Tritone
The tritone is a musical interval spanning three whole tones, or six semitones, which equals half an octave and can manifest as either an augmented fourth (such as C to F♯) or a diminished fifth (such as C to G♭). Renowned for its inherent dissonance and instability, the tritone creates a sense of tension that demands resolution, arising from its complex frequency ratio of approximately 45:32, which contrasts with the simpler ratios of consonant intervals.[1][2] Historically termed diabolus in musica ("the devil in music"), the tritone earned its ominous nickname in the medieval and Renaissance periods due to its jarring effect, which clashed with the serene harmonies of sacred polyphony and was sometimes used symbolically to depict evil or chaos, as in certain representations of the devil in liturgical dramas.[3][4] Despite persistent myths of a Catholic Church ban during the Middle Ages—allegedly to prevent its corrupting influence—no contemporary documents support such a prohibition; the interval was simply avoided in strict contrapuntal rules to maintain harmonic purity, though it occasionally appeared in Gregorian chant and early motets.[5][6][7] By the Baroque and Classical eras, composers began embracing the tritone for dramatic effect, integrating it into fugues, operas, and symphonies to heighten emotional intensity—exemplified in Johann Sebastian Bach's chromatic subjects, Ludwig van Beethoven's tense harmonies in Fidelio, and Franz Liszt's evocative use in the Dante Sonata.[3][8][2] In the 20th century, it became a cornerstone of modernism, with Béla Bartók exploiting its symmetry in quartal harmonies and Alban Berg employing it structurally in atonal works like Wozzeck.[9] In jazz and popular music, the tritone substitution technique—replacing a dominant seventh chord's fifth with its tritone counterpart—adds harmonic surprise and color, as heard in Charlie Parker's "Blues for Alice" and Thelonious Monk's improvisations.[10] Iconic examples include the opening riff of Jimi Hendrix's "Purple Haze," the theme of "The Simpsons," and Leonard Bernstein's deliberate use throughout West Side Story to underscore conflict, transforming the once-feared interval into a versatile element of musical expression across genres.[1][11][12]Fundamentals
Definition and Names
The tritone is a musical interval spanning three adjacent whole tones in the diatonic scale. This interval measures six semitones and divides the octave into two equal parts.[13] Known alternatively as the augmented fourth—for instance, from C to F♯—or the diminished fifth, such as from C to G♭, the tritone represents enharmonically equivalent forms of the same pitch relationship. These names distinguish it from the consonant perfect fourth (five semitones) and perfect fifth (seven semitones), which form the structural foundation of many harmonic progressions.[14] In pitch-class set theory, the tritone is classified as interval class 6 (ic6), reflecting its unique position as the largest non-octave interval class.[15] The term "tritone" derives from the Medieval Latin "tritonus," literally meaning "third tone," a compound of the Greek prefix "tri-" (three) and "tonos" (tone or sound).[13] This etymology directly references the interval's composition of three whole tones.[16]Interval Size in Tuning Systems
The size of a musical interval, including the tritone, is quantified in cents using the formula \text{cents} = 1200 \times \log_2(r), where r is the frequency ratio of the higher to lower pitch.[17] This logarithmic measure divides the equal-tempered octave (1200 cents) into 100 equal parts per semitone, facilitating comparisons across tuning systems.[18] In twelve-tone equal temperament, the dominant modern system, the tritone encompasses exactly six semitones, yielding a ratio of \sqrt{2} \approx 1.41421 and precisely 600 cents. This uniform division ensures enharmonic equivalence between the augmented fourth and diminished fifth, both at 600 cents.[19] Just intonation employs simple integer frequency ratios derived from the harmonic series within the 5-limit (primes up to 5). The augmented fourth uses the ratio 45:32 (\approx 1.40625), measuring about 590 cents, while the diminished fifth uses 64:45 (\approx 1.42222), measuring about 610 cents. These values reflect the system's emphasis on pure consonances like the major third (5:4, 386 cents), but result in asymmetric tritones without enharmonic identity.[20][21] Pythagorean tuning, based on stacked 3:2 fifths (702 cents each), produces an augmented fourth of 729:512 (\approx 1.42383), approximately 612 cents. The complementary diminished fifth is 512:729, or about 588 cents, highlighting the system's bias toward pure fifths at the expense of thirds.[19] Meantone temperaments temper fifths (typically to ~697 cents) to achieve purer major thirds (~386 cents), compressing the chromatic scale and narrowing the augmented fourth while widening the diminished fifth. Quarter-comma meantone, a seminal variant, exemplifies this with an augmented fourth of ~578 cents (-22 cents deviation from equal temperament) and a diminished fifth of ~622 cents (+22 cents deviation).[22] Well temperaments, such as Werckmeister III, distribute irregularities more evenly across keys to enable modulation without extreme dissonance in any mode, keeping tritones near 600 cents but with slight variations by context. In Werckmeister III, for instance, the C-to-F♯ tritone measures ~588 cents (-12 cents deviation), while others range from ~595 to ~605 cents (±5 cents deviation).[23][24] The following table summarizes representative tritone sizes and deviations from equal temperament (600 cents) for key systems, focusing on the augmented fourth where applicable:| Tuning System | Augmented Fourth Ratio | Size (cents) | Deviation (cents) | Notes |
|---|---|---|---|---|
| Equal Temperament | \sqrt{2} | 600 | 0 | Symmetric; enharmonic equivalents identical.[19] |
| Just Intonation | 45:32 | 590 | -10 | Narrower form; diminished fifth counterpart at 610 cents.[20] |
| Pythagorean | 729:512 | 612 | +12 | Wider form; based on pure fifths.[19] |
| Quarter-Comma Meantone | N/A (tempered) | 578 | -22 | Compressed for pure thirds; diminished fifth at 622 cents.[22] |
| Werckmeister III (ex.) | N/A (tempered) | 588 | -12 | Varies by key (e.g., 595–605 cents elsewhere); even distribution.[23] |
