Consonance and dissonance

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"Consonance" redirects here. For literary form, see Literary consonance.

In music, a consonance (Latin consonare, "sounding together") is a harmony, chord, or interval considered stable, as opposed to a dissonance — considered unstable (or temporary, transitional). The strictest definition of consonance may be only those sounds which are pleasant, while the most general definition includes any sounds which are used freely.

Contents

[edit] Consonance

Consonance has been defined variously through:

  • Frequency ratios: with ratios of lower simple numbers being more consonant than those which are higher (Pythagoras). Many of these definitions do not require exact integer tunings, only approximation.
    • Coincidence of harmonics: with consonance being a greater coincidence of harmonics or partials (collectively overtones) (Helmholtz, 1877/1954). By this definition consonance is dependent not only on the quality of the interval between two notes, but on the combined spectral distribution and thus sound quality (timbre) of the harmonic interval (see the entry under critical band).
    • 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 consonances being those intervals which are more likely to be mistaken for unisons, the perfect intervals, because of the multiple estimates of fundamentals, at perfect intervals, 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 wave-forms, lower simple numbers creating shorter or common periods or higher coincidence of neural-firing and thus consonance (Patternson, 1986; Boomsliter & Creel, 1961; Meyer, 1898; Roederer, 1973, p.145-149). Pure tones cause neural-firing exactly with the period or some multiple of the pure tone.

In what is now called the common practice period in Western music, consonant intervals include:

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

Polyphonic cadence he first an imperfect consonance on a weak beat, the second a perfect consonance on a strong beat, such as a major sixth moving to an octave (for instance, the major (imperfect) sixth D-B followed by the perfect octave C-C').

[edit] Dissonance

In music, dissonance is the quality of sounds which 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 important 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, in fact all music with a harmonic or tonal basis — even music which is 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 to a great degree responsible for what many listeners perceive as beauty, emotion, and expressiveness in music.

[edit] 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, in the common practice period, harmony is generally governed by chords, which are collections of notes generally considered to be consonant (though even within this harmonic system there is a hierarchy of chords, with some considered relatively more consonant and some relatively 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), 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 considered more forceful 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 this period.

[edit] Dissonance throughout the 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 (DeLone et al. 1975, p.290).

In early Renaissance music intervals such as the perfect fourth were considered strong 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 evidence by the allowance of their successions argued for by Adam von Fulda, but 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, occurring on weak beats and quickly 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:

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 consonant. 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

One example of baroque dissonance:

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

One example of classical era dissonance:

Dissonance in Mozart's Adagio and Fugue in C Minor, K. 546.

One example of modernist dissonance:

Igor Stravinsky's The Rite of Spring, "Sacrificial Dance" excerpt

[edit] The objective (physical/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 minimized.

For instance, two notes played simultaneously but with slightly different frequencies, produce a beating "wah-wah-wah" sound that is very audible. Some musical styles consider this effect to be objectionable ("out of tune") and go to great lengths to eliminate it. Other musical styles 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 acoustical 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.

Two pitches moving from the interval of a Major 2nd to a unison

This file illustrates the roughness and beat oscillations that gradually reduce as the interval moves towards the unison.
Problems listening to the file? See media help.
  • Frequency ratios: ratios of higher simple numbers are more dissonant than those which are lower (Pythagoras).
Diverging tones
Spectral analysis of two diverging tones
Spectral analysis of two diverging tones

Two diverging sinusoid-like tones

The sinusoidal-like tones are filtered triangle waves, which have strong fundamentals, no even overtones (at 2x the fundamental frequency, 4x, 6x, and so on) and highly attenuated odd overtones (at 3x, 5x, 7x, and so on).
Problems listening to the file? See media help.

Two diverging tones with sawtooth waveform

The sawtooth wave has strong overtones, both even and odd, which make some points of dissonance and consonance more audible.
Problems listening to the file? See media help.

The physical basis for Pythagoras's observation can be seen in the spectral analysis above and heard in the accompanying sound file. At the points where the ratios of the frequencies of the tones are more simple (indicated by arrows near the top of the graph), the overtones as observed in the spectral analysis are more ordered and simple. Most listeners perceive the tone of the interval at these points to be more "pure" or "harmonious".

By contrast, when the ratios of the frequencies are not simple, the overtone situation appears complex and chaotic in the spectral analysis. Most listeners perceive the intervals at these points to be "rougher" or more "disharmonious".

In human hearing, the varying effect of these different 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 which are 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 wave-forms, higher simple numbers creating longer periods or lesser coincidence of neural-firing and thus dissonance (Patternson, 1986; Boomsliter & Creel, 1961; Meyer, 1898; Roederer, 1973, p.145-149). Pure tones cause neural-firing exactly with the period or some multiple of the pure tone.
  • Dissonance is also then defined by the amount of beating between non-common harmonics or partials (collectively overtones) (Helmholtz, 1877/1954). Terhardt (1984) calls this "sensory dissonance". By this definition dissonance is dependent not only on the quality of the interval between two notes, but the harmonics and thus sound quality (timbre) of those notes themselves. 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.

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.

[edit] Modulo 12 maths

The hierarchy of consonant and dissonant intervals in equal temperament music can be arrived at by modular arithmetic. The arithmetic is modulo 12 with the basic unit the semitone. By starting with the unison 0 semitones add the number 7 successively with modulo 12 wrap around. The sequence of intervals is 0,7,2,9,4,11,6,1,8,3,10,5,12. The intervals progress from the most consonant 0 to the most dissonant 6 (the tritone) and then back to the most consonant 12(the octave). The second half of the sequence represents the inverted intervals in reverse order of the first half. The progression can be thought of as generating successively more unrelated notes. The theory provides an objective basis for consonance and dissonance.

[edit] 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.[1]

[edit] See also

[edit] References

  1. ^ Lydian Chromatic Concept of Tonal Organization, George Russell

[edit] Sources (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
  • DeLone et al. (Eds.) (1975). Aspects of Twentieth-Century Music. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0-13-049346-5.
  • 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.]
  • 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.
  • Zwicker, E., Flottorp, G., and Stevens, S. S. (1957). Critical band-width in loudness summation. Journal of the Acoustical Society of America, 29(5): 548-557.

[edit] External links