Tritone

For other uses, see Tritone (disambiguation).

tritone

Inverse	tritone
Name
Other names	augmented fourth, diminished fifth
Abbreviation	TT
Size
Semitones	6
Interval class	6
Just interval	25:18, 45:32, 10:7, ...
Cents
Equal temperament	600
24 equal temperament	600
Just intonation	569, 590, 617, ...

The augmented fourth between C and F♯ forms a tritone.  Play EQ (help·info)  Play 25:18 (help·info)

Tritone: just augmented fourth between C and F♯+  Play 45:32 (help·info).

Tritone: Pythagorean augmented fourth between C and F♯++  Play 729:512 (help·info).

The diminished fifth between C and G♭ forms a tritone.  Play EQ (help·info)  Play 1024:729 (help·info)  Play 64:45 (help·info)  Play 36:25 (help·info)

Lesser septimal tritone on C^[1]  Play 7:5 (help·info).

Greater septimal tritone on C^[1]  Play 10:7 (help·info).

In classical music from Western culture, the tritone ( Play (help·info), tri- "three" and tone) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones. In this context, a tritone may also be defined as any interval spanning six semitones.

Since a chromatic scale is formed by 12 pitches, it contains 12 distinct tritones, each of which starting from a different pitch. According to a widely used naming convention, six of them are classified as augmented fourths, and the other six as diminished fifths. In a diatonic scale there's only one tritone, classified as an augmented fourth. For instance, in the C major diatonic scale the only interval formed by three adjacent tones (F-G, G-A, and A-B) is that from F to B.

In the above-mentioned naming convention, a fourth is an interval encompassing four staff positions, while a fifth encompasses five staff positions (see interval number for more details). The augmented fourth (A4) and diminished fifth (d5) are defined as the intervals produced by widening and narrowing by one chromatic semitone the perfect fourth and fifth, respectively.^[2] They both span six semitones, and they are the inverse of each other, meaning that their sum is exactly equal to one perfect octave (A4 + d5 = P8). In 12-tone equal temperament, the most commonly used tuning system, the A4 is equivalent to a d5, as both have the size of exactly half an octave. In most other tuning systems, they are not equivalent, and neither is equal to half an octave. The d5 is also called semidiapente.

The tritone is often used as the main interval of dissonance in Western harmony, and is important in the study of musical harmony. "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."^[3]

Definitions

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 be also written as follows:

TT = T+T+T

Only if the three tones are of the same size (which is not the case for many tuning systems) can this formula be simplified to:

TT = 3T

This definition, however, has two different interpretations (broad and strict).

Broad interpretation (chromatic scale)

If a chromatic scale is used, with its 12 notes it is possible to define 12 different tritones, each of which starting from a different note. Six of them are A4, and the other six are d5. Therefore, in this case both A4 and d5 are considered to be tritones. Since each whole tone, in a chromatic scale, can be divided into two semitones:

T = S+S

then three tones are equal to six semitones. In this case, we can generalize the definition of tritone as follows:

TT = T+T+T = S+S+S+S+S+S.

This means that a tritone can be also defined as any musical interval spanning six semitones (indeed, both A4 and d5 are intervals spanning 6 semitones).

Only when the semitones (and the tones formed by pairs of semitones) are equal in size can this formula be simplified to:

TT = 3T = 6S.

Strict interpretation (diatonic scale)

In a diatonic scale, whole tones are regarded as incomposite intervals (that is, they do not divide into smaller intervals). Therefore, 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, and it is sometimes called the proper tritone. 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 most of the fourths found in the scale have smaller size (they are perfect fourths).

According to this interpretation, the d5 is not a tritone. Indeed, in a diatonic scale, there's 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 most of the fifths found in the scale have larger size (they are perfect fifths).

Alternative definition

Some contemporary authors define a tritone as any interval spanning exactly half an octave, including both the A4 and d5 as tuned in 12-tone equal temperament. This is not consistent with the above mentioned traditional definition (TT = T+T+T).

In this case, context may resolve the tritone to more an A4, a d5, or a "neutral" interval with no clear conventional classification.

Size in different tuning systems

In 12-tone equal temperament, the A4 is exactly half an octave (i.e., a ratio of √2:1 or 600 cents;  play (help·info)). The inverse of 600 cents is 600 cents. Thus, in this tuning system, the A4 and its inverse (d5) are equivalent.

