This contribution gives a brief overview of a mathematical and physical interpretation of musical consonance and dissonance. In addition to the traditionally accepted Pythagorean explanation of consonance, Helmholtz's approach based on the coincidence of harmonics is also mentioned. An analysis of the controversial minor triad chord shows that Helmholtz's theory is more generally valid. Finally, reference to various neurophysiological, psychological, musicological, and psychoacoustic studies demonstrates that the issue of consonance and dissonance in music is of a distinctly interdisciplinary character.
Von HelmholtzH. L. F.: ‘On the sensation of tone as a physiological basis for the theory of music’, (trans. EllisA. J.); 1954, New York, NY, Dover(first published 1863, in German).
3.
CazdenN.: ‘Hindemith and nature’, Musical Review, 1954, 15, 288.
4.
McleanD.: ‘The science behind Europe's music scale’ (see Note 1).
5.
FishmanY. I., VolkovI. O., NohM. D., GarellP. C., BakkenH., ArezzoJ. C., HowardM. A., and SteinschnederM. ‘Consonance and dissonance of musical chords: neural correlates in auditory cortex of monkeys and humans’, Journal of Neurophysiology, 2001, 86, 2761.
6.
TramoM. J., CarianiP. A., DelguteB., and BraidaL. D.: ‘Neurobiological foundations for the theory of harmony in Western tonal music’, Annals of the New York Academy of Sciences, 2001, 930, 92.
7.
See for exampleLandauL. D. and LifshitzE. M.: ‘Mechanics’, 3rd edn, (trans. SykesJ. B. and BellJ. S.); 1981, Oxford, Butterworth-Heineman, or other university textbooks of theoretical physics.
8.
Capital F is used here to denote a particular frequency value, to avoid confusion with the dimensionless proportional coeYcients fn m.
9.
Von HelmholtzH. L. F.: ‘On the sensation of tone’ (see Note 2).
10.
BloodA. J., ZatorreR. J., BermdezP., and EvansA. C.: ‘Emotional responses to pleasant and unpleasant music correlate with activity in paralimbic brain regions’, Nature Neuroscience, 1999, 2, 382.
11.
ShepardR. N.: ‘Geometrical approximations to the structure of musical pitch’, Psychological Review, 1982, 89, 305.
12.
KrumhanslC. L.: ‘Rhythm and pitch in music cognition’, Psychological Bulletin, 2000, 126, 159.
13.
BalzanoG. J.: ‘The group-theoretic description of 12-fold and microtonal pitch systems’, Computer Music Journal, 1980, 4, (4), 66.
14.
KrantzR. J. and DouthettJ.: ‘Construction and interpretation of equal-tempered scales using frequency ratios, maximally even sets, and P-cycles’, Journal of the Acoustical Society of America, 2000, 107, 2725.
15.
BoonJ. P. and DecrolyO.: ‘Dynamical-systems theory for music dynamics’, Chaos, 1995, 5, 501.
16.
SchellD.: ‘Optimality in musical melodies and harmonic progressions:the travelling musician’, European Journal of Operational Research, 2002, 140, 354.
17.
RameauJ. P.: ‘Treatise on harmony’, (trans. GossetP.); 1971, New York, NY, Dover(first published 1721, in French).
18.
ForteA.: ‘Tonal harmony in concept and practice’; 1962, New York, NY, Holt, Rinehardt andWinston.
19.
ChristensenT.: ‘Eighteenth-century science and the corps sonore: the scientific background to Rameau's principle of harmony’, Journal of Music Theory, 1987, 31, 22.
20.
LerdahlF.: ‘Calculating tonal tension’, Music Perception, 1996, 13, 319.
21.
VosP. G. and PasveerD.: ‘Goodness ratings of melodic openings and closures’, Perception and Psychophysics, 2002, 64, 631.
22.
CrowderR. G.: ‘Perception of the major minor distinction II. Experimental investigations’, Psychomusicology, 1986, 5, 3.
23.
AuhagenW. and VosP. G.: ‘Experimental methods in tonality induction research’, Music Perception, 2000, 17, 417.
24.
BalkwillL. L. and ThompsonW. F.: ‘A cross-cultural investigation of the perception of emotion in music: psychophysical and cultural cues’, Music Perception, 1999, 17, 43.
25.
ParncuttR.: ‘Harmony: a psychoacoustical approach’; 1989, Berlin, Springer-Verlag.
26.
TerhardtE.: ‘Pitch, consonance and harmony’, Journal of the Acoustical Society of America, 1974, 55, 1061.
27.
TerhardtE.: ‘The concept of musical consonance, a link between music and psychoacoustics’, Music Perception, 1984, 1, 276.
28.
ParncüttR. and StrasburgerH. ‘Applying psychoacoustics in composition – harmonic progressions of nonharmonic sonorities’, Perspectives of New Music, 1994, 32, (2), 88.
29.
FishmanY. I., et al.: ‘Consonance and dissonance of musical chords’ (see Note 5).
