Tritone

In music theory, a tritone is a musical interval spanning three whole tones. For instance, the interval from F to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B.

In 12-tone-equal temperament, the tritone divides the octave exactly in half, making it six semitones, or 600 cents.
In traditional functional harmony, the tritone is a harmonic and melodic dissonance and tritones in chords push toward resolution. For instance, the tritone found in diminished triads as well as the dominant, half-diminished, and fully diminished seventh chords push toward resolution to the tonic. On the other hand, the tritone can also be used to avoid tonality altogether, as composer Reginald Smith Brindle explains: "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."

Definition

A tritone is composed of three whole tones. There are two possible interpretations of this, which are discussed below. Under the narrow definition, only augmented fourths are considered tritones, while under the broader definition, augmented fourths and diminished fifths —as well as rarer intervals like doubly augmented third and a doubly diminished sixth—are all considered tritones. The augmented fourth is the interval produced by widening the perfect fourth by one semitone, while the diminished fifth is produced by narrowing the perfect fifth by one semitone.
Under the narrow definition, each of the three whole tones that compose a tritone must be a diatonic step, so only the interval of an augmented fourth is considered a tritone. By this definition, within a diatonic scale there is only one tritone per octave. For instance, in the C major scale, the augmented fourth F–B is the only tritone because it is composed of three major seconds, while its inversion, the diminished fifth B–F, is not considered a tritone because three major seconds above B is E, not F.

More broadly defined, however, a tritone may include any interval spanning six semitones, regardless of scale degree. According to this definition, a diatonic scale contains two tritones for each octave. For instance, the C major scale contains the tritones, F–B and B–F. With this broad definition, a tritone can typically be classified as either an augmented fourth or a diminished fifth, though far rarer spellings of the notes in a tritone may be classified as a doubly augmented third, a doubly diminished sixth, etc.

Dissonance and expressiveness

Ján Haluska wrote:
Harry Partch has written:

In tonal music

In major and minor scales

In major scales, there is an augmented fourth between the fourth and seventh scale degrees.
In natural minor scales, there is a diminished fifth between the second and sixth scale degrees.
In harmonic minor scales, there is a diminished fifth between the second and sixth scale degrees and an augmented fourth between the fourth and seventh scale degrees.
Melodic minor scales, having two forms, contain tritones in different places when ascending and descending. When ascending, there are augmented fourths between the third and sixth scale degrees and between the fourth and seventh scale degrees. When descending, there is a diminished fifth 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 referred to as tritonic. A scale without tritones is called atritonic.

In tonal harmony

contain a diminished fifth between their third and seventh chord factors. Diminished triads also contains a tritone in their construction between their root and fifth. Half-diminished seventh chords contain the same tritone, while fully diminished seventh chords are composed of two superposed tritones a minor third apart. Other chords built on these, such as ninth chords, often include tritones as diminished fifths.
In addition, augmented sixth chords contain tritones spelled as augmented fourths. The Italian and German sixth chords each contain one augmented fourth, while the French sixth chord is composed of two superposed augmented fourths a major second apart.
In traditional functional harmony, the tritone in all of the chords described above push towards resolution, generally resolving by step in contrary motion. This determines the resolution of chords containing tritones; that is, augmented fourths resolve outward to a minor or major sixth, while diminished fifths resolve inward to a major or minor third.
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Historical uses

Classical music

Medieval and Rennaissance periods

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, who suggested that rather than make B a diatonic note, the hexachord be moved and based on C to avoid the F–B tritone altogether. Later theorists such as Ugolino d'Orvieto and Tinctoris advocated the inclusion of B.
From then until the end of the Renaissance, the tritone was regarded as an unstable interval and rejected as a consonance by most theorists. The name has been applied to the interval from at least the early 18th century or the late Middle Ages, though its use is not restricted to the tritone, being that the original found example of the term "diabolus en musica" is "Mi Contra Fa est diabolus en musica", referring to the minor second. Andreas Werckmeister cites this term in 1702 as being used by "the old authorities" for both the tritone and for the clash between chromatically related tones such as F and F, and five years later likewise calls "diabolus in musica" the opposition of "square" and "round" B because these notes represent the juxtaposition of "mi contra fa".
Johann Joseph Fux cites the phrase in his seminal 1725 work Gradus ad Parnassum, Georg Philipp Telemann in 1733 describes, "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 italic=no or 'the devil in music'." 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. However Denis Arnold, in the New Oxford Companion to Music, suggests that the nickname was already applied early in the medieval music itself:
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 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", mi and fa refer to notes from two adjacent hexachords. For instance, in the tritone B–F, B would be mi—the third scale degree in the hard hexachord beginning on G—while F would be fa—the fourth scale degree in the natural hexachord beginning on C.

Common practice period

Later, during the Baroque and Classical periods, composers accepted the tritone, but used it in a specific, controlled way—notably through the principle of the tension-release mechanism of the tonal system. In that system, 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 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.
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 culturally associated with it, such as Franz Liszt's use of the tritone to suggest Hell in his Dante Sonata:
Wagner uses timpani tuned to C and F to convey a brooding atmosphere at the start of the second act of the opera Siegfried.
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. For example, the climax of Hector Berlioz's La damnation de Faust consists of a transition between "huge B and F chords" as Faust arrives in Pandaemonium, the capital of Hell. Musicologist Julian Rushton calls this "a tonal wrench by a tritone".

20th century music

In his early cantata La Damoiselle élue, Debussy uses a tritone to convey the words of the poem by Dante Gabriel Rossetti:
Roger Nichols says that "the bare fourths, the wide spacing, the tremolos, all depict the words—'the light thrilled towards her'—with sudden, overwhelming power." Debussy's String Quartet also features passages that emphasize the tritone:
Later, in twelve-tone music, serialism, and other 20th century compositional idioms, composers considered it a neutral interval. In some analyses of the works of 20th century composers, the tritone plays an important structural role; perhaps the most cited is the axis system, proposed by Ernő Lendvai, in his analysis of the use of tonality in the music of Béla Bartók.
Benjamin Britten's War Requiem features a tritone between C and F♯ as a recurring motif. John Bridcut describes the power of the interval in creating the sombre and ambiguous opening of the War Requiem: