source file: mills2.txt Date: Tue, 16 Jul 1996 07:27:30 -0700 Subject: Post from McLaren From: John Chalmers From: mclaren Subject: Paul Erlich's points about inharmonic tone complexes - post 1 of 2 -- John Chalmers informs me that some folks object to the idea of matching inharmonic timbres to inharmonically-derived tunings. Apparently, your contention is that the inharmonic tone complexes tones will be heard as having a virtual pitch entirely different from that of their component inharmonic partials. Paul Erlich seems typical in these respects. On the surface, this sounds like a good argument. More: Erlich apparently claims that inharmonic tone complex will fail to exhibit spectral fusion and will fall apart into a set of discrete partials if they're highly inharmonic. Thus it would seem pointless to attempt to match a non-just non-equal- tempered tuning to a n-j n-e-t timbre, since the resultant inharmonic timbre will never be heard as having a fundamental pitch related to the pitches of the n-j n-e-t scale from which it is drawn. Is this claim correct? It turns out that it isn't. Paul Erlich strikes me as a smart fellow with some knowledge of acoustics and pyschoacoustics. Alas, as a newly-minted graduate much of Paul's knowledge is book learning only. And, as it turns out, much of what has been written about complexes of inharmonic tones is either wrong, irrelevant to xenharmonic composition, or the result of peculiar and non-musical experimental conditions. This issue is important. James Dashow, Jean-Claude Risset, William Sethares, myself, and a number of other composers have produced a substantial body of compositional work in which non-just non-equal- tempered timbres and tunings are used. If Paul's claims (and some of the available psychoacoustic literature about inharmonic tones) are correct, then the theoretical and musical justification for much of this compositional work collapses. As will be seen, however, there is a great deal of evidence *against* Paul Erlich's view, and many of the psychoacoustic papers which purport to "prove" that inharmonic tone complexes do not audibly fuse are lethally flawed. Let us begin with the alleged evidence *against* the coherent audibility of inharmonic tone complexes: Rayleigh, in his 1876 2-volume text "Acoustics," points out that the fundamental pitch of orchestral timpani is a frequency not present in the partials produced by this instrument. Rayleigh explained this by pointing out that the ear hears some of the upper partials of timpani as belonging to a fragment of an harmonic series; the ear extrapolates from this harmonic-series fragment a lower pitch which is heard as the fundamental though not physically present. Other partials, which are physically present, are heard as "hum" notes which do not contribute to the sense of musical pitch of the note. Rayleigh noted the same phenomenon with regard to bells: here again, the lowest several inharmonic partials are heard as "hum" notes, but do not make musical contribution to the sense of pitch of the bell. 65 years later Schouten demonstrated in an elegant experiment that the human ear often heard a fundamental pitch not physically present in a musical sound. "He had constructed a sort of optical siren (Figure 6-4) by means of which he could produce sounds with various waveforms. Using this, he produced sounds with harmonically related partials. (..) Then, by proper adjustments, he could cancel out the fundamental frequency... I could hear this fundamental frequency come and go, but the pitch of the sound did not change at all. In some way, my ear inferred the proper pitch form the harmonics..." [Pierce, J. R., "The Science of Musical Sound," 2nd ed., 1992, pg. 92] The paper "Pitch of the residue," by J. F. Schouten, R. J. Ritsma and B. L. Cardozo, J. Acoust. Soc. Am., Vol. 34, pp. 1418-1424, 1962, presents these results concisely. "Whenever a sound consists of a small number of frequencies, spaced sufficiently widely, subjective sound analysis allows us to hear each Fourier component as a separate pure tone with corresponding pitch (Ohm's acoustical law). If, however, the frequencies are narrowly spaced, the ear fails partly or completely to analyze the Fourier components into corresponding pure tones and, instead, hears one single percept of sharp timbre. If, moreover, the frequencies are harmonics of one fundamental frequency, the percept may have a decidedly low pitch." [Schouten, J. R., J. Ritsman, B. Lopes Cardozo, "Pitch of the Residue," J. Acoust. Soc. Am., Vol. 34, No. 