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E-grāmata: From Music to Mathematics

3.78/5 (16 ratings by Goodreads)
(College of the Holy Cross)
  • Formāts: EPUB+DRM
  • Izdošanas datums: 15-Feb-2016
  • Izdevniecība: Johns Hopkins University Press
  • Valoda: eng
  • ISBN-13: 9781421419190
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From Music to MathematicsPalielināt
  • Formāts: EPUB+DRM
  • Izdošanas datums: 15-Feb-2016
  • Izdevniecība: Johns Hopkins University Press
  • Valoda: eng
  • ISBN-13: 9781421419190
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Taking a "music first" approach, Gareth E. Roberts's From Music to Mathematics will inspire students to learn important, interesting, and at times advanced mathematics. Ranging from a discussion of the geometric sequences and series found in the rhythmic structure of music to the phase-shifting techniques of composer Steve Reich, the musical concepts and examples in the book motivate a deeper study of mathematics. Comprehensive and clearly written, From Music to Mathematics is designed to appeal to readers without specialized knowledge of mathematics or music. Students are taught the relevant concepts from music theory (notation, scales, intervals, the circle of fifths, tonality, etc.), with the pertinent mathematics developed alongside the related musical topic. The mathematics advances in level of difficulty from calculating with fractions, to manipulating trigonometric formulas, to constructing group multiplication tables and proving a number is irrational. Topics discussed in the book include Rhythm Introductory music theory The science of sound Tuning and temperament Symmetry in music The Bartok controversy Change ringing Twelve-tone music Mathematical modern music The Hemachandra-Fibonacci numbers and the golden ratio Magic squares Phase shifting Featuring numerous musical excerpts, including several from jazz and popular music, each topic is presented in a clear and in-depth fashion. Sample problems are included as part of the exposition, with carefully written solutions provided to assist the reader. The book also contains more than 200 exercises designed to help develop students' analytical skills and reinforce the material in the text. From the first chapter through the last, readers eager to learn more about the connections between mathematics and music will find a comprehensive textbook designed to satisfy their natural curiosity.

Recenzijas

Overall, From Music to Mathematics is a pleasing and well-written book that is accessible for everyone who wants to explore the connections between music and mathematics. Gareth Roberts does a great job of making numerous suggestions on how music can be used to illuminate mathematical concepts... From Music to Mathematics is very enjoyable to read - not only for students, but for anyone who loves music and mathematics. Musicae Scientiae Overall, I strongly recommend this as an excellent basis for teaching. MathSciNet

