Consonance and dissonance


In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Within the Western tradition, consonance is typically associated with sweetness, pleasantness, and acceptability; dissonance is associated with harshness, unpleasantness, or unacceptability although this depends also on familiarity and musical expertise. The terms form a structural dichotomy in which they define each other by mutual exclusion: a consonance is what is not dissonant, and a dissonance is what is not consonant. However, a finer consideration shows that the distinction forms a gradation, from the most consonant to the most dissonant. As Hindemith stressed, "The two concepts have never been completely explained, and for a thousand years the definitions have varied". The term sonance has been proposed to encompass or refer indistinctly to the terms consonance and dissonance.

Definitions

The opposition between consonance and dissonance can be made in different contexts:
In both cases, the distinction mainly concerns simultaneous sounds; if successive sounds are considered, their consonance or dissonance depends on the memorial retention of the first sound while the second sound is heard. For this reason, consonance and dissonance have been considered particularly in the case of Western polyphonic music, and the present article is concerned mainly with this case. Most historical definitions of consonance and dissonance since about the 16th century have stressed their pleasant/unpleasant, or agreeable/disagreeable character. This may be justifiable in a psychophysiological context, but much less in a musical context properly speaking: dissonances often play a decisive role in making music pleasant, even in a generally consonant context—which is one of the reasons why the musical definition of consonance/dissonance cannot match the psychophysiologic definition. In addition, the oppositions pleasant/unpleasant or agreeable/disagreeable evidence a confusion between the concepts of "dissonance" and of "noise".
While consonance and dissonance exist only between sounds and therefore necessarily describe intervals, such as the perfect intervals, which are often viewed as consonant, Occidental music theory often considers that, in a dissonant chord, one of the tones alone is in itself deemed to be the dissonance: it is this tone in particular that needs "resolution" through a specific voice leading procedure. For example, in the key of C Major, if F is produced as part of the dominant seventh chord, it is deemed to be "dissonant" and it normally resolves to E during a cadence, with the G7 chord changing to a C Major chord.

Acoustics and psychoacoustics

Scientific definitions have been variously based on experience, frequency, and both physical and psychological considerations. These include:
Consonances may include:
Dissonances may include:
's classification, from Studies in Counterpoint, of a triad's overall consonance or dissonance through the consonance or dissonance of the three intervals contained within . For example, C-E-G consists of three consonances and is ranked 1 while C-D-B consists of one mild dissonance and two sharp dissonances and is ranked 6.

Physiological basis

Two notes played simultaneously but with slightly different frequencies produce a beating "wah-wah-wah" sound. This phenomenon is used to create the Voix céleste stop in organs. Other musical styles such as Bosnian ganga singing, pieces exploring the buzzing sound of the Indian tambura drone, stylized improvisations on the Middle Eastern mijwiz, or Indonesian gamelan consider this sound an attractive part of the musical timbre and go to great lengths to create instruments that produce this slight "roughness".
Sensory dissonance and its two perceptual manifestations are both closely related to a sound signal's amplitude fluctuations. Amplitude fluctuations describe variations in the maximum value of sound signals relative to a reference point and are the result of wave interference. The interference principle states that the combined amplitude of two or more vibrations at any given time may be larger or smaller than the amplitude of the individual vibrations, depending on their phase relationship. In the case of two or more waves with different frequencies, their periodically changing phase relationship results in periodic alterations between constructive and destructive interference, giving rise to the phenomenon of amplitude fluctuations.
"Amplitude fluctuations can be placed in three overlapping perceptual categories related to the rate of fluctuation. Slow amplitude fluctuations are perceived as loudness fluctuations referred to as beating. As the rate of fluctuation is increased, the loudness appears constant, and the fluctuations are perceived as "fluttering" or roughness. As the amplitude fluctuation rate is increased further, the roughness reaches a maximum strength and then gradually diminishes until it disappears.
Assuming the ear performs a frequency analysis on incoming signals, as indicated by Ohm's acoustic law, the above perceptual categories can be related directly to the bandwidth of the hypothetical analysis filters. For example, in the simplest case of amplitude fluctuations resulting from the addition of two sine signals with frequencies f1 and f2, the fluctuation rate is equal to the frequency difference between the two sines |f1-f2|, and the following statements represent the general consensus:
  1. If the fluctuation rate is smaller than the filter bandwidth, then a single tone is perceived either with fluctuating loudness or with roughness.
  2. If the fluctuation rate is larger than the filter bandwidth, then a complex tone is perceived, to which one or more pitches can be assigned but which, in general, exhibits no beating or roughness.
Along with amplitude fluctuation rate, the second most important signal parameter related to the perceptions of beating and roughness is the degree of a signal's amplitude fluctuation, that is, the level difference between peaks and valleys in a signal. The degree of amplitude fluctuation depends on the relative amplitudes of the components in the signal's spectrum, with interfering tones of equal amplitudes resulting in the highest fluctuation degree and therefore in the highest beating or roughness degree.
For fluctuation rates comparable to the auditory filter bandwidth, the degree, rate, and shape of a complex signal's amplitude fluctuations are variables that are manipulated by musicians of various cultures to exploit the beating and roughness sensations, making amplitude fluctuation a significant expressive tool in the production of musical sound. Otherwise, when there is no pronounced beating or roughness, the degree, rate, and shape of a complex signal's amplitude fluctuations remain important, through their interaction with the signal's spectral components. This interaction is manifested perceptually in terms of pitch or timbre variations, linked to the introduction of combination tones.
"The beating and roughness sensations associated with certain complex signals are therefore usually understood in terms of sine-component interaction within the same frequency band of the hypothesized auditory filter, called critical band."
In human hearing, the varying effect of simple ratios may be perceived by one of these mechanisms:
Generally, the sonance of any given interval can be controlled by adjusting the timbre in which it is played, thereby aligning its partials with the current tuning's notes . The sonance of the interval between two notes can be maximized by maximizing the alignment of the two notes' partials, whereas it can be minimized by mis-aligning each otherwise nearly aligned pair of partials by an amount equal to the width of the critical band at the average of the two partials' frequencies.
Controlling the sonance of more-or-less non-harmonic timbres in real time is an aspect of dynamic tonality. For example, in Sethares' piece , the sonance of intervals is affected both by tuning progressions and timbre progressions.
The strongest homophonic cadence, the authentic cadence, dominant to tonic, is in part created by the dissonant tritone created by the seventh, also dissonant, in the dominant seventh chord, which precedes the tonic.
in C.

