View source for Tritone

{{other uses}}
{{Redirect|Devil's Chord|the Doctor Who episode|The Devil's Chord}}
{{Infobox Interval
|main_interval_name=tritone
|inverse=tritone
|complement=tritone
|other_names=augmented fourth, diminished fifth, the Devil’s interval (obscure)
|abbreviation=TT, A4, d5
|semitones=6
|tones=3
|interval_class=6
|just_interval='''Pythagorean:''' 729:512, 1024:729<br/> '''5-limit:'''&nbsp;25:18,&nbsp;36:25;&nbsp;45:32,&nbsp;64:45<br/> '''7-limit:''' 7:5, 10:7<br/> '''13-limit:''' 13:9, 18:13
|cents_equal_temperament=600
|cents_24T_equal_temperament=
|cents_just_intonation='''Pythagorean:''' 612, 588<br/> '''5-limit:''' 569, 631; 590, 610<br/> '''7-limit:''' 583, 617 <br/> '''13-limit:''' 563, 637}}

In [[music theory]], the '''tritone''' is defined as a [[interval (music)|musical interval]] spanning three adjacent [[Major second|whole tones]] (six [[semitone]]s).<ref>{{cite web |last1=Drabkin |first1=William |title=Tritone |url=https://www.oxfordmusiconline.com/grovemusic/view/10.1093/gmo/9781561592630.001.0001/omo-9781561592630-e-0000028403 |website=[[Oxford Music Online]] |access-date=August 31, 2020 |doi=10.1093/gmo/9781561592630.article.28403 |date=20 January 2001 |isbn=978-1-56159-263-0 |url-access=limited}}</ref> For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B.

Narrowly defined, each of these whole tones must be a step in the [[scale (music)|scale]], so by 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. More broadly, a tritone is also commonly defined as any interval with a width of three whole tones (spanning six [[semitone]]s in the chromatic scale), regardless of [[scale degree]]s. 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''');<ref>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 MS 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.</ref> the latter is decomposed as a semitone B–C, a whole tone C–D, a whole tone D–E, and a semitone E–F, for a total width of three whole tones, but composed as four steps in the scale. In [[Equal temperament#Twelve-tone equal temperament|twelve-equal temperament]], the tritone divides the octave exactly in half as 6 of 12 semitones or 600 of 1,200 [[Cent (music)|cents]].<ref>[https://books.google.com/books?id=02rFSecPhEsC&q=%22A+prominent+element%22 Randel (2003), p. 911]. "A prominent element in the [[whole-tone scale]]...its symmetry with respect to the octave gives it a special role in [[twelve-tone music]] as well."</ref>

In [[classical music]], the tritone is a harmonic and melodic [[Consonance and dissonance|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."<ref>
{{cite book
| last = Smith Brindle
| first = Reginald
| author-link = Reginald Smith Brindle
| title = Serial Composition
| publisher = [[Oxford University Press]]
| year = 1966
| isbn = 0-19-311906-4
| url = https://books.google.com/books?id=gvNLXHpusw8C
| pages = 66 }}</ref> The tritone found in the [[dominant seventh chord]] can also drive the piece of music towards resolution with its tonic. These various uses exhibit the flexibility, ubiquity, and distinctness of the tritone in music.

The condition of having tritones is called [[Anhemitonic scale|tritonia]]; that of having no tritones is [[Anhemitonic scale|atritonia]]{{fact|date=March 2025}}. A [[Scale (music)|musical scale]] or [[Chord (music)|chord]] containing tritones is called [[Anhemitonic scale|tritonic]]; one without tritones is [[Anhemitonic scale|atritonic]].

== Augmented fourth and diminished fifth ==
{{more citations needed section|date=January 2017}}
[[File:Chromatic scale full octave ascending and descending on C.svg|thumb|right|400px|Chromatic scale on C: full octave ascending and descending[[File:ChromaticScaleUpDown.ogg|thumb|center|Play in equal temperament]]]]
[[File:Chromatic scale tritones.png|thumb|right|400px|Full ascending and descending chromatic scale on C, with tritone above each pitch. Pairs of tritones that are [[Inversion (music)#Intervals|inversions]] of each other are marked below.]]
[[File:Enharmonic tritone.png|thumb|Augmented fourth between C and F{{music|sharp}} and the diminished fifth between C and G{{music|b}} are [[enharmonic|enharmonically equivalent]] intervals. Both are 600 [[Cent (music)|cent]]s wide in [[Equal temperament#Twelve-tone equal temperament|12-TET]].[[File:Tritone on C.mid]]]]

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 [[interval (music)#Interval number and quality|naming convention]], six of them are classified as ''augmented fourths'', and the other six as ''diminished fifths''.

