Consonance and dissonance

For the mental phenomena, see Cognitive dissonance.

For the stylistic device often used in poetry, see Literary consonance.

In music, a consonance (Latin com-, "with" + sonare, "to sound") is a harmony, chord, or interval considered stable, as opposed to a dissonance (Latin dis-, "apart" + sonare, "to sound"), which is considered to be unstable (or temporary, transitional). In more general usage, a consonance is a combination of notes that sound pleasant to most people when played at the same time; dissonance is a combination of notes that sound harsh or unpleasant to most people.

Consonance

Consonance has been defined variously through: With ratios of lower simple numbers being more consonant than those that are higher (Pythagoras). Many of these definitions do not require exact integer tunings, only approximation.

• Coincidence of partials: with consonance being a greater coincidence of partials (called harmonics or overtones when occurring in harmonic timbres) (Helmholtz, 1877/1954). 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.^[1] Although Helmholtz's work focused almost exclusively on harmonic timbres and tunings, subsequent work has generalized his findings to embrace non-harmonic tunings and timbres.^{[2][3][4][5][6]}

• Fusion or pattern matching: fundamentals may be perceived through pattern matching of the separately analyzed partials to a best-fit exact-harmonic template (Gerson & Goldstein, 1978) or the best-fit subharmonic (Terhardt, 1974). Harmonics may be perceptually fused into one entity, with consonances being those include:
• Perfect consonances:
  • unisons and octaves
  • perfect fourths⁺ and perfect fifths
• Imperfect consonances:
  • major thirds and minor sixths
  • minor thirds and major sixths

⁺The perfect fourth is considered a dissonance in most classical music when its function is contrapuntal.

Note that in the Western Middle Ages, only the octave and perfect fifth were considered consonant harmonically (see Interval (music) or Just Intonation for the explanation).

"A stable tone combination is a consonance; consonances are points of arrival, rest, and resolution."

—Roger Kamien (2008), p.41^[7]

Dissonance

Ernst Krenek's classification, from Studies in Counterpoint (1940), of a triad's overall consonance or dissonance through the consonance or dissonance of the three intervals contained within^[8]  Play (help·info). For example, C-E-G consists of three consonances (C-E, E-G, C-G) and is ranked 1 while C-D♭-B consists of one mild dissonance (B-D♭) and two sharp dissonances (C-D♭, C-B) and is ranked 6.

"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^[7]

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.

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

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 to be tolerably consonant by the style. (Though there is likely 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. Particular attention is paid to how dissonances are approached (approach by step is less jarring, approach by leap more jarring), and even more to how they are resolved (almost always by step), to how they are placed within the meter and rhythm (dissonances on stronger beats are more emphatic and those on weaker beats less vital), and to how they lie within the phrase (dissonances tend to resolve at phrase's end). In short, dissonance is not used willy-nilly but is used in a very careful, controlled, and well-circumscribed way. The subtle interplay of different levels of dissonance and resolution is vital to understanding the tonal and harmonic language of any style.

Dissonance in traditional music

Sharp dissonant intervals and chords play prominent role in many traditional musical cultures. Vocal polyphonic traditions from Bulgaria, Bosnia-Herzegovina, Albania, Latvia, Georgia, Nuristan, some Vietnamese and Chinese minority singing traditions, Lithuanian sutartines, 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". Joseph Jordania recently suggested that extremely loud group singing/shouting, based on dissonant intevals, augmented by stomping and drumming on external objects, threatening body movements and object throwing, was developed by the forces of natural selection during the early stages of hominid evolution in order to reach the state of the battle trance^[9].

Dissonance in history of Western music

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, p. 290).

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 Gerbert (vol.3, p. 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)

In the common practice period all dissonances were required to be prepared and then resolved, giving way or returning to a consonance. There was also a distinction between melodic and harmonic dissonance. Dissonant melodic intervals then 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 starting with a quite limited definition of consonance and progressing towards 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) 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.

One example of imperfect consonances previously considered dissonances in Guillaume de Machaut's "Je ne cuit pas qu'onques":

Xs mark thirds and sixths  Play (help·info)

One example of baroque dissonance:

A sharply dissonant chord in Bach's Well-Tempered Clavier, Vol. I (Preludio XXI)  Play (help·info)

One example of classical-era dissonance:

Dissonance in Mozart's Adagio and Fugue in C Minor, K. 546.  Play (help·info)

One example of modernist dissonance:

Igor Stravinsky's The Rite of Spring, "Sacrificial Dance" excerpt  Play (help·info)

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 et al., 2007, 2008; Sethares et al., 2009). 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".^[10]

The Middle Ages

According to Johannes de Garlandia:

• Perfect consonance: unisons and octaves
• Mediocre consonance: fourths and fifths
• Imperfect consonance: minor and major thirds
• Perfect dissonance: minor seconds, tritonus, and major sevenths
• Mediocre dissonance: major seconds and minor sixths
• Imperfect dissonance: major sixths and minor sevenths

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 that is very audible. Musical styles such as traditional European classical music consider this effect to be objectionable ("out of tune") and go to great lengths to eliminate it. Other musical styles such as Indonesian gamelan consider this sound to be an attractive part of the musical timbre and go to equally great lengths to create instruments that have this slight "roughness" as a feature of their sound (Vassilakis, 2005).

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 to be 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; Plomp 1964), the above perceptual categories can be related directly to the bandwidth of the hypothetical analysis filters (Zwicker et al. 1957; Zwicker 1961). 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:

a) If the fluctuation rate is smaller than the filter bandwidth, then a single tone is perceived either with fluctuating loudness (beating) or with roughness.

b) 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; Vassilakis 2001). 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 are variables that continue to be 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, 2005, 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.

• Frequency ratios: ratios of higher simple numbers are more dissonant than lower ones (Pythagoras).

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 & Goldstein, 1978) or the best-fit subharmonic (Terhardt, 1974), or harmonics may be perceptually fused into one entity, with dissonances being those intervals less likely to be mistaken for unisons, the imperfect intervals, because of the multiple estimates, at perfect intervals, of fundamentals, for one harmonic tone (Terhardt, 1974). 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. 183–187; Srulovicz & 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 (Patternson, 1986; Boomsliter & Creel, 1961; Meyer, 1898; Roederer, 1973, pp. 145-149). 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, 1877/1954). Terhardt (1984) 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).^[11] 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.^[12]

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).^[13] 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 (ibid., Sethares 2009).

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^{[citation needed]} created by the seventh, also dissonant, in the dominant seventh chord, which precedes the tonic.

Tritone resolution inwards and outwards.  Play inward. (help·info)  Play outwards. (help·info)

Perfect authentic cadence (V-I with roots in the bass and tonic in the highest voice of the final chord): ii-V-I progression in C  Play (help·info).

George Russell's theory

There is some disagreement on this consonance to dissonance chart, stemming from George Russell's Lydian Chromatic Concept of Tonal Organization. The theorist regards the tritone over the tonic as a rather consonant interval, contrary to slightly popular belief.^[14]

See also

• Chord factor
• Dissonant counterpoint
• Limit (music)
• Phonoaesthetics
• Semitone
• Beat (acoustics)
• Roughness (psychophysics)
• Ainulindalë

