Tritone

Last updated
tritone
Inverse tritone
Name
Other namesaugmented fourth, diminished fifth
AbbreviationTT, A4, d5
Size
Semitones 6
Interval class 6
Just interval 25:18, 36:25, 45:32, 64:45, 7:5, 10:7, 13:9...
Cents
Equal temperament 600
24 equal temperament 600
Just intonation 569, 631; 590, 610; 583, 617; 563, 637 ...

In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones. [1] 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. According to this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale. A tritone is also commonly defined as an interval spanning six semitones. 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). [2] In twelve-equal temperament, the tritone divides the octave exactly in half.

Music theory considers the practices and possibilities of music

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 what is otherwise called "rudiments", currently taught as the elements of notation, of key signatures, of time signatures, of rhythmic notation, and so on. [...] The second is the study of writings about music from ancient times onwards. [...] The third is an area of current musicological study that seeks to define processes and general principles in music—a sphere of research that can be distinguished from analysis in that it takes as its starting-point not the individual work or performance but the fundamental materials from which it is built.

In music theory, an interval is the 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 harmonic if it pertains to simultaneously sounding tones, such as in a chord.

Major second musical interval

In Western music theory, a major second is a second spanning two semitones. A second is a musical interval 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 positions. Diminished, minor and augmented seconds are notated on adjacent staff positions as well, but consist of a different number of semitones.

Contents

In classical music, the tritone is a harmonic and melodic dissonance and is important in the study of musical harmony. The tritone can be used to avoid traditional tonality: "Any tendency for a tonality to emerge may be avoided by introducing a note three whole tones distant from the key note of that tonality." [3] Contrarily, the tritone found in the dominant seventh chord helps establish the tonality of a composition. These contrasting uses exhibit the flexibility, ubiquity, and distinctness of the tritone in music.

Classical music broad tradition of Western art music

Classical music is art music produced or rooted in the traditions of Western culture, including both liturgical (religious) and secular music. While a more precise term is also used to refer to the period from 1750 to 1820, this article is about the broad span of time from before the 6th century AD to the present day, which includes the Classical period and various other periods. The central norms of this tradition became codified between 1550 and 1900, which is known as the common-practice period.

Consonance and dissonance categorizations of simultaneous or successive sounds

In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Consonance is associated with sweetness, pleasantness, and acceptability; dissonance is associated with harshness, unpleasantness, or unacceptability.

Harmony aspect of music

In music, harmony considers the process by which the composition of individual sounds, or superpositions of sounds, is analysed by hearing. Usually, this means simultaneously occurring frequencies, pitches, or chords.

The condition of having tritones is called tritonia; that of having no tritones is atritonia. A musical scale or chord containing tritones is called tritonic; one without tritones is atritonic.

Anhemitonic scale

Musicology commonly classifies scales as either hemitonic or anhemitonic. Hemitonic scales contain one or more semitones, while anhemitonic scales do not contain semitones. For example, in traditional Japanese music, the anhemitonic yo scale is contrasted with the hemitonic in scale. The simplest and most commonly used scale in the world is the atritonic anhemitonic "major" pentatonic scale. The whole tone scale is also anhemitonic.

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. Some scales contain different pitches when ascending than when descending, for example, the melodic minor scale.

Chord (music) harmonic set of three or more notes

A chord, in music, is any harmonic set of pitches consisting of multiple notes that are heard as if sounding simultaneously. For many practical and theoretical purposes, arpeggios and broken chords, or sequences of chord tones, may also be considered as chords.

Augmented fourth and diminished fifth

Chromatic scale on C: full octave ascending and descending
Play in equal temperament (help*info)
. Chromatic scale full octave ascending and descending on C.svg
Chromatic scale on C: full octave ascending and descending Loudspeaker.svg Play in equal temperament  .
Full ascending and descending chromatic scale on C, with tritone above each pitch. Pairs of tritones that are inversions of each other are marked below. Chromatic scale tritones.png
Full ascending and descending chromatic scale on C, with tritone above each pitch. Pairs of tritones that are inversions of each other are marked below.
The augmented fourth between C and F# and the diminished fifth between C and G are enharmonically equivalent intervals. Both are 600 cents wide in 12-TET.
Play (help*info)
. Enharmonic tritone.png
The augmented fourth between C and F and the diminished fifth between C and G are enharmonically equivalent intervals. Both are 600 cents wide in 12-TET. Loudspeaker.svg Play  .