Under that convention, a ''fourth'' is an interval encompassing four [[staff position]]s, while a ''fifth'' encompasses five staff positions (see [[Interval (music)#Number|interval number]] for more details).
The augmented fourth (''A4'') and diminished fifth (''d5'') are defined as the intervals produced by [[augmentation (music)|widening]] the [[perfect fourth]] and [[Diminution|narrowing]] the [[perfect fifth]] by one [[chromatic semitone]].<ref>Bruce Benward & Marilyn Nadine Saker (2003). ''Music: In Theory and Practice, Vol. I'', seventh edition (Boston: McGraw-Hill), p.&nbsp;54. {{ISBN|978-0-07-294262-0}}.</ref>
They both span six semitones, and they are the [[Inversion (music)#Intervals|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 [[Enharmonic|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.

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 the rule of interval addition explained in [[Interval (music)#Number|Interval (music)]], 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{{Music|sharp}} (major second), C{{Music|sharp}}–D{{Music|sharp}} (major second), D{{Music|sharp}}–E{{Music|sharp}} (major second), and E{{Music|sharp}}–F{{Music|natural}} ([[diminished second]]).
Notice that the last diminished second is formed by two [[Enharmonic|enharmonically equivalent]] notes (E{{Music|sharp}} and F{{Music|natural}}). 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 position]]s and have different diatonic functions within music theory.

== Definitions ==
{{more citations needed section|date=January 2017}}
[[File:Tritone in the chromatic circle.png|thumb|right|Tritone drawn in the [[chromatic circle]].]]
A tritone (abbreviation: ''TT'') is traditionally defined as a musical interval composed of three [[major second|whole tones]]. As the symbol for whole tone is ''T'', this definition may also be written as follows:

: ''TT = T+T+T''

Only if the three tones are of the same size (which is not the case for many [[tuning system]]s) can this formula be simplified to:

: ''TT = 3T''

This definition, however, has two different interpretations (broad and strict).

=== Broad interpretation (chromatic scale) ===
In a [[chromatic scale]], the 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{{Music|sharp}} and C{{Music|sharp}}–D by using the note C{{Music|sharp}}, which in a chromatic scale lies between C and D. This means that, when a chromatic scale is used, a tritone can be also defined as any musical interval spanning six semitones:

: ''TT = T+T+T = S+S+S+S+S+S''.

According to this definition, with the twelve notes of a chromatic scale it is possible to define twelve different tritones, each starting from a different note and ending six notes above it. Although all of them span six semitones, six of them are classified as ''augmented fourth''s, and the other six as ''diminished fifths''.

=== Strict interpretation (diatonic scale) ===
Within a [[diatonic scale]], whole tones are always formed by adjacent notes (such as C and D) and therefore they are regarded as [[incomposite interval]]s. 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 fourth]]s).

According to this interpretation, the d5 is not a tritone. Indeed, in a diatonic scale, there is only one d5, and this interval does not meet the strict definition of tritone, as it is formed by one semitone, two whole tones, and another semitone:

: ''d5 = S+T+T+S''.

For instance, in the C major diatonic scale, the only d5 is from B to F. It is a ''fifth'' because the notes from B to F are five (B, C, D, E, F). It is ''diminished'' (i.e. narrowed) because it is smaller than most of the fifths found in the scale (they are [[perfect fifth]]s).

== Size in different tuning systems ==
{{multiple image
 | align = right
 | direction = vertical
 | width = 220
 | header = Tritones
 | image1 = Just augmented fourth on C.png
 | caption1 = [[major fourth|just augmented fourth]] between C and F{{music|sharp}}+ – 45:32 (590.22 cents)[[File:Just augmented fourth on C.mid]]
 | image2 = Pythagorean augmented fourth on C.png
 | caption2 = Pythagorean augmented fourth between C and F{{music|sharp}}++ – 729:512 (611.73 cents)[[File:Pythagorean augmented fourth on C.mid]]
 | image3 = Augmented fourth on C.png
 | caption3 = Classic augmented fourth between C and F{{music|sharp}} – 25:18 (568.72 cents)[[File:Classic augmented fourth on C.mid]]
 | image4 = Diminished fifth on C.png
 | caption4 = Classic diminished fifth between C and G{{music|flat}} – 36:25 (631.28 cents) [[File:Classic diminished fifth on C.mid]]
 | image5 = Lesser septimal tritone on C.png
 | caption5 = Lesser septimal tritone between C and G{{music|7}}{{music|b}}<ref name="Fonville">{{cite journal|last1=Fonville|first1=John|author-link=John Fonville|title=Ben Johnston's Extended Just Intonation: A Guide for Interpreters|journal=[[Perspectives of New Music]]|date=1991|volume=29|issue=2|pages=106–137|doi=10.2307/833435|jstor=833435}}</ref> – 7:5 (582.51 cents)[[File:Lesser septimal tritone on C.mid]]
 | image6 = Comparison of tritones.png
 | caption6 = Comparison of intervals near or enharmonic with the tritone
}}