Instruments producing non-harmonic overtone series

Musical instruments like bells and xylophones, called Idiophones, are played such that their relatively stiff, non-trivial mass is excited to vibration by means of a blow. This contrasts with violins, flutes, or drums, where the vibrating medium is a light, supple string, column of air, or membrane. The overtones of the inharmonic series produced by such instruments may differ greatly from that of the rest of the orchestra, and the consonance or dissonance of the harmonic intervals as well.
According to John, the carillon's harmony profile is summarized:
Dissonance has been understood and heard differently in different musical traditions, cultures, styles, and time periods. Relaxation and tension have been used as analogy since the time of Aristotle till the present.
The terms dissonance and consonance are often considered equivalent to tension and relaxation. A cadence is a place where tension is resolved; hence the long tradition of thinking of a musical phrase as consisting of a cadence and a passage of gradually accumulating tension leading up to it.
Various psychological principles constructed through the audience's general conception of tonal fluidity determine how a listener will distinguish an instance of dissonance within a musical composition. Based on one's developed conception of the general tonal fusion within the piece, an unexpected tone played sightly variant to the overall schema will generate a psychological need for resolve. When the consonant is followed thereafter, the listener will encounter a sense of resolution. Within Western music, these particular instances and psychological effects within a composition have come to possess an ornate connotation.
The application of consonance and dissonance "is sometimes regarded as a property of isolated sonorities that is independent of what precedes or follows them. In most Western music, however, dissonances are held to resolve onto following consonances, and the principle of resolution is tacitly considered integral to consonance and dissonance".

