Consonance and dissonance
In music, a consonance (Latin con-, "with" + sonare, "to sound") is a harmony, chord, or interval considered stable (at rest), as opposed to a dissonance (Latin dis-, "apart" + sonare, "to sound"), which is considered 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. This is a cultural concept as musics other than those from the western art music tradition, e.g., Balkan, Arabic, Chinese, do not follow this definition (Rice 2004, pp. 31, 41, 58, 106).
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,[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.[contradiction] 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
|This section does not cite any references or sources. (July 2014)|
Sharp dissonant intervals and chords play 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 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".
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.||”|
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).
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]
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.
One example[clarification needed] of baroque dissonance:
One example[where?] 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 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]).
The Middle Ages
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 + sixth). (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 joigned 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 semitone 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.")
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 minor 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.
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 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).
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 (Patternson 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)
- Beat (acoustics)
- Roughness (psychophysics)
- Butler, David, and Burdette Green (2002). "From Acoustics to Tonpsychologie". In The Cambridge History of Western Music Theory, edited by Thomas Christensen, 246–71. The Cambridge History of Music 3. Cambridge and New York: Cambridge University Press. ISBN 9780521623711.
- Dahlhaus, Carl (1990). Studies in the Origin of Harmonic Tonality, translated by Robert O. Gjerdigan. Princeton: Princeton University Press. ISBN 0-691-09135-8.
- Forte, Allen. Tonal Harmony in Concept and Practice, third edition. New York: Holt, Rinehart and Winston, 1979. ISBN 0-03-020756-8.
- Gerbert, Martin. Scriptores ecclesiastici de musica sacra potissimum: ex variis Italiae, Galliae & Germaniae codicibus manuscriptis, 3 vols. [n.p.]: Typis San-Blasianis, 1784. Reprinted, Hildesheim: G. Olms, 1963.
- Gerson, A., and J. L. Goldstein. 1978. "Evidence for a General Template in Central Optimal Processing for Pitch of Complex Tones". Journal of the Acoustical Society of America 63:498–510.
- Helmholtz, Hermann. 1954 . On the Sensations of Tone, translated by Alexander J. Ellis from the fourth German edition (1887). New York: Dover Publications.
- Helmholtz, H. L. F. (1885 ). 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.]
- Hindemith, Paul (1937–70). Unterweisung im Tonsatz, 3 vols. Mainz: B. Schott's Söhne.
- Johannes de Garlandia. 13th century. De musica mensurabili positio, text on CHTML at Indiana University.
- Jonas, Oswald (1982). Introduction to the Theory of Heinrich Schenker: The Nature of the Musical Work of Art, translated by John Rothgeb. New York: Longman; London: Collier-Macmillan. ISBN 0-582-28227-6. [Translated from Einführung in die Lehre Heinrich Schenkers, das Wesen des musikalischen Kunstwerkes, second edition. Vienna: Universal Edition, 1972. First edition as Das Wesen des musikalischen Kunstwerks: Eine Einführung in die Lehre Heinrich Schenkers. Vienna: Saturn-Verlag, 1934.]
- Kamien, Roger (2008). Music: An Appreciation, sixth brief edition, student edition. Boston: McGraw-Hill Higher Education. ISBN 978-0-07-340134-8.
- Kliewer, Vernon L. (1975). "Melody: Linear Aspects of Twentieth-Century Music", Aspects of Twentieth-Century Music, edited by Gary E. Wittlich, 270–321. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0-13-049346-5.
- Levelt, Willem, and Reiner Plomp. "The Appreciation of Musical Intervals". In Actes du cinquième Congrès international d'esthétique/Proceedings of the Fifth International Congress of Aesthetics, edited by Jan Aler, 901–904. The Hague: Mouton, 1964.
- McAdams. 1983.[full citation needed]
- Milne, Andrew, William A. Sethares, and James Plamondon, "Invariant Fingerings Across a Tuning Continuum", Computer Music Journal 31, no. 4 (Winter 2007): 15-32.
- Milne, Andrew, William A. Sethares, and James Plamondon. 2008. "Tuning Continua and Keyboard Layouts". Journal of Mathematics and Music (Spring).[page needed]
- Myers, Charles S. (1904). "Theories of Consonance and Dissonance". The British Journal of Psychology 1 (June 25): 315–16.