{{stack|[[File:Tritone1.mid|thumb|Tritone]]}}
In twelve-tone equal temperament, the Aug&nbsp;4 is exactly half an [[octave]] (i.e., a ratio of [[square root of 2|{{sqrt|2}}]]:1 or 600&nbsp;[[cent (music)|cents]]. The inverse of 600&nbsp;cents is 600&nbsp;cents. Thus, in this tuning system, the Aug&nbsp;4 and its inverse (dim&nbsp;5) are [[Enharmonic|equivalent]].

The half-octave or equal tempered Aug&nbsp;4 and dim&nbsp;5 are unique in being equal to their own inverse (each to the other). In other [[meantone temperament|meantone]] tuning systems, besides 12&nbsp;tone equal temperament, Aug&nbsp;4 and dim&nbsp;5 are distinct intervals because neither is exactly half an octave. In any meantone tuning near to {{sfrac|2|9}}-comma meantone the Aug&nbsp;4 is near to the ratio&nbsp;7:5 (582.51) and the dim&nbsp;5 to 10:7 (617.49), which is what these intervals are in [[septimal meantone temperament]]. In [[31 equal temperament]], for example, the Aug&nbsp;4 is 580.65&nbsp;cents, whereas the dim&nbsp;5 is 619.35 cents. This is perceptually indistinguishable from septimal meantone temperament.

Since they are the inverse of each other, by definition Aug&nbsp;4 and dim&nbsp;5 always add up (in cents) to exactly one [[octave|perfect octave]]:
: ''Aug&nbsp;4 + dim&nbsp;5 = Perf&nbsp;8''.
On the other hand, two Aug&nbsp;4 add up to six whole tones. In equal temperament, this is equal to exactly one perfect octave:
: ''Aug&nbsp;4 + Aug&nbsp;4 = Perf&nbsp;8''.
In [[quarter-comma meantone]] temperament, this is a [[diesis]] (128:125) less than a perfect octave:
: ''Aug&nbsp;4 + Aug&nbsp;4 = Perf&nbsp;8 − [[diesis]]''.

{{stack|[[File:Just diminished fifth on C.mid|thumb|Just diminished fifth on C]]}}
In [[just intonation]] several different sizes can be chosen both for the Aug&nbsp;4 and the dim&nbsp;5. For instance, in [[5-limit tuning]], the Aug&nbsp;4 is either 45:32<ref name=Partch>{{cite book |author-link=Harry Partch |last=Partch |first=H. |year=1979 |orig-year=1974 |title=Genesis of a Music: An account of a creative work, its Roots and its fulfillments |title-link=Genesis of a Music |edition=2nd |place=New York, NY |publisher=Da Capo Press |page=69 |ISBN=0-306-80106-X }}
:{{cite book |section=''Genesis of a Music'' |year=1979 |title=scanned copy |edition=2nd |type=online |url=https://pearl-hifi.com/06_Lit_Archive/02_PEARL_Arch/Vol_16/Sec_51/4555_Genesis_of_a_Music_2nd_Edn.pdf |via=pearl-hifi.com |access-date=22 July 2021 }}</ref><ref name=Renold>{{cite book |last=Renold |first=Maria |year=2004 |title=Intervals, Scales, Tones, and the Concert Pitch {{nobr|&hairsp; C {{=}} 128 Hz }} |quote=translated from German |lang=en |others=Stevens, Bevis (translator) ; Meuss, Anna R. (additional editing) |place=Forest Row |publisher=Temple Lodge |ISBN=1-902636-46-5 |pages=15–16 }}</ref><ref name=Helmholtz>{{cite book |last=Helmholtz |first=H. |author-link=Hermann von Helmholtz |editor-first=A.J. |editor-last=Ellis |editor-link=Alexander J. Ellis |year=2005 |orig-year=1875, 1st&nbsp;Engl. |title=On the Sensations of Tone as a Physiological Basis for the Theory of Music |title-link=On Sensations of Tone |edition=reprint |page=457 |ISBN=1-4191-7893-8 |quote=Name of interval: ''Just Tritone'', cents in interval: 590, number to an octave: 2&nbsp;; Name of interval: ''Pyth. Tritone'',  cents in interval: 612, number to an octave: 2&nbsp; }}</ref> or 25:18,<ref name="Haluska 24">{{cite book |last=Haluska |first=Ján |year=2003 |title=The Mathematical Theory of Tone Systems |series=Pure and Applied Mathematics Series |volume=262 |place=New York / London |publisher=Marcel Dekker / Momenta |ISBN=0-8247-4714-3 |page={{mvar|xxiv}}  |quote=25:18 classic augmented fourth }}</ref> and the dim&nbsp;5 is either 64:45 or 36:25.<ref name="Haluska 25">{{harvp|Haluska|2003|p=&nbsp;{{mvar|xxv}} }} "36:25 classic diminished fifth".</ref> 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.<ref>{{cite book |last=Paul |first=Oscar |year=1885  |quote=musical interval 'pythagorean major third' |title=A Manual of Harmony for use in Music-Schools and Seminaries, and for Self-Instruction |page=165 |publisher=Theodore Baker |translator-first=((Gustav, Sr.)) |translator-last=Schirmer |translator-link=G. Schirmer, Inc. |url=https://archive.org/details/bub_gb_4WEJAQAAMAAJ |via=archive.org }}</ref>