Since a chromatic scale 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, six of them are classified as augmented fourths, and the other six as diminished fifths.

The chromatic scale is a musical scale with twelve pitches, each a semitone above or below its adjacent pitches. As a result, in 12-tone equal temperament, the chromatic scale covers all 12 of the available pitches. Thus, there is only one chromatic scale.

Under that 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 the perfect fourth and narrowing the perfect fifth by one chromatic semitone. [4] 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 twelve-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 exactly equal to half an octave.

In Western music and music theory, augmentation is the lengthening of a note or interval.

Perfect fourth musical interval

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. For example, the ascending interval from C to the next F is a perfect fourth, because the note F is the fifth semitone above C, and there are four staff positions between C and F. Diminished and augmented fourths span the same number of staff positions, but consist of a different number of semitones.

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. Diminution may also be the compositional device where a melody, theme or motif is presented in shorter note-values than were previously used. Diminution is also the term for the proportional shortening of the value of individual note-shapes in mensural notation, either by coloration or by a sign of proportion. A minor or perfect interval that is narrowed by a chromatic semitone is a diminished interval, and the process may be referred to as diminution.

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, and according to a rule explained elsewhere,[ where? ] 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), 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).

Notice that the latter diminished second is formed by two enharmonically equivalent notes (E and F). On a piano keyboard, 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

Tritone drawn in the chromatic circle. Tritone in the chromatic circle.png
Tritone drawn in the chromatic circle.

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 systems) 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 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 fourths, and the other six as diminished fifths.

Strict interpretation (diatonic scale)

Within a diatonic scale, 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 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 fourths).

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 fifths).

Size in different tuning systems

Tritone: just augmented fourth between C and F#+
Play 45:32 (590.22 cents) (help*info)
. Just augmented fourth on C.png
Tritone: just augmented fourth between C and F+ Loudspeaker.svg Play 45:32 (590.22 cents)  .
Tritone: Pythagorean augmented fourth between C and F#++
Play 729:512 (611.73 cents) (help*info)
. Pythagorean augmented fourth on C.png
Tritone: Pythagorean augmented fourth between C and F++ Loudspeaker.svg Play 729:512 (611.73 cents)  .
Tritone: the classic augmented fourth between C and F#
Play 25:18 (568.72 cents) (help*info) Augmented fourth on C.png
Tritone: the classic augmented fourth between C and F Loudspeaker.svg Play 25:18 (568.72 cents)  
Tritone: the classic diminished fifth between C and G
Play 36:25 (631.28 cents) (help*info) Diminished fifth on C.png
Tritone: the classic diminished fifth between C and G Loudspeaker.svg Play 36:25 (631.28 cents)  
Lesser septimal tritone on between C and G

Play 7:5 (582.51 cents) (help*info)
. Lesser septimal tritone on C.png
Lesser septimal tritone on between C and G 7 rightside up.png Loudspeaker.svg Play 7:5 (582.51 cents)  .

In twelve-tone equal temperament, the A4 is exactly half an octave (i.e., a ratio of 2:1 or 600 cents; Loudspeaker.svg play  ). The inverse of 600 cents is 600 cents. Thus, in this tuning system, the A4 and its inverse (d5) are equivalent.

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 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 29-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 31 equal temperament, for example, the A4 is 619.35 cents, whereas the d5 is 580.65 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:

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 temperament, this is a diesis (128:125) less than a perfect octave:

A4 + A4 = P8 − diesis .

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 [6] [7] [8] or 25:18, [9] and the d5 is either 64:45 Loudspeaker.svg Play   or 36:25. [10] The 64:45 just diminished fifth arises in the C major scale between B and F, consequently the 45:32 augmented fourth arises between F and B. [11]

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 (ratios with the smallest numerator and denominator), 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). [6] [12] [13] 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

Eleventh harmonic between C and F|.
Play 11:8 (551.32 cents) (help*info) Eleventh harmonic on C.png
Eleventh harmonic between C and F. Loudspeaker.svg Play 11:8 (551.32 cents)  

The ratio of the eleventh harmonic, 11:8 (551.318 cents; approximated as F Arabic music notation half sharp.svg 4 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 alphorns may reach the twelfth harmonic and transcriptions of their music usually show the eleventh harmonic sharp (F above C, for example), as in Brahms's First Symphony. [14] This note is often corrected to 4:3 on the natural horn in just intonation or Pythagorean tunings, but the pure eleventh harmonic was used in pieces including Britten's Serenade for tenor, horn and strings . [15] Ivan Wyschnegradsky considered the major fourth a good approximation of the eleventh harmonic.

