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The Computation of Pitch with Vectors

Journal of the Brazilian Computer Society volume 14, pages 65–85 (2008)Cite this article

Abstract

A pitch model is proposed which is supported by a vector representation of tones. First, an algorithm capable of performing the vector addition of the spectral components of two-tone harmonic complexes is introduced which initially converts the amplitude, frequency, and phase (AFP) parameters into coordinates of the here introduced quotient, distance in octaves, and loudness (QOL) tone space. As QOL is isomorphic to the hue, saturation, and value (HSV) color space, a transformation from QOL to the red, green, and blue (RGB) vector space can be formulated so that the vector addition of two pure tones is conceived by analogy with color mixing operations. Since the QOL to RGB transformation is invertible, the resulting RGB vector sum can be transformed back to QOL. Then, by converting QOL coordinates back to AFP parameters, a tone is found whose frequency supposedly corresponds to the pitch evoked by the original two-tone complex. As for complexes having more than two components, the algorithm is to be sequentially applied to pairs of vectors in such a way that initially the first two vector tones are added together, then the resulting vector is added to the third vector tone, and so on.

References

  1. A. Arcela.As árvores de tempos e a configuração genética dos intervalos musicais. PhD thesis, Pontifical Catholic University of Rio de Janeiro, 1984.

  2. A. Arcela. The equilibrium theorem in Bach’s two-part inventions: an audible demonstration. http://www.cic.unb.br/docentes/arcela/equilibrium/, January 2008.

  3. A. Arcela. Out of tune: audible demonstration of the vector-addition tone in the computation of the pitch of some multitone complexes. http://www.cic.unb.br/docentes/arcela/outoftune/, January 2008.

  4. A. Bachem. Tone height and tone chroma as two different pitch qualities.Acta Psychol., 5:80–88, 1950.

    Article  Google Scholar 

  5. J. F. Schouten; R. J. Ritsma; B. Lopes Cardozo. Pitch of the residue.J. Acoust. Soc. Am., 34:1418–1424, 1962.

    Article  Google Scholar 

  6. E. de Boer.On the “residue” in hearing. PhD thesis, University of Amsterdam, 1956.

  7. A. J. M. Houtsma; J. L. Goldstein. The central origin of the pitch of complex tones: evidence from musical interval recognition.J. Acoust. Soc. Am., 69:520–529, 1972.

    Article  Google Scholar 

  8. J. L. Goldstein. An optimum processor theory for the central formation of the pitch of complex tones.J. Acoust. Soc. Am., 54:1496–1516, 1973.

    Article  Google Scholar 

  9. R. Meddis; M. J. Hewitt. Virtual pitch and phase sensitivity of a computer model of the auditory periphery. I: Pitch identification.J. Acoust. Soc. Am., 89:2866–2882, 1991.

    Article  Google Scholar 

  10. A. J. M. Houtsma. Musical pitch of two-tone complexes and predictions by modern pitch theories.J. Acoust. Soc. Am., 66:87–99, 1979.

    Article  Google Scholar 

  11. A. J. M. Houtsma. Pitch of unequal-amplitude dichotic two-tone harmonic complexes.J. Acoust. Soc. Am., 69:1778–1785, 1981.

    Article  Google Scholar 

  12. D. Pressnitzer; R. D. Patterson; K. Krumbholz. The lower limit of melodic pitch.J. Acoust. Soc. Am., 109:2074–2084, 2001.

    Article  Google Scholar 

  13. J. C. R. Licklider. A duplex theory of pitch perception.J. Acoust. Soc. Am., 7:128–133, 1951.

    Google Scholar 

  14. A. H. Munsell. A pigment color system and notation.The American Journal of Psychology, 23:236–244, 1912.

    Article  Google Scholar 

  15. H. Fletcher; W. A. Munson. Loudness, its definition, measurement and calculation.J. Acoust. Soc. Am., 5:82–108, 1933.

    Article  Google Scholar 

  16. G. S. Ohm. Ueber die definition des tones, nebst daran geknüpfter theorie der sirene.Ann. Phys. Chem., 59:513–565, 1843.

    Article  Google Scholar 

  17. R. Meddis; L. O’Mard. Virtual pitch in a computational physiological model.J. Acoust. Soc. Am., 120:3861–3869, 2006.

    Article  Google Scholar 

  18. J. G. Bernstein; A. J. Oxenham. An autocorrelation model with place dependence to account for the effect of harmonic number on fundamental frequency discrimination.J. Acoust. Soc. Am., 117:2816–3831, 2005.

    Article  Google Scholar 

  19. R. Plomp. Pitch of complex tones.J. Acoust. Soc. Am., 41:1526–1533, 1967.

    Article  Google Scholar 

  20. E. Zwicker; B. Scharf. A model of loudness summation.Psychological Review, 72:3–26, 1965.

    Article  Google Scholar 

  21. A. Seebeck. Beobachtungen über einige bedingungen der entstehung von tönen.Ann. Phys. Chem., 53:417–436, 1841.

    Article  Google Scholar 

  22. W. P. Shofner; G. Selas. Pitch strength and Stevens’s power law.Perception & Psychophysics, 64:437–450, 2002.

    Article  Google Scholar 

  23. R. N. Shepard. Structural representations of musical pitch. In D. Deutsch, editor,The Psychology of Music, chapter 11, pages 343–390. Academic, Orlando, 1982.

    Google Scholar 

  24. A. R. Smith. Color gamut transform pairs.ACM SIGGRAPH, Computer Graphics, 12:12–19, 1978.

    Article  Google Scholar 

  25. G. F. Smoorenburg. Pitch perception of twofrequency stimuli.J. Acoust. Soc. Am., 48:924–942, 1970.

    Article  Google Scholar 

  26. S. S. Stevens. The measurement of loudness.J. Acoust. Soc. Am., 27:815–829, 1955.

    Article  Google Scholar 

  27. S. S. Stevens. On the validity of the loudness scale.J. Acoust. Soc. Am., 31:995–1003, 1959.

    Article  Google Scholar 

  28. H. Fastl; G. Stoll. Scaling of pitch strength.Hearing Research, 119:293–301, 1979.

    Article  Google Scholar 

  29. E. Terhardt. Pitch, consonance and harmony.J. Acoust. Soc. Am., 55:1061–1069, 1974.

    Article  Google Scholar 

  30. J. D. Foley; A. van Dam; S. K. Feiner; J. F. Hughes.Computer Graphics: principles and practice, chapter 13, pages 343–363. Addison-Wesley, second edition, 1990.

  31. H. L. F. von Helmholtz.On the Sensations of Tone, chapter 4, pages 58–65. Dover (English translation A. J. Ellis, 1885, 1954), 1877.

  32. F. L. Wightman. The pattern-transformation model of pitch.J. Acoust. Soc. Am., 54:407–416, 1973.

    Article  Google Scholar 

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  1. Department of Computer Science, University of Brasilia Campus Universitário, Asa Norte, Zip 70910-900, Brasilia, DF, Brazil

    Aluizio Arcela

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Arcela, A. The Computation of Pitch with Vectors. J Braz Comp Soc 14, 65–85 (2008). https://doi.org/10.1007/BF03192565

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