Consonance and dissonance

• 1. Introduction

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  Consonance and dissonance are among the oldest arguments in all of music theory, and after a thousand years of debate, no one has fully settled them. German composer and music theorist Paul Hindemith put it plainly: "The two concepts have never been completely explained, and for a thousand years the definitions have varied." What makes one combination of sounds feel restful and another feel tense? Why does a note that sounds harsh in a Bach prelude from the 1700s sound perfectly comfortable in a Claude Debussy piece from the early 1900s? And why do organ builders, Bosnian folk singers, and Indonesian gamelan musicians deliberately seek out the very roughness that Western classical theory has long treated as a problem? The answers reach back to Ancient Greece, run through the physics of vibrating strings, and extend forward to a jazz theorist who rewrote the rules in 1953.

• 2. What the Words Actually Mean

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  Aristoxenus, writing in Ancient Greece, used the term symphonos to describe the intervals of the fourth, the fifth, the octave, and their doublings. Everything else he called diaphonos. That basic split survived into the Latin terminology of the early Middle Ages, when the monk Boethius in the 6th century described consonance as "the blending of a high sound with a low one, sweetly and uniformly arriving to the ears," and dissonance as "the harsh and unhappy percussion of two sounds mixed together."

  Hucbald of Saint Amand, writing around 900, sharpened the definition by insisting on simultaneity. For Hucbald, consonance required two sounds from different sources combining, at the same moment, into "a single musical whole." He named exactly six such consonances: the octave, the fifth, the fourth, the octave-plus-fifth, the octave-plus-fourth, and the double octave.

  The 13th-century theorist Johannes de Garlandia extended this taxonomy into a six-part gradation. Perfect consonances were unisons and octaves. Median consonances were fourths and fifths. Imperfect consonances were minor and major thirds. Perfect dissonances were the semitone, the tritone, and the major seventh. The taxonomy was intricate, but its underlying logic was simple: the less two simultaneous sounds blended into one, the more dissonant they were considered. What the taxonomy could not resolve was whether that judgment was a fact of nature or a habit of culture.

• 3. The Physics of Roughness

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  Two notes played at slightly different frequencies produce a throbbing "wah-wah-wah" effect that acousticians call beating. Organ builders exploit this deliberately to create the Voix celeste stop, a pair of pipes tuned fractionally apart so their slight misalignment shimmers. Bosnian ganga singers, players of the Middle Eastern mijwiz, and musicians of the Indonesian gamelan go even further, designing their instruments to produce exactly this roughness as a prized element of timbre.

  The underlying mechanism is wave interference. When two sound waves with different frequencies meet, their amplitudes add and subtract periodically, creating fluctuations. Slow fluctuations, fewer than roughly 20 per second, are heard as beating. As the rate increases toward somewhere between 20 and 75-150 per second, the loudness stabilises and the sensation becomes fluttering roughness. Beyond that range, the roughness peaks and then fades as the two sounds separate into distinct pitches.

  The ear processes these signals through what researchers call the critical band, a kind of frequency-sorting window in the auditory system. When two partials fall within the same critical band, they interfere and produce roughness. When they fall far enough apart, each is processed cleanly. Researchers Pankovski and Pankovska proposed that dissonance is ultimately a stress response: the brain logs which sound patterns it hears most often, and when an unusual combination arrives, a well-known brainwave signature called the P300 or P3b emerges, flagging what they call an oddball event. On this view, consonance is what the brain has learned to expect, and dissonance is what surprises it.

• 4. Dissonance in the Renaissance and Beyond

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  In Renaissance music, the perfect fourth above the bass was reclassified as a dissonance requiring immediate resolution, a reversal from the medieval system where fourths had been among the most stable intervals. The regola delle terze e seste, or "rule of thirds and sixths," required that imperfect consonances resolve to a perfect one by moving one voice a half step and the other a whole step.

  How many imperfect consonances could be strung together in sequence was itself a matter of debate. Anonymous XIII in the 13th century allowed two or three. Johannes de Garlandia's Optima introductio from the 13th-14th century permitted three, four, or more. Anonymous XI from the 15th century extended the limit to four or five. Adam von Fulda eventually declared the argument settled in favour of freedom: "Although the ancients formerly would forbid all sequences of more than three or four imperfect consonances, we more modern do not prohibit them."

  By the common practice period, the rule was preparation and resolution. Any dissonance had to be approached carefully and then released onto a consonance. Melodic dissonances included the tritone and all augmented and diminished intervals. Harmonic dissonances included the major second, the minor seventh, the minor second, the major seventh, and the tritone in both its enharmonic forms. Theorists during this era observed a long-term drift: only intervals low in the overtone series were consonant in early music, but as centuries passed, intervals higher on the overtone series were gradually accepted. Some 20th-century composers would eventually call this drift the "emancipation of the dissonance."

• 5. Bach, Beethoven, and the Drama of Discord

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  Baroque composers treated dissonance as a rhetorical instrument. In the opening aria of Bach's Cantata BWV 54, Widerstehe doch der Sunde, which translates as "do resist sin," nearly every strong beat carries a dissonance. Albert Schweitzer described the aria as beginning with "an alarming chord of the seventh... meant to depict the horror of the curse upon sin." Gillies Whittaker counted the bass line of the opening four bars and found that of the thirty-two continuo notes, only four coincide with consonances.

  At the close of the St Matthew Passion, conductor John Eliot Gardiner heard Bach insert "the unexpected and almost excruciating dissonance... over the very last chord: the melody instruments insist on B natural, the jarring leading tone, before eventually melting in a C minor cadence."

  Mozart explored dissonance in a different register. His Quartet in C major, K465, opens with an adagio introduction so harmonically unsettled that it earned the work its nickname, the "Dissonance Quartet." Critic Philip Radcliffe described the slow movement of Mozart's Piano Concerto 21, K467, nicknamed "Elvira Madigan," as "a remarkably poignant passage with surprisingly sharp dissonances" that offered "a vivid foretaste of Schumann" and were "equally prophetic of Schubert." Eric Blom added that the movement must have "made Mozart's hearers sit up by its daring modernities."

  Beethoven's finale to Symphony No. 9 opens with what Roger Scruton, following Wagner, called "a huge Schreckensfanfare, a horror fanfare": a B flat inserted into a D minor chord. When the passage returns just before the voices enter, Beethoven adds a diminished seventh chord on top, producing what Scruton called "the most atrocious dissonance that Beethoven ever wrote, a first inversion D-minor triad containing all the notes of the D minor harmonic scale." Robert Schumann later concentrated an entire song's tension into a single bar. In "Auf einer Burg" from his cycle Liederkreis Op. 39, a climactic dissonance arrives in the fourteenth bar, the only chord in the whole song that Schumann marked with an accent. Critic Nicholas Cook noted that the bars leading up to it are "set on a kind of collision course; hence the feeling of tension rising steadily to a breaking point."

• 6. Mahler, Modernism, and the Limit

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  Gustav Mahler's unfinished 10th Symphony, composed around 1910, opens its Adagio with a chord that musicologist Richard Taruskin parsed as a "diminished nineteenth, a searingly dissonant dominant harmony containing nine different pitches." Taruskin noted that Guido Adler had already complained that Mahler's Second and Third Symphonies contained "unprecedented cacophonies," making the 10th's harmonic density even more striking by comparison.

  Wagner had pushed toward this limit across his career. In the scene known as "Hagen's Watch" from the first act of Gotterdammerung, Scruton described the music as conveying "matchless brooding evil," and the passage in bars 9-10 as "a semitonal wail of desolation."

  A work that received its first performance in 1913, three years after Mahler's 10th was drafted, offered another example of modernist dissonance taken to an extreme. Where Baroque composers had used dissonance to resolve, these modernist works refused resolution altogether, pressing toward what would come to be called atonality. In that context, even a major triad could sound dissonant because it no longer fit the prevailing harmonic language.

• 7. Jazz Redraws the Map

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  In 1953, American theorist George Russell published the Lydian Chromatic Concept of Tonal Organization, a framework widely adopted in jazz that challenged the classical hierarchy of consonance and dissonance from an unexpected angle. Russell treated the tritone over the tonic not as the most unstable interval, but as relatively consonant, because he derived it from the Lydian dominant thirteenth chord and from the mathematical ratios of the harmonic series.

  Russell's argument was essentially a return to what he called "harmonic consonance": intervals that correctly reproduce the ratios of the overtone series without octave compression are, in his view, genuinely non-dissonant. On this basis, the septimal minor seventh, the major ninth, the neutral eleventh, the neutral thirteenth, and the diminished fifteenth all qualify as consonant. He extended this reasoning by approximation to the 12-tone equal temperament system used in jazz, granting consonance to the sharp eleventh note, which is the sole pitch difference between the major scale and the Lydian mode.

  In 1980, theorist Dan Haerle extended Russell's framework to revise Paul Hindemith's own gradation table from The Craft of Musical Composition, which had become the de facto standard for ranking intervals by consonance and dissonance. Where Hindemith placed the tritone at maximum dissonance, Haerle moved it to a position just slightly less consonant than the perfect fourth and fifth. Haerle also ranked the minor ninth as the most dissonant interval of all, more dissonant even than the minor second, overturning the older assumption that these two intervals were equivalent because one is simply an octave expansion of the other. The bell, meanwhile, had been quietly operating under its own rules the whole time: on a carillon, the minor third and the tritone are consonant, while the major third and major sixth are dissonant, a profile so unlike the standard model that interval inversion does not even apply.