Antiquity and the middle ages

In Ancient Greece, armonia denoted the production of a unified complex, particularly one expressible in numerical ratios. Applied to music, the concept concerned how sounds in a scale or a melody fit together . The term symphonos was used by Aristoxenus and others to describe the intervals of the fourth, the fifth, the octave and their doublings; other intervals were said diaphonos. This terminology probably referred to the Pythagorean tuning, where fourths, fifths and octaves were directly tunable, while the other degrees could only be tuned by combinations of the preceding. Until the advent of polyphony and even later, this remained the basis of the concept of consonance/dissonance in Occidental theory.
In the early Middle Ages, the Latin term consonantia translated either armonia or symphonia. Boethius characterizes consonance by its sweetness, dissonance by its harshness: "Consonance is the blending of a high sound with a low one, sweetly and uniformly arriving to the ears. Dissonance is the harsh and unhappy percussion of two sounds mixed together ". It remains unclear, however, whether this could refer to simultaneous sounds. The case becomes clear, however, with Hucbald of Saint Amand, who writes: "Consonance is the measured and concordant blending of two sounds, which will come about only when two simultaneous sounds from different sources combine into a single musical whole . There are six of these consonances, three simple and three composite, octave, fifth, fourth, and octave-plus-fifth, octave-plus-fourth and double octave".
According to :
One example of imperfect consonances previously considered dissonances in Guillaume de Machaut's "Je ne cuit pas qu'onques" :
According to Margo :
Stable:
Unstable:
It is worth noting that "perfect" and "imperfect" and the notion of being must be taken in their contemporaneous Latin meanings to understand these terms, such that imperfect is "unfinished" or "incomplete" and thus an imperfect dissonance is "not quite manifestly dissonant" and perfect consonance is "done almost to the point of excess". Also, inversion of intervals and octave reduction were yet unknown during the Middle Ages.
Due to the different tuning systems compared to modern times, the minor seventh and major ninth were "harmonic consonances", meaning that they correctly reproduced the interval ratios of the harmonic series which softened a bad effect. They were also often filled in by pairs of perfect fourths and perfect fifths respectively, forming resonant units characteristic of the musics of the time, where "resonance" forms a complementary trine with the categories of consonance and dissonance. Conversely, the thirds and sixths were tempered severely from pure ratios, and in practice usually treated as dissonances in the sense that they had to resolve to form complete perfect cadences and stable sonorities.
The salient differences from modern conception:
In Renaissance music, the perfect fourth above the bass was considered a dissonance needing immediate resolution. The regola delle terze e seste required that imperfect consonances should resolve to a perfect one by a half-step progression in one voice and a whole-step progression in another. The viewpoint concerning successions of imperfect consonances—perhaps more concerned by a desire to avoid monotony than by their dissonant or consonant character—has been variable. Anonymous XIII allowed two or three, Johannes de Garlandia's Optima introductio three, four or more, and Anonymous XI four or five successive imperfect consonances. Adam von Fulda wrote "Although the ancients formerly would forbid all sequences of more than three or four imperfect consonances, we more modern do not prohibit them."

