Plomp&Levelt vs. Zwicker&Fastl - Little evidence of conflict

["Donald D. Greenwood" <ddg@xxxxxxxxxx>]

Re: Questions and comments of Marc Leman and Bill Sethares concerning the
frequency separations of tones required to yield maximum dissonance or
roughness at 1000 Hz.  They refer to differences in the values obtained by
Plomp and Levelt (1965) and Zwicker and Fastl (1990) at 1 kHz.

I.

Marc Leman and Bill Sethares refer to values of 35 Hz and 70 Hz,
respectively, for the two papers above.  However, there appears to be less
discrepancy than would appear.  I cannot find my copy of P&L 1965, but I have
the full enlarged set of their figures, from which I have extracted values,
past and present (partly in response to the questions referenced above).

In whatever way 35 Hz may be mentioned in the P&L text, maximum roughness at
dyads geometrically centered at 1 kHz occurs in the median curve of
'consonance' in their 1 kHz figure at two identical minima - i.e. at the
delta-f values of ~42 Hz and ~62 Hz (no values between used).  In the curve
that plots the upper quartile values, the two low points are at ~31 and ~42
Hz.  In the lower quartile curve, the lowest point is at 31 Hz and a second
lesser low point is at 62 Hz.  The three curves are a little variable as
expected, but the clear visual center of the roughness trough between unison
and wider separations yielding consonance judgments is closer to 45 or 50 Hz
than 35 Hz (and the mean of those six values happens also to be 45 Hz).

Although the difference at issue is somewhat reduced by substituting a 45 or
50 Hz value that better describes the data, some attention needs also to be
paid to subject differences and variability.  Plomp and Steeneken provide
another experiment that provides data on those questions.

In Plomp and Steeneken (1968) more detailed information was obtained in
respect to the frequency separation required for maximal roughness, on which
greater reliance should be placed.  The data were obtained in experiments
where the lower tone was fixed and the upper tone increased in frequency -
and the subjects were asked to adjust frequency separation to (1) maximum
roughness and also (2) to the absence of interference, in a separate series
of adjustments.  Adjusting for (1) lead to judgments of separations that
corresponded reasonably well (but not without some differences) to the
troughs of P&Ls plots of "consonance" versus frequency separation, and
judgments of (2) lead to adjudged frequency separations that corresponded
quite well to the "shoulder peaks" of the same P&L curves.  Recall that the
two tones were separated around a geometric center frequency in Plomp and
Levelt, rather than upward from a lower tone as in P&S.

The 1968 P&S data were obtained from more subjects (20) under conditions of
improved stimulus delivery, which provided the two tones at a constant
loudness of 60 phons each, through earphones.  In the 1962 and 1965 work of
P&L the tones had been held at the same 65 dB SPL level just outside the ear
canal and perforce changed in level with respect to quiet threshold and
loudness as the lower tone moved lower (and the upper tone higher) in
frequency as they separated.

In the 1968 report of P&S, the 25th and 75th quartile values and median were
reported for the separations yielding the judgment of maximum roughness (and
also for the absence of interference).  Near 1 kHz, the lower quartile for
maximum roughness occurred at ~50 Hz, the median at ~63 Hz, and the upper
quartile at ~76 Hz.  Thus, the lower quartile of 50 Hz value (when tones of
1000 and 1050 Hz were used) is quite close to 45 or 50 Hz, which marks the
visual center of the roughness trough (the corresponding dyad for 45 Hz would
consist of tones of about 977.75 and 1022.75 Hz).

The Z&F value of 70 Hz (centered on a carrier of 1 kHz), which would be
expected to be about 72 Hz if centered at the slightly higher center
frequencies in the P&S experiment, is within the interquartile range of 50 Hz
to 76 Hz for the 20 subjects of P&S, i.e.between median and upper quartile.


Thus, the differences between P&L maximum roughness separations and P&S's
median separations do not seem unexpected considering variability among
subjects and the variability expected in small sample means (or medians) when
results from smaller groups of subjects are compared with those from groups
of 20.

The basic consistency of P&L and P&S data and the comparisons above indicate
that even more in the case of the small difference between Z&F and P&S the
explanation is likely to be a matter mainly of different samples of subjects
and sample sizes.  The difference between an AM complex and a two tone
stimulus may also be relevant, with a real effect - even if small effect, but
determining that would require testing both stimuli on the same (preferably
large) group of subjects.

II.

Concerning the significance of 70 Hz in Bill Hartman's reply, Plomp and
Steeneken's data at frequencies over 1 kHz are relevant (at least in so far
as two tone stimuli are concerned). The lower frequency members of the dyads
ranged from 125 to 8000 Hz.  For dyad pairs whose lower frequency was
constant at 2 kHz, the lower quartile for maximum roughness occurred at ~80
Hz, the median at ~127 Hz, and the upper quartile at ~155 Hz.  Median
judgments of the separations required for maximum roughness were wider than
127 Hz for all higher frequency dyads (though progressively smaller as CB
fractions or as ratios, re geometric center frequency).  Their data from 1
kHz to 8 kHz appear below. Something does happen above 2 kHz in these two
tone data, but the median maximum roughness judgments do not limit at a 70 Hz
frequency separation.

Plomp and Steeneken's frequency separations yielding maximum roughness,
for dyads with lower frequencies at and above 1 kHz, as extracted carefully
from their graph are about:

25th Quartile settings:

1000  50
1400  64
2000  80
2800  80
4000  107
5600  73
8000  115

Median settings:

1000  63            ratio (delta-F/Fgeom) = 0.061
1400  91            ratio (delta-F/Fgeom) = 0.063
2000  127          ratio (delta-F/Fgeom) = 0.062
2800  127          ratio (delta-F/Fgeom) = 0.044
4000  133          ratio (delta-F/Fgeom) = 0.033
5600  167          ratio (delta-F/Fgeom) = 0.029
8000  179          ratio (delta-F/Fgeom) = 0.022

75 Quartile settings:

1000  76
1400  117
2000  155
2800  165
4000  185
5600  200
8000  266

I hope this consultation of the data helps to answer the original question.
There seems to be little evidence of conflict and the usual evidence of
inter-subject differences.

Donald D. Greenwood

Greenwood, D.D. (1990)  A cochlear frequency-position function for several
species - 29 years later. J. Acoust. Soc. Am. 87, 2592-2605.
Greenwood, D.D. (1991)  Critical bandwidth and consonance in relation to
cochlear frequency-position coordinates, Hear. Res. 54, 164-208.
Plomp, R. and Levelt, W.J.M. (1965)  Tonal consonance and critical
bandwidth.  J. Acoust. Soc. Am. 37, 548-560.
Plomp R. and Steeneken, H.J.M. (1968)  Interference between two simple
tones.  J. Acoust. Soc. Am. 43, 883-884