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In
music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...
, the tritone is defined as a musical interval composed of three adjacent whole tones (six
semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...
s). 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 scale, so by this definition, within a
diatonic scale In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...
there is only one tritone for each
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the
C major C major (or the key of C) is a major scale based on C, consisting of the pitches C, D, E, F, G, A, and B. C major is one of the most common keys used in music. Its key signature has no flats or sharps. Its relative minor is A minor and ...
scale. More broadly, a tritone is also commonly defined as any interval with a width of three whole tones (spanning six
semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...
s in the chromatic scale), regardless of scale degrees. According to this definition, a diatonic scale contains two tritones for each octave. For instance, the above-mentioned C major scale contains the tritones F–B (from F to the B above it, also called augmented fourth) and B–F (from B to the F above it, also called diminished fifth, semidiapente, or semitritonus); the latter is decomposed as a semitone B–C, a whole tone C–D, a whole tone D–E, and a semitone E–F, for a total width of three whole tones, but composed as four steps in the scale. In twelve-equal temperament, the tritone divides the octave exactly in half as 6 of 12 semitones or 600 of 1,200 cents. In
classical music Classical music generally refers to the art music of the Western world, considered to be distinct from Western folk music or popular music traditions. It is sometimes distinguished as Western classical music, as the term "classical music" also ...
, the tritone is a harmonic and melodic dissonance and is important in the study of musical
harmony In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However ...
. The tritone can be used to avoid traditional tonality: "Any tendency for a
tonality Tonality is the arrangement of pitches and/or chords of a musical work in a hierarchy of perceived relations, stabilities, attractions and directionality. In this hierarchy, the single pitch or triadic chord with the greatest stability is call ...
to emerge may be avoided by introducing a note three whole tones distant from the key note of that tonality." 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 In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Often, especially in the ...
or chord containing tritones is called
tritonic A tritonic scale is a musical scale or mode with three notes per octave. This is in contrast to a heptatonic (seven-note) scale such as the major scale and minor scale, or a dodecatonic (chromatic 12-note) scale, both common in modern Western ...
; one without tritones is atritonic.


Augmented fourth and diminished fifth

Since a
chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...
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 A naming convention is a convention (generally agreed scheme) for naming things. Conventions differ in their intents, which may include to: * Allow useful information to be deduced from the names based on regularities. For instance, in Manhatta ...
, 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 staff positions, while a ''fifth'' encompasses five staff positions (see
interval number In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or h ...
for more details). The augmented fourth (''A4'') and diminished fifth (''d5'') are defined as the intervals produced by widening the
perfect fourth A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...
and narrowing the
perfect fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval fro ...
by one
chromatic semitone In modern Western tonal music theory an augmented unison or augmented prime is the interval between two notes on the same staff position, or denoted by the same note letter, whose alterations cause them, in ordinary equal temperament, to be one ...
. They both span six semitones, and they are the inverse of each other, meaning that their sum is exactly equal to one
perfect octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
(A4 + d5 = P8). In twelve-tone equal temperament, the most commonly used
tuning system In music, there are two common meanings for tuning: * Tuning practice, the act of tuning an instrument or voice. * Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases. Tuning practice Tun ...
, the A4 is
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...
to a d5, as both have the size of exactly half an
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
. In most other tuning systems, they are not equivalent, and neither is exactly equal to half an octave. 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 In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...
, and according to a rule explained elsewhere, 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 A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...
), 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 In modern Western tonal music theory, a diminished second is the interval produced by narrowing a minor second by one chromatic semitone.Bruce Benward and Marilyn Saker (2003). ''Music: In Theory and Practice, Vol. I'', p. 54. . Specific example ...
). Notice that the last diminished second is formed by two
enharmonically equivalent In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written no ...
notes (E and F). On a
piano keyboard A musical keyboard is the set of adjacent depressible levers or keys on a musical instrument. Keyboards typically contain keys for playing the twelve notes of the Western musical scale, with a combination of larger, longer keys and smaller, sh ...
, these notes are produced by the same key. However, in the above-mentioned naming convention, they are considered different notes, as they are written on different staff positions and have different diatonic functions within music theory.