These ratios are not in all contexts regarded as [[Five-limit tuning#The just ratios|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 Aug&nbsp;4 (about 582.5 cents, also known as [[septimal tritone]]) and 10:7 for the dim&nbsp;5 (about 617.5 cents, also known as Euler's tritone).<ref name=Partch/><ref name="Haluska 23">{{harvp|Haluska|2003|p={{mvar|xxiii}} }} "7:5 septimal or Huygens' tritone, Bohlen-Pierce fourth", "10:7 Euler's tritone".</ref><ref name=Strange>{{cite book |last1=Strange |first1=Patricia |last2=Patricia |first2=Allen |year=2001 |title=The Contemporary Violin: Extended performance techniques |page=147 |ISBN=0-520-22409-4 |quote= ...&nbsp;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. }}</ref>
These ratios are more consonant than 17:12 (about 603.0&nbsp;[[musical cents|cents]]) and 24:17 (about 597.0&nbsp;cents), which can be obtained in 17&nbsp;limit tuning, yet the latter are also fairly common, as they are closer to the equal-tempered value of 600&nbsp;cents.

== Eleventh harmonic ==
<!--[[Eleventh harmonic]] and [[Acoustic scale]] redirect directly here-->
{{stack|[[File:Eleventh harmonic on C.png|thumb|Eleventh harmonic between C and F{{music|up}} – 11:8 (551.32 cents)[[File:Eleventh harmonic on C.mid]]]]}}

The ratio of the eleventh harmonic, 11:8 (551.318 cents; approximated as F{{music|t}}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 [[alphorn]]s may reach the twelfth harmonic and transcriptions of their music usually show the eleventh harmonic sharp (F{{music|#}} above C, for example), as in [[Johannes Brahms|Brahms]]'s [[Symphony No. 1 (Brahms)|First Symphony]].<ref>Monelle, Raymond (2006). ''The Musical Topic: Hunt, Military And Pastoral'', p. 102. {{ISBN|9780253347664}}.</ref> 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 [[Benjamin Britten|Britten]]'s ''[[Serenade for Tenor, Horn and Strings|Serenade for tenor, horn and strings]]''.<ref>Fauvel, John; Flood, Raymond; and Wilson, Robin J. (2006). ''Music And Mathematics'', pp. 21–22. {{ISBN|9780199298938}}.</ref> [[Ivan Wyschnegradsky]] considered the [[major fourth]] a good approximation of the eleventh harmonic.

[[File:Britten - Serenade prologue.png|thumb|center|400px|Use of the eleventh harmonic in the prologue to Britten's [[Serenade for Tenor, Horn and Strings]].[[File:Britten - Serenade prologue.mid]]]]

== Dissonance and expressiveness ==

Ján Haluska wrote:

{{Blockquote|The unstable character of the tritone sets it apart, as discussed in <!--original text "[28]"--><nowiki>[</nowiki>[[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 ratio]]s. 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.<ref>Haluska (2003), p.&nbsp;286.</ref>}}

[[Harry Partch]] has written:

{{blockquote|
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 [[musical temperament|temperament]]-perverted ear could possibly prefer 45/32 to a small-number interval of about the same width.