Papildus informācija

A guided tour of the mathematical principles inherent in music.
Preface xi
Acknowledgments xv
Introduction xvii
1 Rhythm
1(32)
1.1 Musical Notation and a Geometric Property
1(8)
1.1.1 Duration: Geometric sequences
2(2)
1.1.2 Dots: Geometric series
4(5)
1.2 Time Signatures
9(8)
1.2.1 Musical examples
10(3)
1.2.2 Rhythmic repetition
13(4)
1.3 Polyrhythmic Music
17(11)
1.3.1 The least common multiple
19(3)
1.3.2 Musical examples
22(6)
1.4 A Connection with Indian Classical Music
28(5)
References for
Chapter 1
31(2)
2 Introduction to Music Theory
33(48)
2.1 Musical Notation
34(7)
2.1.1 The common clefs
34(3)
2.1.2 The piano keyboard
37(4)
2.2 Scales
41(14)
2.2.1 Chromatic scale
42(2)
2.2.2 Whole-tone scale
44(1)
2.2.3 Major scales
45(4)
2.2.4 Minor scales
49(1)
2.2.5 Why are there 12 major scales?
50(5)
2.3 Intervals and Chords
55(9)
2.3.1 Major and perfect intervals
56(1)
2.3.2 Minor intervals and the tritone
57(2)
2.3.3 Chords
59(5)
2.4 Tonality, Key Signatures, and the Circle of Fifths
64(17)
2.4.1 The critical tonic-dominant relationship
65(2)
2.4.2 Key signatures
67(2)
2.4.3 The circle of fifths
69(3)
2.4.4 Transposition
72(2)
2.4.5 The evolution of polyphony
74(5)
References for
Chapter 2
79(2)
3 The Science of Sound
81(38)
3.1 How We Hear
81(3)
3.1.1 The magnificent ear-brain system
82(2)
3.2 Attributes of Sound
84(4)
3.2.1 Loudness and decibels
84(2)
3.2.2 Frequency
86(2)
3.3 Sine Waves
88(9)
3.3.1 The sine function
89(2)
3.3.2 Graphing sinusoids
91(3)
3.3.3 The harmonic oscillator
94(3)
3.4 Understanding Pitch
97(18)
3.4.1 Residue pitch
98(6)
3.4.2 A vibrating string
104(1)
3.4.3 The overtone series
105(2)
3.4.4 The starting transient
107(1)
3.4.5 Resonance and beats
108(7)
3.5 The Monochord Lab: Length versus Pitch
115(4)
References for
Chapter 3
118(1)
4 Tuning and Temperament
119(46)
4.1 The Pythagorean Scale
119(8)
4.1.1 Consonance and integer ratios
120(2)
4.1.2 The spiral of fifths
122(2)
4.1.3 The overtone series revisited
124(3)
4.2 Just Intonation
127(6)
4.2.1 Problems with just intonation: The syntonic comma
129(2)
4.2.2 Major versus minor
131(2)
4.3 Equal Temperament
133(8)
4.3.1 A conundrum and a compromise
133(2)
4.3.2 Rational and irrational numbers
135(3)
4.3.3 Cents
138(3)
4.4 Comparing the Three Systems
141(3)
4.5 Strahle's Guitar
144(14)
4.5.1 An ingenious construction
145(4)
4.5.2 Continued fractions
149(6)
4.5.3 On the accuracy of Strahle's method
155(3)
4.6 Alternative Tuning Systems
158(7)
4.6.1 The significance of log2(3/2)
158(1)
4.6.2 Meantone scales
159(2)
4.6.3 Other equally tempered scales
161(2)
References for
Chapter 4
163(2)
5 Musical Group Theory
165(38)
5.1 Symmetry in Music
165(17)
5.1.1 Symmetric transformations
166(3)
5.1.2 Inversions
169(4)
5.1.3 Other examples
173(9)
5.2 The Bartok Controversy
182(9)
5.2.1 The Fibonacci numbers and nature
183(1)
5.2.2 The golden ratio
184(1)
5.2.3 Music for Strings, Percussion and Celesta
185(6)
5.3 Group Theory
191(12)
5.3.1 Some examples of groups
192(1)
5.3.2 Multiplication tables
193(2)
5.3.3 Symmetries of the square
195(2)
5.3.4 The musical subgroup of D4
197(4)
References for
Chapter 5
201(2)
6 Change Ringing
203(26)
6.1 Basic Theory, Practice, and Examples
203(13)
6.1.1 Nomenclature
204(1)
6.1.2 Rules of an extent
205(3)
6.1.3 Three bells
208(2)
6.1.4 The number of permissible moves
210(1)
6.1.5 Example: Plain Bob Minimus
211(2)
6.1.6 Example: Canterbury Minimus
213(3)
6.2 Group Theory Revisited
216(13)
6.2.1 The symmetric group Sn
216(2)
6.2.2 The dihedral group revisited
218(3)
6.2.3 Ringing the cosets
221(2)
6.2.4 Example: Plain Bob Doubles
223(4)
References for
Chapter 6
227(2)
7 Twelve-Tone Music
229(24)
7.1 Schoenberg's Twelve-Tone Method of Composition
229(6)
7.1.1 Notation and terminology
230(3)
7.1.2 The tone row matrix
233(2)
7.2 Schoenberg's Suite fur Klavier, Op. 25
235(3)
7.3 Tone Row Invariance
238(15)
7.3.1 Using numbers instead of pitches
241(1)
7.3.2 Further analysis: The symmetric interval property
242(3)
7.3.3 Tritone symmetry
245(3)
7.3.4 The number of distinct tone rows
248(1)
7.3.5 Twelve-tone music and group theory
249(3)
References for
Chapter 7
252(1)
8 Mathematical Modern Music
253(36)
8.1 Sir Peter Maxwell Davies: Magic Squares
253(15)
8.1.1 Magic squares
255(1)
8.1.2 Some examples
256(2)
8.1.3 The magic constant
258(2)
8.1.4 A Mirror of Whitening Light
260(8)
8.2 Steve Reich: Phase Shifting
268(10)
8.2.1 Clapping Music
271(5)
8.2.2 Phase shifts
276(2)
8.3 Xenakis: Stochastic Music
278(5)
8.3.1 A Greek architect
278(1)
8.3.2 Metastasis and the Philips Pavilion
279(1)
8.3.3 Pithoprakta: Continuity versus discontinuity
280(3)
8.4 Final Project: A Mathematical Composition
283(6)
References for
Chapter 8
287(2)
Credits 289(4)
Index 293
Gareth E. Roberts is an associate professor of mathematics at the College of the Holy Cross.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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