| 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 |
Inmusic theory, atritone is amusical interval spanning threewhole tones.[1] For instance, the interval from F 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.
In12-tone-equal temperament, the tritone divides theoctave (which is 12 semitones or 1200cents) exactly in half, making it six semitones, or 600 cents.[2]
In traditionalfunctional harmony, the tritone is a harmonic and melodicdissonance and tritones in chords push towardresolution. For instance, the tritone(s) found indiminished triads as well as thedominant,half-diminished, andfully diminished seventh chords push toward resolution to thetonic. On the other hand, the tritone can also be used to avoidtonality 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."[3]
A tritone is composed of threewhole tones. There are two possible interpretations of this, a narrow definition and a broad definition. Under the narrow definition, onlyaugmentedfourths (often abbreviated as A4) are considered tritones, while under the broad definition, augmented fourths anddiminishedfifths (d5)—as well as rarer intervals likedoubly augmentedthirds and adoubly diminishedsixths—are all considered tritones. The augmented fourth is the interval produced bywidening theperfect fourth by onesemitone (without changing either letter name), while the diminished fifth is produced bynarrowing theperfect fifth by one semitone (without changing either letter name).[4]
Under the narrow definition, each of the three whole tones that compose a tritone must be adiatonic step, so only the interval of an augmented fourth is considered a tritone. By this definition, within a diatonic scale (such as amajor scale) there is only one tritone peroctave. For instance, in the C major scale, the augmented fourth F–B is the only tritone because it is composed of threemajor seconds (F–G, G–A, and A–B), while itsinversion, the diminished fifth B–F, isnot considered a tritone because three major seconds above B is E♯, not F.
Under the broad definition, however, a tritone may includeany interval spanning six semitones, regardless ofscale 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.[5] With this broad definition, a tritone can typically be classified as either an augmented fourth or a diminished fifth, though far rarerspellings of the notes in a tritone may be classified as a doubly augmented third, a doubly diminished sixth, etc.
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 suitablesuperparticular 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.[6]
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 atemperament-perverted ear could possibly prefer 45/32 to a small-number interval of about the same width.
In thePythagorean 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....[7]
Inmajor scales, there is an augmented fourth between the fourth and seventhscale degrees (e.g., F–B inC major).
Innatural minor scales, there is a diminished fifth between the second and sixth scale degrees (e.g., D–A♭ inC minor).
Inharmonic 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 (e.g., D–A♭ and F–B, respectively, in C minor).
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 (e.g., E♭–A and F–B, respectively, in C minor). When descending, there is a diminished fifth between the second and sixth scale degrees (e.g., D–A♭ in C minor).
Supertonic chords using the notes from the natural minor mode thus contain a tritone, regardless of inversion.
Containing tritones, these scales are referred to astritonic. A scale without tritones is calledatritonic.
Dominant seventh chords contain a diminished fifth (tritone) between theirthird andseventhchord factors.Diminished triads also contains a tritone in their construction between theirroot andfifth.Half-diminished seventh chords contain the same tritone, whilefully diminished seventh chords are composed of two superposed tritones aminor third apart. Other chords built on these, such asninth chords, often include tritones as diminished fifths.
In addition,augmented sixth chords contain tritones spelled as augmented fourths. TheItalian andGerman sixth chords each contain one augmented fourth, while theFrench sixth chord is composed of two superposed augmented fourths amajor second apart.
In traditionalfunctional harmony, the tritone(s) in all of the chords described above push towardsresolution, generally resolving bystep incontrary motion. This determines the resolution of chords containing tritones; that is, augmented fourths resolve outward to aminor ormajor sixth (the first measure below), while diminished fifths resolve inward to amajor or minor third (the second measure below).
