Tritone
Encyclopedia
In classical music
from Western culture
, the tritone ' onMouseout='HidePop("7260")' href="/topics/Tri">tri
- "three" and tone
) is traditionally defined as a musical interval composed of three whole tones
. In a chromatic scale
, each whole tone can be further divided into two semitone
s. In this context, a tritone may also be defined as any interval spanning six semitones.
Since a chromatic scale is formed by 12 pitches, it contains 12 distinct tritones, each of which starting from a different pitch. According to a widely used naming convention, six of them are classified as augmented fourths, and the other six as diminished fifths. In a diatonic scale
there's only one tritone, classified as an augmented fourth. For instance, in the C major
diatonic scale the only interval formed by three adjacent tones (F-G, G-A, and A-B) is that from F to B.
In the above-mentioned naming convention, a fourth is an interval encompassing four staff positions, while a fifth encompasses five staff positions (see interval number for more details).
The augmented fourth (A4) and diminished fifth (d5) are defined as the intervals produced by widening
and narrowing
by one chromatic semitone the perfect fourth
and fifth
, respectively.
They both span six semitones, and they are the inverse
of each other, meaning that their sum is exactly equal to one perfect octave (A4 + d5 = P8).
In 12-tone equal temperament
, the most commonly used tuning system, the A4 is equivalent
to a d5, as both have the size of exactly half an octave
. In most other tuning systems, they are not equivalent, and neither is equal to half an octave. The d5 is also called semidiapente.
The tritone is often used as the main interval of dissonance
in Western
harmony, and is important in the study of musical harmony
. "Any tendency for a tonality
to emerge may be avoided by introducing a note three whole tones distant from the key note of that tonality."
. As the symbol for whole tone is T, this definition may be also written as follows:
Only if the three tones are of the same size (which is not the case for many tuning systems) can this formula be simplified to:
This definition, however, has two different interpretations (broad and strict).
is used, with its 12 notes it is possible to define 12 different tritones, each of which starting from a different note. Six of them are A4, and the other six are d5. Therefore, in this case both A4 and d5 are considered to be tritones. Since each whole tone, in a chromatic scale, can be divided into two semitone
s:
then three tones are equal to six semitones. In this case, we can generalize the definition of tritone as follows:
This means that a tritone can be also defined as any musical interval spanning six semitones (indeed, both A4 and d5 are intervals spanning 6 semitones).
Only when the semitones (and the tones formed by pairs of semitones) are equal in size can this formula be simplified to:
, whole tones are regarded as incomposite intervals (that is, they do not divide into smaller intervals). Therefore, in this context the above mentioned "decomposition" of the tritone into six semitones is typically not allowed.
If a diatonic scale is used, with its 7 notes it is possible to form only one sequence of three adjacent whole tones (T+T+T). This interval is an A4, and it is sometimes called the proper tritone. For instance, in the C major
diatonic scale (C-D-E-F-G-A-B-...), the only tritone is from F to B. It is a tritone because F-G, G-A, and A-B are three adjacent whole tones. It is a fourth because the notes from F to B are four (F, G, A, B). It is augmented (i.e., widened) because most of the fourths found in the scale have smaller size (they are perfect fourth
s).
According to this interpretation, the d5 is not a tritone. Indeed, in a diatonic scale, there's only one d5, and this interval does not meet the strict definition of tritone, as it is formed by one semitone, two whole tones, and another semitone:
For instance, in the C major diatonic scale, the only d5 is from B to F. It is a fifth because the notes from B to F are five (B, C, D, E, F). It is diminished (i.e. narrowed) because most of the fifths found in the scale have larger size (they are perfect fifth
s).
In this case, context may resolve the tritone to more an A4, a d5, or a "neutral" interval with no clear conventional classification.
, the A4 is exactly half an octave
(i.e., a ratio of √2
: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
.
The half-octave A4 is unique in being equal to its own inverse. In other meantone
tuning systems, besides 12-tone equal temperament, A4 and d5 are distinct intervals because neither is exactly half an octave. In any meantone tuning near to -comma meantone the A4 will be near to the ratio and the d5 to , which is what these intervals are taken to be in septimal meantone temperament
. In 31 equal temperament
, for example, the A4, a , or tritone proper, is 617.49 cents, whereas a is 582.51 cents. This is perceptually indistinguishable from septimal meantone temperament.
