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Re: critical bands and dissonance

Regarding Marc Leman and Bill Sethares question about critical
bands and sensory dissonance, let me offer the following mini-history.

     The story begins with the work of Harvey Fletcher who linked the
frequency-place coordinates of the Bekesy cochlear map to experimental
data from frequency discrimination and masking experiments (Fletcher,
1953; pp. 168-175).  Fletcher showed that there is a close correspondence
between distances along the basilar membrane and regions of masking.
In pursuing this research, Fletcher defined a hypothetical entity dubbed the
"critical band" to denote frequency-domain regions of roughly equivalent or
proportional behavior (Fletcher, 1940).  Subsequent research by Zwicker
and others established critical bandwidths as an empirical rather than
hypothetical construct.  Most notably, Zwicker, Flottorp, and Stevens (1957)
showed that distance along the basilar membrane accounts for changes in
cumulative loudness as a function of the overall frequency spread of
several tones or a band of noise (see also Scharf, 1961).

     Greenwood (1961b, 1990) extended Fletcher's work by comparing
psychoacoustic measures of critical bandwidth with the frequency-place
coordinates of the Bekesy-Skarstein cochlear map.  Greenwood showed that
there is a linear relationship, with one critical bandwidth being roughly
equivalent to the distance of 1.0 millimeter on the basilar membrane

     Greenwood (1961b) went on to suggest that tonotopic effects might also
account for the perception of sensory dissonance.  Greenwood tested this
hypothesis by comparing perceptual data collected by Mayer (1894) against
the critical bandwidth/cochlear map.  Mayer had collected experimental
data where listeners were instructed to identify the smallest possible
interval free of roughness or dissonance.  This interval is not constant
with respect to log frequency -- as implied in traditional music theory.
Greenwood showed that Mayer's data correspond almost precisely with
changes of critical bandwidth with respect to frequency.

     Plomp and Levelt (1965) extended Greenwood's work linking the
perception of sensory dissonance ("tonal consonance") to the
critical band -- and hence to the mechanics of the basilar membrane.
Plomp and Levelt estimated that pure tones produce maximum sensory
dissonance when they are separated by about 25% of a critical bandwidth.
However, their estiamte was based on a critical bandwidth that is now
considered to be excessively large, especially below about 500 Hz.
Greenwood (1991) has estimated that maximum dissonance arises when
pure tones are separated by about 40% of a critical bandwidth.

     Regretably, the music perception community has overlooked the
seminal work of Donald Greenwood and has misattributed the origin
of the tonal-consonance/critical-band hypothesis to Drs. Plomp and Levelt
(see Greenwood, 1961b; especially pp.1351-1352).  In addition, the
auditory community in general overlooked Greenwood's accurate early
characterization of the size of the critical band.  After the passage of
three decades, the revised ERB function (Glasberg and Moore, 1990) was
virtually identical to Greenwood's 1961 equation -- as acknowledged by
Glasberg and Moore.  A good survey of the relation of consonance and
critical bandwidth to cochlear resolution can be found in Greenwood (1991).

     Subsequent work by my students has drawn attention to physiological
correlates of sensory dissonance (Simpson, 1994) and errors in sensory
dissonance formulae proposed by Kameoka and Kuriyagawa and Hutchinson
and Knopoff (Mashinter, 1995).  Work on sensory dissonance is continuing
in my Ohio State lab.  An up-to-date bibliography can be found at

David Huron



Glasberg, B. R., & Moore, B. C. J.  (1990).
   Derivation of auditory filter shapes from notched noise data.
   Hearing Research, 47, 103-138.

Greenwood, D. D.  (1961a).
   Auditory masking and the critical band.  Journal of the Acoustical
   Society of America, 33 (4), 484-502.

Greenwood, D. D.  (1961b).
   Critical bandwidth and the frequency coordinates of the basilar membrane.
   Journal of the Acoustical Society of America, 33 (4), 1344-1356.

Greenwood, D. D.  (1990).
   A cochlear frequency-position function for several species -- 29 years
   later.  Journal of the Acoustical Society of America, 87 (6), 2592-2605.

Greenwood, D. D.  (1991).
   Critical bandwidth and consonance in relation to cochlear frequency-position
   coordinates.  Hearing Research, 54 (2), 164-208.

Mashinter, K. (1995).
   Discrepancies in theories of sensory dissonance arising from the models
   of Kameoka & Kuriyagawa and Hutchinson & Knopoff. Bachelor of Applied
   Mathematics thesis, University of Waterloo.

Plomp, R., & Levelt, W. J. M.  (1965).
   Tonal consonance and critical bandwidth.  Journal of the Acoustical
   Society of America, 37, 548-560.

Plomp, R., & Steeneken, H. J. M.  (1968).
   Interference between two simple tones.  Journal of the Acoustical
   Society of America, 43, 883-884.

Simpson, J. (1994).
   Cochlear modeling of sensory dissonance and chord roots. Master of
   Applied Science thesis, University of Waterloo.