In the [[Pythagorean interval|Pythagorean ratio]] 81/64 both numbers are multiples of 3 or under, yet because of their excessive largeness the ear certainly prefers 5/4 for this approximate degree, even though it involves a prime number higher than 3. In the case of the 45/32 "tritone" our theorists have gone around their elbows to reach their thumbs, which could have been reached simply and directly and non-"diabolically" via the number 7....<ref name="Partch 115">Partch (1974), p.&nbsp;115.</ref>}}

== Common uses ==
{{more citations needed section|date=January 2017}}

=== 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{{Music|flat}} 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 [[Anhemitonic scale|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 (chord)|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{{music|sharp}} 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 chord]]s, often include tritones (as diminished fifths).

=== Resolution ===
{{thumb|content=<score sound="1">
      \new Staff <<
         \new Voice \relative c' {
             \clef treble \key c \major \time 4/4
             \stemUp b'2 c f e
             }
         \new Voice \relative c' {
             \clef treble \key c \major \time 4/4
             \stemDown f2 e \bar "||" b' c \bar "||"
             }
            >>
</score>}}
In all of the sonorities mentioned above, used in functional harmonic analysis, the tritone pushes towards resolution, generally resolving by [[step (music)|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 music|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.<ref>{{cite journal |last1=Bent |first1=Margaret |title=Accidentals, Counterpoint and Notation in Aaron's ''Aggiunta'' to the ''Toscanello in Musica'' |journal=Journal of Musicology |date=1994 |volume=12 |issue=3 |pages=306–344 [308] |doi=10.2307/764089|jstor=764089 }}</ref>

=== Other uses ===
The tritone is also one of the defining features of the [[Locrian mode]], being featured between the {{music|scale|1}} and fifth scale degrees.

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

== Historical uses ==
{{Image frame|content=<score sound="1"> {
#(set-global-staff-size 14)
\relative c' {
  \clef treble \time 9/8 \key e \major
  \set Score.tempoHideNote = ##t \tempo "Très modéré" 4. = 36
   \override Score.SpacingSpanner #'common-shortest-duration = #(ly:make-moment 1 8)
  \set Staff.midiInstrument = "flute"
  \stemDown cis'4.~(^"Flute"\p cis8~_\markup \italic "doux et expressif" cis16 \set stemRightBeamCount = #1 b \times 2/3 { \set stemLeftBeamCount = #1 ais16 a gis } g8. a16 b bis)
  cis4.~( cis8~ cis16 \set stemRightBeamCount = #1 b \times 2/3 { \set stemLeftBeamCount = #1 ais16 a gis } g8. a16 b bis) \override DynamicLineSpanner.staff-padding = #3
  cis8(\< dis gis e4 gis,8 b4.~\! 
  b8\> b cis ais4)\!
}
}
</score>|width=530|caption=The theme that opens [[Claude Debussy]]'s {{lang|fr|[[Prélude à l'après-midi d'un faune]]}} outlines the tritone between C{{music|#}} and G.}}

The tritone is a restless interval, classed as a [[consonance and dissonance|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 [[hexachord]]al system, who suggested that rather than make B{{music|flat}} a diatonic note, the hexachord be moved and based on C to avoid the F–B tritone altogether. Later theorists such as [[Ugolino of Forlì|Ugolino d'Orvieto]] and [[Johannes Tinctoris|Tinctoris]] advocated the inclusion of B{{music|flat}}.<ref>{{citation | author =Guido d'Arezzo | title = Epistola de ignoto cantu|pages=lines 309–322|no-pp=y}}{{Full citation needed|date=October 2015}}<!--If a particular manuscript copy is being cited, the location, library name, and shelf number are needed; if a modern edition, then the editor, location, and publisher are needed at a minimum.-->{{Failed verification|date=October 2015}}<!--Guido does not mention Ugolino or Tinctoris, nor how could he? Unless this incomplete citation is to a modern annotated edition, it cannot possibly verify this claim.--></ref>

From then until the end of the [[Renaissance]] the tritone was regarded as an unstable interval and rejected as a consonance by most theorists.<ref>{{cite web | last = Drabkin | first = William | title = Tritone | work = [[Grove Music Online]] (subscription access) | publisher = [[Grove Music Online#Oxford Music Online|Oxford Music Online]] | url = http://www.oxfordmusiconline.com/subscriber/article/grove/music/28403 | access-date = 2008-07-21}}</ref>

