||augmented fourth, diminished fifth
7:5, 10:7, 25:18, 45:32, 64:45, 1024:729, 729:512, 36/25...
|24 tone equal temperament
||582.512, 617.488, 568.726, 590.223, 609.777, 588.269, 611.731, 631.283...
The tritone ( Play (help·info), tri- "three" and tone) is a musical interval that spans three whole tones. The tritone is the same as an augmented fourth, which in 12-tone equal temperament is enharmonic to a diminished fifth. It 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.".
Definition and nomenclature
Only the augmented fourth consists of three whole tones in meantone temperament, hence the derivation of the term "tritone". Calling the diminished fifth a "tritone" is parlance. Writers often use the term tritone to mean specifically half of an octave from a given tone, without regard to what system of tuning it may belong to. Two tritones add up to six whole tones, which in meantone temperament is a diesis less than an octave, but in equal temperament, where the diesis is tempered out, it is equal to a perfect octave. A common symbol for tritone is TT. It is also sometimes called a tritonus, the name used in German. An equal-tempered tritone may be heard here.
The equal-tempered tritone (a ratio of √2:1 or 600 cents) is unique in being its own octave inversion. Note that in other meantone tunings, the augmented fourth and the diminished fifth are distinct intervals because neither is exactly half an octave. In any meantone tuning near to 2⁄9 comma meantone the augmented fourth will be near to the ratio 7⁄5 and the diminished fifth to 10⁄7, which is what these intervals are taken to be in septimal meantone temperament. In 31 equal temperament, for example, the diminished fifth, or tritone proper, is 580.6 cents, whereas a 7⁄5 is 582.5 cents.
The half-octave tritone interval is used in the musical/auditory illusion known as the tritone paradox.
The unstable character of the tritone sets it apart, as discussed in  ["Hindemith P.: 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.
—Haluska (2003), p.286
The tritone 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.
The dominant seventh chord contains a 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.
In tonal music the tritone normally resolves inward to a major third:
Tritone resolution inward.
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).
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 tritone is also one of the defining features of the Locrian mode, being featured between the and fifth scale degrees.
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.
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."
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'". 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 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')".
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 iio-V-i. Often, the inversion iio6 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 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. 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. Tritone relations are also important in the music of George Crumb.
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 588.269 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 609.777 cents...
—Maria Renold (2004), p.15-16
^ a b Haluska, Jan (2003). The Mathematical Theory of Tone Systems, p.xxiii. ISBN 0824747143. "7/5 septimal or Huygens' tritone, Bohlen-Pierce fourth", "10/7 Euler's tritone".
^ a b c d Partch (1979). Genesis Of A Music, p.69. ISBN 030680106X.
^ Haluska (2003), p.xxiv. "25:18 classic augmented fourth".
^ a b c d e Renold, Maria (2004). Intervals, scales, tones and the concert pitch, p.15. ISBN 1902636465.
^ a b Haluska (2003), p.xxv. "36/25 classic diminished fifth".
^ Haluska (2003), p.xxvi. "64:45 2nd tritone".
^ a b Fonville, John. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p.121-122, Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991), pp. 106-137.
^ Brindle, Reginald Smith (1966). Serial Composition. Oxford University Press. pp. 66. ISBN 0-19-311906-4. http://books.google.com/books?id=gvNLXHpusw8C.
^ Haluska (2003), p.286.
^ Drabkin, William. "Tritone". Grove Music Online (subscription access). Oxford Music Online. http://www.oxfordmusiconline.com/subscriber/article/grove/music/28403. Retrieved 2008-07-21.
^ 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."
^ 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.
^ Arnold, Denis (1983) « Tritone » in The New Oxford Companion to Music, Volume 1: A-J,Oxford University Press. ISBN 0-19-311316-3
^ Jeppesen, Knud (1992). 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.
^ Persichetti, Vincent (1961). Twentieth-century Harmony: Creative Aspects and Practice. New York: W. W. Norton. ISBN 0-393-09539-8. OCLC 398434.
^ Lendvai, Ernő (1971). Béla Bartók: An Analysis of his Music. introd. by Alan Bush. London: Kahn & Averill. pp. 1–16. ISBN 0900707046. OCLC 240301.
|Numbers in brackets are the number of semitones in the interval.|
Fractional semitones are approximate.