Tritone

Inverse tritone

Name

Other names augmented fourth,

diminished fifth

Abbreviation TT

Size

Semitones 6

Interval class 6

Just interval 25:18, 36:25, 45:32,

64:45, 7:5, 10:7, ...

Cents

Equal

temperament

600

24 equal

temperament

600

Just intonation 569, 631, 590, 610, 583,

617, ...

tritone

From Wikipedia, the free encyclopedia

In music theory, the tritone is strictly defined as amusical interval composed of three adjacent wholetones.[1] For instance, the interval from F up to the Babove it (in short, F–B) is a tritone as it can bedecomposed into the three adjacent whole tones F–G,G–A, and A–B. According to this definition, within adiatonic scale there is only one tritone for each octave.For instance, the above-mentioned interval F–B is theonly tritone formed from the notes of the C major scale.A tritone is also commonly defined as an intervalspanning six semitones. According to this definition, adiatonic scale contains two tritones for each octave. Forinstance, the above-mentioned C major scale containsthe tritones F–B (from F to the B above it, also calledaugmented fourth) and B–F (from B to the F above it,also called diminished fifth, semidiapente, orsemitritonus).[2]

In classical music, the tritone is a harmonic and melodicdissonance and is important in the study of musicalharmony. The tritone can be used to avoid traditionaltonality: "Any tendency for a tonality to emerge may beavoided by introducing a note three whole tones distantfrom the key note of that tonality."[3] Contrarily, thetritone found in the dominant seventh chord helpsestablish the tonality of a composition. These contrasting uses exhibit the flexibility, ubiquity, anddistinctness of the tritone in music.

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

1 Augmented fourth and diminished fifth2 Definitions

2.1 Broad interpretation (chromatic scale)2.2 Strict interpretation (diatonic scale)

3 Size in different tuning systems4 Eleventh harmonic5 Dissonance and expressiveness6 Common uses

6.1 Occurrences in diatonic scales6.2 Occurrences in chords

Tritone - Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Tritone

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Chromatic scale on C: full octave ascending and descending

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.

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 .

6.3 Resolution6.4 Other uses

7 Historical uses8 See also9 Sources10 Further reading11 External links

Since a chromatic scale is formed by 12pitches (each a semitone apart from itsneighbors), it contains 12 distincttritones, each starting from a differentpitch and spanning six semitones.According to a complex but widely usednaming convention, six of them are classified asaugmented fourths, and the other six as diminishedfifths.

Under that convention, a fourth is an intervalencompassing four staff positions, while a fifthencompasses five staff positions (see interval numberfor more details). The augmented fourth (A4) anddiminished fifth (d5) are defined as the intervalsproduced by widening the perfect fourth and narrowingthe perfect fifth by one chromatic semitone.[4] Theyboth span six semitones, and they are the inverse ofeach other, meaning that their sum is exactly equal toone perfect octave (A4 + d5 = P8). In 12-tone equaltemperament, the most commonly used tuningsystem, the A4 is equivalent to a d5, as both have thesize of exactly half an octave. In most other tuningsystems, they are not equivalent, and neither is exactly equal tohalf an octave.

Any augmented fourth can be decomposed into three wholetones. For instance, the interval F–B is an augmented fourth andcan 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 threeadjacent whole tones. The reason is that a whole tone is a majorsecond, and according to a rule explained elsewhere, thecomposition of three seconds is always a fourth (for instance, anA4). To obtain a fifth (for instance, a d5), it is necessary to addanother second. For instance, using the notes of the C major scale, the diminished fifth B–F can bedecomposed into the four adjacent intervals


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Tritone drawn in the chromatic

circle.

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 (diminishedsecond).

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-mentionednaming convention, they are considered different notes, as they are written on different staff positions.

A tritone (abbreviation: TT) is traditionally defined as a musicalinterval composed of three whole tones. As the symbol for wholetone 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 casefor many tuning systems) can this formula be simplified to:

TT = 3T

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

Broad interpretation (chromatic scale)

In a chromatic scale, the interval between any note and the previous or next is a semitone. Using thenotes 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 sixsemitones:

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 twelvedifferent tritones, each starting from a different note and ending six notes above it. Although all ofthem span six semitones, six of them are classified as A4, and the other six as d5.

Strict interpretation (diatonic scale)

Within a diatonic scale, whole tones are always formed by adjacent notes (such as C and D) andtherefore they are regarded as incomposite intervals. In other words, they cannot be divided intosmaller intervals. Consequently, in this context the above-mentioned "decomposition" of the tritone


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Tritone: just augmented fourth

between C and F♯+ Play 45:32

(590.22 cents) .

Tritone: Pythagorean augmented

fourth between C and F♯++

Play 729:512 (611.73 cents) .

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 adjacentwhole 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 threeadjacent whole tones. It is a fourth because the notes from F to B are four (F, G, A, B). It isaugmented (i.e., widened) because it is wider than most of the fourths found in the scale (they areperfect fourths).

According to this interpretation, the d5 is not a tritone. Indeed, in a diatonic scale, there's only one d5,and this interval does not meet the strict definition of tritone, as it is formed by one semitone, twowhole 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 notesfrom B to F are five (B, C, D, E, F). It is diminished (i.e. narrowed) because it is smaller than most ofthe fifths found in the scale (they are perfect fifths).

In 12-tone equal temperament, the A4 is exactly half an octave(i.e., a ratio of √2:1 or 600 cents; play ). The inverse of 600cents is 600 cents. Thus, in this tuning system, the A4 and itsinverse (d5) are equivalent.

The half-octave or equal tempered A4 and d5 are unique in beingequal to their own inverse (each to the other). In other meantonetuning systems, besides 12-tone equal temperament, A4 and d5are distinct intervals because neither is exactly half an octave. Inany meantone tuning near to 2⁄9-comma meantone the A4 is near

to the ratio 7⁄5 (582.51) and the d5 to 10⁄7 (617.49), which is whatthese intervals are in septimal meantone temperament. In 31equal temperament, for example, the A4 is 619.35 cents, whereasthe d5 is 580.65 cents. This is perceptually indistinguishable fromseptimal meantone temperament.

Since they are the inverse of each other, by definition A4 and d5always 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 exactlyone 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.


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Tritone: the classic augmented

fourth between C and F♯ Play

25:18 (568.72 cents)

Tritone: the classic diminished fifth

between C and G♭ Play 36:25

(631.28 cents)

Lesser septimal tritone on

between C and G ♭[5] Play 7:5

(582.51 cents) .

Greater septimal tritone between

C and F♯ [5] Play 10:7 (617.49

cents) .

In just intonation several different sizes can be chosen both forthe A4 and the d5. For instance, in 5-limit tuning, the A4 is either45/32[6][7][8] or 25/18,[9] and the d5 is either 64/45 Play or36/25,[10] or 1024:729 Play . The 64:45 just diminished fiftharises in the C major scale between B and F, consequently the45:32 augmented fourth arises between F and B.[11]

These ratios are not in all contexts regarded as strictly just butthey are the justest possible in 5-limit tuning. 7-limit tuning allowsfor the justest possible ratios (ratios with the smallest numeratorand denominator), namely 7/5 for the A4 (about 582.5 cents, alsoknown 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 moreconsonant than 17/12 (about 603.0 cents) and 24/17 (about 597.0cents), which can be obtained in 17-limit tuning, yet the latter arealso fairly common, as they are closer to the equal-temperedvalue of 600.0 cents.

Known as the lesser undecimal tritone or undecimalsemi-augmented fourth, 11:8 (551.318 cents), the ratio of theeleventh harmonic (F 4 above C1), is found in some just tuningsand on many instruments. For example, very long alphorns mayreach the twelfth harmonic and transcriptions of their musicusually show the eleventh harmonic sharp (F♯ above C, forexample), as in Brahms's First Symphony.[14] This note is oftencorrected to 4:3 on the natural horn in just intonation orPythagorean tunings, but the pure eleventh harmonic was used inpieces including Britten's Serenade for tenor, horn and strings.[15]

Ivan Wyschnegradsky considered the major fourth a goodapproximation of the eleventh harmonic.

Use of the eleventh harmonic in the prologue to Britten's

Serenade for tenor, horn and strings. Play

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


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Eleventh harmonic between C

and F↑. Play 11:8 (551.32

cents)

is assigned the ratio 64/45 or 45/32, depending on themusical context, or indeed some other ratio, it is notsuperparticular, which is in keeping with its unique rolein music.[16]

Although this ratio [45/32] is composed of numberswhich are multiples of 5 or under, they are excessivelylarge for a 5-limit scale, and are sufficient justification,either in this form or as the tempered "tritone," for theepithet "diabolic," which has been used to characterizethe interval. This is a case where, because of thelargeness 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 oftheir excessive largeness the ear certainly prefers 5/4 for this approximate degree, eventhough it involves a prime number higher than 3. In the case of the 45/32, 'tritone' ourtheorists have gone around their elbows to reach their thumbs, which could have beenreached simply and directly and non-'diabolically' via number 7.[17]

Occurrences in diatonic scales

The augmented fourth (A4) occurs naturally between the fourth and seventh scale degrees of themajor scale (for example, from F to B in the key of C major). It is also present in the natural minorscale 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 differentlocations when ascending and descending (when the scale ascends, the tritone appears between thethird and sixth scale degrees and the fourth and seventh scale degrees, and when the scaledescends, the tritone appears between the second and sixth scale degrees). Supertonic chords usingthe notes from the natural minor mode thus contain a tritone, regardless of inversion. Containingtritones, these scales are tritonic.

