music era, composers accepted the tritone, but used it in a specific, controlled way—notably through the principle of the tension-release mechanism of the
It is only with the
—or
In his early cantata
Roger Nichols (1972, p19) says that "the bare fourths, the wide spacing, the tremolos, all depict the words—'the light thrilled towards her'—with sudden, overwhelming power."[29] Debussy's String Quartet also features passages that emphasize the tritone.
The tritone was also exploited heavily in that period as an interval of
Later, in
Tritones also became important in the development of jazz tertian harmony, where triads and seventh chords are often expanded to become 9th, 11th, or 13th chords, and the tritone often occurs as a substitute for the naturally occurring interval of the perfect 11th. Since the perfect 11th (i.e. an octave plus perfect fourth) is typically perceived as a dissonance requiring a resolution to a major or minor 10th, chords that expand to the 11th or beyond typically raise the 11th a semitone (thus giving us an augmented or sharp 11th, or an octave plus a tritone from the root of the chord) and present it in conjunction with the perfect 5th of the chord. Also in jazz harmony, the tritone is both part of the dominant chord and its substitute dominant (also known as the sub V chord). Because they share the same tritone, they are possible substitutes for one another. This is known as a tritone substitution. The tritone substitution is one of the most common chord and improvisation devices in jazz.
In the theory of harmony it is known that a diminished interval needs to be resolved inwards, and an augmented interval outwards. ... and with the correct resolution of the true tritones this desire is totally satisfied. However, if one plays a just diminished fifth that is perfectly in tune, for example, there is no wish to resolve it to a major third. Just the opposite—aurally one wants to enlarge it to a minor sixth. The opposite holds true for the just augmented fourth. ...
These apparently contradictory aural experiences become understandable when the cents of both types of just tritones are compared with those of the true tritones and then read 'crossed-over'. One then notices that the just augmented fourth of 590.224 cents is only 2 cents bigger than the true diminished fifth of 588.270 cents, and that both intervals lie below the middle of the octave of 600.000 cents. It is no wonder that, following the ear, we want to resolve both downwards. The ear only desires the tritone to be resolved upwards when it is bigger than the middle of the octave. Therefore the opposite is the case with the just diminished fifth of 609.776 cents.[8]
See also
- List of meantone intervals
- List of musical intervals
- List of pitch intervals
- Ditone
- Tone
- Hexatonic scale § Tritone scale
- Consecutive fifths § Unequal fifths
- Petrushka chord
- Tritonic scale
References
- ISBN 978-1-56159-263-0. Retrieved August 31, 2020.
- Jacobus Leodiensis, Speculum musicae, Liber secundus, in Jacobi Leodiensis Speculum musicae, edited by Roger Bragard, Corpus Scriptorum de Musica 3/2 ([Rome]: American Institute of Musicology, 1961): 128–31, citations on 192–96, 200, and 229; Jacobus Leodiensis, Speculum musicae, Liber sextus, in Jacobi Leodiensis Speculum musicae, edited by Roger Bragard, Corpus Scriptorum de Musica 3/6 ([Rome]: American Institute of Musicology, 1973): 1–161, citations on 52 and 68; Johannes Torkesey, Declaratio et expositio, London: British Library, Lansdowne MS 763, ff.89v-94v, citations on f.92r,2–3; Prosdocimus de Beldemandis, Tractatus musice speculative, in D. Raffaello Baralli and Luigi Torri, "Il Trattato di Prosdocimo de' Beldomandi contro il Lucidario di Marchetto da Padova per la prima volta trascritto e illustrato", Rivista Musicale Italiana 20 (1913): 731–62, citations on 732–34.
- twelve-tone musicas well."
- ^
ISBN 0-19-311906-4.
- ISBN 978-0-07-294262-0.
- JSTOR 833435.
- ^ . Retrieved July 22, 2021
- ^ ISBN 1-902636-46-5.
- ISBN 1-4191-7893-8. "Cents in interval: 590, Name of Interval: Just Tritone, Number to an Octave: 2.0. Cents in interval: 612, Name of Interval: Pyth. Tritone, Number to an Octave: 2.0."
- ISBN 0-8247-4714-3. "25:18 classic augmented fourth".
- ^ Haluska (2003), p. xxv. "36/25 classic diminished fifth".
- ^ Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction, p. 165. Theodore Baker, trans. G. Schirmer.
- ^ Haluska (2003). p. xxiii. "7:5 septimal or Huygens' tritone, Bohlen-Pierce fourth", "10:7 Euler's tritone".
- ISBN 0-520-22409-4. "...septimal tritone, 10:7; smaller septimal tritone, 7:5;...This list is not exhaustive, even when limited to the first sixteen partials. Consider the very narrow augmented fourth, 13:9....just intonation is not an attempt to generate necessarily consonant intervals."
- ISBN 9780253347664.
- ISBN 9780199298938.
- ^ Haluska (2003), p. 286.
- ^ Partch (1974), p. 115.
- JSTOR 764089.
- ^ Guido d'Arezzo, Epistola de ignoto cantu, lines 309–322[full citation needed][failed verification]
- Oxford Music Online. Retrieved 2008-07-21.
- ^ Randel (2003), p.239.
- ^ Andreas Werckmeister. Harmonologia musica, oder kurze Anleitung zur musicalischen Composition (Frankfurt and Leipzig: Theodor Philipp Calvisius 1702): 6.
- ^ Andreas Werckmeister, Musicalische Paradoxal-Discourse, oder allgemeine Vorstellungen (Quedlinburg: Theodor Philipp Calvisius, 1707): 75–76.
- OCLC 1390982.
... mi contra fa ... welches die alten den Satan in der Music nenneten ... alten Solmisatores dieses angenehme Intervall mi contra fa oder den Teufel in der Music genannt haben.
- .
- ISBN 0-19-311316-3
- ISBN 0-486-27036-X.
- ^ Nichols, R. (1972). Debussy. Oxford University Press.
- ^ Rushton, Julian (1983). The Musical Language of Berlioz. Cambridge University Press. p. 254.
- ^ Rushton, Julian (2001). The Music of Berlioz. Oxford University Press.
- OCLC 398434.
- OCLC 240301.
- ^ "Musical Analysis of the War Requiem". Retrieved 16 March 2016.
- ^ "Britten: War Requiem". YouTube.
- ^ Bridcut, J. (2010), Essential Britten, a pocket guide for the Britten Centenary. London, Faber.
- ^ Kogan, Judith (2017-10-31). "The Unsettling Sound Of Tritones, The Devil's Interval". NPR. Retrieved 2021-11-11.
- ^ Rizzi, Sofia (2019-03-04). "Why did Bernstein build West Side Story around 'The Devil's Interval'?". Classic FM. Retrieved 2021-11-11.
- ^ Dominic Pedler. The Songwriting Secrets of the Beatles. Music Sales Ltd. Omnibus Press. London, 2010 pp. 522–523
- ^ Moskowitz, D. (2010). The Words and Music of Jimi Hendrix. Praeger.
- ^ Chesna, James (26 February 2010). "'Sleeping (In the Fire)': Listening Room fearless leader faces down fear". WJRT-TV/DT. Archived from the original on 29 June 2011. Retrieved 28 February 2010.
- ISBN 978-1118397596
Further reading
- R., Ken (2012). DOG EAR Tritone Substitution for Jazz Guitar, Amazon Digital Services, Inc., ASIN: B008FRWNIW