The half-octave A4 is unique in being equal to its own inverse. 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 will be near to the ratio 7⁄5 and the d5 to 10⁄7, which is what these intervals are taken to be in septimal meantone temperament. In 31 equal temperament, for example, the A4, a 10⁄7, or tritone proper, is 617.49 cents, whereas a 7⁄5 is 582.51 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 (proper tritones) 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 just intonation several different sizes can be chosen both for the A4 and the d5. For instance, in 5-limit tuning, the A4 is either 45/32^[4][5][6] or 25/18,^[7] and the d5 is either 64/45 or 36/25.^[8] These ratios are not in all contexts regarded as strictly just, but they are the justest possible in 5-limit tuning. 7-limit tuning allows for the justest possible ratios, namely 7/5 for the A4 (about 582.5 cents, also known as septimal tritone) and 10/7 for the d5 (about 617.5 cents, also known as Euler's tritone).^[4][9][10] These ratios are more consonant than 17/12 (about 603.0 cents) and 24/17 (about 597.0 cents), which can be obtained in 17-limit tuning, yet the latter are also fairly common, as they are closer to the equal-tempered value of 600.0 cents.

Dissonance and expressiveness

Compared to other commonly occurring intervals like the major second or the minor third, the augmented fourth and the diminished fifth (both two valid enharmonic interpretations of the tritone) are considered awkward intervals to sing. Western composers have traditionally avoided using it explicitly in their melody lines, often preferring to use passing tones or extra note skipping instead of using a direct leap of an augmented fourth or diminished fifth in their melodies. However, as time went by, composers have gradually used the tritone more and more in their music, disregarding its awkwardness and exploiting its expressiveness.^{[citation needed]}

The unstable character of the tritone sets it apart, as discussed in [28] [P. Hindemith. The Crafts 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.^[11]

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.^[12]

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 number 7.^[12]

Common uses

Occurrences in diatonic scales

The proper tritone (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 will thus contain a tritone, regardless of inversion.

Occurrences in chords

The dominant seventh chord contains a (d5) tritone within its tone 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 chord 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

Tritone resolution inward.

In all of the sonorities mentioned above, used in functional harmonic analysis, the tritone pushes towards resolution, generally resolving by step in contrary motion. The inversion of this, a diminished fifth resolving inward to a major third, is often loosely called a tritone as well in modern tonal theory, but functionally and notationally it can only resolve inwards as a kind of fifth and is therefore not reckoned a tritone in baroque and renaissance music theory.

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 dissonance in Western music from the early Middle Ages through to the end of the common practice period. This interval was frequently avoided in medieval ecclesiastical singing because of its dissonant quality. The first explicit prohibition of it seems to occur with the development of Guido of Arezzo's hexachordal system, which made B♭ a diatonic note, namely as the fourth degree of the hexachord on F. From then until the end of the Renaissance the tritone, nicknamed the diabolus in musica, was regarded as an unstable interval and rejected as a consonance by most theorists.^[13]

The name diabolus in musica ("the Devil in music") has been applied to the interval from at least the early 18th century. Johann Joseph Fux cites the phrase in his seminal 1725 work Gradus ad Parnassum, Georg Philipp Telemann in 1733 notes, "mi against fa", which the ancients called "Satan in music", and Johann Mattheson in 1739 writes that the "older singers with solmization called this pleasant interval 'mi contra fa' or 'the devil in music'".^[14] Although the latter two of these authors cite the association with the devil as from the past, there are no known citations of this term from the Middle Ages, as is commonly asserted.^[15] However Denis Arnold, in the The New Oxford Companion to Music, suggests that the nickname was already applied early in the medieval music itself:

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").^[16]

Because of that original symbolic association with the devil and its avoidance, this interval came to be heard in Western cultural convention as suggesting an "evil" connotative meaning in music. Today the interval continues to suggest an "oppressive", "scary", or "evil" sound. However, suggestions that singers were excommunicated or otherwise punished by the Church for invoking this interval are likely fanciful. At any rate, avoidance of the interval for musical reasons has a long history, stretching back to the parallel organum of the Musica Enchiriadis. In all these expressions, including the commonly cited "mi contra fa est diabolus in musica", the "mi" and "fa" refer to notes from two adjacent hexachords. For instance, in the tritone B-F, B would be "mi", that is the third scale degree in the "hard" hexachord beginning on G, while F would be "fa", that is the fourth scale degree in the "natural" hexachord beginning on C.

Later in history with the rise of the Baroque and Classical music era, that interval came to be perfectly accepted, but yet was used in a specific controlled way, notably through the principle of the tension/release mechanism of the tonal system. In that system (which is the fundamental musical grammar of Baroque and Classical music), the tritone is one of the defining intervals of the dominant-seventh chord and two tritones separated by a minor third give the fully diminished seventh chord its characteristic sound. In minor, the diminished triad (comprising two minor thirds which together add up to a tritone) appears on the second scale degree, and thus features prominently in the progression ii^o-V-i. Often, the inversion ii^o6 is used to move the tritone to the inner voices as this allows for stepwise motion in the bass to the dominant root. In three-part counterpoint, free use of the diminished triad in first inversion is permitted, as this eliminates the tritone relation to the bass.^[17]

It is only with the Romantic music and modern classical music that composers started to use it totally freely, without functional limitations notably in an expressive way to exploit the "evil" connotations which are culturally associated to it (e.g., Liszt's use of the tritone to suggest hell in his Dante Sonata). The tritone was also exploited heavily in that period as an interval of modulation for its ability to evoke a strong reaction by moving quickly to distantly related keys. Later on, in twelve-tone music, serialism, and other 20th century compositional idioms it came to be considered as a neutral interval.^[18] In some analyses of the works of 20th century composers, the tritone plays an important structural role; perhaps the most noted is the axis system, proposed by Ernő Lendvai, in his analysis of the use of tonality in the music of Béla Bartók.^[19] Tritone relations are also important in the music of George Crumb.

