Consonance and dissonance
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In music, consonance and dissonance form a structural dichotomy in which the terms define each other by mutual exclusion: a consonance is what is not dissonant, and reciprocally. However, a finer consideration shows that the distinction forms a gradation, from the most consonant to the most dissonant. Consonance and dissonance define a level of sweetness / harshness, pleasantness / unpleasantness, acceptability / unacceptability, of the sounds or intervals under consideration. As Hindemith stressed, "The two concepts have never been completely explained, and for a thousand years the definitions have varied" (Hindemith 1942, p. 85).
The opposition can be made in different contexts:
- In acoustics or psychophysiology, the distinction may be objective. In modern times, it usually is based on the perception of harmonic partials of the sounds considered, to such an extent that the distinction really holds only in the case of harmonic sounds (i.e. sounds with harmonic partials).
- In music, even if the opposition often is founded on the preceding, objective distinction, it more often is subjective, conventional, cultural, and style-dependent. Dissonance can then be defined as a combination of sounds that does not belong to the style under consideration; in recent music, what is considered stylistically dissonant may even correspond to what is said to be consonant in the context of acoustics (e.g. a major triad in atonal music).
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 is heard. For this reason, consonance and dissonance have been considered particularly in the case of polyphonic Occidental 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'. (See also Noise in music, Noise music and Noise (acoustic).)
While consonance and dissonance exist only between sounds and therefore necessarily describe intervals (or chords), Occidental music theory often considers that, in a dissonant chord, one of the tones alone is in itself the dissonance: it is this tone in particular that needs "resolution" through a specific voice leading.
- 1 Consonance
- 2 Dissonance
- 3 Consonance and Dissonance in history of Western music
- 4 Instruments producing non-harmonic overtone series
- 5 Physiological basis of dissonance
- 6 See also
- 7 References
- 8 Further reading
- 9 External links
The definition of consonance has been variously based on experience, frequency, and both physical and psychological considerations (Myers 1904, p. 315). These include:
- Frequency ratios: with ratios of lower simple numbers being more consonant than those that are higher (Pythagoras[full citation needed]). Many of these definitions do not require exact integer tunings, only approximation.[vague]
- Coincidence of partials: with consonance being a greater coincidence of partials (called harmonics or overtones when occurring in harmonic timbres) (Helmholtz 1954 ,[page needed]). By this definition, consonance is dependent not only on the width of the interval between two notes (i.e., the musical tuning), but also on the combined spectral distribution and thus sound quality (i.e., the timbre) of the notes (see the entry under critical band). Thus, a note and the note one octave higher are highly consonant because the partials of the higher note are also partials of the lower note (Roederer 1995, p. 165). Although Helmholtz's work focused almost exclusively on harmonic timbres and also the tunings, subsequent work has generalized his findings to embrace non-harmonic tunings and timbres (Sethares 1992; Sethares 2005[not in citation given][page needed]; Milne, Sethares, and Plamondon 2007,[page needed]; Milne, Sethares, and Plamondon 2008,[page needed]; Sethares et al. 2009,[page needed]).
- Fusion or pattern matching: fundamentals may be perceived through pattern matching of the separately analyzed partials to a best-fit exact-harmonic template (Gerson and Goldstein 1978,[page needed]) or the best-fit subharmonic (Terhardt 1974[page needed]). Harmonics may be perceptually fused into one entity—consonances being those that include:
These may be generalized as simplicity.
