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The Computation of Pitch with Vectors

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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. [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. [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. [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. [4]

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

  5. [5]

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

  6. [6]

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

  7. [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.

  8. [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.

  9. [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.

  10. [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.

  11. [11]

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

  12. [12]

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

  13. [13]

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

  14. [14]

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

  15. [15]

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

  16. [16]

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

  17. [17]

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

  18. [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.

  19. [19]

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

  20. [20]

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

  21. [21]

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

  22. [22]

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

  23. [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.

  24. [24]

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

  25. [25]

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

  26. [26]

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

  27. [27]

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

  28. [28]

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

  29. [29]

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

  30. [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. [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. [32]

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

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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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Keywords

  • pitch computation
  • vector representation of tones
  • two-tone complexes
  • missing undamental