The name {{Language with name/for|la|diabolus in musica|the [[Devil]] in music}} has been applied to the interval from at least the early 18th century, or the late Middle Ages,<ref>[https://books.google.com/books?id=02rFSecPhEsC&q=tritone Randel (2003), p.239].</ref> though its use is not restricted to the tritone, being that the original found example of the term {{lang|la|"diabolus en musica"}} is {{lang|la|"Mi Contra Fa est diabolus en musica"}} (Mi against Fa is the devil in music), referring to the [[minor second]]. [[Andreas Werckmeister]] cites this term in [[1702 in music|1702]] as being used by "the old authorities" for both the tritone and for the clash between chromatically related tones such as F{{Music|natural}} and F{{Music|sharp}},<ref>[[Andreas Werckmeister]]. [https://www.digitale-sammlungen.de/de/view/bsb10527826?page=5 {{lang|de|Harmonologia musica, oder kurze Anleitung zur musicalischen Composition}}] (Frankfurt and Leipzig: Theodor Philipp Calvisius 1702): 6.</ref> and five years later likewise calls {{lang|la|"diabolus in musica"}} the opposition of "square" and "round" B (B{{Music|natural}} and B{{Music|flat}}, respectively) because these notes represent the juxtaposition of {{lang|la|"mi contra fa"}}.<ref>[[Andreas Werckmeister]], [https://www.digitale-sammlungen.de/de/view/bsb10527832?page=5 {{lang|de|Musicalische Paradoxal-Discourse, oder allgemeine Vorstellungen}}] (Quedlinburg: Theodor Philipp Calvisius, 1707): 75–76.</ref> [[Johann Joseph Fux]] cites the phrase in his seminal [[1725 in music|1725]] work {{lang|la|[[Gradus ad Parnassum]]}}, [[Georg Philipp Telemann]] in [[1733 in music|1733]] describes, "mi against fa", which the ancients called "Satan in music"—and [[Johann Mattheson]], in [[1739 in music|1739]], writes that the "older singers with solmization called this pleasant interval {{lang|la|italic=no|'mi contra fa'}} or 'the devil in music'."<ref>{{cite book |last=Reinhold |first=Hammerstein |author-link = Reinhold Hammerstein| title=Diabolus in musica: Studien zur Ikonographie der Musik im Mittelalter|series=Neue Heidelberger Studien zur Musikwissenschaft |volume=6 |year=1974 |publisher=Francke |location=Bern |language=de |oclc=1390982 |pages=7 |quote=...&nbsp;mi contra fa&nbsp;... welches die alten den Satan in der Music nenneten&nbsp;... alten Solmisatores dieses angenehme Intervall mi contra fa oder den Teufel in der Music genannt haben.}}</ref> 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.<ref>{{cite journal |last1=Smith |first1=F. J. |title=Some aspects of the tritone and the semitritone in the ''Speculum Musicae'': the non-emergence of the ''diabolus in musica'' |journal=Journal of Musicological Research |date=1979 |volume=3 |issue=1–2 |pages=63–74 [70] |doi=10.1080/01411897908574507}}</ref> However [[Denis Arnold]], in the ''[[Oxford Companion to Music#The New Oxford Companion to Music|New Oxford Companion to Music]]'', suggests that the nickname was already applied early in the medieval music itself:

{{blockquote|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 {{lang|la|italic=no|"Diabolus in Musica"}} ("the devil in music").<ref>[[Denis Arnold|Arnold, Denis]] (1983). "Tritone". in ''The New Oxford Companion to Music'', Volume 1: A–J, Oxford University Press. {{ISBN|0-19-311316-3}}</ref>}}

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 [[excommunication|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 {{lang|la|[[Musica Enchiriadis]]}}. In all these expressions, including the commonly cited {{lang|la|"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, 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 [[Tonality|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 ii<sup>o</sup>–V–i. Often, the [[Inverted chord|inversion]] ii<sup>o6</sup> 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.<ref>{{cite book |last=Jeppesen |first=Knud |author-link=Knud Jeppesen |translator-first=Glen |translator-last=Haydon |translator-link=Glen Haydon |others=foreword by Alfred Mann |title=Counterpoint: the polyphonic vocal style of the sixteenth century |orig-year=1939 |url=https://archive.org/details/counterpointpoly0000jepp |url-access=registration |year=1992 |publisher=Dover |location=New York |isbn=0-486-27036-X }}</ref>

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, such as [[Franz Liszt]]'s use of the tritone to suggest Hell in his ''[[Dante Sonata]]'':