Since they are the inverse of each other, by definition A4 and d5 always add up to exactly one perfect octave
:
On the other hand, two A4 (proper tritones) add up to six whole tones. In equal temperament, this is equal to exactly one perfect octave:
In quarter-comma meantone
temperament, this is a diesis
(128/125) less than a perfect octave:
In just intonation
several different sizes can be chosen both for the A4 and the d5. For instance, in 5-limit tuning, the A4 is either 45/32
or 25/18,
and the d5 is either 64/45 or 36/25.
These ratios are not in all contexts regarded as strictly just, but they are the justest possible in 5-limit tuning. 7-limit tuning allows for the justest possible ratios, namely 7/5 for the A4 (about 582.5 cents, also known as septimal tritone
) and 10/7 for the d5 (about 617.5 cents, also known as Euler's tritone).
These ratios are more consonant than 17/12 (about 603.0 cents) and 24/17 (about 597.0 cents), which can be obtained in 17-limit tuning, yet the latter are also fairly common, as they are closer to the equal-tempered value of 600.0 cents.
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.
(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.
contains a (d5) tritone within its tone construction: it occurs between the third and seventh above the root
. In addition, augmented sixth chord
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
); the French sixth chord can be viewed as a superposition of two tritones a major second apart.
The diminished chord
also contains a tritone in its construction, deriving its name from the diminished fifth interval (i.e. a tritone). The half-diminished seventh chord
contains the same tritone, while the fully diminished seventh chord
is made up of two superposed tritones a minor third apart.
Other chords built on these, such as ninth chord
s, often include tritones (as diminished fifths).
. The inversion of this, a diminished fifth resolving inward to a major third, is often loosely called a tritone as well in modern tonal theory, but functionally and notationally it can only resolve inwards as a kind of fifth and is therefore not reckoned a tritone in baroque
and renaissance
music theory.
, 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
.
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 hexachord
al 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:
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 hexachord
s. 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 Lendvaï
, 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.
Classical music
Classical music is the art music produced in, or rooted in, the traditions of Western liturgical and secular music, encompassing a broad period from roughly the 11th century to present times...
from Western culture
Western culture
Western culture, sometimes equated with Western civilization or European civilization, refers to cultures of European origin and is used very broadly to refer to a heritage of social norms, ethical values, traditional customs, religious beliefs, political systems, and specific artifacts and...
, the tritone ' onMouseout='HidePop("7260")' href="/topics/Tri">tri
Tri
Tri may refer to:* An intern specializing in Float and Share research.* tri-, a numerical prefix meaning three* El Tri or El Tricolor, are the nickname of the Mexico national football team.* El Tri, Mexican rock group...
- "three" and tone
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
) is traditionally defined as a musical interval composed of three whole tones
Major second
In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...
. In a chromatic scale
Chromatic scale
The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
, each whole tone can be further divided into two semitone
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....
s. In this context, a tritone may also be defined as any interval spanning six semitones.
Since a chromatic scale is formed by 12 pitches, it contains 12 distinct tritones, each of which starting from a different pitch. According to a widely used naming convention, six of them are classified as augmented fourths, and the other six as diminished fifths. In a diatonic scale
Diatonic scale
In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...
there's only one tritone, classified as an augmented fourth. For instance, in the C major
C major
C major is a musical major scale based on C, with pitches C, D, E, F, G, A, and B. Its key signature has no flats/sharps.Its relative minor is A minor, and its parallel minor is C minor....
diatonic scale the only interval formed by three adjacent tones (F-G, G-A, and A-B) is that from F to B.
In the above-mentioned naming convention, a fourth is an interval encompassing four staff positions, while a fifth encompasses five staff positions (see interval number for more details).
The augmented fourth (A4) and diminished fifth (d5) are defined as the intervals produced by widening
Augmentation (music)
In Western music and music theory, the word augmentation has three distinct meanings. Augmentation is a compositional device where a melody, theme or motif is presented in longer note-values than were previously used...
and narrowing
Diminution
In Western music and music theory, diminution has four distinct meanings. Diminution may be a form of embellishment in which a long note is divided into a series of shorter, usually melodic, values...
by one chromatic semitone the perfect fourth
Perfect fourth
In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...
and fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
, respectively.