Use of the eleventh harmonic in the prologue to Britten's Serenade for tenor, horn and strings.
Play (help*info) Britten - Serenade prologue.png
Use of the eleventh harmonic in the prologue to Britten's Serenade for tenor, horn and strings. Loudspeaker.svg Play  

Dissonance and expressiveness

The unstable character of the tritone sets it apart, as discussed in [28] [Paul Hindemith. The Craft of Musical Composition, Book I. Associated Music Publishers, New York, 1945]. It can be expressed as a ratio by compounding suitable superparticular ratios. Whether it is assigned the ratio 64/45 or 45/32, depending on the musical context, or indeed some other ratio, it is not superparticular, which is in keeping with its unique role in music. [16]

Although this ratio [45/32] is composed of numbers which are multiples of 5 or under, they are excessively large for a 5-limit scale, and are sufficient justification, either in this form or as the tempered "tritone", for the epithet "diabolic", which has been used to characterize the interval. This is a case where, because of the largeness of the numbers, none but a temperament-perverted ear could possibly prefer 45/32 to a small-number interval of about the same width. [17]

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 number 7. [17]

Common uses

Occurrences in diatonic scales

The augmented fourth (A4) occurs naturally between the fourth and seventh scale degrees of the major scale (for example, from F to B in the key of C major). It is also present in the natural minor scale as the interval formed between the second and sixth scale degrees (for example, from D to A in the key of C minor). The melodic minor scale, having two forms, presents a tritone in different locations when ascending and descending (when the scale ascends, the tritone appears between the third and sixth scale degrees and the fourth and seventh scale degrees, and when the scale descends, the tritone appears between the second and sixth scale degrees). Supertonic chords using the notes from the natural minor mode thus contain a tritone, regardless of inversion. Containing tritones, these scales are tritonic.

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 addition, augmented sixth chords, some of which are enharmonic to dominant seventh chords, contain tritones spelled as augmented fourths (for example, the German sixth, from A to D in the key of A minor); the French sixth chord can be viewed as a superposition of two tritones a major second apart.

The diminished triad also contains a tritone in its construction, deriving its name from the diminished-fifth interval (i.e. a tritone). The half-diminished seventh chord contains the same tritone, while the fully diminished seventh chord is made up of two superposed tritones a minor third apart.

Other chords built on these, such as ninth chords, often include tritones (as diminished fifths).

Resolution

In all of the sonorities mentioned above, used in functional harmonic analysis, the tritone pushes towards resolution, generally resolving by step 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 (early 16th-century) music theory. [18]

Tritone

Other uses

The tritone is also one of the defining features of the Locrian mode, being featured between the Scale deg 1.svg and fifth scale degrees.

The half-octave tritone interval is used in the musical/auditory illusion known as the tritone paradox.

Historical uses

Tritone
The theme that opens Claude Debussy's Prélude à l'après-midi d'un faune outlines the tritone between C and G.

The tritone is a restless interval, classed as a dissonance in Western music from the early Middle Ages through to the end of the common practice period. This interval was frequently avoided in medieval ecclesiastical singing because of its dissonant quality. The first explicit prohibition of it seems to occur with the development of Guido of Arezzo's hexachordal system, who suggested that rather than make B a diatonic note, the hexachord be moved and based on C to avoid the F–B tritone altogether. Later theorists such as Ugolino d'Orvieto and Tinctoris advocated for the inclusion of B. [19]

From then until the end of the Renaissance the tritone was regarded as an unstable interval and rejected as a consonance by most theorists. [20]

The name diabolus in musica ("the Devil in music") has been applied to the interval from at least the early 18th century, though its use is not restricted to the tritone. Andreas Werckmeister cites this term in 1702 as being used by "the old authorities" for both the tritone and for the clash between chromatically related tones such as F and F, [21] 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". [22] Johann Joseph Fux cites the phrase in his seminal 1725 work Gradus ad Parnassum , Georg Philipp Telemann in 1733 describes, "mi against fa", which the ancients called "Satan in music"—and Johann Mattheson, in 1739, writes that the "older singers with solmization called this pleasant interval 'mi contra fa' or 'the devil in music'." [23] 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. [24] However Denis Arnold, in the New Oxford Companion to Music , suggests that the nickname was already applied early in the medieval music itself:

It seems first to have been designated as a "dangerous" interval when Guido of Arezzo developed his system of hexachords and with the introduction of B flat as a diatonic note, at much the same time acquiring its nickname of "Diabolus in Musica" ("the devil in music"). [25]

That original symbolic association with the devil and its avoidance led to Western cultural convention seeing the tritone as suggesting "evil" in music. However, stories that singers were excommunicated or otherwise punished by the Church for invoking this interval are likely fanciful. At any rate, avoidance of the interval for musical reasons has a long history, stretching back to the parallel organum of the Musica Enchiriadis . In all these expressions, including the commonly cited "mi contra fa est diabolus in musica", 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. 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. [26]

It is only with the Romantic music and modern classical music that composers started to use it totally freely, without functional limitations notably in an expressive way to exploit the "evil" connotations culturally associated with it (e.g., Franz Liszt's use of the tritone to suggest Hell in his Dante Sonata :

Liszt, "Apres une lecture du Dante" from Annees de Pelerinage. Listen Liszt apres une lecture du Dante.png
Liszt, "Après une lecture du Dante" from Années de Pèlerinage. Listen

—or Wagner'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 .

Wagner, Prelude to Act 2 of Siegfried. Listen Siegfried Act 2 prelude.png
Wagner, Prelude to Act 2 of Siegfried. Listen

In his early cantata La Damoiselle élue , Debussy uses a tritone to convey the words of the poem by Dante Gabriel Rossetti.

Debussy, La Damoiselle Elue, Figure 30. Link to passage Debussy la Damoiselle Fig 30.png
Debussy, La Damoiselle Élue, Figure 30. Link to passage

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." [27] Debussy's String Quartet also features passages that emphasise the tritone:

Debussy, String Quartet, 2nd movement, bars 140-147. Link to passage Debussy Quartet 2nd movement, bars 140-7.png
Debussy, String Quartet, 2nd movement, bars 140–147. Link to passage

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, in twelve-tone music, serialism, and other 20th century compositional idioms, composers considered it a neutral interval. [28] In some analyses of the works of 20th century composers, the tritone plays an important structural role; perhaps the most cited is the axis system, proposed by Ernő Lendvai, in his analysis of the use of tonality in the music of Béla Bartók. [29] Tritone relations are also important in the music of George Crumb [ citation needed ] and Benjamin Britten, whose War Requiem features a tritone between C and F♯ as a recurring motif. [30] John Bridcut (2010, p. 271) describes the power of the interval in creating the sombre and ambiguous 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'." [31] George Harrison uses tritones on the downbeats of the opening phrases of the Beatles songs "The Inner Light", "Blue Jay Way" and "Within You Without You", creating a prolonged sense of suspended resolution. [32] Perhaps the most striking use of the interval in rock music of the late 1960s can be found in Jimi Hendrix'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'." [33]

Tritone substitution: F# may substitute for C , and vice versa, because they both share E and B/A# and due to voice leading considerations.
Play (help*info) Tritone substitution.png
Tritone substitution: F may substitute for C , and vice versa, because they both share E and B/A and due to voice leading considerations. Loudspeaker.svg Play  

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 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 one of the most common chord and improvisation devices in jazz.

In the theory of harmony it is known that a diminished interval needs to be resolved inwards, and an augmented interval outwards. ... and with the correct resolution of the true tritones this desire is totally satisfied. However, if one plays a just diminished fifth that is perfectly in tune, for example, there is no wish to resolve it to a major third. Just the opposite—aurally one wants to enlarge it to a minor sixth. The opposite holds true for the just augmented fourth. ...
These apparently contradictory aural experiences become understandable when the cents of both types of just tritones are compared with those of the true tritones and then read 'crossed-over'. One then notices that the just augmented fourth of 590.224 cents is only 2 cents bigger than the true diminished fifth of 588.270 cents, and that both intervals lie below the middle of the octave of 600.000 cents. It is no wonder that, following the ear, we want to resolve both downwards. The ear only desires the tritone to be resolved upwards when it is bigger than the middle of the octave. Therefore the opposite is the case with the just diminished fifth of 609.776 cents. ... [7]