- Rice, Timothy (2004). Music in Bulgaria. Oxford and New York: Oxford University Press. ISBN 0-19-514148-2.
- Roederer, Juan G. (1995). The Physics and Psychophysics of Music: An Introduction (third ed.). New York: Springer-Verlag. ISBN 0-387-94366-8.
- Russell, George. 2008. "George Russell's Lydian Chromatic Concept of Tonal Organization", fourth edition. Brookline, MA: Concept Publishing. ISBN 9780846426004 (vol. 1)
- Schuijer, Michiel (2008). Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts. Eastman Studies in Music 60. Rochester: University of Rochester Press. ISBN 978-1-58046-270-9.
- Schulter, Margo (1997). "Thirteenth-Century Polyphony". Medieval Music & Arts Foundation. Todd M. McComb. Retrieved 1 Jan 2015.
- Schulter, Margo (1997). "Pythagorean Tuning". Medieval Music & Arts Foundation. Todd M. McComb. Retrieved 1 Jan 2015.
- Schulter, Margo (1997). "Multi-voice combinations". Medieval Music & Arts Foundation. Todd M. McComb. Retrieved 1 Jan 2015.
- Schulter, Margo (1997). "Pythagorean tuning and Gothic polyphony". Medieval Music & Arts Foundation. Todd M. McComb. Retrieved 1 Jan 2015.
- Sethares, William. A. "Relating Tuning and Timbre", Experimental Musical Instruments, September 1992. (Archive from 10 June 2010, accessed 29 July 2014).
- Sethares, William A. (2005). Tuning, Timbre, Spectrum, Scale (second ed.). London: Springer. ISBN 1-85233-797-4.
- Sethares, William A., Andrew Milne, Stefan Tiedje, Anthony Prechtl, and James Plamondon. "Spectral Tools for Dynamic Tonality and Audio Morphing". Computer Music Journal 33, no. 2 (Summer 2009): 71–84.
- Stein, Erwin. 1953. Orpheus in New Guises. London: Rockliff. Reprinted Westport, Conn.: Hyperion Press, 1979. ISBN 978-0-88355-765-5.
- Terhardt, Ernst (1974). "On the Perception of Periodic Sound Fluctuations (Roughness)". Acustica 30 (4): 201–13.
- Studio 224. 1980.[full citation needed].
- Tymoczko, Dimitri (2011). A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. Oxford and New York: Oxford University Press. [ISBN missing]
- Vassilakis, Panteleimon Nestor (2001). Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance. Doctoral Dissertation. University of California, Los Angeles.
- Vassilakis, Panteleimon Nestor (2005). "Auditory Roughness as Means of Musical Expression". Selected Reports in Ethnomusicology, 12: 119–44.
- Vassilakis, Panteleimon Nestor, and K. Fitz (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, Eberhard (1961). "Subdivision of the Audible Frequency into Critical Bands". Journal of the Acoustical Society of America 33 (2 (May)): 248–249. Bibcode:1961ASAJ...33..248Z. doi:10.1121/1.1908630.
- Zwicker, Eberhard; Flottorp, G.; Stevens, S. S. (1957). "Critical Band-width in Loudness Summation". Journal of the Acoustical Society of America 29 (5): 548–557. Bibcode:1957ASAJ...29..548Z. doi:10.1121/1.1908963.
- 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.
- Jeppesen, Knud (1946). The Style of Palestrina and the Dissonance, second revised and enlarged edition, translated by Margaret Hamerik with linguistic alterations and additions by Annie I. Fausboll. Copenhagen: E. Munksgaard; Oxford: Oxford University Press. Reprinted, with corrections, New York: Dover Publications, 1970. ISBN 9780486223865.
- Sethares, William 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 )
- Tenney, James. (1988). A History of "Consonance" and "Dissonance". White Plains, NY: Excelsior; New York: Gordon and Breach.
|Wikiquote has quotations related to: Consonance and dissonance|
|Wikimedia Commons has media related to Consonance and dissonance.|
- Atlas of Consonance
- Octave Frequency Sweep, Consonance/Dissonance
- Consonance and Dissonance—Index to Notes by David Huron at Ohio State University School of Music
- The Keyboard Tuning of Domenico Scarlatti
- index of Dissonance for any musical scales in LucyTuning and meantone-type tunings