References

1. ^ Juan G. Roederer (1995). The Physics and Psychophysics of Music. p. 165. ISBN 0387943668. 
2. ^ Sethares, W.A., Relating Tuning and Timbre, Experimental Musical Instruments, September 1992.
3. ^ William A. Sethares (2005). Tuning, Timbre, Spectrum, Scale. ISBN 1852337974. 
4. ^ Milne, A., Sethares, W.A. and Plamondon, J., Invariant Fingerings Across a Tuning Continuum, Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 15-32.
5. ^ Milne, A., Sethares, W.A. and Plamondon, J., Tuning Continua and Keyboard Layouts, Journal of Mathematics and Music, Spring 2008.
6. ^ Sethares, W.A., Milne, A., Tiedje, S., Prechtl, A., and Plamondon, J., Spectral Tools for Dynamic Tonality and Audio Morphing, Computer Music Journal, Spring 2009.
7. ^ ^{a b} Kamien, Roger (2008). Music: An Appreciation, 6th Brief Edition, p.41. ISBN 978-0-07-340134-8.
8. ^ Schuijer, Michael (2008). Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts, p.138. ISBN 978-1-58046-270-9.
9. ^ Joseph Jordania. Why do People Sing? Music in Human Evolution. Logos, 2011
10. ^ Stein, E., 1953, Orpheus in New Guises, ISBN 9780883557655.
11. ^ William A. Sethares (2005). Tuning, Timbre, Spectrum, Scale. p. 43. ISBN 1852337974. 
12. ^ Juan G. Roederer (1995). The Physics and Psychophysics of Music. p. 106. ISBN 0387943668. 
13. ^ William A. Sethares (2005). Tuning, Timbre, Spectrum, Scale. p. 1. ISBN 1852337974. 
14. ^ Lydian Chromatic Concept of Tonal Organization, George Russell

Further reading (partial list)

• Burns, Edward M. (1999). "Intervals, Scales, and Tuning", in The Psychology of Music second edition. Deutsch, Diana, ed. San Diego: Academic Press. ISBN 0-12-213564-4.
• Dahlhaus, Carl. Gjerdingen, Robert O. trans. (1990). Studies in the Origin of Harmonic Tonality. Princeton University Press. ISBN 0-691-09135-8.
• Helmholtz, H. L. F. (1885 [1954]). On the Sensations of Tone as a Physiological Basis for the Theory of Music. 2nd English edition. New York: Dover Publications. [Die Lehre von den Tonempfindungen, 1877. 4th German edition, trans. A. J. Ellis.]
• Kliewer, Vernon (1975). "Melody: Linear Aspects of Twentieth-Century Music", Aspects of Twentieth-Century Music. Delone, et al. (eds.). Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0-13-049346-5.
• Sethares, W. A. (1993). "Local consonance and the relationship between timbre and scale". Journal of the Acoustical Society of America, 94(1): 1218. (A non-technical version of the article is available at [1])
• Tenney, James. (1988). A History of "Consonance" and "Dissonance". White Plains, NY: Excelsior; New York: Gordon and Breach.
• Terhardt, E. (1974). "On the perception of periodic sound fluctuations (roughness)". Acustica 30 (4): 201–213. 
• Vassilakis, P.N. (2001). Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance. Doctoral Dissertation. University of California, Los Angeles.
• Vassilakis, P.N. (2005). "Auditory roughness as means of musical expression". Selected Reports in Ethnomusicology, 12: 119-144.
• Vassilakis, P.N. and Fitz, K. (2007). SRA: A Web-based Research Tool for Spectral and Roughness Analysis of Sound Signals. Supported by a Northwest Academic Computing Consortium grant to J. Middleton, Eastern Washington University.
• Zwicker, E. (1961). "Subdivision of the audible frequency into critical bands". Journal of the Acoustical Society of America 33 (2): 248–249. doi:10.1121/1.1908630. 
• Zwicker, E.; Flottorp, G.; Stevens, S. S. (1957). "Critical band-width in loudness summation". Journal of the Acoustical Society of America 29 (5): 548–557. doi:10.1121/1.1908963. 

External links

• Atlas of Consonance
• Consonance and Dissonance — Index to Notes by David Huron at Ohio State University School of Music
  • Bibliography of Consonance and Dissonance
• The Keyboard Tuning of Domenico Scarlatti
• "Music has teeth! (Dissonance in music)" A simple explanation of dissonance with short examples