Music and Measurement

Music and Measurement: On the Eidetic Principles of Harmony and Motion Jeffrey C. Kalb, Jr. Released into the public domain: The Feast of St. Cecilia November 22, 2016 Beatae Mariae Semper Virgini, Mediatrici, Coredemptrici, et Advocatae ταὐτὸν γὰρ ποιοῦσι τοῖς ἐν τῇ ἀστρονομίᾳ· τοὺς γὰρ ἐν ταύταις ταῖς συμφωνίαις ταῖς ακουομέναις ἀριθμοὺς ζητοῦσιν, ἀλλ᾽ οὐκ εἰς προβλήματα ἀνίασιν ἐπισκοπεῖν, τίνες ξύμφωνοι ἀριθμοὶ καὶ τίνες οὔ, καὶ διὰ τί ἑκάτεροι.1 For they do the same thing as those in astronomy: they seek the numbers in these harmonies being heard, but they do not ascend to contemplate the problems of which numbers harmonize and which do not, and the reason for each. ― Plato, Republic Index 1. Musica Abscondita 1 1.1 The Definition of Music 3 1.2 Mathematical, Physical, and Intermediate Sciences of Music 5 Figure 1: The Decomposition of a Periodic Waveform into Harmonics 5 1.3 Against the Beat Theory of Harmony 8 Figure 2: The Formation of Beats 9 Figure 3: Standing Waves Formed on a Vibrating String 10 1.4 The Classical Theory of Number and Magnitude 12 1.5 Theoretical Logistic 13 1.6 Common Axioms 15 1.7 Symbolic Mathematics 17 1.8 The Stepwise Symbolic Origin of Algebra 19 1.9 Symbolic Abstraction and the Problem of Measurement 21 1.10 The Act of Measurement 22 Figure 4: Failure in Measurement 22 1.11 Fraction and Meter 23 1.12 A Critique of Metrical Theories of Harmony 24 1.13 The Meaning of Exponents 26 Figure 5: The Classical and Modern Expressions of Area 26 1.14 The Algebraic Division of Operation 28 1.15 Algebra as Confused Music 29 Table 1: The Harmonic Intervals of Simon Stevin 31 2. Musica Instrumentalis 33 2.1 The Logistical Basis of Equality 35 2.2 How Division and Partition Differ 36 2.3 The Derivation of Eidetic Ratios from Arithmetical Species 37 2.4 Bateman’s Rule for Eidetic Ratios 39 Table 2: The First Sixteen Eidetic Ratios 40 2.5 The Octave (2/1) 41 2.6 The Perfect Fourth (4/3) and Perfect Fifth (3/2) 43 Table 3: A Comparison of Perfect Fourths and Perfect Fifths 44 2.7 The Major Third (44/35) and Minor Sixth (35/22) 44 Table 4: A Comparison of Major Thirds and Minor Sixths 45 2.8 The Major Sixth (176/105) and Minor Third (105/88) 45 Table 5: A Comparison of Major Sixths and Minor Thirds 45 2.9 The Tritone 46 Table 6: A Comparison of Tritones 47 2.10 The Minor Seventh and Major Second 47 i Table 7: A Comparison of Minor Sevenths and Major Seconds 48 2.11 The Weak Minor Third and Weak Major Sixth 48 Table 8: A Comparison of Weak Major Sixths and Minor Thirds 49 2.12 Sixteenth-Order Species 49 Table 9: Basic Intervals and their Shadow Intervals 50 2.13 The Variety of Eidetic Hierarchies 51 2.14 The Major Tetrad 51 Table 10: Intervals of the Ω2-Ω4-Ω8-Ω32 Eidetic Hierarchy 52 Table 11: Intervals of the Ω2-Ω6-Ω12-Ω36 Eidetic Hierarchy 53 Figure 6: Intervals Formed from Two Simple Eidetic Hierarchies 54 2.15 The Minor Tetrad 55 Figure 7: Major and Minor Eidetic Triads 56 2.16 The Diatonic Scale 56 Figure 8: Major and Natural Minor Scales 57 2.17 A Short Account of Key Coloration 58 2.18 On the Ptolemaic Critique of Aristoxenus 58 2.19 The Aesthetic Criteria of Harmony 60 3. Musica Mundana et Humana 63 3.1 Uniform States of Motion 65 3.2 Mathematics and Motion 67 3.3 Aristoxenus on Intervallic Motion 68 3.4 Bergson’s Critique of Modern Kinematics 69 Figure 9: The Calculation of Tangential Slope 70 3.5 The Attribution of Spatial Properties to States of Motion 72 3.6 A Metrical Paradox in Relativity Theory 73 3.7 The Role of Ensembles in the Science of Meter 74 Figure 10: The Generation of Equal and Juxtaposed Measures 75 3.8 Physical and Statistical Time 76 Table 12: An Infinite Metrical Ensemble 77 3.9 Quantum Physics, Statistical Mechanics, and Eidetic Theory 77 3.10 Mathematical Constants in Physics 79 Figure 11: A Comparison of the Natural Logarithm and the Harmonic Series 79 3.11 Quantitative Operation and Quantity as Such 80 3.12 The Analogy of Physical Natures 81 3.13 A Summary of the Thomistic Doctrine of Corporeal Substance 81 3.14 Musical Harmony and the Immortal Soul 83 3.15 The Algebraic Foundation of Mechanical Philosophy 84 3.16 Prime Matter and the Principle of Individuation 87 3.17 Music and Ethics 89 End Notes 91 ii 1. Musica Abscondita 1 2 1.1 The Definition of Music It is widely accepted that one ought to define a subject’s matter and form before proceeding to its investigation, but this supposes in the reader a framework in which to locate and interpret the author’s definition. A mathematical theory of music presents difficulties in this regard. Indeed, few musicians today would take seriously the assertion of Leibniz that “music is the hidden arithmetical exercise of the mind unaware that it is counting.” (Musica est exercitium arithmeticae occultum nescientis se numerare animi.2) Even if this definition does