Subject: Plomp&Levelt vs. Zwicker&Fastl - Little evidence of conflict From: "Donald D. Greenwood" <ddg(at)neteze.com> Date: Tue, 5 Oct 1999 14:32:04 -0700Re: 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