Acoustic Foundations
Connection to Harmonics
The tritone finds its acoustic origins in the harmonic series, where it emerges as an approximation of the interval between the fundamental tone and the eleventh partial. In the spectrum of a vibrating string or air column producing ideal harmonic overtones, the partials occur at integer multiples of the fundamental frequency, yielding simple frequency ratios that underpin consonant intervals. The eleventh partial, at a 11:1 ratio to the fundamental, reduces octave-wise to 11:8 (approximately 551 cents when measured from the fundamental), positioning it as the first overtone-derived interval resembling an augmented fourth—a tritone variant that is notably flatter than the equal-tempered standard of 600 cents.[25] This 11:8 ratio, known as the undecimal tritone, illustrates how the tritone arises naturally beyond the more stable lower partials (such as the perfect fifth at 3:2 from the third partial), but its appearance in the series highlights the increasing complexity and potential dissonance as partial numbers rise. In just intonation systems, which prioritize small-integer ratios for diatonic harmony, the tritone is instead approximated by 45:32 (about 590 cents) for the augmented fourth or its inversion 64:45 (about 610 cents) for the diminished fifth, bridging the acoustic ideal with practical scalar contexts without relying directly on the eleventh partial.[26] While pure sinusoidal tones generate perfectly harmonic series, real musical instruments introduce inharmonic deviations in their partials due to material properties like string stiffness or bore irregularities, which stretch higher overtones and subtly shift interval tunings, including the tritone, away from theoretical ratios. These inharmonicity effects are particularly pronounced in piano strings and brass instruments, where the eleventh partial may deviate by several cents, influencing the tritone's spectral alignment in performance.[27]Dissonance and Perceptual Qualities
The tritone is perceived as acoustically dissonant primarily due to the inharmonicity between the overtones of the two notes, where partials do not align periodically, leading to a sense of instability. Specifically, in the case of an augmented fourth (e.g., C to F♯), the fifth partial of the lower note (a major third above its second octave) lies in close proximity to the fourth partial of the upper note (its second octave), causing subtle beating and roughness that contributes to the interval's tense quality.[28] This clashing of overtones disrupts the smooth fusion expected in consonant intervals, as the combined spectrum lacks the regular periodicity found in simpler ratios like the perfect fifth.[29] Psychoacoustic models further explain the tritone's perceptual dissonance through concepts of sensory roughness and harmonicity. Helmholtz's theory posits that dissonance arises from the beating of nearby partials within the critical bandwidth, producing an unpleasant "roughness" sensation; while the tritone exhibits less intense beating than minor seconds, its overall dissonance stems from low harmonicity, where the tones fail to reinforce each other's overtones coherently.[30] Building on this, Sethares' sensory dissonance function quantifies this effect by summing pairwise dissonances between all partials across a range of intervals, revealing that the tritone (approximately √2 in equal temperament) yields higher dissonance values than most consonant intervals for harmonic timbres, approximating an eleventh harmonic relationship that lacks strong reinforcement.[28] These models emphasize that the tritone's instability is rooted in auditory processing rather than purely cultural factors. The perception of the tritone's tension varies culturally, with empirical studies showing that while Western listeners consistently rate it as dissonant and unpleasant, non-Western groups such as indigenous Amazonian populations exhibit indifference to dissonant intervals like the tritone, rating them as equally pleasant as consonant ones, suggesting that aesthetic aversion is culturally influenced.[31] For instance, research using isolated intervals shows the tritone evokes higher emotional arousal and lower pleasantness ratings compared to consonants in Western participants, linked to increased autonomic responses such as skin conductance, in both musicians and non-musicians.[30] Interval recognition tasks further indicate that the tritone is consistently identified as unstable in Western contexts, with brain imaging revealing heightened activity in auditory cortex regions associated with dissonance processing.[32]Musical Roles
Appearances in Scales and Chords
In the major scale, the tritone occurs between the fourth and seventh scale degrees, creating a dissonant interval within the otherwise consonant diatonic framework. For example, in C major, this spans from F to B, dividing the octave into two equal parts of six semitones each.[33] Within the natural minor scale, the tritone appears between the second and sixth scale degrees, contributing to the mode's characteristic tension. In C minor, for instance, it forms from D to A♭, again encompassing six semitones.[34] The dominant seventh chord features a tritone between its third and seventh, which defines its pull toward resolution and distinguishes it from the major triad. In the chord of G7, this interval lies between B and F, enharmonically an augmented fourth or diminished fifth.[35] Augmented triads incorporate a tritone between the root and the augmented fifth, resulting from stacking two major thirds and producing symmetrical voicing possibilities. A C augmented triad, for example, includes the tritone from C to F♯.[36] Fully diminished seventh chords stack four minor thirds, yielding two interlocking tritones: one between the root and fifth, and another between the third and seventh. In a B diminished seventh chord (B-D-F-A♭), the tritones are B to F and D to A♭.[37] The single tritone in the diatonic collection shifts position relative to the tonic across the seven modes, influencing each mode's stability and color. The following table summarizes these positions:| Mode | Tritone Between Degrees |
|---|---|
| Ionian | 4 and 7 |
| Dorian | 3 and 6 |
| Phrygian | 2 and 5 |
| Lydian | 1 and 4 |
| Mixolydian | 3 and 7 |
| Aeolian | 2 and 6 |
| Locrian | 1 and 5 |