They both span six semitones, and they are the inverse
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
of each other, meaning that their sum is exactly equal to one perfect octave (A4 + d5 = P8).
In 12-tone equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
, the most commonly used tuning system, the A4 is equivalent
Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...
to a d5, as both have the size of exactly half an octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
. In most other tuning systems, they are not equivalent, and neither is equal to half an octave. The d5 is also called semidiapente.
The tritone is often used as the main interval of dissonance
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...
in Western
Western world
The Western world, also known as the West and the Occident , is a term referring to the countries of Western Europe , the countries of the Americas, as well all countries of Northern and Central Europe, Australia and New Zealand...
harmony, and is important in the study of musical harmony
Harmony
In music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...
. "Any tendency for a tonality
Tonality
Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron and was borrowed by François-Joseph Fétis in 1840...
to emerge may be avoided by introducing a note three whole tones distant from the key note of that tonality."
Definitions
A tritone (abbreviation: TT) is traditionally defined as a musical interval composed of three whole tonesMajor second
In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...
. As the symbol for whole tone is T, this definition may be also written as follows:
- TT = T+T+T
Only if the three tones are of the same size (which is not the case for many tuning systems) can this formula be simplified to:
- TT = 3T
This definition, however, has two different interpretations (broad and strict).
Broad interpretation (chromatic scale)
If a chromatic scaleChromatic scale
The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
is used, with its 12 notes it is possible to define 12 different tritones, each of which starting from a different note. Six of them are A4, and the other six are d5. Therefore, in this case both A4 and d5 are considered to be tritones. Since each whole tone, in a chromatic scale, can be divided into two semitone
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....
s:
- T = S+S
then three tones are equal to six semitones. In this case, we can generalize the definition of tritone as follows:
- TT = T+T+T = S+S+S+S+S+S.
This means that a tritone can be also defined as any musical interval spanning six semitones (indeed, both A4 and d5 are intervals spanning 6 semitones).
Only when the semitones (and the tones formed by pairs of semitones) are equal in size can this formula be simplified to:
- TT = 3T = 6S.
Strict interpretation (diatonic scale)
In a diatonic scaleDiatonic scale
In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...
, whole tones are regarded as incomposite intervals (that is, they do not divide into smaller intervals). Therefore, in this context the above mentioned "decomposition" of the tritone into six semitones is typically not allowed.
If a diatonic scale is used, with its 7 notes it is possible to form only one sequence of three adjacent whole tones (T+T+T). This interval is an A4, and it is sometimes called the proper tritone. For instance, in the C major
C major
C major is a musical major scale based on C, with pitches C, D, E, F, G, A, and B. Its key signature has no flats/sharps.Its relative minor is A minor, and its parallel minor is C minor....
diatonic scale (C-D-E-F-G-A-B-...), the only tritone is from F to B. It is a tritone because F-G, G-A, and A-B are three adjacent whole tones. It is a fourth because the notes from F to B are four (F, G, A, B). It is augmented (i.e., widened) because most of the fourths found in the scale have smaller size (they are perfect fourth
Perfect fourth
In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...
s).
According to this interpretation, the d5 is not a tritone. Indeed, in a diatonic scale, there's only one d5, and this interval does not meet the strict definition of tritone, as it is formed by one semitone, two whole tones, and another semitone:
- d5 = S+T+T+S.
For instance, in the C major diatonic scale, the only d5 is from B to F. It is a fifth because the notes from B to F are five (B, C, D, E, F). It is diminished (i.e. narrowed) because most of the fifths found in the scale have larger size (they are perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
s).
Alternative definition
Some contemporary authors define a tritone as any interval spanning exactly half an octave, including both the A4 and d5 as tuned in 12-tone equal temperament. This is not consistent with the above mentioned traditional definition (TT = T+T+T).In this case, context may resolve the tritone to more an A4, a d5, or a "neutral" interval with no clear conventional classification.
Size in different tuning systems
In 12-tone equal temperamentEqual temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
, the A4 is exactly half an octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
(i.e., a ratio of √2
Square root of 2
The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...
:1 or 600 cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
; ). The inverse of 600 cents is 600 cents. Thus, in this tuning system, the A4 and its inverse (d5) are equivalent
Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...
.
