Mathematics and Music : Composition, Perception, and Performance

334 Pages·2013·13.51 MB·English

by  Don

General Mathematics M
A
At first glance, mathematics and music seem to be from separate worlds—
T
one from science, one from art. But in fact, the connections between the two
H
go back thousands of years, such as Pythagoras’s ideas about how to quantify
changes of pitch for musical tones (musical intervals). Mathematics and Music: E
Composition, Perception, and Performance explores the many links between
M
mathematics and different genres of music, deepening your understanding of
music through mathematics. A
In an accessible way, the text teaches the basics of reading music and explains T
how various patterns in music can be described with mathematics. The authors I
extensively use the powerful time-frequency method of spectrograms to analyze C
the sounds created in musical performance. Numerous examples of music
S
notation assist you in understanding basic musical scores. The text also provides

mathematical explanations for musical scales, harmony, and rhythm and includes A
a concise introduction to digital audio synthesis. N
D
Features

• Provides thorough coverage of spectrograms for analyzing the sound of M
recorded music
• Gives equal emphasis to mathematics and music U
• Requires only high school mathematics and no prior experience reading
S
music
I
• Discusses harmony and rhythm, including the mathematical similarities of
C
pitch and rhythm
• Presents an in-depth yet accessible treatment of the Geometry of Harmony
as found in the Tonnetz
• Explains the principal methods of audio synthesis W
• Offers video demos, links to articles, music software, and musical scores on A
L
the book’s CRC Press web page
K
E
Along with helping you master some fundamental mathematics, this book gives
R
you a deeper appreciation of music by showing how music is informed by both its
•
mathematical and aesthetic structures.
D
O
N
K13012
K13012_Cover.indd 1 3/4/13 10:33 AM
MATHEMATICS
MUSIC
AND
COMPOSITION, PERCEPTION,
AND PERFORMANCE
K13012_FM.indd 1 2/28/13 12:52 PM
K13012_FM.indd 2 2/28/13 12:52 PM
MATHEMATICS
MUSIC
AND
COMPOSITION, PERCEPTION,
AND PERFORMANCE
JAMES S. WALKER • GARY W. DON
UNIVERSITY OF WISCONSIN
EAU CLAIRE, USA
K13012_FM.indd 3 2/28/13 12:52 PM
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2013 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20130410
International Standard Book Number-13: 978-1-4822-0850-4 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been
made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid-
ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright
holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this
form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may
rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti-
lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy-
ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the
publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://
www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,
978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For
organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
(cid:2) (cid:2)
(cid:2) (cid:2)
ToAngela
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) (cid:2)
Contents
Preface
Abouttheauthors
1 Pitch,Frequency,andMusicalScales 1
1.1 PitchandFrequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 InstrumentalTones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 PureTonesCombiningtoCreateanInstrumentalTone . . . . . . . . . . . . 4
1.2 Overtones,PitchEquivalence,andMusicalScales . . . . . . . . . . . . . . . . . . . 6
1.2.1 PitchEquivalence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 MusicalScales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 The12-ToneEqual-TemperedScale . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 MusicalScaleswithintheChromaticScale. . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1 TheC-MajorScale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.2 OtherMajorScales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.3 ScalesandClockArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.4 RelationbetweenJustandEqual-TemperedTunings . . . . . . . . . . . . . 20
1.5 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.1 HalfStepsandLogarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.2 Cents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 BasicMusicalNotation 33
2.1 StaffNotation,Clefs,andNotePositions . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.1 TrebleClefStaff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.2 BassClefStaff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.3 GrandStaff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 TimeSignaturesandTempo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.1 TimeSignatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.2 Tempo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.3 RhythmicEmphasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 KeySignaturesandtheCircleofFifths . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3.1 CircleofFifths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3.2 CircleofFifthsforNaturalMinorScales . . . . . . . . . . . . . . . . . . . 48
3 SomeMusicTheory 51
3.1 IntervalsandChords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 MelodicandHarmonicIntervals . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 DiatonicMusic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 LevelsofImportanceofNotesandChords . . . . . . . . . . . . . . . . . . 58
3.2.2 MajorKeys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.3 ChordProgressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.4 RelationofMelodytoChords . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.5 MinorKeys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.6 Chromaticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 DiatonicTransformations—ScaleShifts . . . . . . . . . . . . . . . . . . . . . . . 68
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) (cid:2)
3.4 DiatonicTransformations—Inversions,Retrograde . . . . . . . . . . . . . . . . . . 76
3.4.1 DiatonicScaleInversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4.2 Retrograde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5 ChromaticTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.1 Transpositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.2 ChromaticInversionsandRetrograde . . . . . . . . . . . . . . . . . . . . . 85
3.5.3 ChromaticTransformationsandChords . . . . . . . . . . . . . . . . . . . . 87
