===== The formalization and abstraction of paradigmatic psychoacoustical properties of agglomerates of sounds of definite pitch as theoretical grounds for a new discipline of Harmony =====
{{  :roughnesschart_v1.png}}
**Research conducted at the State University of Maringá, Paraná, Brazil, from 2014 to 2019.**\\
\\
This research project, which was conducted from 2014 to 2019 at the [[http://www.uem.br | State University of Maringá]], Paraná, Brazil, with funding from a two-year fellowship grant (2014-2016) given by the [[http://www.fappr.pr.gov.br | Araucaria Foundation for the Scientific and Technological Development of the State of Paraná]], aimed to construct the theoretical grounds for a new discipline of Harmony by means of the theoretical formalization and abstraction of paradigmatic psychoacoustical properties of agglomerates of sounds of definite pitch. The research investigated and defined paradigmatic psychoacoustical concepts for this type of sounds, their qualification and mathematical quantification, as well as the enumeration and a qualitative and taxonomical study of a Harmonic Universe defined by these studies. The research included the computational implementation of the quantitative mathematical modellings developed, creating Computer-Assisted Composition software components that aim at the fast taxonomical identification of a harmonic combination and the survey of its properties and comparative interrelationships with other combinations. Among the expected direct results of this research are: a) the development and formalization of a more appropriate theoretical body for the learning of Music Harmony in a way which is prepared and atuned to the musical and aesthetic directives of the 21st century and of a globalized world; and b) to make viable the creation of software components to aid contemporary composers in their task of researching new harmonic languages, serving them as basic theoretical grounds. The methodology for this research included three stages: 1) definition and quantitative mathematical formalization of archetypal psychoacoustical concepts, which includes a bibliographical revision of previous modern and historical theoretical works on the subject, as well as modern psychoacoustical research; 2) definition of a Harmonic Universe and its taxonomical study; and 3) the final formalization of the research, including the programming of software components and scientific papers.
\\
==== Objectives ====

  - The investigation and definition of archetypal psychoacoustical concepts capable of being used as grounds for the design of horizontal and vertical mixtures of sounds of definite pitch;
  - The qualification of these archetypal psychoacoustical concepts, and the research, choice and formalization of a quantification method for these concepts, according to mathematical modelling methods;
  - The enumeration as well as the qualitative and taxonomical study of several paradigmatic harmonic combinations, grounded by the formalization of the basic psychoacoustical concepts studied;
  - The computational implementation of the mathematical methods developed for the modelling of the paradigmatic musical concepts studied, creating possible software components of the "Computer-Assisted Composition" type that aim at the fast taxonomical identification of a harmonic combination and the survey of its properties and comparative interrelationships with other combinations.
\\
==== Bibliography ====
BARBOUR, J. M.. Tuning and Temperament: A Historical Survey. East Lansing: Michigan State College Press, 1951.\\
BITTENCOURT, Marcus Alessi. Sketches for the foundations of a contemporary experimental treatise on Harmony. In: Anais do II Encontro Internacional de Teoria e Análise Musical. São Paulo: UNESP-USP-UNICAMP, 2011.\\
BLACKBURN, Bonnie J..On Compositional Process in the Fifteenth Century. Journal of the American Musicological Society. Los Angeles: University of California Press, Vol. 40, No. 2. (Summer), pp. 210-284, 1987.\\
COSTÈRE, Edmond. Lois et Styles des Harmonies Musicales. Paris: Presses Universitaires de France, 1954.\\
FORTE, Allen. The Structure of Atonal Music. New Haven: Yale University Press, 1973.\\
HELMHOLTZ, H. L. F.. On the Sensations of Tone as a Physiological Basis for the Theory of Music. Translated by A. J. Ellis. 2nd edition. New York: Dover Publications, Inc., 1954 [1885].\\
HINDEMITH, Paul. The Craft of Musical Composition, Book 1: Theoretical Part. Associated Music Publishers, Inc., 1945.\\
HURON, D.. Tone and Voice: A Derivation of the Rules of Voice-Leading from Perceptual Principles. Music Perception. Los Angeles: University of California Press, Fall, Vol. 19, No. 1, pp. 1-64, 2001.\\
HUTCHINSON, W. and KNOPOFF, L.. The Acoustic Component of Western Consonance. Interface-Journal of New Music Research. Amsterdam: Routledge, Vol. 7, No. 1, pp. 1-29, 1978.\\
KAMEOKA, A. & KURIYAGAWA, M.. “Consonance theory, part I: Consonance of dyads.” Journal of the Acoustical Society of America. Vol. 45, No. 6, pp. 1451–1459, 1969.\\
KAMEOKA, A. & KURIYAGAWA, M.. “Consonance theory, part II: Consonance of complex tones and its computation method.” Journal of the Acoustical Society of America. Vol. 45, No. 6, pp. 1460–1469, 1969.\\
LEVY, Ernst. A Theory of Harmony. Albany: State University of New York Press, 1985.\\
MOORE, B. C. J.; PETERS, R. W.; GLASBERG, B. R.. Thresholds for the detection of inharmonicity in complex tones. Journal of The Acoustical Society of America. Melville: Acoustical Society of America, vol. 77, pp. 1861-1867, 1985.\\
OETTINGEN, Arthur von.. Harmoniesystem in Dualer Entwicklung. Leipzig: W. Glaser, 1866.\\
PARNCUTT. R.. Revision of Terhardt's Psychoacoustical Model of the Root(s) of a Musical Chord. Music Perception. Los Angeles: University of California Press, Vol. 6, No. 1 (Fall), pp. 65-93, 1988.\\
PARNCUTT. R..  Harmony: A Psychoacoustical Approach. Springer-Verlag, 1989.\\
PARNCUTT. R..  A model of the perceptual root(s) of a chord accounting for voicing and prevailing tonality. In: LEMAN, M. (Ed.). Music, Gestalt, and computing: Studies in cognitive and systematic musicology. Berlin: Springer-Verlag, 1997, pp. 181-199.\\
PARNCUTT. R..  Tonal implications of harmonic and melodic Tn-types. In T. Klouche & T. Noll (Eds.), Mathematics and computing in music (pp. 124-139). Berlin: Springer-Verlag, 2009.\\
PLOMP, R.; LEVELT, W. J. M.. “Tonal Consonance and Critical Bandwidth.” Journal of the Acoustical Society of America. Melville: Acoustical Society of America, Vol. 38, pp. 548-560, 1965.\\
RAHN, J.. Basic Atonal Theory. New York: Schirmer Books, 1980.\\
RIEMANN, Hugo. Harmony Simplified ; or, The theory of the tonal functions of chords. London: Augener & Co., 1903 [1893].\\
SETHARES, W. A.. Tuning, Timbre, Spectrum, Scale. London: Springer-Verlag, 1998.\\
VASSILAKIS, P. N.. Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance.  Doctoral Dissertation. Los Angeles: University of California, Los Angeles; Systematic Musicology, 2001.\\
ZARLINO, Gioseffo. Le Istitutioni Harmoniche. New York: Broude Brothers, 1965 [1571].\\
ZWICKER, E.. Subdivision of the audible frequency range into critical bands (frequenzgruppen). The Journal of the Acoustical Society of America 33 (2), p. 248, 1961.\\
\\
----