The half-octave A4 is unique in being equal to its own inverse. In other meantone
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
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 will be near to the ratio and the d5 to , which is what these intervals are taken to be in septimal meantone temperament
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 amount of fifths for 12 equal temperament to those as flat as...
. In 31 equal temperament
31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
, for example, the A4, a , or tritone proper, is 617.49 cents, whereas a is 582.51 cents. This is perceptually indistinguishable from septimal meantone temperament.
Since they are the inverse of each other, by definition A4 and d5 always add up to exactly one perfect octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
:
- A4 + d5 = P8.
On the other hand, two A4 (proper tritones) add up to six whole tones. In equal temperament, this is equal to exactly one perfect octave:
- A4 + A4 = P8.
In quarter-comma meantone
Quarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
temperament, this is a diesis
Diesis
In classical music from Western culture, a diesis is either an accidental , or a comma type of musical interval, usually defined as the difference between an octave and three justly tuned major thirds , equal to 128:125 or about 41.06 cents...
(128/125) less than a perfect octave:
- A4 + A4 = P8 - diesisDiesisIn classical music from Western culture, a diesis is either an accidental , or a comma type of musical interval, usually defined as the difference between an octave and three justly tuned major thirds , equal to 128:125 or about 41.06 cents...
.
In just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
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
or 25/18,
and the d5 is either 64/45 or 36/25.
These ratios are not in all contexts regarded as strictly just, but they are the justest possible in 5-limit tuning. 7-limit tuning allows for the justest possible ratios, namely 7/5 for the A4 (about 582.5 cents, also known as septimal tritone
Septimal tritone
The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7...
) and 10/7 for the d5 (about 617.5 cents, also known as Euler's tritone).
These ratios are more consonant than 17/12 (about 603.0 cents) and 24/17 (about 597.0 cents), which can be obtained in 17-limit tuning, yet the latter are also fairly common, as they are closer to the equal-tempered value of 600.0 cents.
Dissonance and expressiveness
Compared to other commonly occurring intervals like the major second or the minor third, the augmented fourth and the diminished fifth (both two valid enharmonic interpretations of the tritone) are considered awkwardConsonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...
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.
Occurrences in diatonic scales
The proper tritone (A4) occurs naturally between the fourth and seventh scale degrees of the major scaleMajor scale
In music theory, the major scale or Ionian scale is one of the diatonic scales. It is made up of seven distinct notes, plus an eighth which duplicates the first an octave higher. In solfege these notes correspond to the syllables "Do, Re, Mi, Fa, Sol, La, Ti/Si, ", the "Do" in the parenthesis at...
(for example, from F to B in the key of C major
C major
C major is a musical major scale based on C, with pitches C, D, E, F, G, A, and B. Its key signature has no flats/sharps.Its relative minor is A minor, and its parallel minor is C minor....
). It is also present in the natural minor scale
Minor scale
A minor scale in Western music theory includes any scale that contains, in its tonic triad, at least three essential scale degrees: 1) the tonic , 2) a minor-third, or an interval of a minor third above the tonic, and 3) a perfect-fifth, or an interval of a perfect fifth above the tonic, altogether...
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
C minor is a minor scale based on C, consisting of the pitches C, D, E, F, G, A, and B. The harmonic minor raises the B to B. Changes needed for the melodic and harmonic versions of the scale are written in with naturals and accidentals as necessary.Its key signature consists of three flats...
). 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
Supertonic
In music or music theory, the supertonic is the second degree or note of a diatonic scale, one step above the tonic. In music theory, the supertonic chord is symbolized by the Roman numeral ii in a major scale, indicating that the chord is a minor chord , or ii in a natural minor scale, indicating...
chords using the notes from the natural minor mode will thus contain a tritone, regardless of inversion.
Occurrences in chords
The dominant seventh chordDominant seventh chord
In music theory, a dominant seventh chord, or major minor seventh chord,is a chord composed of a root, major third, perfect fifth, and minor seventh. It can be also viewed as a major triad with an additional minor seventh...
contains a (d5) tritone within its tone construction: it occurs between the third and seventh above the root
Root (chord)
In music theory, the root of a chord is the note or pitch upon which a triadic chord is built. For example, the root of the major triad C-E-G is C....
. In addition, augmented sixth chord
Augmented sixth chord
In music theory, an augmented sixth chord contains the interval of an augmented sixth above its "root" or bass tone . This chord has its origins in the Renaissance, further developed in the Baroque, and became a distinctive part of the musical style of the Classical and Romantic periods.-Resolution...