8, September 1962, pg. 1418.] Schouten et al. go on to point out that "The phenomenon of the reside necessarily leads to a hypothetical pitch extractor different from and subsequent to the analyzer. As a consequence of the pitch shifts, the operation of the pitch extraction in the frequency domain is highly improbable. Therefore, the hypothetical pitch extractor probably operates in the time domain (e.g., with delay-line tehcniques)." [ibid., pg. 1424] Further evidence for the existence of periodicity (or "virtual") pitch is found in "Periodicity Pitch for Interrupted White Noise--Fact or Artifact?" by Irwin Pollack, J. Acoust. Soc. Am., Vol 45, no. 1, 1969, pp. 237-238. The author concludes that switching transients cannot provide a reasonable explanation for the perceived ability of listeners to accurately match the pitch of the interrupted noise despite the fact that "The long-term spectrum of an interrupted white noise shows no sepctral peaks at the frequency of interruption." Consequently "The results, therefore, suggest that the periodicity pitch of interrupted noise is factual, no artifactual." [Ibid., pg. 238] The book "Frequency Analysis and Periodicity Detection in Hearing," ed. R. Plomp and G. F. SMoorenburg, A. W. Sithoff, Leiden, 1970, adduced further evidence in favor of virtual pitch as the primary pitch extractor in the human auditory system. Franz Bilsen concludes "The perceptual similarity between a sound with its repetition and a pure periodic signal can be represented in a general model for the perceptibility of pitch and timbre. Like the explanation of repteition pitch, time separation pitch, and peridocity pitch, the model finds its description in the time domain." [Bilsen, F., "Repetition Pitch: Its Implications for Hearing Theory and Room Acoustics," Frequency Analysis and Periodicity Detection In Hearing--Ed. R. Plomp and G. F. SMoorenburg, A. W. Sithoff, Leiden, 1970, pp. 291-299] Schouten himself points that the earlier (Helmholtz) explanation of fundamental perception by nonlinear production of difference tones between higher harmonics fails: "The case of the missing fundamental was solved for the time being (the 1860s-1920s) by a modelmaker's brainwave that the nonlinearity of the ear could produce the missing fundamental as a difference tone between the higher harmonics. Both Fletcher (1929) and von Bekesy (1934), now as observers, proved this point by the method of best beats. Sadly enough, the method of best beats is unreliable unless one knows "who beats whom." (..) In a way the discovery of the residue confirmed Seebeck's "wider interpretation" that the high harmonics might contribute somehow to the fundamental tone, except that it is not the fundamental tone itself which is enhanced but a different subject component: the residue. "As a modelmaker, the author (Schouten) ascribed the low pitch of the residue to the *period* in the time pattern of the joint harmonics striking a particular area of the basilar membrane (Schouten, 1940s). This amounted to a revival of many older assumptions, except that the initial quasi-Fourier analysis was left standing in full glory. "If a set of harmonics is shifted collectively over a small distance in the frequency scale, the pitch of the residue shifts *in proportion* to the constituent frequencies (Schouten, 1940c). This crucial experiment (now called the first effect of pitch shift) kills two birds with one stone. It implies that pitch is determined neither by the *spacing* of the harmonics, nor by the *time* envelope of the wave pattern since both remain invariant. Hence, in the realm of modelmaking, it must be anouther form of periodicity detection in the time domain. This notion goes back to some etent to Seebeck's (1844a) idea of 'the periodic recurrence of an equal or similar state of movement' and to Hermann's (1912) 'intermittence tones.'" [Schouten, J., "The Residue Revisited," Frequency Analysis and Periodicity Detection in Hearing, Ed. R. Plomp and G. F. Smoorenburg, A. W. Sithoff, Leiden, 1970, pp. 41-58] These papers provided apparently convincing proof of the residue's primacy in determining perceived pitch. Such was the power of the residue pitch hypothesis that, from the late 1960s to the present, Ernst Terhardt produced an influential series of papers based on his model of virtual pitch detection: "Zur Tonho"henwarnehmung von Kla"ngen I. Psykoakustische Grundlagen" [Concerning the perceived pitch height of sounds: I. Psychoacoustic Foundations]" Acustica, Vol. 26, No. 4, 1972, pp. 175-186] and "Zur