Occurrences in chords

The dominant seventh chord in root position contains a diminished fifth (tritone) within its pitchconstruction: it occurs between the third and seventh above the root. In addition, augmented sixthchords, some of which are enharmonic to dominant seventh chords, contain tritones spelled asaugmented fourths (for example, the German sixth, from A to D♯ in the key of A minor); the Frenchsixth 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 thediminished-fifth interval (i.e. a tritone). The half-diminished seventh chord contains the same tritone,


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Tritone resolution inward

( Play ), and outwards ( Play ).

The theme opening Claude Debussy's Prélude à

l'après-midi d'un faune outlines a tritone (between C♯and G) Play

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 functionalharmonic analysis, the tritone pushes towards resolution,generally resolving by step in contrary motion. This determinesthe resolution of chords containing tritones.

The augmented fourth resolves outward to a minor or major sixth.The inversion of this, a diminished fifth, resolves inward to a majoror minor third. The diminished fifth is often called a tritone inmodern tonal theory, but functionally and notationally it can onlyresolve inwards as a diminished fifth and is therefore notreckoned a tritone—that is, an interval composed of three adjacent whole tones—in mid-renaissance(early 16th-century) music theory.[18]

Other uses

The tritone is also one of the defining features of the Locrian mode, being featured between the andfifth scale degrees.

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

The tritone is a restless interval, classed as adissonance in Western music from the earlyMiddle Ages through to the end of the commonpractice period. This interval was frequentlyavoided in medieval ecclesiastical singingbecause of its dissonant quality. The firstexplicit prohibition of it seems to occur with thedevelopment of Guido of Arezzo's hexachordalsystem, who suggested that rather than makeB♭ a diatonic note, the hexachord be movedand based on C to avoid the F-B tritonealtogether. Later theorists such as Ugolino d'Orvieto and Tinctoris advocated for the inclusion ofB♭.[19] From then until the end of the Renaissance the tritone was regarded as an unstable intervaland 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 theearly 18th century, though its use is not restricted to the tritone. Andreas Werckmeister cites this termin 1702 as being used by "the old authorities" for both the tritone and for the clash betweenchromatically related tones such as F and F♯,[21] and five years later likewise calls "diabolus inmusica" the opposition of "square" and "round" B (B♮ and B♭, respectively) because these notes


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represent the juxtaposition of "mi contra fa".[22] Johann Joseph Fux cites the phrase in his seminal1725 work Gradus ad Parnassum, Georg Philipp Telemann in 1733 describes, "mi against fa," whichthe ancients called "Satan in music"—and Johann Mattheson, in 1739, writes that the "...older singerswith solmization called this pleasant interval 'mi contra fa' or 'the devil in music'."[23] Although thelatter two of these authors cite the association with the devil as from the past, there are no knowncitations of this term from the Middle Ages, as is commonly asserted.[24] However Denis Arnold, inthe New Oxford Companion to Music, suggests that the nickname was already applied early in themedieval music itself:

It seems first to have been designated as a "dangerous" interval when Guido of Arezzodeveloped 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 inmusic").[25]

That original symbolic association with the devil and its avoidance led to Western cultural conventionseeing the tritone as suggesting "evil" in music. However, stories that singers were excommunicatedor otherwise punished by the Church for invoking this interval are likely fanciful. At any rate,avoidance of the interval for musical reasons has a long history, stretching back to the parallelorganum of the Musica Enchiriadis. In all these expressions, including the commonly cited "mi contrafa est diabolus in musica", the "mi" and "fa" refer to notes from two adjacent hexachords. Forinstance, in the tritone B–F, B would be "mi", that is the third scale degree in the "hard" hexachordbeginning on G, while F would be "fa", that is the fourth scale degree in the "natural" hexachordbeginning on C.

Later, with the rise of the Baroque and Classical music era, composers accepted the tritone, but stilldidn't use it in a specific, controlled way—notably through the principle of the tension-releasemechanism of the tonal system. In that system (which is the fundamental musical grammar ofBaroque and Classical music), the tritone is one of the defining intervals of the dominant-seventhchord and two tritones separated by a minor third give the fully diminished seventh chord itscharacteristic sound. In minor, the diminished triad (comprising two minor thirds, which together addup to a tritone) appears on the second scale degree—and thus features prominently in theprogression iio–V–i. Often, the inversion iio6 is used to move the tritone to the inner voices as thisallows for stepwise motion in the bass to the dominant root. In three-part counterpoint, free use of thediminished 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 totallyfreely, without functional limitations notably in an expressive way to exploit the "evil" connotationsculturally associated with it (e.g., Franz Liszt's use of the tritone (https://www.youtube.com/watch?v=_PrsflaFB58) to suggest Hell in his Dante Sonata:


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Liszt, "Après une lecture du Dante" from Années de Pèlerinage

- or Wagner’s use of timpani tuned to C and F sharp to convey a brooding atmosphere at the start ofthe second act (https://www.youtube.com/watch?v=qt8VeUzG3Zc) of the opera Siegfried.

Wagner, Prelude to Act 2 of Siegfried

In his early cantata La Damoiselle Élue, Debussy uses a tritone to convey the words of the poem byDante Gabriel Rosetti.

Debussy, La Damoiselle Élue, Figure 30

Roger Nichols (1972, p19) says that "the bare fourths, the wide spacing, the tremolos, all depict thewords – ‘the light thrilled towards her’ – with sudden, overwhelming power.” [27] Debussy’s StringQuartet also features passages that emphasise the tritone:


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Debussy, String Quartet, 2nd movement, bars 140-7

The tritone was also exploited heavily in that period as an interval of modulation for its ability to evokea 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 someanalyses 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 oftonality in the music of Béla Bartók.[29] Tritone relations are also important in the music of GeorgeCrumb and Benjamin Britten, whose War Requiem features a tritone between C and F♯ as a recurringmotif.[30] John Bridcut (2010, p. 271) describes the power of the interval in creating the sombre andambiguous opening of the War Requiem (https://www.youtube.com/watch?v=rsSMCq7pl_k): “Theidea that the chorus and orchestra are confident in their wrong-headed piety is repeatedly disputed bythe 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 arrivalof the boys singing 'Te decet hymnus’.”[31] George Harrison uses tritones on the downbeats of theopening phrases of the Beatles songs "The Inner Light", "Blue Jay Way" and "Within You WithoutYou", creating a prolonged sense of suspended resolution.[32] Perhaps the most striking use of theinterval in rock music of the late 1960s can be found in Jimi Hendrix's song "Purple Haze"(https://www.youtube.com/watch?v=Hj1EI0vCvFw). According to Dave Moskowitz (2010, p.12),Hendrix "ripped into 'Purple Haze' by beginning the song with the sinister sounding tritone intervalcreating an opening dissonance, long described as 'The Devil in Music'."[33]

Tritones also became important in the development of jazz tertian harmony, where triads and seventhchords are often expanded to become 9th, 11th, or 13th chords, and the tritone often occurs as asubstitute for the naturally occurring interval of the perfect 11th. Since the perfect 11th (i.e. an octaveplus perfect fourth) is typically perceived as a dissonance requiring a resolution to a major or minor10th, chords that expand to the 11th or beyond typically raise the 11th a semitone (thus giving us anaugmented or sharp 11th, or an octave plus a tritone from the root of the chord) and present it inconjunction with the perfect 5th of the chord. Also in jazz harmony, the tritone is both part of thedominant chord and its substitute dominant (also known as the sub V chord). Because they share the


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Tritone substitution: F♯7 may substitute for C7,

and vice versa, because they both share E♮and B♭/A♯ and due to voice leading

considerations. Play

same tritone, they are possible substitutes for oneanother. This is known as a tritone substitution. Thetritone substitution is one of the most common chordand improvisation devices in jazz.

In the theory of harmony it is known that adiminished interval needs to be resolvedinwards, and an augmented intervaloutwards. ...and with the correct resolutionof the true tritones this desire is totallysatisfied. However, if one plays a justdiminished fifth that is perfectly in tune, forexample, there is no wish to resolve it to amajor third. Just the opposite—aurally onewants to enlarge it to a minor sixth. Theopposite holds true for the just augmentedfourth....These apparently contradictory auralexperiences become understandable whenthe cents of both types of just tritones arecompared with those of the true tritonesand then read 'crossed-over'. One thennotices that the just augmented fourth of590.224 cents is only 2 cents bigger thanthe true diminished fifth of 588.270 cents,and that both intervals lie below the middleof the octave of 600.000 cents. It is nowonder that, following the ear, we want toresolve both downwards. The ear onlydesires the tritone to be resolved upwardswhen it is bigger than the middle of theoctave. Therefore the opposite is the casewith the just diminished fifth of 609.776cents....[7]

List of meantone intervalsList of musical intervalsList of pitch intervalsHexatonic scale#Tritone scaleConsecutive fifths#Unequal fifthsPetrushka chord

Don Michael Randel (2003). The Harvard Dictionary of Music: Fourth Edition. Harvard University Press.1.