See also

Sources

  1. Don Michael Randel (2003). The Harvard Dictionary of Music: Fourth Edition. Harvard University Press. ISBN   0-674-01163-5.
  2. E.g., Jacobus Leodiensis, Speculum musicae, Liber secundus, in Jacobi Leodiensis Speculum musicae, edited by Roger Bragard, Corpus Scriptorum de Musica 3/2 ([Rome]: American Institute of Musicology, 1961): 128–31, citations on 192–96, 200, and 229; Jacobus Leodiensis, Speculum musicae, Liber sextus, in Jacobi Leodiensis Speculum musicae, edited by Roger Bragard, Corpus Scriptorum de Musica 3/6 ([Rome]: American Institute of Musicology, 1973): 1-161, citations on 52 and 68; Johannes Torkesey, Declaratio et expositio, London: British Library, Lansdowne 763, ff.89v-94v, citations on f.92r,2–3; Prosdocimus de Beldemandis, Tractatus musice speculative, in D. Raffaello Baralli and Luigi Torri, "Il Trattato di Prosdocimo de' Beldomandi contro il Lucidario di Marchetto da Padova per la prima volta trascritto e illustrato", Rivista Musicale Italiana 20 (1913): 731–62, citations on 732–34.
  3. Smith Brindle, Reginald (1966). Serial Composition. Oxford University Press. p. 66. ISBN   0-19-311906-4.
  4. Bruce Benward & Marilyn Nadine Saker (2003). Music: In Theory and Practice, Vol. I, seventh edition (Boston: McGraw-Hill), p. 54. ISBN   978-0-07-294262-0.
  5. Fonville, John. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p. 121–22, Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991), pp. 106–37.
  6. 1 2 Partch, Harry. (1974). Genesis of a Music: An Account of a Creative Work, Its Roots and Its Fulfillments, second edition, enlarged (New York: Da Capo Press): p. 69. ISBN   0-306-71597-X (cloth); ISBN   0-306-80106-X (pbk).
  7. 1 2 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. ISBN   1-902636-46-5.
  8. Helmholtz, Hermann von (2005). On the Sensations of Tone as a Physiological Basis for the Theory of Music, p. 457. ISBN   1-4191-7893-8. "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."
  9. Haluska , Ján (2003), The Mathematical Theory of Tone Systems, Pure and Applied Mathematics Series 262 (New York: Marcel Dekker; London: Momenta), p. xxiv. ISBN   0-8247-4714-3. "25:18 classic augmented fourth".
  10. Haluska (2003), p. xxv. "36/25 classic diminished fifth".
  11. Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction , p.165. Theodore Baker, trans. G. Schirmer.
  12. Haluska (2003). p. xxiii. "7:5 septimal or Huygens' tritone, Bohlen-Pierce fourth", "10:7 Euler's tritone".
  13. Strange, Patricia and Patricia, Allen (2001). The contemporary violin: Extended performance techniques, p. 147. ISBN   0-520-22409-4. "...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."
  14. Monelle, Raymond (2006). The Musical Topic: Hunt, Military And Pastoral, p.102. ISBN   9780253347664.
  15. Fauvel, John; Flood, Raymond; and Wilson, Robin J. (2006). Music And Mathematics, p.21-22. ISBN   9780199298938.
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  17. 1 2 Partch (1974), p. 115. ISBN   0-306-80106-X.
  18. Margaret Bent, ""Accidentals, Counterpoint, and Notation in Aaron’s Aggiunta to the Toscanello", Journal of Musicology 12: "Aspects of Musical Language and Culture in the Renaissance: A Birthday Tribute to James Haar" (1994): 306–44. Citation on 308.
  19. Guido d'Arezzo, Epistola de ignoto cantu, lines 309–22[ full citation needed ][ not in citation given ]
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  23. Reinhold, Hammerstein (1974). Diabolus in musica: Studien zur Ikonographie der Musik im Mittelalter. Neue Heidelberger Studien zur Musikwissenschaft (in German). 6. Bern: Francke. p. 7. OCLC   1390982. ...mi contra fa ... welches die alten den Satan in der Music nenneten" "...alten Solmisatores dieses angenehme Intervall mi contra fa oder den Teufel in der Music genannt haben.
  24. F. J. Smith, "Some Aspects of the Tritone and the Semitritone in the Speculum Musicae: The Non-Emergence of the Diabolus in Music," Journal of Musicological Research 3 (1979), pp. 63–74, at 70.
  25. Arnold, Denis (1983) « Tritone » in The New Oxford Companion to Music, Volume 1: A–J,Oxford University Press. ISBN   0-19-311316-3
  26. Jeppesen, Knud (1992) [1939]. Counterpoint: the polyphonic vocal style of the sixteenth century. trans. by Glen Haydon, with a new foreword by Alfred Mann. New York: Dover. ISBN   0-486-27036-X.
  27. Nichols, R. (1972) Debussy. Oxford University Press.
  28. Persichetti, Vincent (1961). Twentieth-century Harmony: Creative Aspects and Practice. New York: W. W. Norton. ISBN   0-393-09539-8. OCLC   398434.
  29. Lendvai, Ernő (1971). Béla Bartók: An Analysis of his Music. introd. by Alan Bush. London: Kahn & Averill. pp. 1–16. ISBN   0-900707-04-6. OCLC   240301.
  30. "Musical Analysis of the War Requiem" . Retrieved 16 March 2016.
  31. Bridcut, J. (2010), Essential Britten, a pocket guide for the Britten Centenary. London, Faber.
  32. Dominic Pedler. The Songwriting Secrets of the Beatles. Music Sales Ltd. Omnibus Press. London, 2010 pp. 522–523
  33. Moskowitz, D. (2010) The Words and Music of Jimi Hendrix. Praeger.