The tritone is a restless interval, classed as adissonance in Western music from the earlyMiddle Ages through to the end of thecommon 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 ofGuido of Arezzo'shexachordal 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 asUgolino d'Orvieto andTinctoris advocated the inclusion of B♭.[8]
From then until the end of theRenaissance, the tritone was regarded as an unstable interval and rejected as a consonance by most theorists.[9] The namediabolus in musica (Latin for 'theDevil in music') has been applied to the interval from at least the early 18th century or the late Middle Ages,[10] 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" ("Mi againstFa is the devil in music"), referring to theminor 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♯,[11] and five years later likewise calls"diabolus in musica" the opposition of "square" and "round" B (B♮ and B♭, respectively) because these notes represent the juxtaposition of"mi contra fa".[12]
Johann Joseph Fux cites the phrase in his seminal 1725 workGradus ad Parnassum,Georg Philipp Telemann in 1733 describes, "mi against fa", which the ancients called "Satan in music"—andJohann Mattheson, in 1739, writes that the "older singers with solmization called this pleasant interval'mi contra fa' or 'the devil in music'."[13] 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.[14] HoweverDenis Arnold, in theNew Oxford Companion to Music, suggests that the nickname was already applied early in themedieval 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").[15]
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 wereexcommunicated 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 parallelorganum of theMusica Enchiriadis. In all these expressions, including the commonly cited"mi contra fa est diabolus in musica",mi andfa refer to notes from two adjacent hexachords. For instance, in the tritone B–F, B would be mi—the thirdscale degree in the hard hexachord beginning on G—while F would be fa—the fourth scale degree in the natural hexachord beginning on C.
Later, during theBaroque andClassical periods, composers accepted the tritone, but used it in a specific, controlled way—notably through the principle of the tension-release mechanism of thetonal system. In that system, the tritone is one of the defining intervals of thedominant seventh chord and two tritones separated by a minor third give thefully diminished seventh chord its characteristic sound. In minor, thediminished triad appears on the second scale degree—and thus features prominently in the progressioniio–V–i. Often, theinversion iio6 is used to move the tritone to the inner voices as this allows forstepwise 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.[16]
It is only with theRomantic music andmodern 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 asFranz Liszt's use of the tritone to suggest Hell in hisDante Sonata:
Wagner uses timpani tuned to C and F♯ to convey a brooding atmosphere at the start of the second act of the operaSiegfried.
The tritone was also exploited heavily in that period as an interval ofmodulation for its ability to evoke a strong reaction by moving quickly todistantly related keys. For example, the climax ofHector Berlioz'sLa damnation de Faust consists of a transition between "huge B and F chords" as Faust arrives inPandaemonium, the capital of Hell.[17] MusicologistJulian Rushton calls this "a tonal wrench by a tritone".[18]
In his early cantataLa Damoiselle élue,Debussy uses a tritone to convey the words of the poem byDante Gabriel Rossetti:
Roger Nichols (1972, p. 19) says that "the bare fourths, the wide spacing, the tremolos, all depict the words—'the light thrilled towards her'—with sudden, overwhelming power."[19] Debussy'sString Quartet also features passages that emphasize the tritone:
Later, intwelve-tone music,serialism, and other 20th century compositional idioms, composers considered it a neutral interval.[20] In some analyses of the works of 20th century composers, the tritone plays an important structural role; perhaps the most cited is theaxis system, proposed byErnő Lendvai, in his analysis of the use of tonality in the music ofBéla Bartók.[21]
Benjamin Britten'sWar Requiem features a tritone between C and F♯ as a recurringmotif.[22]John Bridcut (2010, p. 271) describes the power of the interval in creating the sombre and ambiguous opening of theWar Requiem:
The idea that the chorus and orchestra are confident in their wrong-headed piety is repeatedly disputed by the music. From the instability of the opening tritone—that unsettling interval between C and F sharp—accompanied by the tolling of warning bells... eventually resolves into a major chord for the arrival of the boys singing "Te decet hymnus."[23]
Leonard Bernstein uses the tritone as a basis for much of his musicalWest Side Story.[24][25] As Timothy Judd writes, "It's the first interval we hear in the opening Prologue. It returns prominently in 'Maria' and 'Cool.' It even opens each verse of the comically sardonic 'Gee, Officer Krupke.' If there is amotivic counterweight, it is the yearning sunlight of the expansiveminor seventh, heard in 'Somewhere' and 'I Have a Love.'"[26]
George Harrison uses tritones on the downbeats of the opening phrases ofthe Beatles songs "The Inner Light", "Blue Jay Way", and "Within You Without You", creating a prolonged sense of suspended resolution.[27] Perhaps the most striking use of the interval in rock music of the late 1960s can be found inJimi Hendrix's song "Purple Haze". According to Dave Moskowitz (2010, p. 12), Hendrix "ripped into 'Purple Haze' by beginning the song with thesinister sounding tritone interval creating an opening dissonance, long described as 'The Devil in Music'."[28] The opening riff of "Black Sabbath", the first song onBlack Sabbath'seponymous debut album, is an inversion of a tritone;[29] the album, and this song in particular, are considered to mark the birth ofheavy metal music.[30]
Tritones also became important in the development ofjazz harmony, where triads and seventh chords are often expanded to becomeextended chords (likeninth andeleventh chords), and the tritone often occurs as a substitute for the naturally occurring interval of theperfect eleventh. Since the perfect eleventh is typically perceived as a dissonance requiring a resolution to amajor orminor tenth, chords that expand to the eleventh or beyond typically raise the eleventh a semitone (resulting in an augmented orsharp eleventh, or an octave plus a tritone from the root of the chord) and present it in conjunction with the perfect fifth of the chord.
