Attention, Perception, & Psychophysics

, Volume 76, Issue 8, pp 2508–2521 | Cite as

Fractionation of pitch intervals: An axiomatic study testing monotonicity, commutativity, and multiplicativity in musicians and non-musicians

  • Florian KattnerEmail author
  • Wolfgang Ellermeier


Direct ratio scaling (e.g., magnitude estimation; Stevens, 1956, American Journal of Psychology) is a widely used approach in psychophysics resting on the assumption that participants are able to refer to sensations with numbers that are valid on a ratio scale. Only recently, the necessary conditions of commutativity and multiplicativity have been formulated (Narens, 1996, Journal of Mathematical Psychology) and tested empirically, e.g., for the sensation of loudness. The present investigation evaluated these properties for the ratio production of pitch intervals. Musically trained (n = 10) and untrained (n = 11) participants adjusted intervals defined by two ascending pure-tone frequencies to given fractions (1/3, 1/2, 2/3), starting either from a 12 or 17-semitone standard pitch interval. The results show that the axioms of commutativity and multiplicativity held for most of the participants, irrespective of musical training. Furthermore, all participants produced larger frequency intervals in response to larger ratio numbers used in the instructions (monotonicity), but only musically trained participants were sensitive to the size of the standard interval (thus producing strictly increasing magnitudes). Overall, the results indicate that pitch intervals are ratio-scalable. However, restrictions must be made, especially for non-musical listeners, and when an octave is exceeded.


Direct scaling Ratio production Pitch Commutativity Multiplicativity Monotonicity 


