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Comments by Ernest G. McClain
Comments
363
relevant to either musical or philosophic thought remains an open question. The reference to Von Btkesy’s ‘triangles of syntony’ is particularly puzzling and calls for documentation. The phrase and the sentences into which it leads are unclear. I have found no mention of ‘triangles of syntony’ in any of Von B&k&y’s writings known to me; it certainly does not occur in the Nobel Prize winner’s main text, Experiments in Hearing. Nor is the phrase intelligible to two experts I consulted, one of them a former close collaborator of Von . Btkesy. Siegmund Levarie City University of New York, New York, USA
Comments by Ernest G. McClain Tonal ontology It is hard to believe that the ‘unique’ aspect of Ansermet’s phenomenology of music is represented by a banal major hypothesis (that there is a qualitative difference between visual and audial perception) and a pathetic conclusion (that some still-to-be-discovered ‘meta-logarithmic’ formula will rescue us from conceptual confusion). Is it really necessary to ‘defend’ the proposition that ‘music does not function in the same kind of “space” that Newtonian-Kantian physics does’? Does it matter, except as a convenience, how semitones are measured? Should a pheonomenology which prohibits separating ‘the sonic thing . . . and human perception’ pledge its future to a new kind of logarithm which, if discovered, would only land us back in rejected Cartesian dualism? Quite mysteriously, Piguet affirms the tone A to be ‘musically in the middle of the octave D-D’ where the phenomenologically ambiguous tritone actually reigns jealously, and he wipes out the spatial distinction between ‘the ascending fifth and the descending fourth’ which a phenomenologist might be expected to treasure. Clearly, everything is awry. Apparently anything valuable in Ansermet’s work has been lost in the process of compression and translation. A charitable response to Piguet’s version of Ansermet requires us to assume that good intentions have gone astray, that English is unkind to both men. Defining satisfactorily the ontological status of tonal objects has defied the best minds of all ages. The notion of ratio, as old as the earliest mathematics of Babylon (c. 1800 BC) served the Greeks very well for describing ‘laboratory’ measurements of pitch perception; modern logarithms (including French savarts and English cents) are conveniences which add nothing to historical awareness that the octave 2: I is the harmonic matrix within which larger numbers define smaller intervals. But Aristotle was acutely aware that the ‘sonic thing’ is not, phenomenologically, a number, and that it lacks the conceptual exactness of number. His pupil Aristoxenus, ‘The Musician’ to the ancient world, developed his own Harmonics by appealing to the ear and its musical sense rather than to Pythagorean ratios. By limiting their attention to a few central octaves, ancient phenomenologists (Aristotle’s Problems, Book XIX contains many discussions on tonal phenomenology) spared themselves the discovery that these ratios fail at the upper and lower limits of our hearing. We can only wonder whether the wide range Aristoxenus attributes to the family of Greek auloi was great enough to call his attention to a more subtle problem: melodies change character considerably when transposed even to adjacent octaves; bemused tolerance accepts a melody as being somehow ‘the same’
364
Comments
when shifted several octaves, but we know it is not so even though all octaves demonstrably enjoy the same ratios. Our conceptually abstract interval measuring system is in subtle conflict with our ‘spatial’ interval feeling except in a few central octaves; ratio theory fails to do justice to our (often sublimal) sensitivity to ‘absolute’ pitch. Musicians solve this dilemma comfortably by treating ‘harmony’ and ‘orchestration’ as two different aspects (even to giving them quite separate courses of instruction), but pseudophilosophers, stuck with a single, simplistic descriptive apparatus, can be marooned in the never-never land reserved for the unmusical. But this is only a small part of the problem for tonal phenomenology. The conceptual ‘yardstick’ with which we measure the pitch continuum not only must be stretched at both ends (while we pretend not to notice), and (in Plato’s metaphor) be simultaneously ‘bent round into a circle’ (of equivalent octaves for tuning theory, but of phenomenologically exotically different octaves for composers and audience), but it must suffer - in any actual musical context - further degradation from our harmonic sense, which has its own notions about who one’s ‘closest’ relatives are, and from our agogic feeling, which affirms the superiority of rhythm and thus radically limits the worth of all of our concerns with pitch. The classical statement of the philosophical muddle posed by the presence of a cyclic octave within the linear pitch continuum - the ‘same’ tones recur in ‘different’ octaves is Plato’s. In the creation myth of his Timaeus, describing how the ‘world-soul’ is constructed linearly from Pythagorean tone ratios, he postulates a ‘form of Existence composed of Sameness and Difference’, which confounds logical distinctions. The creator, he writes, has forced ‘the nature of Difference, hard as it was to mingle, into union with Sameness, mixing them together with Existence’ (Timaeus 35a). He is thinking here of the powers of 3, which define successive tones in a sequence of fifths, hence constituting the aspect of invariance, or Sameness, and of the powers of 2 which transpose them to D$krent octaves. His creator, who produces only a ‘single model’, will later bend these numbers ‘round into a circle’ of cyclic octaves to satisfy Plato’s notions of the economy of deity. For him, tonal Existence was an interplay of octaves and fifths, of ‘numbers in motion’ (McClain, 1978). For musicians, the octave remains ‘the basic miracle of music’ (Levarie & Levy, 1968). Herman Weyl, a great mathematician, in his formal exposition of the laws of symmetry, points out that although ‘all musicians agree that underlying the emotional element of music is a strong formal element, . . . we have probably not yet discovered the appropriate mathematical tools’. But he did not share Ansermet’s naive faith in a meta-logarithmic answer, for he perceived a deeper problem. Weyl cites his own difficulty in recognizing ‘a melody played backwards’; he is questioning whether the laws of symmetry which he is developing for transformations in space can be appropriately applied to a melody which ‘changes its character to a considerable degree if played backward’, hence losing its ontological right to be thought of merely as an ‘inversion in time’ (Weyl, 1952). The many-splendored ‘dimensionality’ of even the simplest musical experiences defy summation by any kind of mathematical formula, which at best can encode only limited aspects of feeling, thus falsifying the ‘global’ experience. ‘Sound’, Whitehead points out, ‘though voluminous, is very vague as to the dimensions of its volumes, as between three or fifteen, for instance’ (Whitehead, 1938). He may not be thinking of anything so complicated as an actual musical composition, whose ‘dimensions’ depend largely on the awareness and the memory and the anticipation of the listener.
Comments
365
Thus for twenty-five centuries some of the greatest minds have been warning us of the difficulties in any phenomenology of tone. If it turns out that Ansermet has nothing of interest to add to the historical dialogue, at least he must be credited with wrestling with matters of profound complexity and importance. Tone in general and the octave in particular remain paradigmatic tests for any general theory of mind-brain functions. Ernest C. McClain Box 127. Belkont. Vermont 05730, USA References Levarie, S.& Levy, E. (1968). Tone: A Study in Musical Acoustics. Kent: Kent State University Press. McClain, E. G. (1978). The Pythagorean Plato. Stony Brook: Nicolas Hays. Weyl, H. (1952). Symmetry. Princeton: Princeton University Press. Whitehead, A. N. (1938, 1966). Modes of Thought. New York: The Free Press.
Comments by J. H. M. Whiteman This paper by Piguet is almost exclusively concerned with two topics of psychophysics: (I) The relationship between the perception of a musical interval made by sounds of two pitches, heard in the mind, and the calculated ratio of the corresponding two vibrationfrequencies, detectable in matter; and (2) the philosophical position taken by the phenomenology of Husserl and Merleau-Ponty on audial psychophysics, as opposed to that taken by scientific dualism. In regard to the first topic, the paper refers to the perception of musical intervals as ‘music’ (though intensity, timbre, melodic and rhythmic Gestalt, etc., are not considered), and to the vibration-frequency in a material object as ‘sound’ (see paragraphs 5, 11, and 16). Also confusing, I found, is the use of the term ‘natural numbers’ for real or rational numbers, and the frequent vague or unorthodox use of the epithets logarithmic and hyperbolic. As I have not read Ansermet’s 1961 book, I shall not try to distinguish between the views of Ansermet and those of Piguet in reporting Ansermet, but will merely contrast what I think is the true state of affairs with what is presented in the paper. ( 1) Logarithmic formulae in the measurement of nature. Any measuring of nature must begin with the construction of isometry tools such as equally graduated straight-edges or circles and proceed to the setting up of some correspondence with the phenomenon to be measured (Whiteman, 1967, especially chp. 6). Often, having obtained a measurement P in this way for some phenomenon, it is convenient to introduce a derived measure Q = flP), e.g. P2, or ep. Thus for the slide rule, Q = log P, so that if Q, and Q, are added, then P, and PI are multiplied. Nearly all natural laws link derived measures or their rates of change. It is important to remark, in connection with any functional relation of the kind Q = f(P), that any statement that Q and P constitute different (topological) ‘spaces’ (as in para. 12) means no more than that Q and P stand for ‘points’ or entities to which different operational rules are applicable. Either the same thing is being given a measure in two ways; or the measures for two different things are being correlated by a mathematical formalism. By itself, the equation carries no ontological significance.
363
relevant to either musical or philosophic thought remains an open question. The reference to Von Btkesy’s ‘triangles of syntony’ is particularly puzzling and calls for documentation. The phrase and the sentences into which it leads are unclear. I have found no mention of ‘triangles of syntony’ in any of Von B&k&y’s writings known to me; it certainly does not occur in the Nobel Prize winner’s main text, Experiments in Hearing. Nor is the phrase intelligible to two experts I consulted, one of them a former close collaborator of Von . Btkesy. Siegmund Levarie City University of New York, New York, USA
Comments by Ernest G. McClain Tonal ontology It is hard to believe that the ‘unique’ aspect of Ansermet’s phenomenology of music is represented by a banal major hypothesis (that there is a qualitative difference between visual and audial perception) and a pathetic conclusion (that some still-to-be-discovered ‘meta-logarithmic’ formula will rescue us from conceptual confusion). Is it really necessary to ‘defend’ the proposition that ‘music does not function in the same kind of “space” that Newtonian-Kantian physics does’? Does it matter, except as a convenience, how semitones are measured? Should a pheonomenology which prohibits separating ‘the sonic thing . . . and human perception’ pledge its future to a new kind of logarithm which, if discovered, would only land us back in rejected Cartesian dualism? Quite mysteriously, Piguet affirms the tone A to be ‘musically in the middle of the octave D-D’ where the phenomenologically ambiguous tritone actually reigns jealously, and he wipes out the spatial distinction between ‘the ascending fifth and the descending fourth’ which a phenomenologist might be expected to treasure. Clearly, everything is awry. Apparently anything valuable in Ansermet’s work has been lost in the process of compression and translation. A charitable response to Piguet’s version of Ansermet requires us to assume that good intentions have gone astray, that English is unkind to both men. Defining satisfactorily the ontological status of tonal objects has defied the best minds of all ages. The notion of ratio, as old as the earliest mathematics of Babylon (c. 1800 BC) served the Greeks very well for describing ‘laboratory’ measurements of pitch perception; modern logarithms (including French savarts and English cents) are conveniences which add nothing to historical awareness that the octave 2: I is the harmonic matrix within which larger numbers define smaller intervals. But Aristotle was acutely aware that the ‘sonic thing’ is not, phenomenologically, a number, and that it lacks the conceptual exactness of number. His pupil Aristoxenus, ‘The Musician’ to the ancient world, developed his own Harmonics by appealing to the ear and its musical sense rather than to Pythagorean ratios. By limiting their attention to a few central octaves, ancient phenomenologists (Aristotle’s Problems, Book XIX contains many discussions on tonal phenomenology) spared themselves the discovery that these ratios fail at the upper and lower limits of our hearing. We can only wonder whether the wide range Aristoxenus attributes to the family of Greek auloi was great enough to call his attention to a more subtle problem: melodies change character considerably when transposed even to adjacent octaves; bemused tolerance accepts a melody as being somehow ‘the same’
364
Comments
when shifted several octaves, but we know it is not so even though all octaves demonstrably enjoy the same ratios. Our conceptually abstract interval measuring system is in subtle conflict with our ‘spatial’ interval feeling except in a few central octaves; ratio theory fails to do justice to our (often sublimal) sensitivity to ‘absolute’ pitch. Musicians solve this dilemma comfortably by treating ‘harmony’ and ‘orchestration’ as two different aspects (even to giving them quite separate courses of instruction), but pseudophilosophers, stuck with a single, simplistic descriptive apparatus, can be marooned in the never-never land reserved for the unmusical. But this is only a small part of the problem for tonal phenomenology. The conceptual ‘yardstick’ with which we measure the pitch continuum not only must be stretched at both ends (while we pretend not to notice), and (in Plato’s metaphor) be simultaneously ‘bent round into a circle’ (of equivalent octaves for tuning theory, but of phenomenologically exotically different octaves for composers and audience), but it must suffer - in any actual musical context - further degradation from our harmonic sense, which has its own notions about who one’s ‘closest’ relatives are, and from our agogic feeling, which affirms the superiority of rhythm and thus radically limits the worth of all of our concerns with pitch. The classical statement of the philosophical muddle posed by the presence of a cyclic octave within the linear pitch continuum - the ‘same’ tones recur in ‘different’ octaves is Plato’s. In the creation myth of his Timaeus, describing how the ‘world-soul’ is constructed linearly from Pythagorean tone ratios, he postulates a ‘form of Existence composed of Sameness and Difference’, which confounds logical distinctions. The creator, he writes, has forced ‘the nature of Difference, hard as it was to mingle, into union with Sameness, mixing them together with Existence’ (Timaeus 35a). He is thinking here of the powers of 3, which define successive tones in a sequence of fifths, hence constituting the aspect of invariance, or Sameness, and of the powers of 2 which transpose them to D$krent octaves. His creator, who produces only a ‘single model’, will later bend these numbers ‘round into a circle’ of cyclic octaves to satisfy Plato’s notions of the economy of deity. For him, tonal Existence was an interplay of octaves and fifths, of ‘numbers in motion’ (McClain, 1978). For musicians, the octave remains ‘the basic miracle of music’ (Levarie & Levy, 1968). Herman Weyl, a great mathematician, in his formal exposition of the laws of symmetry, points out that although ‘all musicians agree that underlying the emotional element of music is a strong formal element, . . . we have probably not yet discovered the appropriate mathematical tools’. But he did not share Ansermet’s naive faith in a meta-logarithmic answer, for he perceived a deeper problem. Weyl cites his own difficulty in recognizing ‘a melody played backwards’; he is questioning whether the laws of symmetry which he is developing for transformations in space can be appropriately applied to a melody which ‘changes its character to a considerable degree if played backward’, hence losing its ontological right to be thought of merely as an ‘inversion in time’ (Weyl, 1952). The many-splendored ‘dimensionality’ of even the simplest musical experiences defy summation by any kind of mathematical formula, which at best can encode only limited aspects of feeling, thus falsifying the ‘global’ experience. ‘Sound’, Whitehead points out, ‘though voluminous, is very vague as to the dimensions of its volumes, as between three or fifteen, for instance’ (Whitehead, 1938). He may not be thinking of anything so complicated as an actual musical composition, whose ‘dimensions’ depend largely on the awareness and the memory and the anticipation of the listener.
Comments
365
Thus for twenty-five centuries some of the greatest minds have been warning us of the difficulties in any phenomenology of tone. If it turns out that Ansermet has nothing of interest to add to the historical dialogue, at least he must be credited with wrestling with matters of profound complexity and importance. Tone in general and the octave in particular remain paradigmatic tests for any general theory of mind-brain functions. Ernest C. McClain Box 127. Belkont. Vermont 05730, USA References Levarie, S.& Levy, E. (1968). Tone: A Study in Musical Acoustics. Kent: Kent State University Press. McClain, E. G. (1978). The Pythagorean Plato. Stony Brook: Nicolas Hays. Weyl, H. (1952). Symmetry. Princeton: Princeton University Press. Whitehead, A. N. (1938, 1966). Modes of Thought. New York: The Free Press.
Comments by J. H. M. Whiteman This paper by Piguet is almost exclusively concerned with two topics of psychophysics: (I) The relationship between the perception of a musical interval made by sounds of two pitches, heard in the mind, and the calculated ratio of the corresponding two vibrationfrequencies, detectable in matter; and (2) the philosophical position taken by the phenomenology of Husserl and Merleau-Ponty on audial psychophysics, as opposed to that taken by scientific dualism. In regard to the first topic, the paper refers to the perception of musical intervals as ‘music’ (though intensity, timbre, melodic and rhythmic Gestalt, etc., are not considered), and to the vibration-frequency in a material object as ‘sound’ (see paragraphs 5, 11, and 16). Also confusing, I found, is the use of the term ‘natural numbers’ for real or rational numbers, and the frequent vague or unorthodox use of the epithets logarithmic and hyperbolic. As I have not read Ansermet’s 1961 book, I shall not try to distinguish between the views of Ansermet and those of Piguet in reporting Ansermet, but will merely contrast what I think is the true state of affairs with what is presented in the paper. ( 1) Logarithmic formulae in the measurement of nature. Any measuring of nature must begin with the construction of isometry tools such as equally graduated straight-edges or circles and proceed to the setting up of some correspondence with the phenomenon to be measured (Whiteman, 1967, especially chp. 6). Often, having obtained a measurement P in this way for some phenomenon, it is convenient to introduce a derived measure Q = flP), e.g. P2, or ep. Thus for the slide rule, Q = log P, so that if Q, and Q, are added, then P, and PI are multiplied. Nearly all natural laws link derived measures or their rates of change. It is important to remark, in connection with any functional relation of the kind Q = f(P), that any statement that Q and P constitute different (topological) ‘spaces’ (as in para. 12) means no more than that Q and P stand for ‘points’ or entities to which different operational rules are applicable. Either the same thing is being given a measure in two ways; or the measures for two different things are being correlated by a mathematical formalism. By itself, the equation carries no ontological significance.