Further reading

Related Research Articles

Pythagorean tuning

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈702 cents wide.

Enharmonic (in modern musical notation and tuning) note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently

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. Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. For example, in twelve-tone equal temperament, the notes C and D are enharmonic notes. Namely, they are the same key on a keyboard, and thus they are identical in pitch, although they have different names and different roles in harmony and chord progressions. Arbitrary amounts of accidentals can produce further enharmonic equivalents, such as B, although these are much rarer and have less practical use.

Perfect fifth musical interval

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.

Wolf interval Dissonant musical interval

In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including most meantone temperaments.

Semitone musical interval

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 notes in a 12-tone scale. For example, C is adjacent to C; the interval between them is a semitone.

Comma (music) small musical interval, the difference between two tunings of the same note

In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning system, and another F tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B and A are both approximated by the same interval although they are a septimal kleisma apart.

Augmented second musical interval

In classical music from Western culture, an augmented second is an interval that, in equal temperament, is sonically equivalent to a minor third, spanning three semitones, and is created by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and the interval from C to D is an augmented second, spanning three semitones.

Quarter-comma meantone, or 14-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), with respect to its just intonation used in Pythagorean tuning ; the result is . The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

53 equal temperament musical tuning system with 53 pitches equally-spaced on a logarithmic scale

In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps. Play  Each step represents a frequency ratio of 2153, or 22.6415 cents, an interval sometimes called the Holdrian comma.

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 19 equal temperament, with 31 equal temperament being a more or less optimal tuning for both the 5- and 7-limits. Meantone temperament represents a frequency ratio of approximately 5 by means of four fifths, so that the major third, for instance C-E, is obtained from two tones in succession. Septimal meantone represents the frequency ratio of 56 (7*23) by ten fifths, so that the interval 7:4 is reached by five successive tones. Hence C-A, not C-B, represents a 7:4 interval in septimal meantone.

Pythagorean interval

In musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 (equivalent to 31/21) and the perfect fourth with ratio 4/3 (equivalent to 22/31) are Pythagorean intervals.

Rothenberg propriety

In diatonic set theory, Rothenberg propriety is an important concept, lack of contradiction and ambiguity, in the general theory of musical scales which was introduced by David Rothenberg in a seminal series of papers in 1978. The concept was independently discovered in a more restricted context by Gerald Balzano, who termed it coherence.

Diminished second musical interval

In modern Western tonal music theory, a diminished second is the interval produced by narrowing a minor second by one chromatic semitone. It is enharmonically equivalent to a perfect unison. Thus, it is the interval between notes on two adjacent staff positions, or having adjacent note letters, altered in such a way that they have no pitch difference in twelve-tone equal temperament. An example is the interval from a B to the C immediately above; another is the interval from a B to the C immediately above.

Diminished third musical interval

In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A to C, and from A to C are diminished thirds, two semitones wide. Being diminished, it is considered a dissonant interval.

Diatonic and chromatic

Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.

Regular diatonic tuning

A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.

Septimal may refer to:

Augmented seventh musical interval

In classical music from Western culture, an augmented seventh is an interval produced by widening a major seventh by a chromatic semitone. For instance, the interval from C to B is a major seventh, eleven semitones wide, and both the intervals from C to B, and from C to B are augmented sevenths, spanning twelve semitones. Being augmented, it is classified as a dissonant interval. However, it is enharmonically equivalent to the perfect octave.

Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.