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 3. J. S. 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