Consonance and dissonance

Consonance and dissonance

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In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, a consonance (Latin com-, "with" + sonare, "to sound") is a harmony
Harmony
In music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...

, chord
Chord (music)
A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...

, or interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

 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
Pythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...

). Many of these definitions do not require exact integer tunings, only approximation.
  • Coincidence of partials: with consonance being a greater coincidence of partial
    Harmonic series (music)
    Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...

    s (called harmonic
    Harmonic
    A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

    s or overtone
    Overtone
    An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...

    s when occurring in harmonic timbre
    Timbre
    In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the...

    s) (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
    Musical tuning
    In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...

    ), but also on the combined spectral distribution and thus sound quality (i.e., the timbre
    Timbre
    In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the...

    ) 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. 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.

  • 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:
    • unison
      Unison
      In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...

      s and octave
      Octave
      In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

      s
    • perfect fourth
      Perfect fourth
      In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...

      s+ and perfect fifth
      Perfect fifth
      In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

      s
  • Imperfect consonances:
    • major third
      Major third
      In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

      s and minor sixth
      Minor sixth
      -Subminor sixth:In music, a subminor sixth or septimal sixth is an interval that is noticeably narrower than a minor sixth but noticeably wider than a diminished sixth.The sub-minor sixth is an interval of a 14:9 ratio or alternately 11:7....

      s
    • minor third
      Minor third
      In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...

      s and major sixth
      Major sixth
      In classical music from Western culture, a sixth is a musical interval encompassing six staff positions , and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the largest of the two...

      s

+The perfect fourth is considered a dissonance in most classical music when its function is contrapuntal
Counterpoint
In music, counterpoint is the relationship between two or more voices that are independent in contour and rhythm and are harmonically interdependent . It has been most commonly identified in classical music, developing strongly during the Renaissance and in much of the common practice period,...

.

Note that in the Western Middle Ages, only the octave and perfect fifth were considered consonant harmonically (see Interval (music)
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

 or Just Intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

 for the explanation).

Dissonance


In Western music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, dissonance is the quality of sounds that seems "unstable" and has an aural "need" to "resolve
Resolution (music)
Resolution in western tonal music theory is the move of a note or chord from dissonance to a consonance .Dissonance, resolution, and suspense can be used to create musical interest...

" to a "stable" consonance
Consonance
Consonance is a stylistic device, most commonly used in poetry and songs, characterized by the repetition of the same consonant two or more times in short succession, as in "pitter patter" or in "all mammals named Sam are clammy".Consonance should not be confused with assonance, which is the...

. Both consonance and dissonance are words applied to harmony
Harmony
In music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...

, chord
Chord (music)
A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...

s, and interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

s and, by extension, to melody
Melody
A melody , also tune, voice, or line, is a linear succession of musical tones which is perceived as a single entity...

, tonality
Tonality
Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron and was borrowed by François-Joseph Fétis in 1840...

, and even rhythm
Rhythm
Rhythm may be generally defined as a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions." This general meaning of regular recurrence or pattern in time may be applied to a wide variety of cyclical natural phenomena having a periodicity or...

 and metre
Metre (music)
Meter or metre is a term that music has inherited from the rhythmic element of poetry where it means the number of lines in a verse, the number of syllables in each line and the arrangement of those syllables as long or short, accented or unaccented...

. 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
Harmony
In music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...

 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
Bulgaria
Bulgaria , officially the Republic of Bulgaria , is a parliamentary democracy within a unitary constitutional republic in Southeast Europe. The country borders Romania to the north, Serbia and Macedonia to the west, Greece and Turkey to the south, as well as the Black Sea to the east...

, Bosnia-Herzegovina, Albania
Albania
Albania , officially known as the Republic of Albania , is a country in Southeastern Europe, in the Balkans region. It is bordered by Montenegro to the northwest, Kosovo to the northeast, the Republic of Macedonia to the east and Greece to the south and southeast. It has a coast on the Adriatic Sea...

, Latvia
Latvia
Latvia , officially the Republic of Latvia , is a country in the Baltic region of Northern Europe. It is bordered to the north by Estonia , to the south by Lithuania , to the east by the Russian Federation , to the southeast by Belarus and shares maritime borders to the west with Sweden...

, Georgia
Georgia (country)
Georgia is a sovereign state in the Caucasus region of Eurasia. Located at the crossroads of Western Asia and Eastern Europe, it is bounded to the west by the Black Sea, to the north by Russia, to the southwest by Turkey, to the south by Armenia, and to the southeast by Azerbaijan. The capital of...