"A stable tone combination is a consonance; consonances are points of arrival, rest, and resolution."— Roger Kamien 2008, p. 41
An unstable tone combination is a dissonance; its tension demands an onward motion to a stable chord. Thus dissonant chords are "active"; traditionally they have been considered harsh and have expressed pain, grief, and conflict.— Roger Kamien 2008, p. 41
In Western music, dissonance is the quality of sounds that seems unstable and has an aural need to resolve to a stable consonance. Both consonance and dissonance are words applied to harmony, chords, and intervals and, by extension, to melody, tonality, and even rhythm and metre. Although there are physical and neurological facts important to understanding the idea of dissonance, the precise definition of dissonance is culturally conditioned—definitions of and conventions of usage related to dissonance vary greatly among different musical styles, traditions, and cultures. Nevertheless, the basic ideas of dissonance, consonance, and resolution exist in some form in all musical traditions that have a concept of melody, harmony, or tonality.[contradictory] Dissonance being the complement of consonance it may be defined, as above, as non-coincidence of partials, lack of fusion or pattern matching, or as complexity.
Additional confusion about the idea of dissonance is created by the fact that musicians and writers sometimes use the word dissonance and related terms in a precise and carefully defined way, more often in an informal way, and very often in a metaphorical sense ("rhythmic dissonance"). For many musicians and composers, the essential ideas of dissonance and resolution are vitally important ones that deeply inform their musical thinking on a number of levels.
Despite the fact that words like unpleasant and grating are often used to explain the sound of dissonance, all music with a harmonic or tonal basis—even music perceived as generally harmonious—incorporates some degree of dissonance. The buildup and release of tension (dissonance and resolution), which can occur on every level from the subtle to the crass, is partially responsible for what listeners perceive as beauty, emotion, and expressiveness in music.
Dissonance and musical style
The concept of dissonance does not belong to the domain of harmony as it is presented us by Nature [harmonic series], but is derived from voice leading [guidelines], which is an essential constituent of Art.
Understanding a particular musical style's treatment of dissonance—what is considered dissonant and what rules or procedures govern how dissonant intervals, chords, or notes are treated—is key in understanding that particular style. For instance, harmony is generally governed by chords, which are collections of notes defined as tolerably consonant by the style. (There is likely, however, to be a hierarchy of chords, with some considered more consonant and some more dissonant.) Any note that does not fall within the prevailing harmony is considered dissonant. A given style typically pays attention to how its musical structure approaches dissonance (in steps is less jarring, a leap is more jarring), and even more to how they resolve (almost always by step), to how they fit within the meter and rhythm (dissonances on strong beats are more emphatic, those on weaker beats less vital), and to how they lie within the phrase (dissonances tend to resolve at phrase's end).
Dissonance in traditional music
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Sharp dissonant intervals and chords play a prominent role in many traditional musical cultures. Vocal polyphonic traditions from Bulgaria, Serbia, Bosnia-Herzegovina, Albania, Latvia, Georgia, Nuristan, some Vietnamese and Chinese minority singing traditions, Lithuanian sutartinės, some polyphonic traditions from Flores and Melanesia are predominantly based on the use of sharp dissonant intervals and chords. The most prominent dissonance in most of these cultures is the interval of the neutral second (which is between the minor and major seconds). This interval is known to create the maximum sharpness and is known in German ethnomusicology under the term "Schwebungsdiaphonie".
Consonance and Dissonance in history of Western music
|“||When we consider musical works we find that the triad is ever-present and that the interpolated dissonances have no other purpose than to effect the continuous variation of the triad.||”|
|— Lorenz Mizler 1739 (quoted in Forte 1979, p. 136)|
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 (Kliewer 1975, p. 290).
Antiquity and 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 (in this sense, it could also concern the tuning of a scale) (Philip 1966, pp. 123–24). 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 (ratios 4:3, 3:2 and 2:1) were directly tunable, while the other degrees (other 3-prime ratios) could only be tuned by combinations of the preceding (Aristoxenus 1902, pp. 188–206 See Tenney 1988, pp. 11–12). Until the advent of polyphony and even later, this remained the basis of the concept of consonance/dissonance (symphonia/diaphonia) in Occidental theory.