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
A minor is a minor scale based on A, consisting of the pitches A, B, C, D, E, F, and G. The harmonic minor scale raises the G to G...
); the French sixth chord can be viewed as a superposition of two tritones a major second apart.
The diminished chord
Diminished chord
A diminished triad chord or diminished chord is a triad consisting of two minor thirds above the root — if built on C, a diminished chord would have a C, an E and a G. It resembles a minor triad with a lowered fifth....
also contains a tritone in its construction, deriving its name from the diminished fifth interval (i.e. a tritone). The half-diminished seventh chord
Half-diminished seventh chord
In music theory, the half-diminished seventh chord is created by taking the root, minor third, diminished fifth and minor seventh of any major scale; for example, C half-diminished is . Its consecutive intervals are minor 3rd, minor 3rd, major 3rd...
contains the same tritone, while the fully diminished seventh chord
Diminished seventh chord
A diminished seventh chord is a four note chord that comprises a diminished triad plus the interval of a diminished seventh above the root. Thus it is , or enharmonically , of any major scale; for example, C diminished-seventh would be , or enharmonically...
is made up of two superposed tritones a minor third apart.
Other chords built on these, such as ninth chord
Ninth chord
A ninth chord is a chord that encompasses the interval of a ninth when arranged in close position with the root in the bass.A dominant ninth is a dominant chord with a ninth. A ninth chord, as an extended chord, typically includes the seventh along with the basic triad structure. Thus, a Cmaj9...
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 in contrary motionContrary motion
In music theory, contrapuntal motion is the general movement of two melodic lines with respect to each other. In traditional four-part harmony, it is important that lines maintain their independence, an effect which can be achieved by the judicious use of the four types of contrapuntal motion:...
. The inversion of this, a diminished fifth resolving inward to a major third, is often loosely called a tritone as well in modern tonal theory, but functionally and notationally it can only resolve inwards as a kind of fifth and is therefore not reckoned a tritone in baroque
Baroque
The Baroque is a period and the style that used exaggerated motion and clear, easily interpreted detail to produce drama, tension, exuberance, and grandeur in sculpture, painting, literature, dance, and music...
and renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...
music theory.
Other uses
The tritone is also one of the defining features of the Locrian modeLocrian mode
The Locrian mode is either a musical mode or simply a diatonic scale. Although the term occurs in several classical authors on music theory, including Cleonides and Athenaeus , there is no warrant for the modern usage of Locrian as equivalent to Glarean's Hyperaeolian mode, in either classical,...
, 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
Tritone paradox
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 populations tend to favor one of a limited set of different spots...
.
Historical uses
The tritone is a restless interval, classed as a dissonanceConsonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...
in Western music from the early Middle Ages
Middle Ages
The Middle Ages is a periodization of European history from the 5th century to the 15th century. The Middle Ages follows the fall of the Western Roman Empire in 476 and precedes the Early Modern Era. It is the middle period of a three-period division of Western history: Classic, Medieval and Modern...
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
Guido of Arezzo
Guido of Arezzo or Guido Aretinus or Guido da Arezzo or Guido Monaco or Guido d'Arezzo was a music theorist of the Medieval era...
's hexachord
Hexachord
In music, a hexachord is a collection of six pitch classes including six-note segments of a scale or tone row. The term was adopted in the Middle Ages and adapted in the twentieth-century in Milton Babbitt's serial theory.-Middle Ages:...
al 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
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...
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
Devil
The Devil is believed in many religions and cultures to be a powerful, supernatural entity that is the personification of evil and the enemy of God and humankind. The nature of the role varies greatly...