, Nuristan, some Vietnamese
Vietnamese people
The Vietnamese people are an ethnic group originating from present-day northern Vietnam and southern China. They are the majority ethnic group of Vietnam, comprising 86% of the population as of the 1999 census, and are officially known as Kinh to distinguish them from other ethnic groups in Vietnam...

 and Chinese
Chinese people
The term Chinese people may refer to any of the following:*People with Han Chinese ethnicity ....

 minority singing traditions, Lithuanian sutartines, some polyphonic traditions from Flores
Flores
Flores is one of the Lesser Sunda Islands, an island arc with an estimated area of 14,300 km² extending east from the Java island of Indonesia. The population was 1.831.000 in the 2010 census and the largest town is Maumere. Flores is Portuguese for "flowers".Flores is located east of Sumbawa...

 and Melanesia
Melanesia
Melanesia is a subregion of Oceania extending from the western end of the Pacific Ocean to the Arafura Sea, and eastward to Fiji. The region comprises most of the islands immediately north and northeast of Australia...

 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
Neutral second
A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:...

 (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
Joseph Jordania
Joseph Jordania is an Australian-Georgian ethnomusicologist and evolutionary musicologist. In some early publications his name was spelled as Zhordania...

 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
Natural selection
Natural selection is the nonrandom process by which biologic traits become either more or less common in a population as a function of differential reproduction of their bearers. It is a key mechanism of evolution....

 during the early stages of hominid evolution in order to reach the state of the battle trance
Battle trance
Battle trance is a term denoting a specific altered state of consciousness that characterizes the psychological state of combatants during a combat situation. In this state, combatants do not feel fear or pain , and all the individual members of group are acting as one collective organism...

.

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
Renaissance music
Renaissance music is European music written during the Renaissance. Defining the beginning of the musical era is difficult, given that its defining characteristics were adopted only gradually; musicologists have placed its beginnings from as early as 1300 to as late as the 1470s.Literally meaning...

, 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
Common practice period
The common practice period, in the history of Western art music , spanning the Baroque, Classical, and Romantic periods, lasted from c. 1600 to c. 1900.-General characteristics:...

 all dissonances were required to be prepared
Preparation (music)
In a nonharmonic tone, preparation is, in the move of a pitch or chord from a consonance to a dissonance, the consonant pitch or chord which precedes the dissonance...

 and then resolved
Resolution (music)
Resolution in western tonal music theory is the move of a note or chord from dissonance to a consonance .Dissonance, resolution, and suspense can be used to create musical interest...

, giving way or returning to a consonance. There was also a distinction between melodic and harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

 dissonance. Dissonant melodic intervals then included the tritone
Tritone
In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...

 and all augmented
Augmentation (music)
In Western music and music theory, the word augmentation has three distinct meanings. Augmentation is a compositional device where a melody, theme or motif is presented in longer note-values than were previously used...

 and diminished
Diminution
In Western music and music theory, diminution has four distinct meanings. Diminution may be a form of embellishment in which a long note is divided into a series of shorter, usually melodic, values...

 intervals. Dissonant harmonic intervals included:
  • minor second
    Minor second
    In modern Western tonal music theory a minor second is the interval between two notes on adjacent staff positions, or having adjacent note letters, whose alterations cause them to be one semitone or half-step apart, such as B and C or C and D....

     and major seventh
    Major seventh
    In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the major seventh is one of two commonly occurring sevenths. It is qualified as major because it is the larger of the two...

  • augmented fourth and diminished fifth (enharmonically equivalent, tritone
    Tritone
    In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...

    )


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
Overtone
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...

 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
Emancipation of the dissonance
The emancipation of the dissonance was a concept or goal put forth by composer Arnold Schoenberg and others, including his pupil Anton Webern. The phrase first appears in Schoenberg's 1926 essay "Opinion or Insight?". It may be described as a metanarrative to justify atonality...

" (the words of Arnold Schoenberg
Arnold Schoenberg
Arnold Schoenberg was an Austrian composer, associated with the expressionist movement in German poetry and art, and leader of the Second Viennese School...

) by some 20th-century composers. Early 20th-century American composer Henry Cowell
Henry Cowell
Henry Cowell was an American composer, music theorist, pianist, teacher, publisher, and impresario. His contribution to the world of music was summed up by Virgil Thomson, writing in the early 1950s:...

 viewed tone cluster
Tone cluster
A tone cluster is a musical chord comprising at least three consecutive tones in a scale. Prototypical tone clusters are based on the chromatic scale, and are separated by semitones. For instance, three adjacent piano keys struck simultaneously produce a tone cluster...

s 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
Guillaume de Machaut
Guillaume de Machaut was a Medieval French poet and composer. He is one of the earliest composers on whom significant biographical information is available....

's "Je ne cuit pas qu'onques":
One example of baroque dissonance:
One example of classical-era dissonance:
One example of modernist dissonance:
The West's progressive embrace of increasingly dissonant intervals occurred almost entirely within the context of harmonic
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...

 timbre
Timbre
In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the...

s, 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
Dynamic tonality
Dynamic tonality is tonal music which uses real-time changes in tuning and timbre to perform new musical effects such as polyphonic tuning bends, new chord progressions, and temperament modulations, with the option of consonance. The performance of dynamic tonality requires an isomorphic keyboard...