In the early Middle Ages, the Latin term consonantia translated either armonia or symphonia. Boethius (6th century) characterizes consonance by its sweetness, dissonance by its harshness: "Consonance (consonantia) is the blending (mixtura) of a high sound with a low one, sweetly and uniformly (suauiter uniformiterque) arriving to the ears. Dissonance is the harsh and unhappy percussion (aspera atque iniocunda percussio) of two sounds mixed together (sibimet permixtorum)" (Boethius n.d., f. 13v.). It remains unclear, however, whether this could refer to simultaneous sounds. The case becomes clear, however, with Hucbald of Saint Amand (c900), who writes: "Consonance (consonantia) is the measured and concordant blending (rata et concordabilis permixtio) of two sounds, which will come about only when two simultaneous sounds from different sources combine into a single musical whole (in unam simul modulationem conveniant) […]. There are six of these consonances, three simple and three composite, […] octave, fifth, fourth, and octave-plus-fifth, octave-plus-fourth and double octave" (Hucbald n.d., p. 107; translated in Babb 1978, p. 19).
According to Johannes de Garlandia (13th century):
- Perfect consonance: unisons and octaves. (Perfecta dicitur, quando due voces junguntur in eodem tempore, ita quod una, secundum auditum, non percipitur ab alia propter concordantiam, et dicitur equisonantiam, ut in unisono et diapason. — "[Consonance] is said perfect, when two voices are joined at the same time, so that the one, by audition, cannot be distinguished from the other because of the concordance, and it is called equisonance, as in unison and octave.")
- Median consonance: fourths and fifths. (Medie autem dicuntur, quando duo voces junguntur in eodem tempore; que neque dicuntur perfecte, neque imperfecte, sed partim conveniunt cum perfectis, et partim cum imperfectis. Et sunt due species, scilicet diapente et diatessaron. — "Consonances are said median, when two voices are joined at the same time, which neither can be said perfect, nor imperfect, but which partly agree with the perfect, and partly with the imperfect. And they are of two species, namely the fifth and the fourth.")
- Imperfect consonance: minor and major thirds. (Imperfect consonances are not formally mentioned in the treatise, but the quotation above concerning median consonances does refer to imperfect consonances, and the section on consonances concludes: Sic apparet quod sex sunt species concordantie, scilicet: unisonus, diapason, diapente, diatessaron, semiditonus, ditonus. — "So it appears that there are six species of consonances, that is: unison, octave, fifth, fourth, minor third, major third." The last two appear as imperfect consonances by elimination.)
- Imperfect dissonance: major sixth (tone + fifth) and minor seventh (minor third + fifth). (Imperfecte dicuntur, quando due voces junguntur ita, quod secundum auditum vel possunt aliquo modo compati, tamen non concordant. Et sunt due species, scilicet tonus cum diapente et semiditonus cum diapente. — [Dissonances] are said imperfect, when two voices are joined so that by audition although they can to some extent match, nevertheless they do not concord. And there are two species, namely tone plus fifth and minor third plus fifth.")
- Median dissonance: tone and minor sixth (semitone + fifth). (Medie dicuntur, quando due voces junguntur ita, quod partim conveniunt cum perfectis, partim cum imperfectis. Et iste sunt due species, scilicet tonus et simitonium cum diapente. — [Dissonances] are said median when two voices are joined so that they partly match the perfect, partly the imperfect. And they are of two species, namely tone and semitone plus fifth.")
- Perfect dissonance: semitone, tritone, major seventh (major third + fifth). (Here again, the perfect dissonances can only be deduced by elimination from this phrase: Iste species dissonantie sunt septem, scilicet: semitonium, tritonus, ditonus cum diapente; tonus cum diapente, semiditonus cum diapente; tonus et semitonium cum diapente. — These species of dissonances are seven: semitone, tritone, major third plus fifth; tone plus fifth, minor third plus fifth; tone and semitone plus fifth.")
One example of imperfect consonances previously considered dissonances[clarification needed] in Guillaume de Machaut's "Je ne cuit pas qu'onques" (Machaut 1926, p. 13, Ballade 14, "Je ne cuit pas qu'onques a creature", mm. 27–31):
According to Margo Schulter (1997a):
- Purely blending: unisons and octaves
- Optimally blending: fourths and fifths
- Relatively blending: minor and major thirds
- Relatively tense: major seconds, minor sevenths, and major sixths
- Strongly discordant: minor seconds, tritonus, and major sevenths, and often minor sixths
It is worth noting that "perfect" and "imperfect" and the notion of being (esse) must be taken in their contemporaneous Latin meanings (perfectum, imperfectum) 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 (major second in some sense equivalent to minor seventh) and octave reduction (minor ninth in some sense equivalent to minor second) 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 (Schulter 1997b).[clarification needed] They were also often filled in by pairs of perfect fourths and perfect fifths respectively, forming resonant (blending) units characteristic of the musics of the time (Schulter 1997c), where "resonance" forms a complementary trine with the categories of consonance and dissonance.[clarification needed] Conversely, the thirds and sixths were tempered severely from pure ratios[clarification needed], and in practice usually treated as dissonances in the sense that they had to resolve to form complete perfect cadences and stable sonorities (Schulter 1997d).
- parallel fourths and fifths were acceptable and necessary, open fourths and fifths inside octaves were the characteristic stable sonority in 3 or more voices,
- minor sevenths and major ninths were fully structural,
- tritones—as a deponent[clarification needed] sort of fourth or fifth—were sometimes stacked with perfect fourths and fifths,
- thirds and sixths (and tall stacks thereof) were not the sort of intervals upon which stable harmonies were based,
- final cadential consonances of fourth, fifths, and octaves need not be the target of "resolution" on a beat-to-beat (or similar) time basis: minor sevenths and major ninths may move to octaves forthwith, or sixths to fifths (or minor sevenths), but the fourths and fifths within might become "dissonant" 5/3, 6/3, or 6/4 chordioids[clarification needed], continuing the succession of non-consonant sonorities for timespans limited only by the next cadence.
In early Renaissance music, intervals such as the perfect fourth were considered dissonances that must be immediately resolved. The regola delle terze e seste ("rule of thirds and sixths") 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 (Dahlhaus 1990, p. 179). Anonymous 13 allowed two or three, the Optima introductio three or four, and Anonymous 11 (15th century) four or five successive imperfect consonances. By the end of the 15th century, imperfect consonances were no longer "tension sonorities" but, as evidenced by the allowance of their successions argued for by Adam von Fulda, independent sonorities; according to Fulda (Gerbert 1784, 3:353), "Although older scholars once would forbid all sequences of more than three or four imperfect consonances, we who are more modern allow them." (ibid, p. 92)[clarification needed]
Common practice period
In the common practice period, musical style required preparation for all dissonances, followed by and then 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:
- Minor second and major seventh
- Augmented fourth and diminished fifth (enharmonically equivalent, tritone)
Thus, Western musical history can be seen as progressing from a limited definition of consonance to an ever-wider definition of consonance. 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" (the words of Arnold Schoenberg)[this quote needs a citation] by some 20th-century composers. Early-20th-century American composer Henry Cowell viewed tone clusters as the use of higher and higher overtones.
Despite the fact that this idea of the historical progression towards the acceptance of ever greater levels of dissonance is somewhat oversimplified and glosses over important developments in the history of Western music, the general idea was attractive to many 20th-century modernist composers and is considered a formative meta-narrative of musical modernism.
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 Gardiner (2013, 427) 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."
Albert Schweizer 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" (Schweizer 1905, 53). Gillies Whittaker (1959, 368) 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.”
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. (See also False relation.)
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 Radcliffe (1978, 52) 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.” (The passage can be heard at 0:41 on the linked recording.)
