The structure of recognizable diatonic tunings:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton, N.J.
Princeton Univ. Pr.
1985
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 318 S. |
| ISBN: | 0691091293 |
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MARC
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| 245 | 1 | 0 | |a The structure of recognizable diatonic tunings |
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Record in the Search Index
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| adam_text | EASLEY BLACKWOOD
The Structure of
Recognizable
Diatonic Tunings
PRINCETON UNIVERSITY PRESS
Contents
Preface
Chapter I Fundamental Properties of Musical Intervals 3
Introduction 3
1 The unique determination of musical intervals by frequency
ratios 3
2 Pure tuning, notes, and pitches 5
3 To determine the beat frequency of an impure interval 7
4 The unique prime factorization theorem 8
5 Basic intervals and the interval size convention 11
6 Sizes of the basic intervals in terms of cents 13
7 To express an interval, given its ratio, as a combination of
the first three basic intervals IS
8 To find the size of an interval in cents, given its ratio, and
vice versa 16
Chapter II The Diatonic Scale in Pythagorean Tuning 22
Introduction 22
1 Definition of diatonic scale 23
2 Certain intervals as determined uniquely by the coefficient of v 23
3 Combinations of intervals expressed as int (Q 26
4 The family of Pythagorean diatonic intervals 28
5 Order of notes in the diatonic scale 29
6 Distribution of the adjacent intervals in the diatonic scale 30
Chapter III Names and Distributional Patterns of the
Diatonic Intervals 31
Introduction 31
1 Congruences and residues 31
2 The diatonic scale regarded as a sequence of least residues 35
3 The conventional names of intervals 36
4 The generating array viewed as a broken circle of fifths 38
5 The two modalities of diatonic intervals 40
6 The conventional names of notes and intervals 43
7 Pythagorean thirds and the syntonic comma 46
vi CONTENTS
Chapter IV Extended Pythagorean Tuning 49
Introduction 49
1 The sharp and the flat 49
2 Frequencies of pitches in extended Pythagorean tuning 51
3 Chromatic intervals and their names S3
4 Exceptions to the interval size convention S7
5 The Pythagorean comma 58
6 Major keys, minor keys, and their signatures 59
7 Key signatures and transposition 65
Chapter V The Diatonic Scale in Just Tuning 67
Introduction 67
1 Pure tuning of the primary triads 68
2 The diatonic intervals in just tuning 69
3 Rectification of the second-degree triad 73
4 Distribution of impure intervals within various just tunings 75
5 Rectification of the dominant seventh chord and the seventh-
degree triad 81
6 Pure tuning of the major dominant ninth and secondary
seventh chords 85
Chapter VI Extended Just Tuning 91
Introduction 91
1 Just tuning of the family of major keys 92
2 The diesis, the schisma, and the diaschisma 94
3 The schismatic major third 97
4 Just tuning of the family of minor keys 100
5 Minor seconds, chromatic semitones, and commas 103
6 Augmented triads, diminished sevenths, half-diminished
sevenths, and secondary dominants 106
7 The fifth and sixth basic intervals and their relation to altered
dominant seventh chords and whole-tone scales 110
8 Resume of the intervals and frequencies of just tuning 115
Chapter VII Musical Examples in Just Tuning 129
Introduction 129
1 Guillaume de Machaut (1284-1370), Kyrie from Messe de
nostre Dame 129
2 Orlando di Lasso (1530-1594), motet Ave regina coelorum 133
3JS Bach (1685-1750), Prelude in Eb major, Book 1,
Well-Tempered Clavier 139
4 Cesar Franck (1822-1890), Symphony in D minor 142
5 Other examples and conclusions 150
CONTENTS vii
Chapter VIII The Diatonic Scale in Meantone Tuning 154
Introduction 154
1 The meantone perfect fifth 154
2 The diatonic meantone intervals expressed as int (i) 156
3 The major scale and the names of the diatonic meantone
intervals 159
4 Diatonic seventh chords and the major scale 162
Chapter IX Extended Meantone Tuning 165
Introduction 165
1 Names of notes and numerical data 165
2 Chromatic intervals in meantone tuning 169
3 The meantone chromatic scale 172
4 Major keys, minor keys, triads, and wolves 175
5 Procedure for putting a harpsichord or an organ in meantone
tuning 176
6 Musical examples in meantone tuning 179
7 Werckmeister s tuning 187
Chapter X The General Family of Recognizable Diatonic
Tunings
Introduction 193
1 Change in size of the diatonic intervals relative to the amount
of tempering applied to the perfect fifth 194
2 The range of recognizability and its relation to the size of the
perfect fifth 195
3 Expressions for intervals and for the range of recognizability
in terms of the major second and the minor second 204
4 The nature and character of a diatonic tuning as a function
of the ratio of the size of the major second to the size of the
minor second 208
5 Expressions for intervals and definitions of notes in terms of R 212
6 Silbermann s one-sixth comma temperament 215
Chapter XI Equal Tunings and Closed Circles of Fifths 221
Introduction 221
1 Necessary and sufficient conditions for a particular extended
diatonic tuning eventually to produce a closed circle of fifths 222
2 Diatonic tunings and irrational numbers 224
3 The equal tuning theorem 226
4 Representation of certain equal tunings by the conventional
musical notation 228
5 The circle of fifths theorem 233
6 Diatonic and chromatic intervals in 12-note equal tuning 239
viii CONTENTS
7 Recommended procedure for putting a piano in 12-note
equal tuning 244
8 Application of the conventional musical notation to the
diatonic tunings where R=f and R=3 247
9 Equal tunings that do not contain recognizable diatonic
scales 254
10 Behavior of the circle of fifths when R is a fraction not in its
lowest terms 257
Chapter XII The Diatonic Equal Tunings 261
Introduction 261
1 The diatonic equal tunings regarded as temperaments 261
2 Representation of certain of the commas of just tuning in the
diatonic equal tunings 265
3 Numerical data pertaining to the diatonic equal tunings of
fewer than thirty-six notes 267
4 Distribution of various versions of recognizable and
unrecognizable diatonic scales within the equal tunings 278
5 Equal tunings that are successively closer approximations to
Pythagorean tuning, meantone tuning, and Silbermann s tuning 282
6 Equal tunings that contain approximations to just tuning 292
7 Enharmonic modulations and modulating sequences 306
Index 315
|
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| isbn | 0691091293 |
| language | English |
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| physical | VIII, 318 S. |
| publishDate | 1985 |
| publishDateSearch | 1985 |
| publishDateSort | 1985 |
| publisher | Princeton Univ. Pr. |
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| spelling | Blackwood, Easley Verfasser aut The structure of recognizable diatonic tunings Princeton, N.J. Princeton Univ. Pr. 1985 VIII, 318 S. txt rdacontent n rdamedia nc rdacarrier Intervallen (muziek) gtt Stemmingen (muziek) gtt Musik Music Acoustics and physics Musical intervals and scales Musical temperament Tuning Diatonik (DE-588)4212082-2 gnd rswk-swf Stimmung Musik (DE-588)4122368-8 gnd rswk-swf Diatonik (DE-588)4212082-2 s Stimmung Musik (DE-588)4122368-8 s DE-604 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000395482&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Blackwood, Easley The structure of recognizable diatonic tunings Intervallen (muziek) gtt Stemmingen (muziek) gtt Musik Music Acoustics and physics Musical intervals and scales Musical temperament Tuning Diatonik (DE-588)4212082-2 gnd Stimmung Musik (DE-588)4122368-8 gnd |
| subject_GND | (DE-588)4212082-2 (DE-588)4122368-8 |
| title | The structure of recognizable diatonic tunings |
| title_auth | The structure of recognizable diatonic tunings |
| title_exact_search | The structure of recognizable diatonic tunings |
| title_full | The structure of recognizable diatonic tunings |
| title_fullStr | The structure of recognizable diatonic tunings |
| title_full_unstemmed | The structure of recognizable diatonic tunings |
| title_short | The structure of recognizable diatonic tunings |
| title_sort | the structure of recognizable diatonic tunings |
| topic | Intervallen (muziek) gtt Stemmingen (muziek) gtt Musik Music Acoustics and physics Musical intervals and scales Musical temperament Tuning Diatonik (DE-588)4212082-2 gnd Stimmung Musik (DE-588)4122368-8 gnd |
| topic_facet | Intervallen (muziek) Stemmingen (muziek) Musik Music Acoustics and physics Musical intervals and scales Musical temperament Tuning Diatonik Stimmung Musik |
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