3.6 WebResources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 SpectrogramsandMusicalTones 93
4.1 MusicalGesturesinSpectrograms . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 MathematicalModelforMusicalTones . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.1 BasicTrigonometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.2 ModelingPureTones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3 ModelingInstrumentalTones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3.1 Beating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4 BeatingandDissonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.4.1 SomeUsesofDissonanceinMusic . . . . . . . . . . . . . . . . . . . . . . 108
4.5 EstimatingAmplitudeandFrequency . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5.1 HowtheEstimatingIsDone . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.6 WindowingtheWaveform:Spectrograms . . . . . . . . . . . . . . . . . . . . . . . 118
4.7 ADeeperStudyofAmplitudeEstimation . . . . . . . . . . . . . . . . . . . . . . . 122
4.7.1 MoreonRectangularWindowing . . . . . . . . . . . . . . . . . . . . . . . 126
4.7.2 MoreonBlackmanWindowing . . . . . . . . . . . . . . . . . . . . . . . . 128
5 SpectrogramsandMusic 131
5.1 Singing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.1.1 AnOperaticPerformancebyRenéeFleming . . . . . . . . . . . . . . . . . 131
5.1.2 AnOperaticPerformancebyLucianoPavarotti . . . . . . . . . . . . . . . . 135
5.1.3 ABluesPerformancebyAliciaKeys . . . . . . . . . . . . . . . . . . . . . 135
5.1.4 AChoralPerformancebySweetHoneyintheRock . . . . . . . . . . . . . . 136
5.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 Instrumentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.2.1 JazzTrumpet:LouisArmstrong . . . . . . . . . . . . . . . . . . . . . . . . 141
5.2.2 Beethoven,Goodman,andHendrix . . . . . . . . . . . . . . . . . . . . . . 143
5.2.3 HarmonicsinStringedInstruments . . . . . . . . . . . . . . . . . . . . . . 145
5.3 Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.3.1 RoyHargrove’sStrasbourg/St.Denis . . . . . . . . . . . . . . . . . . . . . 153
5.3.2 TheBeatles’TomorrowNeverKnows . . . . . . . . . . . . . . . . . . . . . 154
5.3.3 APortionofMortonFeldman’sTheRothkoChapel . . . . . . . . . . . . . . 155
5.3.4 APortionofaRaviShankarComposition . . . . . . . . . . . . . . . . . . . 156
5.3.5 MusicalIllusions:LittleBoyandTheDevil’sStaircase . . . . . . . . . . . . 157
5.3.6 TheFinaleofStravinsky’sFirebirdSuite . . . . . . . . . . . . . . . . . . . 161
5.3.7 DukeEllington’sJacktheBear . . . . . . . . . . . . . . . . . . . . . . . . 162
5.3.8 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6 AnalyzingPitchandRhythm 167
6.1 GeometryofPitchOrganizationandTranspositions . . . . . . . . . . . . . . . . . . 167
6.1.1 PitchClasses,Intervals,Chords,andScales . . . . . . . . . . . . . . . . . . 167
6.1.2 Transpositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1.3 ClockArithmeticFormallyDefined . . . . . . . . . . . . . . . . . . . . . . 170
6.2 GeometryofChromaticInversions . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) (cid:2)
(cid:2) (cid:2)
6.3 CyclicRhythms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.3.1 CyclicRhythminDownintheValley . . . . . . . . . . . . . . . . . . . . . 177
6.3.2 CyclicRhythminDrumming. . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.3.3 TimeTranspositionsofCyclicRhythms . . . . . . . . . . . . . . . . . . . . 178
6.3.4 Afro-LatinClaveRhythms . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.3.5 Phasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.4 RhythmicInversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.4.1 InversionofCyclicRhythms . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.4.2 RhythmicandPitchTransformationGroups . . . . . . . . . . . . . . . . . . 189
6.5 ConstructionofScalesandCyclicRhythms . . . . . . . . . . . . . . . . . . . . . . 192
6.5.1 TheEuclideanAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.5.2 ConstructingMusicalScales . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.5.3 ConstructingCyclicRhythms . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.6 ComparingMusicalScalesandCyclicRhythms . . . . . . . . . . . . . . . . . . . . 198
6.6.1 IntervalFrequenciesandMusicalScales . . . . . . . . . . . . . . . . . . . . 198
6.6.2 MeasuringDissonanceforaScale . . . . . . . . . . . . . . . . . . . . . . . 201
6.6.3 IntervalFrequenciesforCyclicRhythms . . . . . . . . . . . . . . . . . . . 202
6.7 Serialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.7.1 PitchSerialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.7.2 MusicalMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.7.3 TotalSerialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7 AGeometryofHarmony 217
7.1 Riemann’sChromaticInversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.2 ANetworkofTriadicChords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.2.1 MusicalExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.3 EmbeddingPitchClasseswithintheTonnetz . . . . . . . . . . . . . . . . . . . . . . 228
7.3.1 GeometryofAcousticConsonanceandDissonance . . . . . . . . . . . . . . 229
7.3.2 AnalyzingScales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.4 OtherChordalTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.4.1 UsingtheTonnetztoDefineMoreTransformations . . . . . . . . . . . . . . 233
7.4.2 ModelingChordProgressionsinDiatonicMusic . . . . . . . . . . . . . . . 235
7.4.3 ExplainingtheQualitativeDifferenceinModes . . . . . . . . . . . . . . . . 237
8 AudioSynthesisinMusic 241
8.1 CreatingNewMusicfromSpectrograms . . . . . . . . . . . . . . . . . . . . . . . . 241
8.2 PhaseVocoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.2.1 ABasicExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.2.2 ImogenHeap’sHideandSeek . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.3 TimeStretchingandTimeShrinking . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.4 MIDISynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
A ExerciseSolutions 257
B MusicSoftware 295
B.1 AUDACITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
B.1.1 ConfiguringAUDACITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
B.1.2 LoadingandDisplayingaMusicFile . . . . . . . . . . . . . . . . . . . . . 295
B.1.3 MusicFilesfromCDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
B.2 MUSESCORE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
B.2.1 DifferentSoundfontsforMUSESCORE . . . . . . . . . . . . . . . . . . . . 296
(cid:2) (cid:2)
(cid:2) (cid:2)