not adequately capture the true mathematical character of music, it certainly specifies music either as an application of mathematics to physical phenomena or as a mathematical discipline in its own right. Concert music, however, has run far beyond mathematical theory, and is commonly classed among the fine arts, rather than among the liberal arts of the quadrivium. Eduard Hanslick’s opinion better captures the ethos of today’s musician: Perhaps more out of caution than from need, we may add in conclusion that the musically beautiful has nothing to do with mathematics. This notion, which laymen (sensitive authors among them) cherish concerning the role of mathematics in music, is a remarkably vague one. Not content that the vibrations of tones, the spacing of intervals, and consonances and dissonances can be traced back to mathematical proportions, they are also convinced that the beauty of a musical work is based on number. The study of harmony and counterpoint is considered a kind of cabala which teaches compositional calculus. Even though mathematics provides an indispensable key for the investigation of the physical aspects of musical art, its importance with regard to completed musical works ought not to be overrated. In a musical composition, be it the most beautiful or the ugliest, nothing at all is mathematically worked out. The creations of the imagination are not sums. All monochord experiments, acoustic figures, proportions of intervals, and the like, are irrelevant. The domain of aesthetics begins where these elementary relationships, however important, have left off. Mathematics merely puts in order the rudimentary material for artistic treatment and operates secretly in the simplest relations.3 Excessive mathematical claims regarding music―claims that take the promise for the deed―produce an impatience with mathematical theory that is rather excusable, if regrettable. For the assertion of Leibniz may be taken to denigrate musical practice, as if mathematicians had easily achieved in the realm of discursive thought what the whole tradition of concert music has accomplished only through genius and untiring effort. Such an interpretation would be unjust to Leibniz no less than to these musicians. The lack of conscious mathematical application, duly recognized by Leibniz, in no way excludes a thoroughgoing mathematical structure to the art. It simply means that, in view of the complexity of composition, mathematics can only describe, rather than prescribe, musical practice. Consider sculpture. The sculptor, for all his technique, cannot escape the mathematical demands of solid geometry, and all his efforts―conscious or unconscious―tend toward a definite geometric form, however complex, however expressive of his culture. This sensible geometric form persists 3 in the medium, and is the only true artistic subject, its shape and proportions being the indispensable components of any aesthetic response. In like manner, musical composition is the molding and direction of sound in time, a motion wholly reducible to mathematical relations. Whatever the impulse or intent of the artist, no matter how much meaning accrues to music from personal, cultural, political, or spiritual concerns, the potential forms of music, though infinite in number, can be fully specified through mathematics. If, as Hanslick claims, there is a properly musical aesthetic, it is surely mathematical. The music theorist must consequently strive to understand how mathematical form satisfies aesthetic criteria, and how it acts as a substrate to which human mores and motives may be subsequently related. Sheet music, a symbolic shorthand that directs the making of music, but not its hearing, is a valuable tool, yet by no means an ultimate description. The fault lies with mathematics, which has long been unmoored from the metaphysical distinctions that lend themselves to aesthetic analysis. A reconciliation cannot be achieved, therefore, by a critique of musical practice, but only by an instauration of modern mathematics, a sifting and reconstruction on more solid principles. How are calculations involving measurable quantities, rather than numbers, to be justified scientifically? This problem of theoretical logistic, first enunciated by Plato4 and lately resurrected by Jacob Klein5, can be solved by recourse to infinite ensembles of natural numbers belonging to a common arithmetical species (εἴδη). The conditions of musical harmony then follow naturally from this new eidetic theory. Music may consequently be defined as the art of measurement and the science of the measurable. By way of comparison, arithmetic may be defined as the art of numeration and the science of the numerable, and geometry as the art of extension and the science of the extended.

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