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
1725 in music
- Events :*Giovanni Battista Pergolesi goes to Naples to study under Gaetano Greco. Domenico Scarlatti is also in Naples at this time.*Nineteen-year-old Giovanni Battista Martini is appointed chapel-master of the Franciscan church at Bologna'...
work Gradus ad Parnassum, Georg Philipp Telemann
Georg Philipp Telemann
Georg Philipp Telemann 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 Hildesheim, Telemann entered the University of Leipzig to study law, but eventually...
in 1733
1733 in music
- Events :*Susannah Maria Arne makes her stage début in the first opera written by her brother, Thomas.*Wilhelm Friedemann Bach is appointed organist of St Sophia's Church, Dresden.*Jean-Marie Leclair becomes musical director to King Louis XV of France....
notes, "mi against fa", which the ancients called "Satan in music", and Johann Mattheson
Johann Mattheson
Johann Mattheson was a German composer, writer, lexicographer, diplomat and music theorist.Mattheson was born and died in Hamburg. He was a close friend of George Frideric Handel, although he nearly killed him in a sudden quarrel, during a performance of Mattheson's opera Cleopatra in 1704...
in 1739
1739 in music
-Events:*March 27 Johann Sebastian Bach performs the Brockes-Passion TWV 5: 1 at St. Nicholas Church, Leipzig.*September – Leopold Mozart is expelled from the Benedictine University of Salzburg for poor attendance....
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 ArezzoGuido of ArezzoGuido of Arezzo or Guido Aretinus or Guido da Arezzo or Guido Monaco or Guido d'Arezzo was a music theorist of the Medieval era...
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
Excommunication
Excommunication is a religious censure used to deprive, suspend or limit membership in a religious community. The word means putting [someone] out of communion. In some religions, excommunication includes spiritual condemnation of the member or group...
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
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 may be sung on the same text, the melody may be followed in parallel motion , or a combination of...
of the Musica Enchiriadis
Musica enchiriadis
Musica enchiriadis is an anonymous musical treatise from the 9th century. It is the first surviving attempt to establish a system of rules for polyphony in classical music. The treatise was once attributed to Hucbald, but this is no longer accepted. Some historians once attributed it to Odo of...
. 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 hexachord
Hexachord
In music, a hexachord is a collection of six pitch classes including six-note segments of a scale or tone row. The term was adopted in the Middle Ages and adapted in the twentieth-century in Milton Babbitt's serial theory.-Middle Ages:...
s. 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
Tonality
Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron and was borrowed by François-Joseph Fétis in 1840...
. 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
Romantic music
Romantic music or music in the Romantic Period is a musicological and artistic term referring to a particular period, theory, compositional practice, and canon in Western music history, from 1810 to 1900....
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
Liszt
Liszt is a Hungarian surname. Notable persons with that surname include:* Franz Liszt , Hungarian composer and pianist* Adam Liszt , father of Franz Liszt* Anna Liszt , mother of Franz Liszt...
's use of the tritone to suggest hell in his Dante Sonata
Dante Sonata
Après une Lecture de Dante: Fantasia quasi Sonata is a piano sonata in one movement, completed by Hungarian composer Franz Liszt in 1849. It was first published in 1856 as part of the second volume of Années de Pèlerinage...
). The tritone was also exploited heavily in that period as an interval of modulation
Modulation (music)
In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature. Modulations articulate or create the structure or form of many pieces, as well as add interest...
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
Axis system
In music, the axis system is a system of analysis originating in the work of Ernő Lendvaï, which he developed in his analysis of the music of Béla Bartók....
, proposed by Ernő Lendvaï
Erno Lendvai
Ernő Lendvai was one of the first theorists to write on the appearance of the golden section and Fibonacci series and how these are implemented in Bartók's music...
, in his analysis of the use of tonality in the music of Béla Bartók
Béla Bartók
Béla Viktor János Bartók was a Hungarian composer and pianist. He is considered one of the most important composers of the 20th century and is regarded, along with Liszt, as Hungary's greatest composer...
. Tritone relations are also important in the music of George Crumb
George Crumb
George Crumb is an American composer of contemporary classical music. He is noted as an explorer of unusual timbres, alternative forms of notation, and extended instrumental and vocal techniques. Examples include seagull effect for the cello , metallic vibrato for the piano George Crumb (born...
.
Jazz
Jazz is a musical style that originated at the beginning of the 20th century in African American communities in the Southern United States. It was born out of a mix of African and European music traditions. From its early development until the present, jazz has incorporated music from 19th and 20th...
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
Tritone substitution
In classical music, a substitute dominant is "a chord sufficiently akin to the dominant to be reasonably set against the tonic, and yet remote enough to give a chromatically expressive, large-scale dissonance to the structure"...
. The tritone substitution is one of the most common chord and improvisation devices in jazz.
See also
- List of meantone intervals
- List of musical intervals
- Hexatonic scale#Tritone scale