Tonho"henwarnehmung von Kla"ngen II. Ein Funktionschema," Acustica, Vol. 26, No. 4., [Concerning the perceived pitch height of sounds: II. A functional model]. Terhardt's theory essentially relegates the residue or "virtual" pitch to the role of a "secondary sensation," ascribing the primary mechanism of pitch detetction to the well-worn place theory, which accords the operation of the basilar membrane primacy in determining pitch. "The pitch of simple tones and the "residue" pitch of complex tones exhibit quite different properties. The pitch of simple tones may be xplained by the principles of the 'classical' place theory. The 'residue' pitch is closely related to the pure tone pitches of partials. It is concluded that the 'residue' pitch may be regarded as a 'seondary sensation,' derived from the pure tone pitches of dominant partials." [Terhardt, E., op. cit., pg. 174] Terhardt went on publish a series of papers in which he elaborated the "residue" pitch as a secondary sensation: "Pitch consonacne, and harmony," J. Acoust. Soc. Am., 1974, Vol. 55, pp. 1061-1069; "On the perception of periodoic slund fluctations (roughenss)," Acustica, 1974, vol. 30, pp. 201-213. "Ein psychoakusitche begru"ndetes Konzept der Kusikalisches Konsonanz," Acustica, 1976, Vol. 36, pp. 121- 137, "The two-component theory of musical consonance, in "Psychophysics and physiology of hearing," ed. E. F. Evans & J. P. Wilson, London: Academic press, 1977; "Psychoacoustic evaluation of musical sounds," Perception & Psychophysics, Vol. 23, pp. 483-492; "Algorithm for extraction of pitch and pitch salience from complex tonal signals," J. Acoust. Soc. Am., J. Acoust. Soc. Am., Vol. 71, No. 3, March 1982, pp. 679-688; Terhardt, E., Stoll, G. and Seewann, M., "Pitch of complex signals according to virtual- pitch theory: tests, examples, and predictions," J. Acoust. Soc. Am. Vol. 71, pp. 671-678, 1982. This would all seem to be very convincing. Clearly Erlich *must* be correct...right? Inharmonic tones complexes exhibit either a virtual pitch which has nothing to do with the frequencies of their component partials, or the inharmonic tone simply falls apart and fails to fuse into a single acoustic percept... don't they? As it turns out, not so. The first cracks in apparently impregnable facade of the residue pitch hypothesis began to appear during de Boer's research. In 1976, de Boer's monumental study (he calls it "a textbook inside a book") of the residue pitch defined the field: "On the 'Residue' and Auditory Pitch Perception," in Handbook of Sensory Physiology, Volume 5, No. 3, Springer-Verlag, Berlin-Heidelberg- New York, 1976, pp. 481-583] This survey of the residue pitch phenomenon drew on de Boer's doctoral dissertation and provided an unparalleled overview: from Seebeck's acoustic siren to Schouten's and Ritsma's work, to the evidence from chopped noise (Pollock, op. cit.), to Smoorenburg's own work. Here, however, doubt begins to creep in about the universal validity of the residue hypothesis as the basis of pitch detection. "Must we say goodbye to the 'residue'? We might be inclined to say: yes. In the course of this treatise, we witnessed the gradual change of meaning of tjhe term 'resideu,' a change that finally led to a new and less restrictive definition (see e.g., Section E. 10). At the same time the musical aspects of 'residue pitch' grew more and more important. (..) What is also new is the ambiguity involved in residue pitch. This was recognized rather reluctantly at first (Section E. 6); later, it was considered an essential feature of the mechanism (Section E 14). How essential it is did not become completely clear before the work on two-tone complexes and musical intervals (Section H. 2). In last instance, even the harmonic number n of the lower partial appeared to be esentially ambiguous (Section H. 6). The ambiguity involves one other concept that is, in the opinion of the present reviewer, of the greatest importance. (..) When we listen to a sound, we do not perceive this or that aspect of the sound all the time; what we perceive depends completely on our training. (..) Hence, the fact that the result of a particular experiment turns out to be extremely clear-cut does not necessarily imply that ever naive listener is able to perceive the signals in the same way. It is not too difficult to associate a residue pitch with a two- tone complex presented monaurally. But to do the same with a two-tone complex presented dichotically is quite anotehr matter. (..) We may conclude that we must be extremely