{{thumb|align=center|caption=Liszt, {{lang|fr|italic=no|"Après une lecture du Dante"}} from {{lang|la|Années de Pèlerinage}}.|content=
<score override_ogg="Liszt, apres une lecture de Dante from Annees de Pelerinage.wav"> {
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\new Staff { \key f \major \tempo "Andante maestoso"
  R1 R r2 \clef F << { <a, c fis>2\(
  \stemDown <bes des g> <ces ees aes> \stemUp <ees ees'>2..\) q16 s1 } \\
  { \stemNeutral s2*3 <aes bes>2\<_\markup\italic{poco rit.} <g bes>\>
  <ees aes ces ees>1\arpeggio\!^\fermata } >>
}
\new Staff { \clef F \key f \major
  \slashedGrace { <a a'>8 } <a a'>2^^ \slashedGrace { <ees ees'>8 } <ees ees'>2^^
  \slashedGrace { <a, a'>8 } <a a'>4^^ \slashedGrace { <ees ees'>8 } <ees ees'>4^^
  \slashedGrace { <a, a'>8 } <a a'>4^^ \slashedGrace { <ees ees'>8 } <ees ees'>4^^~q1~q~
  q4 <fes fes'>8. ( <ges ges'>16 <fes fes'>4 <ees ees'> )
  <aes ees' aes>1\arpeggio_\fermata %<!-- can't get fermata on the barline to work here -->
}
>> } }</score>}}

{{clear}}

{{thumb|content=[[File:Siegfried Act 2 prelude.png|400px]][[File:Wagner, Prelude to Act 2 of Siegfried.wav]]
|caption=Wagner, Prelude to Act 2 of ''Siegfried''.}}

—or [[Wagner]]'s use of timpani tuned to C and F{{music|sharp}} to convey a brooding atmosphere at the start of the second act of the opera ''[[Siegfried (opera)|Siegfried]]''.

{{thumb|content=[[File:Debussy la Damoiselle Fig 30.png|400px]][[File:Debussy, la Damoiselle Elue, Fig 30.wav]]
|caption=Debussy, {{lang|fr|La Damoiselle élue}}, Figure 30.}}

In his early cantata {{lang|fr|[[La Damoiselle élue]]}}, [[Debussy]] uses a tritone to convey the words of the poem by [[Dante Gabriel Rossetti]].

{{thumb|content=[[File:Debussy Quartet 2nd movement, bars 140-7.png|400px]][[File:Debussy Quartet 2nd movement, bars 140-7.wav]]
|caption=Debussy, String Quartet, 2nd movement,<br />bars 140&ndash;147.}}

[[Roger Nichols (musical scholar)|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."<ref>Nichols, R. (1972). ''Debussy''. Oxford University Press.</ref> Debussy's String Quartet also features passages that emphasize the tritone.

The tritone was also exploited heavily in that period as an interval of [[modulation (music)|modulation]] for its ability to evoke a strong reaction by moving quickly to [[distantly related key]]s. For example, the climax of [[Hector Berlioz]]'s {{lang|fr|[[La damnation de Faust]]}} (1846) consists of a transition between "huge B and F chords" as Faust arrives in [[Pandæmonium (Paradise Lost)|Pandaemonium]], the capital of Hell.<ref>{{cite book |first=Julian |last=Rushton |title=The Musical Language of Berlioz |page=254 |publisher= Cambridge University Press |year=1983}}</ref> Musicologist [[Julian Rushton]] calls this "a tonal wrench by a tritone".<ref>{{cite book |first=Julian |last=Rushton |title=The Music of Berlioz | publisher=Oxford University Press |year=2001}}</ref>