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 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
tuning system In music, there are two common meanings for tuning: * Tuning practice, the act of tuning an instrument or voice. * Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases. Tuning practice Tun ...
s) can this formula be simplified to: : ''TT = 3T'' This definition, however, has two different interpretations (broad and strict).


Broad interpretation (chromatic scale)

In a
chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...
, 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 fourth''s, and the other six as ''diminished fifths''.


Strict interpretation (diatonic scale)

Within a
diatonic scale In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...
, 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 C major (or the key of C) is a major scale based on C, consisting of the pitches C, D, E, F, G, A, and B. C major is one of the most common keys used in music. Its key signature has no flats or sharps. Its relative minor is A minor and ...
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 fourth A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...
s). 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 fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval fro ...
s).


Size in different tuning systems

In twelve-tone equal temperament, the A4 is exactly half an
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
(i.e., a ratio of :1 or 600 cents. The inverse of 600 cents is 600 cents. Thus, in this tuning system, the A4 and its inverse (d5) are
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...
. 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 Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. Me ...
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 -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 music, septimal meantone temperament, also called ''standard septimal meantone'' or simply ''septimal meantone'', refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the am ...
. In
31 equal temperament In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET (31 tone ET) or 31- EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps (equ ...
, 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 In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
: : ''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 Quarter-comma meantone, or -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80 ...
temperament, this is a diesis (128:125) less than a perfect octave: : ''A4 + A4 = P8 − diesis''. In
just intonation In music, just intonation or pure intonation is the tuning of musical intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to ...
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 Partch, Harry. (1979
974 Year 974 ( CMLXXIV) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. Events By place Europe * Battle of Danevirke: Emperor Otto II defeats the rebel forces of King Harald I, who ha ...
. ''
Genesis of a Music ''Genesis of a Music'' is a book first published in 1949 by microtonal composer Harry Partch (1901–1974). Partch first presents a polemic against both equal temperament and the long history of stagnation in the teaching of music; according ...
: An Account of a Creative Work, Its Roots and Its Fulfillments'', 2nd ed. (New York: Da Capo Press): p. 69. (pbk). https://pearl-hifi.com/06_Lit_Archive/02_PEARL_Arch/Vol_16/Sec_51/4555_Genesis_of_a_Music_2nd_Edn.pdf. Retrieved July 22, 2021
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. .Helmholtz, Hermann von (2005). ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', p. 457. . "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." or 25:18,Haluska , Ján (2003), ''The Mathematical Theory of Tone Systems'', Pure and Applied Mathematics Series 262 (New York: Marcel Dekker; London: Momenta), p. xxiv. . "25:18 classic augmented fourth".and the d5 is either 64:45 or 36:25.Haluska (2003), p. xxv. "36/25 classic diminished fifth". The 64:45 just diminished fifth arises in the C
major scale The major scale (or Ionian mode) is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at double ...
between B and F, consequently the 45:32 augmented fourth arises between F and B. These ratios are not in all contexts regarded as strictly just but they are the justest possible in 5-limit tuning.
7-limit 7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14 ...
tuning allows for the justest possible ratios (ratios with the smallest numerator and denominator), namely 7:5 for the A4 (about 582.5 cents, also known as
septimal tritone A septimal tritone is a tritone (about one half of an octave) that involves the factor seven. There are two that are inverses. The lesser septimal tritone (also Huygens' tritone) is the musical interval with ratio 7:5 (582.51 cents). The greater ...
) and 10:7 for the d5 (about 617.5 cents, also known as Euler's tritone).Haluska (2003). p. xxiii. "7:5 septimal or Huygens' tritone, Bohlen-Pierce fourth", "10:7 Euler's tritone".Strange, Patricia and Patricia, Allen (2001). ''The contemporary violin: Extended performance techniques'', p. 147. . "...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." 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.


Eleventh harmonic

The ratio of the eleventh harmonic, 11:8 (551.318 cents; approximated as F4 above C1), known as the lesser undecimal tritone or undecimal semi-augmented fourth, is found in some just tunings and on many instruments. For example, very long
alphorn The alphorn or alpenhorn or alpine horn is a labrophone, consisting of a straight several-meter-long wooden natural horn of conical bore, with a wooden cup-shaped mouthpiece. Traditionally the Alphorn was made of one single piece, or two par ...
s may reach the twelfth harmonic and transcriptions of their music usually show the eleventh harmonic sharp (F above C, for example), as in
Brahms Johannes Brahms (; 7 May 1833 – 3 April 1897) was a German composer, pianist, and conductor of the mid-Romantic period. Born in Hamburg into a Lutheran family, he spent much of his professional life in Vienna. He is sometimes grouped with ...
's First Symphony. This note is often corrected to 4:3 on the
natural horn The natural horn is a musical instrument that is the predecessor to the modern-day (French) horn (differentiated by its lack of valves). Throughout the seventeenth and eighteenth century the natural horn evolved as a separation from the trump ...
in just intonation or Pythagorean tunings, but the pure eleventh harmonic was used in pieces including
Britten Edward Benjamin Britten, Baron Britten (22 November 1913 – 4 December 1976, aged 63) was an English composer, conductor, and pianist. He was a central figure of 20th-century British music, with a range of works including opera, other ...
's ''
Serenade for tenor, horn and strings The ''Serenade for Tenor, Horn and Strings'', Op. 31, is a song cycle written in 1943 by Benjamin Britten for tenor, solo horn and a string orchestra. Composed during the Second World War at the request of the horn player Dennis Brain, it is a ...
''.
Ivan Wyschnegradsky Ivan Alexandrovich Wyschnegradsky; Is also transliterated as Vïshnegradsky, Wyshnegradsky, Wischnegradsky, Vishnegradsky, or Wishnegradsky (after he emigrated to France, he used "Wyschnegradsky" as spelling for his surname)., group=n ( ; Septe ...
considered the major fourth a good approximation of the eleventh harmonic.


Dissonance and expressiveness

Ján Haluska wrote:
Harry Partch Harry Partch (June 24, 1901 – September 3, 1974) was an American composer, music theorist, and creator of unique musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century com ...
has written:
... Although this ratio 5/32is 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 In psychology, temperament broadly refers to consistent individual differences in behavior that are biologically based and are relatively independent of learning, system of values and attitudes. Some researchers point to association of temperam ...
-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....Partch (1974), p. 115.


Common uses


Occurrences in diatonic scales

The augmented fourth (A4) occurs naturally between the fourth and seventh scale degrees of the
major scale The major scale (or Ionian mode) is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at double ...
(for example, from F to B in the key of
C major C major (or the key of C) is a major scale based on C, consisting of the pitches C, D, E, F, G, A, and B. C major is one of the most common keys used in music. Its key signature has no flats or sharps. Its relative minor is A minor and ...
). It is also present in the natural
minor scale In music theory, the minor scale is three scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending) – rather than just two as with the major scale, which also ...
as the interval formed between the second and sixth scale degrees (for example, from D to A in the key of
C minor C minor is a minor scale based on C, consisting of the pitches C, D, E, F, G, A, and B. Its key signature consists of three flats. Its relative major is E major and its parallel major is C major. The C natural minor scale is: : Cha ...
). 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 In music, the supertonic is the second degree () of a diatonic scale, one whole step above the tonic. In the movable do solfège system, the supertonic note is sung as ''re''. The triad built on the supertonic note is called the supertonic ch ...
chords using the notes from the natural minor mode thus contain a tritone, regardless of inversion. Containing tritones, these scales are
tritonic A tritonic scale is a musical scale or mode with three notes per octave. This is in contrast to a heptatonic (seven-note) scale such as the major scale and minor scale, or a dodecatonic (chromatic 12-note) scale, both common in modern Western ...
.