cautious in our conclusions." [de Boer, E., "On the 'Residue' and Auditory Pitch Perception," Handbook of Sensory Physiology, Vol. 5, No. 3, Springer-Verlag, New York, 1976, pp. 572-574] R. J. Ritsma admits as much when he points out "The pitch behaviour of inharmonic complexes cannot be described precisely by P = f/n or by the concept of subharmonics of a dominant frequency component predicting a pitch P according to equation 1. " [Ritsma, R. J., "Periodicity Detection," in Frequency Analysis and Periodicity Detection in Hearing, ed. R. Plomp and G. F. Smoorenburg, A. W. Sithoff, Leiden, 1970, pp. 251-266] Further doubts about the validity of the results of tests on residue pitch were created by Elizabeth Cohen's and Mathews' and Pierce's work in the late 1970s and early 1908s. "In fact, in single tones stretched A = 2.4, one tended to hear the partials as separated sounds rather than as fused into a tone of a single pitch. We believe that such fusion depends on the phenomenon of residue or periodicity pitch. This has been noted by Cohen. She has observed further than the degree of fusion of a stretched tone depends on the envelope of the tone, and is greatest for an exponentially decreasing amplitude which gives a 'struck' quality." [Mathews, M., and Pierce, J. R., "Harmony and Inhamonic Partials," Rapports IRCAM No. 28, 1980, pg. 12] These results are at first glance mixed: some of Mathews and Pierce's results appear to support the residue pitch hypothesis, while other results--the fact that inharmonic sounds *can* be made to fuse with the right amplitude envelope--tend to contravene Erlich's claim. However, further on the paper strikes a profound blow to the residue hypothesis: "Subjects can identify keys of both stretched and unstretched matierals in an XMXMt test." The significance of this fact should not be underestimated. "If we can detect 'keys' and some form of finality within a cadence or progressions within inharmonic tones, then some of the theories of harmony in the past must not have been as cogent as some of their proponents have thought them to be." [Roads, C., "An Interview WIth Max Mathews," Computer Music Journal, Vol. 4, No. 4, 1980, pp. 21-22] It is impossible to explain key sense and a sense of finality in harmonic progressions using inharmonic tones with the residue pitch hypothesis. Clearly, other auditory mechanisms must be at work. This should not be surprising: many other researchers have run into a brick wall when they tried to explain all ear/brain phenomena on the basis of a single proposed mechanism of pitch detection. Elizabeth Cohen found similarly mixed results, and raised even further doubts about the purported "uselessness" of inharmonic tones for harmony in her series of experiments. For one thing, Cohen notes that "Throughout history musical isntrument ave included not only sources with integral multiple partials but also thouse sources with inharmonic partials. Examples of the latter are stiff strings, bars, and bells. (..) We already possess convincing musical examples illustrating that sources with inharmonic partials have musical potential outside the confines of western harmony. The musical legacy of the Balinese gamelan is an outstanding example. In addition, recent compositions such as Inharmonique (Jean Claude Risset) and Stria (John Chowning) have demonstrated that inharmonic materials can render musical ideas with exquisite sensitivity within a more "western" context." [COhen, E., "Some Effects of Inharmonic Partials On Interval Perpcetion," Music Perception, VOl. 1, No. 3, Spring 1984, pp. 323-349] Cohen searched for a mechanism which could explain why "Inharmonic tones are not conventionally part of the infra-structure of western harmony, but may be used in the same manner as harmonic tones under certain conditions." Cohen notes that a number of her test subjects had no difficulty perceiving harmonic progressions in inharmonic tones, and perceives as "octaves" intervals far from the 2:1..."Subject GWM shows no desire to cling to an octave based on a learned or innate preference for a physical ratio of 2:1." [op. cit., pg. 337] She found similar results for inharmonic "fifths": "The data suggests that GWM, JWG and JMS tune by matching partials." Cohen concludes that "The nature of an instrument having noninteger partials suggests a need to develop an interval sense on something other than simple integer ratios or fundamental doubling. It is suggested here that GWM relied on an interval sense that was based on recognizing the consonance