  1. ANSI. (1973). American national psychoacoustical terminology - S3.2. American National Standards Institute (ANSI), New YorkGoogle Scholar
  2. Augustin, T. (2006). Stevens’ direct scaling methods and the uniqueness problem. Psychometrika, 71, 469–481CrossRefGoogle Scholar
  3. Augustin, T., & Maier, K. (2008). Empirical evaluation of the axioms of multiplicativity, commutativity, and monotonicity in ratio production of area. Acta Psychologica, 129, 208–216PubMedCrossRefGoogle Scholar
  4. Beck, J., & Shaw, W.A. (1961). The scaling of pitch by the method of magnitude estimation. American Journal of Psychology, 74, 242–251PubMedCrossRefGoogle Scholar
  5. Beck, J. (1962). Magnitude estimations of pitch. Journal of the Acoustical Society of America, 34, 92–98CrossRefGoogle Scholar
  6. Brainard, D.H. (1997). The psychophysics toolbox. Spatial Vision, 10, 443-446CrossRefGoogle Scholar
  7. Cohen, J., & Hansel, C.E.M., Sylvester, J.D. (1954). Interdependence of temporal and auditory judgments. Nature, 174, 642PubMedCrossRefGoogle Scholar
  8. Deutsch, D. (1969). Music recognition. Psychological Review, 76, 300–307PubMedCrossRefGoogle Scholar
  9. Ellermeier, W., & Faulhammer, G. (2000). Empirical evaluation of axioms fundamental to Stevens’s ratio-scaling approach: I. Loudness production. Perception and Psychophysics, 62, 1505–1511PubMedCrossRefGoogle Scholar
  10. Fletcher, H., & Munson, W.A. (1933). Loudness, its definition, measurement, and calculation. Journal of the Acoustical Society of America, 5, 82–108CrossRefGoogle Scholar
  11. Garner, W.R., & Hakem, H.W. (1951). The amount of information in absolute judgments. Psychological Review, 58, 446–459PubMedCrossRefGoogle Scholar
  12. Gescheider, G.A. (1997). Psychphysics: The fundamentals, 3rd edn. Lawrence Erlbaum, New JerseyGoogle Scholar
  13. Greenwood, D.D. (1997). The Mel Scale’s disqualifying bias and a consistency of pitch-difference equisections in 1956 with equal cochlear distances and equal frequency ratios. Hearing Research, 103, 199–224PubMedCrossRefGoogle Scholar
  14. Idson, W.L., & Massaro, D.A. (1978). Bidimensional model of pitch in the recognition of melodies. Perception and Psychophysics, 24, 551–565PubMedCrossRefGoogle Scholar
  15. ISO (1998). Acoustics - Reference zero for the calibration of audiometric equipment - Part 1: Reference equivalent threshold sound pressure levels for pure tones and supraaural earphones (ISO 389-1). International Organization for Standardization (ISO), GenevaGoogle Scholar
  16. Krumhansl, C.L., & Shepard, R.N. (1979). Quantification of the hierarchy of tonal functions within a diatonic context. Journal of Experimental Psychology: Human Perception and Performance, 5, 579–594PubMedGoogle Scholar
  17. Luce, R.D. (2002). A psychophysical theory of intensity proportions, joint presentations, and matches. Psychological Review, 109, 520-532CrossRefGoogle Scholar
  18. Luce, R.D. (2004). Symmetric and asymmetric matching of joint presentations. Psychological Review, 111, 446–454PubMedCrossRefGoogle Scholar
  19. McKenna, F.P. (1985). Another look at the “new psychophysics”. British Journal of Psychology, 76, 109–129CrossRefGoogle Scholar
  20. Merkel, J. (1888). Die Anhängigkeit zwischen Reiz und Empfindung. Philosophische Studien, 4, 541–594Google Scholar
  21. Narens, L. (1996). A theory of ratio magnitude estimation. Journal of Mathematical Psychology, 40, 109-129CrossRefGoogle Scholar
  22. Narens, L. (2002). The irony of measurement by subjective estimations. Journal of Mathematical Psychology, 46, 769–788CrossRefGoogle Scholar
  23. Painton, S.W., & Cullinan, W.L., Mencke, E.O. (1977). Individual pitch functions and pitch-duration cross-dimensional matching. Perception and Psychophysics, 21, 469–476CrossRefGoogle Scholar
  24. Parker, S., & Schneider, B. (1974). Nonmetric scaling of loudness and pitch using similarity and difference estimates. Perception and Psychophysics, 15, 238–242CrossRefGoogle Scholar
  25. Peißner, M. (1999). Experimente zur direkten Skalierbarkeit von gesehenen Helligkeiten (Unpublished master’s thesis). Universität RegensburgGoogle Scholar
  26. Pelli, D.G. (1997). The VideoToolbox software for visual psychophysics: Transforming numbers into movies. Spatial Vision, 10, 437-442PubMedCrossRefGoogle Scholar
  27. Plack, C.J., Oxenhamm, A.J. (2005). Psychophysics of pitch. In A. N. Popper & R. Fay (Eds.), Pitch: Neural coding and perception (pp. 7–55). New York: SpringerGoogle Scholar
  28. Plomp, R. (1967). Pitch of complex tones. Journal of the Acoustical Society of America, 41, 1526–1533PubMedCrossRefGoogle Scholar
  29. Rasch, R., & Plomp, R. (1999). The psychology of music. In D. Deutsch (Ed.), (2nd ed., pp. 89-112). New York: Academic PressGoogle Scholar
  30. Richardson, L.F. (1929). Imagery, conation, and cerebral conductance. Journal of General Psychology, 2, 324–352CrossRefGoogle Scholar
  31. Russo, F.A., & Thompson, W.F. (2005). The subjective size of melodic intervals over a two-octave range. Psychonomic Bulletin and Review, 12, 1068–1075PubMedCrossRefGoogle Scholar
  32. Shepard, R.N. (1964). Circularity of judgments of relative pitch. Journal of the Acoustical Society of America, 36, 2346–2353CrossRefGoogle Scholar
  33. Shepard, R.N. (1981). Psychological relations and psychophysical scales: On the status of “direct” psychological measurement. Journal of Mathematical Psychology, 24, 21–57CrossRefGoogle Scholar
  34. Siegel, J.A. (1964). A replication of the mel scale of pitch. American Journal of Psychology, 78, 615–620CrossRefGoogle Scholar
  35. Steingrimsson, R., & Luce, D.R. (2007). Empirical evaluation of a model of global psychophysical judgments: IV. Forms for the weighting function. Journal of Mathematical Psychology, 51, 29–44CrossRefGoogle Scholar
  36. Steingrimsson, R., & Luce, R.D. (2005). Evaluating a model of global psychophysical judgments - II: Behavioral properties linking summations and productions. Journal of Mathematical Psychology, 49, 308-319CrossRefGoogle Scholar
  37. Stevens, S.S. (1946). On the theory of scales of measurement. Science, 103, 677–680CrossRefGoogle Scholar
  38. Stevens, S.S. (1956). The direct estimation of sensory magnitudes - loudness. American Journal of Psychology, 69, 1–25.Google Scholar
  39. Stevens, S.S. (1957). On the psychophysical law. Psychological Review, 65, 153–181CrossRefGoogle Scholar
  40. Stevens, S.S. (1971). Issues in psychophysical measurement. Psychological Review, 78, 426–450CrossRefGoogle Scholar
  41. Stevens, S.S. (1975). Psychophysics: Introduction to its perceptual, neural and social prospects. Wiley, New YorkGoogle Scholar
  42. Stevens, S.S., & Galanter, E.H. (1957). Ratio scales and category scales for a dozen perceptual continua. Journal of Experimental Psychology, 54, 377–411PubMedCrossRefGoogle Scholar
  43. Stevens, S.S., & Volkman, J. (1940). The relation of pitch to frequency: A revised scale. American Journal of Psychology, 53, 329–353CrossRefGoogle Scholar
  44. Stevens, S.S., Volkman, J., Newman, E.B. (1937). A scale for the measurement of the psychological magnitude pitch. Journal of the Acoustical Society of America, 8, 185–190Google Scholar
  45. Young, R.W. (1939). Terminology for logarithmic frequency units. Journal of the Acoustical Society of America, 11, 134–139.Google Scholar
  46. Zimmer, K. (2005). Examining the validity of numerical ratios in loudness fractionation. Perception and Psychophysics, 67, 569–579PubMedCrossRefGoogle Scholar

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© Psychonomic Society, Inc. 2014

Authors and Affiliations

  1. 1.Technische Universität DarmstadtDarmstadtGermany
  2. 2.Institute of PsychologyTechnische Universität DarmstadtDarmstadtGermany

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