Part I. Preliminaries: Introduction 1. Beginnings, and the problem of measurement Part II. Empirical Harmonics: 2. Empirical harmonics before Aristoxenus 3. The early empiricists in their cultural and intellectual contexts 4. Interlude on Aristotle's account of a science and its methods 5. Aristoxenus: the composition of the Elementa harmonica 6. Aristoxenus: concepts and methods in Elementa harmonica Book 1 7. Elementa harmonica Books 2-3: the science reconsidered 8. Elementa harmonica Book 3 and its missing sequel 9. Contexts and purposes of Aristoxenus' harmonics Part III. Mathematical Harmonics: 10. Pythagorean harmonics in the fifth century: Philolaus 11. Developments in Pythagorean harmonics: Archytas 12. Plato 13. Aristotle on the harmonic sciences 14. Systematising mathematical harmonics: the Sectio canonis 15. Quantification under attack: Theophrastus' critique Postscript: The later centuries.

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The Science of Harmonics in Classical Greece

The transmission of ancient music theory into the Middle Ages

Introduction 1. Songs alive 2. How (not) to write a motet: the exemplary In virtute/Decens 3. Motet visions of an apocalyptic statue 4. Interlude: Nebuchadnezzar's dream 5. Ars nova and division Epilogue: the poetics of representation Appendices: 1. Philippe de Vitry, In virtute/Decens: texts, translations, and music 2. Philippe de Vitry, Cum statua/Hugo: texts, translations, and music 3. Philippe de Vitry, Phi millies/O creator: texts, translations, and music 4. Anonymous, Post missarum/Post misse: texts and translations 5. Anonymous, Fortune/Ma dolour: texts and translations 6. Anonymous, Amer/Durement: texts and translations 7. Philippe de Vitry, Firmissime/Adesto: texts and translations 8. Anonymous, Beatius/Cum humanum: texts and translations.

The Monstrous New Art: Divided Forms in the Late Medieval Motet

The modern theoretical concept of harmonic progression was  rst clearly articulated by Rameau. Yet in his Traite, this concept is still demonstrably linked (via Zarlino) with an older notion of harmonic progression: the late-medieval contrapuntal doctrine that “imperfect” dyadic sonorities “seek” or “require” resolution to speci c “perfect” dyads. This doctrine  rst appears in Marchetto of Padua’s Lucidarium (1317/18). Its explanation, the general principle that “the imperfect seeks its perfection,” is a scholastic formulation of a basic principle of Aristotelian physics: natural motions are directed toward a predetermined goal or “end” (telos), which is the “perfection” of the moved thing. The late-medieval application of this principle to counterpoint thus emerges as the probable origin of the concept of harmonic progression and, more broadly, of the widely accepted view of music as embodying “directed motion.” This content downloaded from 157.55.39.255 on Tue, 24 May 2016 05:37:24 UTC All use subject to http://about.jstor.org/terms

https://www.jstor.org/stable/pdfplus/10.1525/mts.2001.23.2.139.pdf

“The Imperfect Seeks Its Perfection”: Harmonic Progression, Directed Motion, and Aristotelian Physics

In Chapter 6 of The Manual of Harmonics (early second century CE), Nicomachus of Gerasa narrates the legendary story of Pythagoras passing by the blacksmith’s shop, during which in an epiphany of sonorous revelation, he discovered the correlation of sounding intervals and their numerical ratios. According to Nicomachus, Pythagoras perceived from the striking of the hammers on the anvils the consonant intervals of the octave, fifth, and fourth, and the dissonant interval of the whole tone separating the fifth and fourth. Experimenting in the smithy with various factors that might have influenced the interval differences he heard (force of the hammer blows, shape of the hammer, material being cast), he concluded that it was the relative weight of the hammers that engendered the differences in the sounding intervals, and he attempted to verify his conclusion by comparing the sounds of plucked strings of equal tension and lengths, proportionally weighted according to the ratios of the intervals. Physical and logical incongruities or misrepresentations in Nicomachus’s narrative aside, the parable became a fixture of neo-Pythagorean discourse because of its metaphoric resonance: it encapsulated the essence of Pythagorean understanding of number as material or corporeal, and it venerated Pythagoras as the discoverer of the mathematical ratios underlying the science of harmonics. The parable also established a frame of reference in music-theoretical thought in the association between music and number, or more accurately, music theory and mathematical models, since it is not through number alone but through the more fundamental notions of universality and truth embedded in Pythagorean and Platonic mathematics and philosophy that one can best begin to apprehend the broad range of interrelationships between music theory and mathematics.

Music theory and mathematics

This collection of essays argues that early music theory is more interesting than commonly held and that its broad and loose definition in antiquity and the Middle Ages allows for the inclusion of a plethora of intellectual pursuits under the heading of "Music Theory".

Music theory and its sources : antiquity and the Middle Ages

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Music theory and its sources : antiquity and the Middle Ages