determined by coincident partials and associating this sound as a particular interval." Cohen further points out that "If the composer fuses the sound by controlling the temporal evolution of the partials, then the composer can be assured that conventional relationships and impressions can be made to hold. If the composer wishes to exploit the inharmonicity of the stretched tones, yet maintain certain consonance relations, then the knowledge that interaction of the partials can successfully determine interval size will surely be a useful tool." [op. cit., pg. 347] This would appear to strike the death blow for Paul Erlich's argument. We are now in a position to see why earlier experiments with inharmonic tones failed to produce a sense of spectral fusion, and why earlier experiments (pre-1970s) with inharmonic tones usually exhibited a residue pitch unrelated to the frequencies of the partials: [1] Early experiments with inharmonic tones used crude analog electronic tone production circuits which generated a set of unrelated iharmonic partials. Typically, FM or AM was used, or digital counter circuits. By contrast, the use of digital computers to generate inharmonic tone-complexes in the 1970s allowed researchers to generate inharmonic tones all of whose partials were related. Clearly this had a profound audible effect on test subjects. [2] Early experiments used extremely "electronc- sounding" lifeless inharmonic tones with very simple amplitude envelopes. As John Chowning has pointed out (and as can be heard clearly in his composition Phone), even perfectly harmonic sounds often fail to fuse into a single percept if the envelopes of the different partials do not share a common vibrato and tremolo. The effect can be vividly heard in Chowning's composition: a set of partials appears from silence and while harmonic, sounds like an unrelated set of partials: but when each partials slowly begins to exhibit common vibrato, the set of "unrelated" tones gradually fuse into the unified percept of a human voice. The same effect was clearly at work in early experiments with inharmonic tones. Later experiments, like Cohen's, use common amplitude and frequency information "built into" the envelopes of each inharmonic partial to encourage the tones to audibly fuse--and experiments from the 1970s onward show that such inharmonic tones, so modified, do indeed fuse. [3] Early experiments with inharmonic tones use arbitrarily stretched partials. Later, more sophisticated experiments, (as we shall see) use sets of partials which exhibit the properties of an "inharmonic series" of tones which are related to some coherent generating method. The ear can hear this and places the inharmonic partials within the inharmonic series, producing a complete and convincing sense of finality in inharmonic cadences. As the final nail in the coffin of Terhardt's apparently all-embracing theory, note the article "Reivsion of Terhardt's Psychoacoustical Model of the Root(s) Of a Musical Chord," R. Parncutt, Music Perception, Vol. 6, No. 1, Fall 1988, pp. 65-94: "The predictions of Terhardt's octave- generalized model of the root of a musical chord occasionally disagree with music theory (notably, in the case of the minor triad.)" Thus Terhardt's theory (like so many others which purported to give 'all the answers' about how the ear hears) is now crumbling, and various desperate attempts are being made to shore it up. This tells us (as we should have known all along) that the ear uses more than the virtual pitch mechanism to ascertain fundamental pitch of real-world musical sounds. It also tells us that the operation of the ear is complex and context-sensitive, and that therefore an inharmonic tone progression which offers the ear alternate cues can and often will be heard as an inharmonic progression within a coherent yet inharmonic series of timbres. The second and final post in this series will discuss the overwhelming evidence *against* Erlich's claim, and *in favor of* the ability of even the most naive listener to hear inharmonic progressions of inharmonic tones as exhibiting a sense of harmonic closure, and a fundamental pitch related to the fundamental of the inharmonic tone complex.. Ideas fundamental to the extension of microtonality into the realm of timbre. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 16 Jul 1996 16:31 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id HAA00444; Tue, 16 Jul 1996 07:30:58 -0700 Date: Tue, 16 Jul 1996 07:30:58 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu