Tritone | Understanding Western Music Theory

In the realm of Western music theory, the tritone is a particularly intriguing and historically controversial musical interval. Strictly defined, a tritone spans three whole tones, or six semitones, making it exactly half of an octave in the twelve-tone equal temperament (12-TET) system.

Tritone | Understanding Western Music Theory

For example, the interval from F to B (written as F–B) is a tritone. This interval can be broken down into three adjacent whole tones:

• F–G

• G–A

• A–B

Diatonic vs Chromatic Context

In the diatonic scale (such as the C major scale), only one tritone exists per octave. In the key of C major, the F–B interval is the lone tritone. However, under the broader definition, where a tritone is simply any interval spanning six semitones, there are two tritones per octave within a diatonic context:

The Role of the Tritone in Harmony

Historically, the tritone was viewed as dissonant, sometimes even associated with the devil in medieval times (e.g., diabolus in musica). In classical music, its dissonant nature gives it an essential role in harmonic tension and resolution.

• To disrupt tonality: A tritone can be used to avoid establishing a tonal centre. For instance, introducing a note three whole tones away from the key’s tonic helps obscure a clear sense of key.

• To establish tonality: Paradoxically, the tritone plays a critical role in dominant seventh chords, which are central to functional harmony. These chords help reinforce tonality by resolving naturally to the tonic.

This dual capacity—to both create and destroy tonality—makes the tritone uniquely flexible, distinctive, and omnipresent in Western music.

Terminology: Tectonic and Atritonic

A musical scale or chord may be described based on the presence or absence of tritones:

Augmented Fourth and Diminished Fifth

Let’s delve deeper into the two primary manifestations of the tritone:

• Augmented Fourth (A4)
  Encompasses four staff positions, and is formed by raising a perfect fourth by a semitone.

• Diminished Fifth (d5)
  Encompasses five staff positions, formed by lowering a perfect fifth by a semitone.

Both intervals span six semitones and are enharmonically equivalent in 12-TET. For example:

Although these two intervals sound identical on modern pianos, they are written differently in notation and can behave differently in voice leading and harmonic analysis.

Tritones in the Chromatic Scale

A chromatic scale divides the octave into 12 semitones, meaning it contains 12 distinct tritones. Each one begins on a different note and spans six semitones.

Half of these are considered augmented fourths, and the other half diminished fifths.

Chromatic Scale Example:

If we construct a chromatic scale starting on C, the tritones include:

• C–F♯ (A4)

• C♯–G (A4)

• D–G♯ (A4)

• etc.

And inversely:

• F♯–C (d5)

• G–C♯ (d5)

• G♯–D (d5)

• etc.

Mathematical Structure and Definitions

Strict Definition:

A tritone is composed of three whole tones (whole steps):
Tritone (TT) = T + T + T

In equal temperament tuning:

Each tone (T) = 2 semitones (S), hence:
TT = 3T = 6S

Decomposing Tritones

Augmented Fourth (F–B):

• F–G (T)

• G–A (T)

• A–B (T)
  → Total = 3T = Tritone

Diminished Fifth (B–F):
Using diatonic steps:

• B–C (S)

• C–D (T)

• D–E (T)

• E–F (S)
  → Total = 2S + 2T = Tritone

Using chromatic steps:

• B–C♯ (T)

• C♯–D♯ (T)

• D♯–E♯ (T)

• E♯–F (diminished second)
  → Still 3T = Tritone

It’s important to note that E♯ and F are enharmonically equivalent in sound but written differently in musical notation. This distinction affects theoretical analysis.

The tritone is far more than a simple interval; it is a window into the complex interplay between harmony, notation, acoustics, and history in Western music. From its ambiguous tonality to its dual identity as both an augmented fourth and a diminished fifth, it represents both dissonance and possibility.

Strict Interpretation (Diatonic Scale)

• In diatonic theory, whole tones (T) are seen as incomposite; they can’t be broken down into smaller intervals.

• The tritone in this system is only the augmented fourth (A4), formed by three adjacent whole tones (T+T+T).

  • Example: In C major, F to B is the only tritone (F–G–A–B).

• The diminished fifth (d5) is not considered a tritone here because it comprises a different interval structure: S+T+T+S.

  • Example: B to F (d5) in C major.

 

Size in Different Tuning Systems

Equal Temperament (12-TET)

• A4 and d5 are both exactly 600 cents, forming a symmetrical pair (each the inverse of the other).

• A4 + A4 = 6 whole tones = 1 octave (P8).

Meantone Temperament

• In quarter-comma meantone, two A4s are slightly less than an octave by a small interval called a diesis (128/125).

Just Intonation

• Various ratios exist:

  • A4: 45/32 or 25/18

  • d5: 64/45, 36/25, 1024/729

• These are derived based on context, especially the C major scale.

7-limit Tuning (Septimal)

• A4: 7/5 (~582.5 cents), called septimal tritone

• d5: 10/7 (~617.5 cents), called Euler’s tritone

• These are more consonant than some other theoretical ratios (like 17/12 or 24/17).

31-Tone Equal Temperament

• A4: ~619.35 cents

• d5: ~580.65 cents

• This is perceptually close to septimal meantone.

Eleventh Harmonic (Undecimal Tritone)

• Ratio 11/8 (~551.32 cents), known as the lesser undecimal tritone or semi-augmented fourth.

• Found in:

  • Alphorns

  • Natural horn repertoire (Brahms, Britten)

  • Just intonation systems

 

Dissonance and Expressiveness

• Tritones are considered dissonant and awkward to sing or include in melodies.

• Historically avoided by Western composers in melodic lines.

• Instead, composers used:

  • Passing notes

  • Indirect skips

• Over time, composers began to embrace the expressiveness of the tritone.

• Hindemith recognized its unstable and distinctive character, resisting simple classification via superparticular ratios (like 5/4 or 3/2).

• Complex ratios (e.g., 45/32) add to the tritone’s “diabolical” reputation.

 

Historical Perspective

• Some theorists critique the use of large-number ratios like 45/32 in 5-limit tuning, preferring smaller-number consonances.

• The Pythagorean tritone (81/64), despite using only primes 2 and 3, is considered less aurally satisfying than ratios involving higher primes like 5 or 7.

• Emphasis is placed on simplicity and musical practicality rather than numerical complexity.

The Tritone in Music Theory and Practice

Occurrences in Diatonic Scales

• Major Scale:
  The tritone (augmented fourth, A4) naturally occurs between the 4th and 7th scale degrees.
  Example: F to B in C major.

• Natural Minor Scale:
  The tritone appears between the 2nd and 6th scale degrees.
  Example: D to A♭ in C minor.

• Melodic Minor Scale:

  • Ascending: Tritones occur between the 3rd–6th and 4th–7th degrees.

  • Descending: Tritone appears between the 2nd–6th degrees.

  • Implication: Supertonic chords in natural minor inherently contain a tritone.

• Locrian Mode:
  The defining interval is the tritone between the 1st and 5th degrees.

Occurrences in Chords

• Dominant Seventh Chords (V7):
  Contain a tritone between the 3rd and 7th (e.g., B–F in G7).

• Augmented Sixth Chords:

  • German Sixth: Contains a tritone (e.g., A to D♯ in A minor).

  • French Sixth: Seen as two tritones a major second apart.

• Diminished Chords:

  • Diminished Triad: Contains a tritone (diminished fifth).

  • Half-Diminished Seventh: Contains one tritone.

  • Fully Diminished Seventh: Contains two tritones, spaced a minor third apart.

• Extended Chords (e.g., 9th, 11th, 13th):
  Often include tritones for harmonic richness and dissonance.

Resolution of the Tritone

• General Rule: Tritones resolve by step in contrary motion.

• Augmented Fourth (A4): Resolves outward to a minor or major sixth.

• Diminished Fifth (d5): Resolves inward to a major or minor third.

• Historical View:

  • Tritone as “mi contra fa” avoided in Renaissance music.

  • Only considered a “true tritone” if it spans three whole tones.

  • In just intonation, listener perception affects resolution direction (based on cents value).

Historical and Cultural Usage

• Medieval/Renaissance:
  Tritone viewed as dissonant; avoided in church music. Called “Diabolus in Musica” (the Devil in Music).

  • Term appeared in 18th-century writings (Werckmeister, Fux, Mattheson).

  • Avoidance often due to the conflict between B♮ (mi) and F (fa) across hexachords.

• Baroque/Classical Eras:

  • Tritone became structurally essential in functional harmony (especially in dominant and diminished chords).

  • Used for tension-resolution (e.g., ii°6–V–i progressions in minor keys).

• Romantic and Modern Classical:

  • Tritone used expressively (e.g., Liszt’s Dante Sonata to depict Hell).

  • Modulatory use to distant keys.

  • In 12-tone and serial music, tritones became harmonically neutral.

Tritone in Jazz and Contemporary Music

• Jazz Harmony:

  • Tritone is vital in dominant and substitute dominant chords (e.g., F♯7 can substitute for C7).

  • Tritone Substitution: Built on the flat fifth of the original dominant chord; enhances chromaticism and smooth voice leading.

• Chord Extensions:

  • In extended chords (11th, 13th), the perfect 11th is often raised to a ♯11 (tritone) to avoid dissonance and promote brightness.

• 20th-Century Usage:

  • Integral in tonal axis theories (e.g., Bartók via Ernő Lendvaï), works of George Crumb, and The Beatles (e.g., “Blue Jay Way”, “Within You Without You”).

 

Acoustic and Psychoacoustic Considerations

• Cents Comparison in Just Intonation:

  • Just Augmented Fourth: ~590.2 cents.

  • True Diminished Fifth: ~588.3 cents.

  • These lie just under 600 cents (half an octave).

  • Our ears often want to resolve these downward, unless the interval exceeds the midpoint of the octave.

 

See also:

1. List of meantone intervals
2. List of musical intervals
3. List of pitch intervals
4. Hexatonic scale#Tritone scale
5. Consecutive fifths#Unequal fifths
6. Petrushka chord

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