Roger Scruton (2009, 101) alludes to Wagner’s description of this chord as introducing “a huge Schrekensfanfare–horror fanfare.” When this passage returns later in the same movement (just before the voices enter) 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 Cook (1987, p. 242) 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 (bars 9-14) “are set on a kind of collision course; hence the feeling of tension rising steadily to a breaking point.”
Another example of a cumulative build-up of dissonance from the early 20th century (1910) can be found in the Adagio that opens Mahler’s unfinished 10th Symphony:
Taruskin (2005, 23) parses this chord (in bars 206 and 208) 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 (described above as the "coincidence of partials") to embrace non-harmonic timbres and their related tunings, consonance has recently been "emancipated" from harmonic timbres and their related tunings (Milne, Sethares, and Plamondon 2007,[page needed]; Milne, Sethares, and Plamadon 2008,[page needed]; Sethares et al. 2009,[page needed]). 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" (Stein 1953,[page needed]).
Neo-classic harmonic consonance theory
George Russell, 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 (Russell 2008, p. 1).
In effect, he returns to a Medieval consideration of "harmonic consonance": that intervals when not subject to octave equivalence (at least not by contraction) 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 (untempered tritone), half-flat thirteenth note, and half-flat fifteenth note must necessarily be consonant. Octave equivalence (minor ninth in some sense equivalent to minor second, etc.) 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 (minor) 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 (approximating the harmonic eleventh), that accidental being the sole pitch difference between the Major scale and the Lydian mode.
(In another sense, that Lydian scale representing the provenance of the tonic chord (with major seventh and sharp fourth) replaces or supplements the Mixolydian scale of the dominant chord (with minor seventh and natural fourth) as the source from which to derive extended tertian harmony.)
Dan Haerle, in his 1980 The Jazz Language (Studio 224 1980, p. 4), 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 (Hindemith 1937–70, 1:[page needed]). 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 (Tymoczko 2011, p. 106).
Instruments producing non-harmonic overtone series
Musical instruments like bells and xylophones, called Idiophones, are played such that their relatively stiff, non-trivial[clarification needed] 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 (Gouwens 2009, p. 3).
- Consonant: major third, tritone, minor sixth, perfect fourth, perfect fifth, and possibly minor seventh or even major second
- Dissonant: major third, major sixth
- Variable upon individual instrument: major seventh
- Interval inversion does not apply.
Physiological basis of dissonance
Musical styles are similar to languages, in that certain physical, physiological, and neurological facts create bounds that greatly affect the development of all languages. Nevertheless, different cultures and traditions have incorporated the possibilities and limitations created by these physical and neurological facts into vastly different, living systems of human language. Neither the importance of the underlying facts nor the importance of the culture in assigning a particular meaning to the underlying facts should be understated.
For instance, two notes played simultaneously but with slightly different frequencies produce a beating "wah-wah-wah" sound. Musical styles such as traditional European classical music consider this effect objectionable ("out of tune") and go to great lengths to eliminate it. Other musical styles such as Indonesian gamelan consider this sound an attractive part of the musical timbre and go to equally great lengths to create instruments that produce this slight "roughness" (Vassilakis 2005,[page needed]).
Sensory dissonance and its two perceptual manifestations (beating and roughness) are both closely related to a sound signal's amplitude fluctuations. Amplitude fluctuations describe variations in the maximum value (amplitude) 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 (waves) at any given time may be larger (constructive interference) or smaller (destructive interference) than the amplitude of the individual vibrations (waves), 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 (≈≤20 per second) 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 (≈≥75–150 fluctuations per second, depending on the frequency of the interfering tones).
Assuming the ear performs a frequency analysis on incoming signals, as indicated by Ohm's acoustic law (see Helmholtz 1885; Levelt and Plomp 1964,[page needed]), the above perceptual categories can be related directly to the bandwidth of the hypothetical analysis filters (Zwicker, Flottorp, and Stevens 1957,[page needed]; Zwicker 1961,[page needed]). 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:
- If the fluctuation rate is smaller than the filter bandwidth, then a single tone is perceived either with fluctuating loudness (beating) or with roughness.
