Music and the Making of Modern Science
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Among Continental scholars who advanced and reconsidered Newtonian physics, Leon- hard Euler was probably the greatest and surel ...
134 Chapter 9 larger musical forms. Euler ’ s interest in music encompassed many aspects of contemporary composition and practic ...
Euler: The Mathematics of Musical Sadness 135 pleasing. In the whole discussion, I have necessarily had a metaphysical basis, wh ...
136 Chapter 9 accounts did not seek to bridge these two; with few exceptions to this day, he was a lone pioneer of mathematical ...
Euler: The Mathematics of Musical Sadness 137 Euler illustrates his reasoning with a diagram ( figure 9.1 ) showing “ the pulses ...
138 Chapter 9 Figure 9.1 Euler ’ s diagram visualizing the relative agreeableness of various simple ratios of sound pulsation, f ...
Euler: The Mathematics of Musical Sadness 139 But when we calculate the degrees associated with major versus minor triads, in ge ...
140 Chapter 9 evoke joy, and something else can please and bring sadness. ”^20 His paradoxical context points to Aristotle ’ s d ...
Euler: The Mathematics of Musical Sadness 141 mental concepts ( figure 9.3 ). To the more complex of these harmonies, Euler adds ...
142 Chapter 9 A B Figure 9.3 (a) Euler ’ s musical illustration of the first eight species of harmony, according to his degrees ...
Euler: The Mathematics of Musical Sadness 143 in which the ever-recurrent 2s seem to echo the initial 2 from which 2 is drawn. T ...
144 Chapter 9 only as a result of that special skill, combined with his mathematical acumen, could he have achieved such a facto ...
Euler: The Mathematics of Musical Sadness 145 Euler went on to conduct many other inquiries into amicable and perfect numbers, a ...
146 Chapter 9 position alone, and investigates the properties of position; it does not take magnitudes into consideration, nor d ...
Euler: The Mathematics of Musical Sadness 147 Figure 9.5 Euler ’ s 1752 illustrations of various polyhedra. ...
148 Chapter 9 a “ vertex ” by Legendre (about 1794). If a solid polyhedron (closed but not necessarily regular) is bounded by pl ...
Euler: The Mathematics of Musical Sadness 149 Euler thus discovered not just the first important insights that later grew into t ...
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Besides his enormous achievements in mathematics, Euler was deeply involved in many areas of physics. His early work on music ha ...
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