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Re: Frequency to Mel Formula

On 23 July, Dick Lyon began his email to the list with the statement “ I would like to understand the history of the Mel scale, formulas for it, and its relation to other scales;”.   

To the best of my knowledge there have been a total of three Mel scale studies and only two of these were based on half-pitch judgments, unlike the Sone scale that has involved many studies (almost all using half-loudness judgments).

Stevens, Volkmann, and Newman (1937) reported the first Mel scale study using half-pitch judgments for standards from 125 to 12,000 Hz.  In the next study, Stevens and Volkmann (1940) bisected intervals between higher and lower frequencies over three ranges from 40 to 1,000, 200 to 6,500, and 300 to 12,000 Hz.  The third study by Siegel (1965) obtained half-pitch judgments for standard frequencies from 92 to 9,200 Hz. 
The two half-pitch studies are in quite good agreement, while the bisection procedure differed from the other two.  This discussion will concentrate on the data obtained in the half-pitch studies.
In Wever’s book, Theory of Hearing (1949, pages 338-339), he stated that the data of Stevens et al. indicated that for low tones, which Wever considered to be frequencies up to about 5 kHz, “two intervals of pitch that seem equal will embrace the same numbers of cycles.”  In other words, within this range according to Wever, half pitch was equal to half frequency.  While this is only approximate, Siegel’s data are closer to this octave relation.  Since his study is not generally known, the data he reported are given as follows, with the standard frequencies employed followed by the percent judged half in parentheses:  92 Hz (60%), 165 (54%), 510 (50%), 920 (46%), 1650 (48%), 5100 (47%), 9200 (43%).  [If anyone wants a copy of a table showing the half-pitch frequencies along with a figure published by Siegel that compares his and Stevens, Volkmann, and Newman’s data, I will be pleased to send them a PDF file].

It can be seen that when both standard and comparison frequencies lie within the familiar range of orchestral instruments, half pitch appears to be approximately half frequency.  A basis for this relation may be found in Helmholtz’s statement that “The musical scale is at it were the divided rod, by which we measure progressions in pitch, as rhythm measures progression in time.” (1954, page 252).  He pointed out that the octave is the basic unit of musical scales and that this interval is employed in the music of virtually all cultures.  He also stated that the octave is the only interval that is completely consonant:  The harmonics of a higher octave are also harmonics of a lower one, so that they blend perfectly with no discordant beats.  A discussion concerning the use of the octave as a possible basis for half-pitch judgments is discussed more fully in a chapter by Warren (2008).  

Incidentally, on 23 July, Don Greenwood’s email (using the Re: Auditory Digest thread) stated that Mel studies by Stevens and his colleagues appeared to be biased (as he indicated, he had worked on the Mel scale in Stevens’s lab).  He went on to say that “If anyone wants a Mel scale they should do it over, controlling carefully for order bias and using plenty of subjects - more than in the past - and using both musicians and non-musicians…”.

If anyone does intend to do this, I would suggest the single-judgment procedure which avoids experimental biases (and certainly uses plenty of subjects) as used for half-loudness judgments and described by Warren (2008) in the chapter dealing with both the Sone and Mel scales. 

As to the use of musicians as participants, it might be relevant that in their half-pitch experiment, Stevens et al. reported that one of their five participants was a trained musician: he “reported an inability to disregard octaves and other musical intervals when setting the second tone at half the pitch of the first.”  There is more direct evidence involving other sensory scales indicating that, participants having special familiarity with a physical scale associated with the stimulus, use that scale in judging half-sensory magnitude.  Thus, it has been reported from three different labs that listeners having experience using sound-level meters could not avoid using half decibels as half loudness, and they could not be used as participants (Ham & Parkinson, 1932; Laird, Taylor & Wille, 1932; and Rowley & Studebaker, 1969).  For example, 30 dB would seem to be half as loud as 60 dB even when trying consciously to ignore the physical scale.  A similar observation was reported by Warren and Warren (1958) in their study of half- brightness judgments of visual fields.  Undergraduate participants chose one-quarter luminance as half brightness.  However, an individual working in visual physiology, who had considerable experience using an illuminometer, chose one-half luminance even though he tried to judge subjective rather than physical magnitude.

Ham, L.B., and Parkinson, J.S.  1932.  Loudness and intensity relations,” Journal of the Acoustical Society of America, 3, 511-534.  

Helmholtz, H.L.F.  1954.  On the Sensations of Tone as a Physiological Basis for the Theory of Music.  New York:  Dover, 1954.  (Reprint of 2nd English edition of 1885, A.J. Ellis, translator, based upon the 3rd German edition (1870) and rendered conformal with the 4th German edition (1877)).  

Laird, D.A., Taylor, E., Wille, H.H., Jr.  1932.  “The apparent reduction of loudness,” Journal of the Acoustical Society of America, 3, 393-401.  

Rowley, R.R., and Studebaker, G.A.  1969.  “Monaural loudness-intensity relationships for a 1,000-Hz tone,” Journal of the Acoustical Society of America, 45, 1186-1192.  

Siegel, R.J.  1965.  “A Replication of the Mel Scale of Pitch,” American Journal of Psychology, 78 (4), 615-620.  

Stevens, S.S., and Volkmann, J. 1940.  “The Relation of Pitch to Frequency: A Revised Scale,” American Journal of Psychology, 53 (3), 329-353.  

Stevens, S.S., Volkmann, J., and Newman, E.B.  1937.  “A scale for the measurement of the psychological magnitude of pitch,” Journal of the Acoustical Society of America, 8, 185-190.  

Warren, R.M.  2008.  Auditory Perception:  An Analysis and Synthesis.  Cambridge, UK: Cambridge University Press (see Chapter 4 “Judging Auditory Magnitude:  The Sone Scale of Loudness and the Mel Scale of Pitch,” pp. 107-125).  

Warren, R.M., and Warren, R.P.  1958.  “Basis for judgments of relative brightness,” Journal of the Optical Society of America, 48, 445-450.  

Wever, E.G.  1949. Theory of Hearing.  New York:  Wiley.

Dick Warren

Richard M. Warren 
Research Professor 
  and Distinguished Professor Emeritus
Department of Psychology
University of Wisconsin-Milwaukee
PO Box 413
Milwaukee, WI  53201