Later, in [[twelve-tone music]], [[serial music|serialism]], and other 20th century compositional idioms, composers considered it a neutral interval.<ref>{{cite book |last=Persichetti |first=Vincent |author-link=Vincent Persichetti |title=Twentieth-century Harmony: Creative Aspects and Practice |url=https://archive.org/details/isbn_9780393095395 |url-access=registration |year=1961 |publisher=W. W. Norton |location=New York |isbn=0-393-09539-8 |oclc=398434}}</ref> 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]].<ref>{{cite book |last=Lendvai |first=Ernő |author-link=Ernő Lendvai |title=Béla Bartók: An Analysis of his Music |others=introd. by [[Alan Bush]] |year=1971 |publisher=Kahn & Averill |location=London |isbn=0-900707-04-6 |oclc=240301 |pages=1–16}}</ref> Tritone relations are also important in the music of [[George Crumb]]{{Citation needed|date=March 2016}}<!--The cited source mentions only Britten, not Crumb.--> and [[Benjamin Britten]], whose ''[[War Requiem]]'' features a tritone between C and F♯ as a recurring motif.<ref>{{Cite web |url=http://www.cco.caltech.edu/~tan/Britten/req1.html |title=Musical Analysis of the War Requiem |access-date=16 March 2016}}</ref> John Bridcut (2010, p.&nbsp;271) describes the power of the interval in creating the sombre and ambiguous opening of the ''War Requiem'':<ref>{{cite web|url=https://www.youtube.com/watch?v=rsSMCq7pl_k|title=Britten: War Requiem|date=29 August 2014 |via=YouTube}}</ref> "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&nbsp;... eventually resolves into a major chord for the arrival of the boys singing 'Te decet hymnus'."<ref>Bridcut, J. (2010), ''Essential Britten, a pocket guide for the Britten Centenary''. London, Faber.</ref> [[Leonard Bernstein]] uses the tritone harmony as a basis for much of ''[[West Side Story]]''.<ref name=":0">{{Cite news|last=Kogan|first=Judith|date=2017-10-31|title=The Unsettling Sound Of Tritones, The Devil's Interval|language=en|work=NPR|url=https://www.npr.org/2017/10/31/560843189/the-unsettling-sound-of-tritones-the-devils-interval|access-date=2021-11-11}}</ref><ref>{{Cite web|last=Rizzi|first=Sofia|date=2019-03-04|title=Why did Bernstein build West Side Story around 'The Devil's Interval'?|url=https://www.classicfm.com/composers/bernstein-l/bernstein-west-side-story-tritone/|access-date=2021-11-11|website=Classic FM|language=en}}</ref> [[George Harrison]] uses tritones on the downbeats of the opening phrases of [[the Beatles]] songs "[[The Inner Light (song)|The Inner Light]]", "[[Blue Jay Way]]", and "[[Within You Without You]]", creating a prolonged sense of suspended resolution.<ref>Dominic Pedler. ''The Songwriting Secrets of the Beatles''. Music Sales Ltd. [[Omnibus Press]]. London, 2010 pp. 522–523</ref> 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.&nbsp;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'."<ref>Moskowitz, D. (2010). ''The Words and Music of Jimi Hendrix''. Praeger.</ref> The opening riff of "[[Black Sabbath (song)|Black Sabbath]]", the first song on [[Black Sabbath]]'s eponymous debut album, is an inversion of a tritone;<ref>{{cite web|url=http://abclocal.go.com/wjrt/story?section=news/entertainment/listening_room&id=7274202|archive-url=https://web.archive.org/web/20110629021833/http://abclocal.go.com/wjrt/story?section=news%2Fentertainment%2Flistening_room&id=7274202|url-status=dead|archive-date=29 June 2011|title='Sleeping (In the Fire)': Listening Room fearless leader faces down fear|last=Chesna|first=James|date=26 February 2010|publisher=[[WJRT-TV|WJRT-TV/DT]]|access-date=28 February 2010}}</ref> the album, and this song in particular, are considered to mark the birth of [[heavy metal music]].<ref>William Irwin, ''Black Sabbath and Philosophy: Mastering Reality'' (Hoboken: Wiley-Blackwell, 2012), {{ISBN|978-1118397596}}</ref>

[[File:Tritone substitution.png|thumb|upright=1.4|right|[[Tritone substitution]]: F{{music|#}}<sup>7</sup> may substitute for C<sup>7</sup>, and vice versa, because they both share E{{music|natural}} and B{{music|flat}}/A{{music|sharp}} and due to [[voice leading]] considerations.[[File:Tritone substitution.mid]]]]

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.

{{blockquote|
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.<ref name="Renold" />}}

== See also ==
* [[List of meantone intervals]]
* [[Interval (music)|List of musical intervals]]
* [[List of pitch intervals]]
* [[Ditone]]
* [[Whole tone|Tone]]
* {{slink|Hexatonic scale|Tritone scale}}
* {{slink|Consecutive fifths|Unequal fifths}}
* [[Petrushka chord]]
* [[Tritonic scale]]

== References ==
{{reflist|30em}}

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

== External links ==
* {{webarchive |url=https://web.archive.org/web/20080106220501/http://www.cameron.edu/~lloydd/webdoc1.html |date=January 6, 2008 |title=Tritone paradox and Shepard Tones }}
* [http://news.bbc.co.uk/2/hi/uk_news/magazine/4952646.stm BBC News Magazine article about the tritone]
* [https://www.theguardian.com/music/2007/oct/12/popandrock.classicalmusicandopera Satan's all-time greatest hit: Will Hodgkinson on the devil's interval]

{{Intervals}}

[[Category:Augmented fourths]]
[[Category:Diminished fifths]]
[[Category:The Devil in classical music]]