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ISBN 0-674-01163-5 (http://www.hup.harvard.edu/catalog.php?isbn=9780674011632).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, "IlTrattato di Prosdocimo de' Beldomandi contro il Lucidario di Marchetto da Padova per la prima voltatrascritto e illustrato", Rivista Musicale Italiana 20 (1913): 731–62, citations on 732–34.

2.

Smith Brindle, Reginald (1966). Serial Composition. Oxford University Press. p. 66. ISBN 0-19-311906-4.3. 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.

4.

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.

5.

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); ISBN0-306-80106-X (pbk).

6.

Renold, Maria (2004). Intervals, Scales, Tones and the Concert Pitch C=128Hz, translated from theGerman by Bevis Stevens, with additional editing by Anna R. Meuss (Forest Row: Temple Lodge):p. 15–16. ISBN 1-902636-46-5.

7.

Helmholtz, Hermann von (2005). On the Sensations of Tone as a Physiological Basis for the Theory ofMusic, p. 457. ISBN 1-4191-7893-8. "Cents in interval: 590, Name of Interval: Just Tritone, Number to anOctave: 2.0. Cents in interval: 612, Name of Interval: Pyth. Tritone, Number to an Octave: 2.0."

8.

Haluska , Ján (2003), The Mathematical Theory of Tone Systems, Pure and Applied Mathematics Series262 (New York: Marcel Dekker; London: Momenta), p. xxiv. ISBN 0-8247-4714-3. "25:18 classicaugmented fourth".

9.

Haluska (2003), p. xxv. "36/25 classic diminished fifth".10. Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction(http://books.google.com/books?id=4WEJAQAAMAAJ&dq=musical+interval+%22pythagorean+major+third%22&source=gbs_navlinks_s), p.165. Theodore Baker,trans. G. Schirmer.

11.

Haluska (2003). p. xxiii. "7/5 septimal or Huygens' tritone, Bohlen-Pierce fourth", "10/7 Euler's tritone".12. 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 notexhaustive, 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."

13.

Monelle, Raymond (2006). The Musical Topic: Hunt, Military And Pastoral, p.102. ISBN 9780253347664.14. Fauvel, John; Flood, Raymond; and Wilson, Robin J. (2006). Music And Mathematics, p.21-22. ISBN9780199298938.

15.

Haluska (2003), p. 286.16. Partch (1974), p. 115. ISBN 0-306-80106-X.17. Margaret Bent, ""Accidentals, Counterpoint, and Notation in Aaron’s Aggiunta to the Toscanello", Journalof Musicology 12: "Aspects of Musical Language and Culture in the Renaissance: A Birthday Tribute toJames Haar" (1994): 306–44. Citation on 308.

18.

Guido d'Arezzo, Epistola de ignoto cantu, lines 309–2219. Drabkin, William. "Tritone". Grove Music Online (subscription access). Oxford Music Online. Retrieved2008-07-21.

20.

Andreas Werckmeister. Harmonologia musica, oder kurze Anleitung zur musicalischen Composition(http://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10527826_00005.html) (Frankfurt andLeipzig: Theodor Philipp Calvisius 1702): 6.

21.

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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 diesesangenehme Intervall mi contra fa oder den Teufel in der Music genannt haben."F. J. Smith, "Some Aspects of the Tritone and the Semitritone in the Speculum Musicae: TheNon-Emergence of the Diabolus in Music," Journal of Musicological Research 3 (1979), pp. 63–74, at 70.

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

Tritone paradox and Shepard Tones (https://web.archive.org/web/20080106220501/http://www.cameron.edu/~lloydd/webdoc1.html) at the Wayback Machine (archived January 6, 2008)BBC News Magazine article about the tritone (http://news.bbc.co.uk/2/hi/uk_news/magazine/4952646.stm)Satan's all-time greatest hit: Will Hodgkinson on the devil's interval (http://www.theguardian.com/music/2007/oct/12/popandrock.classicalmusicandopera)"Why is the Augmented 4th the "chord of evil" that was banned in Renaissance church music?(http://www.guardian.co.uk/notesandqueries/query/0,,-1767,00.html)", Guardian.co.uk.

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