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 vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
. In addition,
augmented sixth chord In music theory, an augmented sixth chord contains the interval of an augmented sixth, usually above its bass tone. This chord has its origins in the Renaissance, was further developed in the Baroque, and became a distinctive part of the musi ...
s, 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 A minor is a minor scale based on A, with the pitches A, B, C, D, E, F, and G. Its key signature has no flats and no sharps. Its relative major is C major and its parallel major is A major. The A natural minor scale is: : Changes ...
); the French sixth chord can be viewed as a superposition of two tritones a major second apart. The
diminished triad In music theory, a diminished triad (also known as the minor flatted fifth) is a triad consisting of two minor thirds above the root. It is a minor triad with a lowered ( flattened) fifth. When using chord symbols, it may be indicated by the s ...
also contains a tritone in its construction, deriving its name from the diminished-fifth interval (i.e. a tritone). The
half-diminished seventh chord In music theory, the half-diminished seventh chord (also known as a half-diminished chord or a minor seventh flat five chord) is a seventh chord composed of a root note, together with a minor third, a diminished fifth, and a minor seventh (1,  ...
contains the same tritone, while the fully
diminished seventh chord The diminished seventh chord is a four-note chord (a seventh chord) composed of a root note, together with a minor third, a diminished fifth, and a diminished seventh above the root: (1, 3, 5, 7). For example, the diminished seventh ...
is made up of two superposed tritones a minor third apart. Other chords built on these, such as
ninth chord In music theory, a ninth chord is a chord that encompasses the interval of a ninth when arranged in close position with the root in the bass. Heinrich Schenker and also Nikolai Rimsky-Korsakov allowed the substitution of the dominant seve ...
s, 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
step Step(s) or STEP may refer to: Common meanings * Stairs#Step, Steps, making a staircase * Walking * Dance move * Military step, or march ** Marching Arts Films and television * Steps (TV series), ''Steps'' (TV series), Hong Kong * Step (film), ' ...
in contrary motion. This determines the resolution of chords containing tritones. 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 tonal theory, but functionally and notationally it can only resolve inwards as a diminished fifth and is therefore not reckoned a tritone—that is, an interval composed of three adjacent whole tones—in mid-
renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
(early 16th-century) music theory.


Other uses

The tritone is also one of the defining features of the
Locrian mode The Locrian mode is the seventh mode of the major scale. It is either a musical mode or simply a diatonic scale. On the piano, it is the scale that starts with B and only uses the white keys from there. Its ascending form consists of the key note, ...
, 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 __NOTOC__ The tritone paradox is an auditory illusion in which a sequentially played pair of Shepard tones separated by an interval of a tritone, or half octave, is heard as ascending by some people and as descending by others. Different popula ...
.


Historical uses

The tritone is a restless interval, classed as a dissonance in Western music from the early
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
through to the end of the
common practice In European art music, the common-practice period is the era of the tonal system. Most of its features persisted from the mid-Baroque period through the Classical and Romantic periods, roughly from 1650 to 1900. There was much stylistic evoluti ...
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 Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
the tritone was regarded as an unstable interval and rejected as a consonance by most theorists. The name ''diabolus in musica'' () 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" (Mi against Fa is the devil in music).
Andreas Werckmeister Andreas Werckmeister (November 30, 1645 – October 26, 1706) was a German organist, music theorist, and composer of the Baroque era. He was amongst the earliest advocates of equal temperament, and through this advocacy was highly influential to ...
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 (B and B, respectively) because these notes represent the juxtaposition of "mi contra fa".
Johann Joseph Fux Johann Joseph Fux (; – 13 February 1741) was an Austrian composer, music theorist and pedagogue of the late Baroque era. His most enduring work is not a musical composition but his treatise on counterpoint, '' Gradus ad Parnassum'', which has ...
cites the phrase in his seminal 1725 work ''
Gradus ad Parnassum The Latin phrase ''gradus ad Parnassum'' means "steps to Parnassus". It is sometimes shortened to ''gradus''. The name ''Parnassus'' was used to denote the loftiest part of a mountain range in central Greece, a few kilometres north of Delphi, of wh ...
'',
Georg Philipp Telemann Georg Philipp Telemann (; – 25 June 1767) was a German Baroque composer and multi-instrumentalist. Almost completely self-taught in music, he became a composer against his family's wishes. After studying in Magdeburg, Zellerfeld, and Hild ...
in 1733 describes, "mi against fa", which the ancients called "Satan in music"—and
Johann Mattheson Johann Mattheson (28 September 1681 – 17 April 1764) was a German composer, singer, writer, lexicographer, diplomat and music theorist. Early life and career The son of a prosperous tax collector, Mattheson received a broad liberal education ...
, 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 Denis Midgley Arnold (Sheffield, 15 December 1926 – Budapest, 28 April 1986) was a British musicologist. Biography After being employed in the extramural department of Queen's University, Belfast, he became a Lecturer in Music at the Univ ...
, in the ''
New Oxford Companion to Music ''The Oxford Companion to Music'' is a music reference book in the series of Oxford Companions produced by the Oxford University Press. It was originally conceived and written by Percy Scholes and published in 1938. Since then, it has undergon ...
'', 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").
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 Excommunication is an institutional act of religious censure used to end or at least regulate the communion of a member of a congregation with other members of the religious institution who are in normal communion with each other. The purpose ...
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 ''Organum'' () is, in general, a plainchant melody with at least one added voice to enhance the harmony, developed in the Middle Ages. Depending on the mode and form of the chant, a supporting bass line (or '' bourdon'') may be sung on the sam ...
of the ''
Musica Enchiriadis ''Musica enchiriadis'' is an anonymous musical treatise of the 9th century. It is the first surviving attempt to set up a system of rules for polyphony in western art music. The treatise was once attributed to Hucbald, but this is no longer accept ...
''. 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, 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
tonal system The tonal system is a base 16 system of notation (predating the widespread use of hexadecimal in computing), arithmetic, and metrology proposed in 1859 by John W. Nystrom.Nystrom, John W.''Project of a New System of Arithmetic, Weight, Measur ...
. 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 Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...
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 Romantic music is a stylistic movement in Western Classical music associated with the period of the 19th century commonly referred to as the Romantic era (or Romantic period). It is closely related to the broader concept of Romanticism—the ...
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 Franz Liszt, in modern usage ''Liszt Ferenc'' . Liszt's Hungarian passport spelled his given name as "Ferencz". An orthographic reform of the Hungarian language in 1922 (which was 36 years after Liszt's death) changed the letter "cz" to simpl ...
's use of the tritone to suggest Hell in his ''
Dante Sonata ''Après une lecture du Dante: Fantasia quasi Sonata'' (French for ''After a Reading of Dante: Fantasia quasi Sonata''; also known as the Dante Sonata) is a piano sonata in one movement, completed by Hungarian composer Franz Liszt in 1849. It was ...
'': —or
Wagner Wilhelm Richard Wagner ( ; ; 22 May 181313 February 1883) was a German composer, theatre director, polemicist, and conductor who is chiefly known for his operas (or, as some of his mature works were later known, "music dramas"). Unlike most op ...
's use of timpani tuned to C and F to convey a brooding atmosphere at the start of the second act of the opera ''
Siegfried Siegfried is a German-language male given name, composed from the Germanic elements ''sig'' "victory" and ''frithu'' "protection, peace". The German name has the Old Norse cognate ''Sigfriðr, Sigfrøðr'', which gives rise to Swedish ''Sigfrid' ...
''. In his early cantata ''
La Damoiselle élue ''La Damoiselle élue'' (''The Blessed Damozel''), L. 62, is a cantata for soprano soloist, 2-part children's choir, 2-part female (contralto) choir (with contralto solo), and orchestra, composed by Claude Debussy in 1887–1888 based on a text b ...
'',
Debussy (Achille) Claude Debussy (; 22 August 1862 – 25 March 1918) was a French composer. He is sometimes seen as the first Impressionist composer, although he vigorously rejected the term. He was among the most influential composers of the ...
uses a tritone to convey the words of the poem by
Dante Gabriel Rossetti Gabriel Charles Dante Rossetti (12 May 1828 – 9 April 1882), generally known as Dante Gabriel Rossetti (), was an English poet, illustrator, painter, translator and member of the Rossetti family. He founded the Pre-Raphaelite Brotherhoo ...
. 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." Debussy's String Quartet also features passages that emphasize the tritone. The tritone was also exploited heavily in that period as an interval of