. 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
Arnold Schoenberg
Arnold Schoenberg was an Austrian composer, associated with the expressionist movement in German poetry and art, and leader of the Second Viennese School...

 described as the "crisis of tonality".

The Middle Ages


According to Johannes de Garlandia
Johannes de Garlandia (music theorist)
Johannes de Garlandia was a French music theorist of the late ars antiqua period of medieval music...

:
  • 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
Beat (acoustics)
In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies....

 "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
Gamelan
A gamelan is a musical ensemble from Indonesia, typically from the islands of Bali or Java, featuring a variety of instruments such as metallophones, xylophones, drums and gongs; bamboo flutes, bowed and plucked strings. Vocalists may also be included....

 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
Beat (acoustics)
In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies....

 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
Ohm's acoustic law
Ohm's acoustic law, sometimes called the acoustic phase law or simply Ohm's law, states that a musical sound is perceived by the ear as a set of a number of constituent pure harmonic tones.The law was proposed by physicist Georg Ohm in 1843...

 (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 partial
    Harmonic series (music)
    Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...

    s (called harmonic
    Harmonic
    A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

    s or overtone
    Overtone
    An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...

    s when occurring in harmonic timbre
    Timbre
    In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the...

    s) (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
    Critical bands
    The term critical band, introduced by Harvey Fletcher in the 1940s, referred to the frequency bandwidth of the then loosely defined auditory filter....

     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). 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.


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). 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
Dynamic tonality
Dynamic tonality is tonal music which uses real-time changes in tuning and timbre to perform new musical effects such as polyphonic tuning bends, new chord progressions, and temperament modulations, with the option of consonance. The performance of dynamic tonality requires an isomorphic keyboard...

. 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
Cadence (music)
In Western musical theory, a cadence is, "a melodic or harmonic configuration that creates a sense of repose or resolution [finality or pause]." A harmonic cadence is a progression of two chords that concludes a phrase, section, or piece of music...

, the authentic cadence, dominant to tonic (D-T, V-I or V7-I), is in part created by the dissonant tritone
Tritone
In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...

created by the seventh, also dissonant, in the dominant seventh chord, which precedes the tonic
Tonic (music)
In music, the tonic is the first scale degree of the diatonic scale and the tonal center or final resolution tone. The triad formed on the tonic note, the tonic chord, is thus the most significant chord...

.

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
Lydian chromatic concept of tonal organization
The Lydian Chromatic Concept of Tonal Organization was written by George Russell and is the founding text of the Lydian Chromatic Concept , or Lydian Chromatic Theory . The work postulates that all music is based on the tonal gravity of the Lydian mode.-Deriving Lydian:Russell believed that...

. The theorist regards the tritone over the tonic as a rather consonant interval, contrary to slightly popular belief.

See also

  • Chord factor
    Factor (chord)
    In music, a factor or chord factor is a member or component of a chord. These are named root, third, fifth, sixth, seventh, ninth, eleventh, thirteenth, and so on, for their generic interval above the root....

  • Dissonant counterpoint
  • Limit (music)
    Limit (music)
    In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...

  • Phonoaesthetics
  • Semitone
    Semitone
    A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....

  • Beat (acoustics)
    Beat (acoustics)
    In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies....

  • Roughness (psychophysics)
    Roughness (psychophysics)
    The perceived roughness of a sound is simply how rough it sounds. In an experiment to measure and compare the roughness of different sounds, listeners are presented with different sounds and asked to rate their roughness, for example on a rating scale....

  • Ainulindalë
    Ainulindalë
    The Ainulindalë is the first part of the fantasy work The Silmarillion by J. R. R. Tolkien. In Tolkien's legendarium, the Ainur are Eä's divine beings. In Heaven, before Time, they compose a Great Music. This Music is revealed to be the template, or blueprint, commensurable with the entire history...


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.
    William Sethares
    William A. Sethares is an American music theorist and professor of Electrical engineering and who is known primarily for his contributions to music theory, including dynamic tonality and a formalization of the source of consonance....

     (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 http://eceserv0.ece.wisc.edu/~sethares/papers/consance.html)
  • Tenney, James
    James Tenney
    James Tenney was an American composer and influential music theorist.-Biography:Tenney was born in Silver City, New Mexico, and grew up in Arizona and Colorado. He attended the University of Denver, the Juilliard School of Music, Bennington College and the University of Illinois...

    . (1988). A History of "Consonance" and "Dissonance". White Plains, NY: Excelsior; New York: Gordon and Breach.
  • 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.

External links