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Re: two cycles

An illuminating contribution to understanding this paradox (paradox because accurate frequency measurement of a short stimulus seems to contradict the time-frequency tradeoff rules of Heisenberg/Gabor) is that of J.O. Nordmark (1968, 1970), who credits Kneser (1948) for pointing out that we use the same word 'frequency' for two distinct concepts (see quotation below).

Sorry for the plug, but I'd also recommend my chapter on 'Pitch perception models' in the recent book on pitch by Plack, Oxenham, Fay and Popper. It addresses many such issues related to pitch perception and periodicity analysis.

A final word of caution: the fact that subjects agree to report two-period sinusoidal stimuli as 'pitch-like' does not imply that they are actually measuring the period. The fact that they can perform accurate frequency discrimination on such stimuli also does not imply that the discrimination is based on pitch as determined by measurement of the fundamental period. They could be using other cues (such as timbre) that co-vary with frequency. To narrow the interpretation down to pitch (or abstraction of fundamental frequency from other possibly covarying cues) might require comparing pitch between complex stimuli that differ in timbre across intervals. I'm not sure if this has been done.


Author = {Nordmark, J.O.},
Title = {Mechanisms of frequency discrimination},
Journal = {J. Acoust. Soc. Am.},
Volume = {44},
Pages = {1533-1540},
Note = {"...Kneser (1948) was the first to point out that there are two separate
meanings of frequency in common use, each measuring and defining
different physical quantities. The first of these,
phase frequency, he defines as the reciprocal of the time interval between
two events of equal phase. The accuracy with which this can be determined
is, in principle, independent of the duration of the waveform under
investigation and therefore not subject to the uncertainty relation
between time and frequency. The second, group frequency, is the concept
used in the Fourier analysis of periodic waves. For a time function of
limited duration, an analysis will yield a series of sine and cosine waves
grouped around the phase frequency. No exact value can be given the group
frequency, which is thus subject to the uncertainty relation."},
Year = {1968} }

   Author = {Nordmark, J.O.},
   Title = {Time and frequency analysis},
   BookTitle = {Foundations of modern auditory theory},
   Editor = {Tobias, J.V.},
   Publisher = {Academic Press},
   Address = {New York},
   Pages = {55-83},
 Year = {1970} }

   Author = {Kneser},
   Title = {Bemerkungen über Definition und Messung der Frequenz},
   Journal = {Archiv der Elektrishen Übertragung},
   Volume = {2},
   Pages = {167-169},
 Year = {1948} }

   Author = {de Cheveigné, A.},
   Title = {Pitch perception models},
   BookTitle = {Pitch - Neural coding and perception},
   Editor = {Plack, C.J. and Oxenham, A. and Fay, R.R. and Popper, A.N.},
   Publisher = {Springer},
   Address = {New York},
   Pages = {169-233},
 Year = {2005} }

Hi list,

In many of the papers published by Georg von Békésy he makes the statement that the fundamental frequency was determined by the auditory system "even when the stimulus was only two cycles" in length. In at least one of his publications {The Missing Fundamental and Periodicity Detection in Hearing JASA 1972 512) 631-637) he attributes this to his own experiments and to a paper by Savart.[Annalen der Physik und Chemie 1840 53 ( ) 555-561 in german]. It is quite true that Savart found that the fundamental was determined in two cycles but it was published in an earlier paper by Savart (Ueber die Empfindlichkeit des Gehörorgans Felix Savart Annalen der Physik und Chemie 1830 20( ) 290-304 in german)and no mention of the two cycles is mentioned in the citation by Békésy. Actually from a historical point of view the original paper by Felix Savart was published in French in: Annales de chimie et de Physique 1830 44 ( ) 337-352 in French.

Can anyone point me to literature which shows how the auditory system performs the "two cycle" feat, or to papers which show how such a "two cycle" feat might be acomplished mathematically or to any papers of a more recent origin which discusses this ability.


Fred Herzfeld, MIT '54
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Brunswick, Ga.31525