modulation In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the ''carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informatio ...
for its ability to evoke a strong reaction by moving quickly to
distantly related key In music, a closely related key (or close key) is one sharing many common tones with an original key, as opposed to a distantly related key (or distant key). In music harmony, there are six of them: five share all, or all except one, pitches w ...
s. For example, the climax of Hector Berlioz's ''
La damnation de Faust ''La damnation de Faust'' (English: ''The Damnation of Faust''), Op. 24 is a work for four solo voices, full seven-part chorus, large children's chorus and orchestra by the French composer Hector Berlioz. He called it a "''légende dramatique'' ...
'' (1846) consists of a transition between "huge B and F chords" as Faust arrives in Pandaemonium, the capital of Hell. Musicologist
Julian Rushton Julian Gordon Rushton (born 22 May 1941) is an English musicologist, born in Cambridge. He has contributed the entry on Mozart in ''The New Grove Dictionary of Opera'' and several other articles in ''The New Grove Dictionary of Music and Musicians' ...
calls this "a tonal wrench by a tritone." Later, in
twelve-tone music The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law o ...
,
serialism In music, serialism is a method of Musical composition, composition using series of pitches, rhythms, dynamics, timbres or other elements of music, musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, thou ...
, 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 __NOTOC__ Ernő Lendvai (6 February 1925 – 31 January 1993) was one of the first music theorists to write on the appearance of the golden section and Fibonacci series and how these are implemented in Bartók's music. He also formulated the a ...
, in his analysis of the use of tonality in the music of
Béla Bartók Béla Viktor János Bartók (; ; 25 March 1881 – 26 September 1945) was a Hungarian composer, pianist, and ethnomusicologist. He is considered one of the most important composers of the 20th century; he and Franz Liszt are regarded as H ...
. Tritone relations are also important in the music of George Crumb and
Benjamin Britten Edward Benjamin Britten, Baron Britten (22 November 1913 – 4 December 1976, aged 63) was an English composer, conductor, and pianist. He was a central figure of 20th-century British music, with a range of works including opera, other ...
, whose ''
War Requiem The ''War Requiem'', Op. 66, is a large-scale setting of the Requiem composed by Benjamin Britten mostly in 1961 and completed in January 1962. The ''War Requiem'' was performed for the consecration of the new Coventry Cathedral, which was b ...
'' features a tritone between C and F♯ as a recurring motif. John Bridcut (2010, p. 271) describes the power of the interval in creating the sombre and ambiguou
opening of the ''War 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'."
Leonard Bernstein Leonard Bernstein ( ; August 25, 1918 – October 14, 1990) was an American conductor, composer, pianist, music educator, author, and humanitarian. Considered to be one of the most important conductors of his time, he was the first America ...
uses the tritone harmony as a basis for much of ''
West Side Story ''West Side Story'' is a musical conceived by Jerome Robbins with music by Leonard Bernstein, lyrics by Stephen Sondheim, and a book by Arthur Laurents. Inspired by William Shakespeare's play ''Romeo and Juliet'', the story is set in the mid-1 ...
''.
George Harrison George Harrison (25 February 1943 – 29 November 2001) was an English musician and singer-songwriter who achieved international fame as the lead guitarist of the Beatles. Sometimes called "the quiet Beatle", Harrison embraced Indian c ...
uses tritones on the downbeats of the opening phrases of
the Beatles The Beatles were an English Rock music, rock band, formed in Liverpool in 1960, that comprised John Lennon, Paul McCartney, George Harrison and Ringo Starr. They are regarded as the Cultural impact of the Beatles, most influential band of al ...
songs " The Inner Light", " Blue Jay Way" and "
Within You Without You "Within You Without You" is a song by the English rock band the Beatles from their 1967 album ''Sgt. Pepper's Lonely Hearts Club Band''. Written by lead guitarist George Harrison, it was his second composition in the Indian classical style, aft ...
", creating a prolonged sense of suspended resolution. Perhaps the most striking use of the interval in rock music of the late 1960s can be found in
Jimi Hendrix James Marshall "Jimi" Hendrix (born Johnny Allen Hendrix; November 27, 1942September 18, 1970) was an American guitarist, singer and songwriter. Although his mainstream career spanned only four years, he is widely regarded as one of the most ...
's song " Purple Haze". According to Dave Moskowitz (2010, p. 12), Hendrix "ripped into 'Purple Haze' by beginning the song with the sinister sounding tritone interval creating an opening dissonance, long described as 'The Devil in Music'.". The opening riff of
Black Sabbath Black Sabbath were an English rock music, rock band formed in Birmingham in 1968 by guitarist Tony Iommi, drummer Bill Ward (musician), Bill Ward, bassist Geezer Butler and vocalist Ozzy Osbourne. They are often cited as pioneers of heavy met ...
, the first song on
Black Sabbath Black Sabbath were an English rock music, rock band formed in Birmingham in 1968 by guitarist Tony Iommi, drummer Bill Ward (musician), Bill Ward, bassist Geezer Butler and vocalist Ozzy Osbourne. They are often cited as pioneers of heavy met ...
's eponymous debut album, is an inversion of a tritone; the album, and this song in particular, are considered to mark the birth of
heavy metal music Heavy metal (or simply metal) is a genre of rock music that developed in the late 1960s and early 1970s, largely in the United Kingdom and United States. With roots in blues rock, psychedelic rock and acid rock, heavy metal bands developed a ...
.William Irwin, ''Black Sabbath and Philosophy: Mastering Reality'' (Hoboken: Wiley-Blackwell, 2012), Tritones also became important in the development of
jazz Jazz is a music genre that originated in the African-American communities of New Orleans, Louisiana in the late 19th and early 20th centuries, with its roots in blues and ragtime. Since the 1920s Jazz Age, it has been recognized as a major ...
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 a common chord substitution found in both jazz and classical music. Where jazz is concerned, it was the precursor to more complex substitution patterns like Coltrane changes. Tritone substitutions are sometimes used i ...
. The tritone substitution is one of the most common chord and improvisation devices in jazz.