Common practice period

In the common practice period, musical style required preparation for all dissonances, followed by a resolution to a consonance. There was also a distinction between melodic and harmonic dissonance. Dissonant melodic intervals included the tritone and all augmented and diminished intervals. Dissonant harmonic intervals included:
Early in history, only intervals low in the overtone series were considered consonant. As time progressed, intervals ever higher on the overtone series were considered as such. The final result of this was the so-called "emancipation of the dissonance" by some 20th-century composers. Early-20th-century American composer Henry Cowell viewed tone clusters as the use of higher and higher overtones.
Composers in the Baroque era were well aware of the expressive potential of dissonance:
Bach uses dissonance to communicate religious ideas in his sacred cantatas and Passion settings. At the end of the St Matthew Passion, where the agony of Christ’s betrayal and crucifixion is portrayed, John Eliot hears "a final reminder of this comes in the unexpected and almost excruciating dissonance Bach inserts over the very last chord: the melody instruments insist on B natural—the jarring leading tone—before eventually melting in a C minor cadence." In the opening aria of Cantata BWV 54, Widerstehe doch der Sünde, nearly every strong beat carries a dissonance:
Albert Schweitzer says that this aria “begins with an alarming chord of the seventh… It is meant to depict the horror of the curse upon sin that is threatened in the text". Gillies points out that “The thirty-two continuo quavers of the initial four bars support four consonances only, all the rest are dissonances, twelve of them being chords containing five different notes. It is a remarkable picture of desperate and unflinching resistance to the Christian to the fell powers of evil.”
According to H.C. Robbins Landon, the opening movement of Haydn’s Symphony No. 82, "a brilliant C major work in the best tradition" contains "dissonances of barbaric strength that are succeeded by delicate passages of Mozartean grace." :
Mozart's music contains a number of quite radical experiments in dissonance. The following comes from his Adagio and Fugue in C Minor, K. 546:
Mozart’s Quartet in C major, K465 opens with an adagio introduction that gave the work its nickname, the “Dissonance Quartet”:
There are several passing dissonances in this adagio passage, for example on the first beat of bar 3. However the most striking effect here is implied, rather than sounded explicitly. The A flat in the first bar is contradicted by the high A natural in the second bar, but these notes do not sound together as a discord.
An even more famous example from Mozart comes in a magical passage from the slow movement of his popular "Elvira Madigan" Piano Concerto 21, K467, where the subtle, but quite explicit dissonances on the first beats of each bar are enhanced by exquisite orchestration:
Philip speaks of this as “a remarkably poignant passage with surprisingly sharp dissonances." Radcliffe says that the dissonances here "have a vivid foretaste of Schumann and the way they gently melt into the major key is equally prophetic of Schubert." Eric says that this movement must have "made Mozart's hearers sit up by its daring modernities... There is a suppressed feeling of discomfort about it."
The finale of Beethoven’s Symphony No. 9 opens with a startling discord, consisting of a B flat inserted into a D minor chord:
Roger alludes to Wagner’s description of this chord as introducing “a huge Schreckensfanfare—horror fanfare.” When this passage returns later in the same movement the sound is further complicated with the addition of a diminished seventh chord, creating, in Scruton’s words “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’s song ‘Auf Einer Burg’from his cycle Liederkreis Op. 39, climaxes on a striking dissonance in the fourteenth bar. As Nicholas points out, this is “the only chord in the whole song that Schumann marks with an accent.” Cook goes on to stress that what makes this chord so effective is Schumann’s placing of it in its musical context: “in what leads up to it and what comes of it.” Cook explains further how the interweaving of lines in both piano and voice parts in the bars leading up to this chord “are set on a kind of collision course; hence the feeling of tension rising steadily to a breaking point.” ]
Richard Wagner made increasing use of dissonance for dramatic effect as his style developed, particularly in his later operas. In the scene known as "Hagen’s Watch" from the first act of Götterdämmerung, according to the music conveys a sense of "matchless brooding evil", and the excruciating dissonance in bars 9–10 below it constitute "a semitonal wail of desolation". ]
Another example of a cumulative build-up of dissonance from the early 20th century can be found in the Adagio that opens Gustav Mahler’s unfinished 10th Symphony :
parses this chord as a “diminished nineteenth… a searingly dissonant dominant harmony containing nine different pitches. Who knows what Guido Adler, for whom the second and Third Symphonies already contained ‘unprecedented cacophonies’, might have called it?”
One example of modernist dissonance comes from a work that received its first performance in 1913, three years after the Mahler:
The West's progressive embrace of increasingly dissonant intervals occurred almost entirely within the context of harmonic timbres, as produced by vibrating strings and columns of air, on which the West's dominant musical instruments are based. By generalizing Helmholtz's notion of consonance to embrace non-harmonic timbres and their related tunings, consonance has recently been "emancipated" from harmonic timbres and their related tunings. Using electronically controlled pseudo-harmonic timbres, rather than strictly harmonic acoustic timbres, provides tonality with new structural resources such as Dynamic tonality. These new resources provide musicians with an alternative to pursuing the musical uses of ever-higher partials of harmonic timbres and, in some people's minds, may resolve what Arnold Schoenberg described as the "crisis of tonality".

Neo-classic harmonic consonance theory

, in his 1953 Lydian Chromatic Concept of Tonal Organization, presents a slightly different view from classical practice, one widely taken up in Jazz. He regards the tritone over the tonic as a rather consonant interval due to its derivation from the Lydian dominant thirteenth chord.
In effect, he returns to a Medieval consideration of "harmonic consonance": that intervals when not subject to octave equivalence and correctly reproducing the mathematical ratios of the harmonic series are truly non-dissonant. Thus the harmonic minor seventh, natural major ninth, half-sharp eleventh note, half-flat thirteenth note, and half-flat fifteenth note must necessarily be consonant. Octave equivalence is no longer unquestioned.
Note that most of these pitches exist only in a universe of microtones smaller than a halfstep; notice also that we already freely take the flat seventh note for the just seventh of the harmonic series in chords. Russell extends by approximation the virtual merits of harmonic consonance to the 12TET tuning system of Jazz and the 12-note octave of the piano, granting consonance to the sharp eleventh note, that accidental being the sole pitch difference between the Major scale and the Lydian mode.
replaces or supplements the Mixolydian scale of the dominant chord
Dan Haerle, in his 1980 The Jazz Language, extends the same idea of harmonic consonance and intact octave displacement to alter Paul Hindemith's Series 2 gradation table from The Craft of Musical Composition. In contradistinction to Hindemith, whose scale of consonance and dissonance is currently the de facto standard, Haerle places the minor ninth as the most dissonant interval of all, more dissonant than the minor second to which it was once considered by all as octave-equivalent. He also promotes the tritone from most-dissonant position to one just a little less consonant than the perfect fourth and perfect fifth.
For context: unstated in these theories is that musicians of the Romantic Era had effectively promoted the major ninth and minor seventh to a legitimacy of harmonic consonance as well, in their fabrics of 4-note chords.