Tritone substitution: F♯7 may substitute for C7, and vice versa, because they both share E♮ and B♭/A♯ and due to voice leading considerations

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 half step (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....^[5]

See also

• List of meantone intervals
• List of musical intervals
• Hexatonic scale#Tritone scale

Footnotes

1. ^ ^{a b} Fonville, John. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p. 121–22, Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991), pp. 106–37.
2. ^ Bruce Benward & Marilyn Nadine Saker (2003). Music: In Theory and Practice, Vol. I, seventh edition (Boston: McGraw-Hill), p. 54. ISBN 978-0-07-294262-0.
3. ^ Brindle, Reginald Smith (1966). Serial Composition. Oxford University Press. pp. 66. ISBN 0-19-311906-4. http://books.google.com/books?id=gvNLXHpusw8C. 
4. ^ ^{a b} Partch, Harry. (1974). Genesis Of A Music: An Account of a Creative Work, Its Roots and Its Fulfillments, second edition, enlarged (New York: Da Capo Press): p. 69. ISBN 030671597X (cloth); ISBN 030680106X (pbk).
5. ^ ^{a b} Renold, Maria (2004). Intervals, Scales, Tones and the Concert Pitch C=128Hz, translated from the German by Bevis Stevens, with additional editing by Anna R. Meuss (Forest Row: Temple Lodge): p. 15–16. ISBN 1902636465.
6. ^ Helmholtz, Hermann von (2005). On the Sensations of Tone as a Physiological Basis for the Theory of Music, p. 457. ISBN 1419178938. "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."
7. ^ Haluska , Ján (2003), The Mathematical Theory of Tone Systems, Pure and Applied Mathematics Series 262 (New York: Marcel Dekker; London: Momenta), p. xxiv. ISBN 0824747143. "25:18 classic augmented fourth".
8. ^ Haluska (2003), p. xxv. "36/25 classic diminished fifth".
9. ^ Haluska (2003). p. xxiii. "7/5 septimal or Huygens' tritone, Bohlen-Pierce fourth", "10/7 Euler's tritone".
10. ^ Strange, Patricia and Patricia, Allen (2001). The contemporary violin: Extended performance techniques, p. 147. ISBN 0520224094. "...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."
11. ^ Haluska (2003), p. 286.
12. ^ ^{a b} Partch (1974), p. 115. ISBN 030680106X.
13. ^ Drabkin, William. "Tritone". Grove Music Online (subscription access). Oxford Music Online. http://www.oxfordmusiconline.com/subscriber/article/grove/music/28403. Retrieved 2008-07-21. 
14. ^ Reinhold, Hammerstein (1974) (in German). Diabolus in musica: Studien zur Ikonographie der Musik im Mittelalter. Neue Heidelberger Studien zur Musikwissenschaft. 6. Bern: Francke. pp. 7. 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." 
15. ^ F. J. Smith, "Some Aspects of the Tritone and the Semitritone in the Speculum Musicae: The Non-Emergence of the Diabolus in Music," Journal of Musicological Research 3 (1979), pp. 63-74, at 70.
16. ^ Arnold, Denis (1983) « Tritone » in The New Oxford Companion to Music, Volume 1: A-J,Oxford University Press. ISBN 0-19-311316-3
17. ^ Jeppesen, Knud (1992) [1939]. Counterpoint: the polyphonic vocal style of the sixteenth century. trans. by Glen Haydon, with a new foreword by Alfred Mann. New York: Dover. ISBN 048627036X. http://books.google.com/books?id=OcSVGkug58gC. 
18. ^ Persichetti, Vincent (1961). Twentieth-century Harmony: Creative Aspects and Practice. New York: W. W. Norton. ISBN 0-393-09539-8. OCLC 398434. 
19. ^ Lendvai, Ernő (1971). Béla Bartók: An Analysis of his Music. introd. by Alan Bush. London: Kahn & Averill. pp. 1–16. ISBN 0900707046. OCLC 240301. 

External links

• Tritone paradox and Shepard Tones at the Wayback Machine (archived January 6, 2008)
• BBC News Magazine article about the tritone
• Satan's all-time greatest hit: Will Hodgkinson on the devil's interval
• Tritone article by JazzPianoLessons.com