See also

*
List of meantone intervals The following is a list of intervals of extended meantone temperament. These intervals constitute the standard vocabulary of intervals for the Western common practice era. Here 12-EDO refers to the size of the interval in 12 equal divisions of th ...
* List of musical intervals *
List of pitch intervals Below is a list of intervals expressible in terms of a prime limit (see Terminology), completed by a choice of intervals in various equal subdivisions of the octave or of other intervals. For commonly encountered harmonic or melodic intervals ...
*
Ditone In music, a ditone (, from , "of two tones") is the interval of a major third. The size of a ditone varies according to the sizes of the two tones of which it is compounded. The largest is the Pythagorean ditone, with a ratio of 81:64, also cal ...
* Tone * * *
Petrushka chord The Petrushka chord is a recurring polytonal device used in Igor Stravinsky's ballet '' Petrushka'' and in later music. These two major triads, C major and F major – a tritone apart – clash, "horribly with each other", when sounded together ...
*
Tritonic scale A tritonic scale is a musical scale or mode with three notes per octave. This is in contrast to a heptatonic (seven-note) scale such as the major scale and minor scale, or a dodecatonic (chromatic 12-note) scale, both common in modern Wester ...


References


Further reading

*R., Ken (2012). ''DOG EAR Tritone Substitution for Jazz Guitar'', Amazon Digital Services, Inc., ASIN: B008FRWNIW


External links

*
BBC News Magazine article about the tritone

Satan's all-time greatest hit: Will Hodgkinson on the devil's interval
{{Intervals Augmented fourths Diminished fifths The Devil in classical music