- 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 (Terhardt 1974,[page needed]; Vassilakis 2001,[page needed]). 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 (Vassilakis 2001; Vassilakis 2005; Vassilakis 2007).
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.
This file illustrates the roughness and beat oscillations that gradually reduce as the interval moves towards the unison.
|Problems playing this file? See media help.|
- Frequency ratios: ratios of higher simple numbers are more dissonant than lower ones (Pythagoras[full citation needed]).
In human hearing, the varying effect of simple ratios may be perceived by one of these mechanisms:
- Fusion or pattern matching: fundamentals may be perceived through pattern matching of the separately analyzed partials to a best-fit exact-harmonic template (Gerson and Goldstein 1978,[page needed]) or the best-fit subharmonic (Terhardt 1974,[page needed]), or harmonics may be perceptually fused into one entity, with dissonances being those intervals less likely mistaken for unisons, the imperfect intervals, because of the multiple estimates, at perfect intervals, of fundamentals, for one harmonic tone (Terhardt 1974,[page needed]). By these definitions, inharmonic partials of otherwise harmonic spectra are usually processed separately (Hartmann et al., 1990), unless frequency or amplitude modulated coherently with the harmonic partials (McAdams 1983). For some of these definitions, neural firing supplies the data for pattern matching; see directly below (e.g., Moore 1989, pp. 183–87; Srulovicz and Goldstein 1983).
- Period length or neural-firing coincidence: with the length of periodic neural firing created by two or more waveforms, higher simple numbers creating longer periods or lesser coincidence of neural firing and thus dissonance (Patterson 1986,[page needed]; Boomsliter and Creel 1961,[page needed]; Meyer 1898,[page needed]; Roederer 1973, pp. 145–49). Purely harmonic tones cause neural firing exactly with the period or some multiple of the pure tone.
- Dissonance is more generally defined by the amount of beating between partials (called harmonics or overtones when occurring in harmonic timbres) (Helmholtz 1954 ,[page needed]). Terhardt 1984,[page needed] calls this "sensory dissonance". By this definition, dissonance is dependent not only on the width of the interval between two notes' fundamental frequencies, but also on the widths of the intervals between the two notes' non-fundamental partials. Sensory dissonance (i.e., presence of beating and/or roughness in a sound) is associated with the inner ear's inability to fully resolve spectral components with excitation patterns whose critical bands overlap. If two pure sine waves, without harmonics, are played together, people tend to perceive maximum dissonance when the frequencies are within the critical band for those frequencies, which is as wide as a minor third for low frequencies and as narrow as a minor second for high frequencies (relative to the range of human hearing) (Sethares 2005, p. 43). If harmonic tones with larger intervals are played, the perceived dissonance is due, at least in part, to the presence of intervals between the harmonics of the two notes that fall within the critical band (Roederer 1995, p. 106).
Generally, the sonance (i.e., a continuum with pure consonance at one end and pure dissonance at the other) 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 (or vice versa) (Sethares 2005, p. 1). The sonance of the interval between two notes can be maximized (producing consonance) by maximizing the alignment of the two notes' partials, whereas it can be minimized (producing dissonance) 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 (Sethares 2005, p. 1; Sethares 2009,[page needed]).
Controlling the sonance of more-or-less non-harmonic timbres in real time is an aspect of dynamic tonality. For example, in Sethares' piece C To Shining C (discussed here), the sonance of intervals is affected both by tuning progressions and timbre progressions.
The strongest homophonic (harmonic) cadence, the authentic cadence, dominant to tonic (D-T, V-I or V7-I), is in part created by the dissonant tritone created by the seventh, also dissonant, in the dominant seventh chord, which precedes the tonic.
- Chord factor
- Dissonant counterpoint
- Limit (music)
- Tone cluster
- Beat (acoustics)
- Roughness (psychophysics)
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