Also, in jazz harmony, the tritone is both part of the dominant chord and itssubstitute dominant (also known as the sub V chord). Because they share the same tritone, they are possible substitutes for one another, which is known as atritone 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 thetrue tritones this desire is totally satisfied. However, if one plays ajust 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 thejust 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.[31]
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In12-tone equal temperament, the tritone is exactly half of anoctave (i.e., a ratio of√2:1, or 600cents. This means that the augmented fourth and diminished fifth are the unique intervals that are each others'inverses.
In othermeantone tuning systems, the augmented fourth and diminished fifth are distinct intervals because neither is exactly half of an octave. In any meantone tuning near to2/9-comma meantone the augmented fourth is approximately the ratio 7:5 (582.51), while the diminished fifth is approximately 10:7 (617.49), which is what these intervals are inseptimal meantone temperament.
In31-tone equal temperament, for example, the augmented fourth is 580.65 cents, while the diminished fifth is 619.35 cents. This is perceptually indistinguishable from septimal meantone temperament.
Since they are the inverse of each other, by definition, the augmented fourth and diminished fifth add up to exactly oneoctave (i.e., 600 cents + 600 cents = 1200 cents):
On the other hand, two augmented fourth add up to sixwhole tones. In equal temperament, this is equal to exactly one octave:
Inquarter-comma meantone temperament, however, this is adiesis (128:125) less than an octave:
Injust intonation, several different sizes can be chosen both for the augmented fourth and the diminished fifth. For instance, in5-limit tuning, the augmented fourth is either 45:32[32][31][33] or 25:18,[34] and the diminished fifth is either 64:45 or 36:25.[35] The 64:45 just diminished fifth arises in the Cmajor scale between B and F, resulting in the 45:32 augmented fourth arising between F and B.[36]
These ratios are not in all contexts regarded asstrictly just but they are the justest possible in 5-limit tuning.[further explanation needed]Seven-limit tuning allows for the justest possible ratios (ratios with the smallest numerator and denominator), namely 7:5 for the augmented fourth (about 582.5 cents, also known asseptimal tritone) and 10:7 for the diminished fifth (about 617.5 cents, also known asEuler's tritone).[32][37][38] 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 cents.
The ratio of the eleventhharmonic, 11:8 (551.318 cents; approximated as F
4 above C1 inscientific pitch notation), known as the lesserundecimal tritone orundecimal semi-augmented fourth, is found in somejust tunings and on many instruments.
For example, very longalphorns may reach the twelfth harmonic and transcriptions of their music usually show the eleventh harmonic sharp (F♯ above C, for example), as inBrahms'sFirst Symphony.[40] This note is often corrected to 4:3 on thenatural horn in just intonation or Pythagorean tunings, but the pure eleventh harmonic was used in pieces such asBritten'sSerenade for Tenor, Horn and Strings.[41]Ivan Wyschnegradsky considered themajor fourth a good approximation of the eleventh harmonic.
... 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.
{{cite book}}:ISBN / Date incompatibility (help)translated from German
Name of interval:Just Tritone, cents in interval: 590, number to an octave: 2 ; Name of interval:Pyth. Tritone, cents in interval: 612, number to an octave: 2
25:18 classic augmented fourth
musical interval 'pythagorean major third'
... 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.
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