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The Mathematical Theory of Tone Systems : Chapman & Hall/CRC Pure and Applied Mathematics - Jan Haluska

The Mathematical Theory of Tone Systems

Chapman & Hall/CRC Pure and Applied Mathematics

Hardcover Published: 19th December 2003
ISBN: 9780824747145
Number Of Pages: 380

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The Mathematical Theory of Tone Systems patterns a unified theory defining the tone system in functional terms based on the principles and forms of uncertainty theory. This title uses geometrical nets and other measures to study all classes of used and theoretical tone systems, from Pythagorean tuning to superparticular pentatonics. Hundreds of examples of past and prevalent tone systems are featured. Topics include Fuzziness and Sonance, Wavelets and Nonspecificity, Pitch Granulation and Ambiguity, Equal Temperaments, Mean Tone Systems.
Well Tempered Systems, Ptolemy Systems, and more. Appendices include extended lists of tone systems and a catalogue of historical organs with subsemitones.

Prefacep. iii
List of tablesp. xvi
List of figuresp. xviii
List of symbolsp. xxi
List of intervalsp. xxiii
Fundamentalsp. 1
Tone systems and uncertainty theoryp. 3
Introductionp. 3
A piece of metaphysicsp. 3
Practical manners of the tone system descriptionp. 4
Current trendsp. 6
Americap. 7
Europep. 9
Africap. 13
Uncertainty-based information theoryp. 16
Classical harmonic analysis theoryp. 16
Significance of uncertaintyp. 20
Uncertainty forms and principlesp. 24
Fuzziness and sonancep. 27
Dissonance functionsp. 28
Relatedness tone system [left and right arrow] timbrep. 33
Direction: timbre to tone systemp. 33
Direction: tone system to timbrep. 34
Wavelets and nonspecificityp. 39
Time-frequency planep. 39
Gabor's time-frequency atomsp. 41
Lienard's time-frequency atomsp. 43
Wigner-Ville transformp. 44
Definition of the transformp. 44
Computation of Wigner-Ville transformsp. 45
Decomposition problemp. 49
A pseudodifferential calculusp. 49
Instantaneous frequencyp. 51
Wigner-Ville transform of asymptotic tonesp. 53
Return to the problem of optimal decomposition of tones into time-frequency atomsp. 54
Pitch granulation and ambiguityp. 57
Garbuzov zones: strifep. 57
An classical experimentp. 57
Psychological modelp. 61
Geometric netsp. 64
Z-chain conditionp. 64
Discrete geometric netsp. 68
Various net representations of a tone systemp. 71
The harmony-melody uncertaintyp. 72
Special Systemsp. 73
Equal temperamentsp. 75
Algebraic language and harmonyp. 75
Formal language for the theory of harmonyp. 75
Tone systems and chordsp. 77
Structure of chords and pointed chordsp. 79
Representations of harmonic structuresp. 81
Chord and tone rows enumerationsp. 87
Chordsp. 88
Tone rowsp. 90
Computation of numbers of fixed elementsp. 91
Measuring equal temperamentsp. 92
Continued fraction methodp. 92
Increasing number of keys per octavep. 94
Fifths of equal temperamentsp. 97
Equal temperaments in practicep. 98
Music of nationsp. 98
Present equal temperament inspired systemsp. 108
Mean tone systemsp. 117
Pythagorean Systemp. 117
Pythagorean minor and major semitonesp. 117
Images in the planep. 120
Fuzziness and beatsp. 123
Pythagorean System worldwidep. 126
Indian systemsp. 126
Europep. 129
Tone systems of Chinap. 131
On two algorithms in music acousticsp. 133
Description of the algorithmsp. 134
Isomorphism of the fifth and third tuningsp. 136
Pipe organs with subsemitones, 1468-1721p. 141
The Petzval's keyboardp. 146
Overview of meantonesp. 148
Era of meantonesp. 148
Meantone zonesp. 150
Further examples of mean-tone systemsp. 156
Anatomy of the Pythagorean whole tonep. 159
The second level: algebraic numbersp. 161
Semitone metric spacep. 168
Searching in transcendental numbersp. 169
Adaptive Pythagorean systemp. 171
Well tempered systemsp. 175
Well temperament periodsp. 175
Historical well temperamentsp. 175
More than 12p. 183
A revival of interest in the 20th centuryp. 192
Temperature and mistuningp. 192
Three types of temperaments--examplesp. 194
Harmonic mean based measuresp. 195
Possibility of free transpositionsp. 197
Basic musical intervals sound as purep. 198
We cannot avoid uncertaintyp. 198
Well tempered tone systemsp. 199
Symmetry and octave equivalencep. 199
The tempered fifth approximationsp. 200
Basic law of temperingp. 201
A formalization of well tempered tone systemsp. 202
The Petzval's tone systemsp. 202
The Petzval's Tone Systems Ip. 204
The Petzval's Tone Systems IIp. 210
Bimeasures and the last square methodp. 213
Optimal temperaturesp. 214
Optimal temperatures IIp. 216
12 and 10 granulationsp. 219
The major scale extensionsp. 219
Unimodular matrices and diatonic scalesp. 219
TDS geometric netsp. 224
Construction of generated tone systemsp. 229
Comment to superparticular ratiosp. 231
Classification of diatonic scalesp. 232
Application to partial monounary algebrasp. 237
10 granulationp. 245
Gamelanp. 245
Superparticular pentatonicsp. 249
Pacific Ocean region, the fuzzy tone systemsp. 257
Ptolemy Systemp. 261
Tetrachordsp. 261
The set of all tetrachordsp. 262
Tetrachord latticep. 265
Superparticular tetrachordsp. 266
Symmetry: Slendro versus 12 granulationp. 269
Ancient Greecep. 277
Ptolemy System in the 20th centuryp. 283
Ptolemy Systemp. 284
Set of all superparticular ratiosp. 286
Commasp. 291
Aestetics of ratios of small natural numbersp. 291
Mean tone aestheticsp. 293
Approximations of temperamentsp. 294
Pythagorean approximation of Just Intonationp. 297
Comma 32 805/32 768 (Schizma)p. 302
Bibliographyp. 306
Appendicesp. 319
Extended lists of tone systemsp. 319
Tetrachords (2 periods)p. 319
Pentatonicsp. 325
Heptatonicsp. 328
6, 8, 9, 10, 11 tones per octavep. 341
Dodekatonicsp. 350
More than 12 tones per octavep. 358
Historical organs with subsemitones, 1468-1721p. 369
Chronological overviewp. 370
Bibliographical sourcesp. 372
Indexp. 375
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780824747145
ISBN-10: 0824747143
Series: Chapman & Hall/CRC Pure and Applied Mathematics
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 380
Published: 19th December 2003
Publisher: Taylor & Francis Inc
Country of Publication: US
Dimensions (cm): 22.9 x 15.2  x 2.85
Weight (kg): 0.72
Edition Number: 1

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