| Contents |
I. Fundamental properties of musical intervals. Introduction ; The unique determination of musical intervals by frequency ratios ; Pure tuning, notes, and pitches ; To determine the beat frequency of an impure interval ; The unique prime factorization theorem ; Basic intervals and the interval size convention ; Sizes of the basic intervals in terms of cents ; To express an interval, given its ratio, as a combination of the first three basic intervals ; To find the size of an interval in cents, given its ratio, and vice versa -- II. The diatonic scale in Pythagorean tuning. Introduction ; Definition of diatonic scale ; Certain intervals as determined uniquely by the coefficient of v̄ ; Combinations of intervals expressed as int (i) ; The family of Pythagorean diatonic intervals ; Order of notes in the diatonic scale ; Distribution of the adjacent intervals in the diatonic scale -- III. Names and distributional patterns of the diatonic intervals. Introduction ; Congruences and residues ; The diatonic scale regarded as a sequence of least residues ; The conventional names of intervals ; The generating array viewed as a broken circle of fifths ; The two modalities of diatonic intervals ; The conventional names of notes and intervals ; Pythagorean thirds and the syntonic comma -- IV. Extended Pythagorean tuning. Introduction ; The sharp and the flat ; Frequencies of pitches in extended Pythagorean tuning ; Chromatic intervals and their names ; Exceptions to the interval size convention ; The Pythagorean comma ; Major keys, minor keys, and their signatures ; Key signatures and transposition -- V. The diatonic scale in just tuning. Introduction ; Pure tuning of the primary triads ; The diatonic intervals in just tuning ; Rectification of the second-degree triad ; Distribution of impure intervals within various just tunings ; Rectification of the dominant seventh chord and the seventh-degree triad ; Pure tuning of the major dominant ninth and secondary seventh chords -- VI. Extended just tuning. Introduction ; Just tuning of the family of major keys ; The diesis, the schisma, and the diaschisma ; The schismatic major third ; Just tuning of the family of minor keys ; Minor seconds, chromatic semitones, and commas ; Augmented triads, diminished sevenths, half-diminished sevenths, and secondary dominants ; The fifth and sixth basic intervals and their relation to altered dominant seventh chords and whole-tone scales ; Resume of the intervals and frequencies of just tuning -- |
| Contents |
VII. Musical examples in just tuning. Introduction ; Guillaume de Machaut (1284-1370), Kyrie from Messe de nostre Dame ; Orlando di Lasso (1530-1594), motet Ave regina coelorum ; J. S. Bach (1685-1750), Prelude in E [flat] major, Book 1, Well-Tempered Clavier ; César Franck (1822-1890), Symphony in D minor ; Other examples and conclusions -- VIII. The diatonic scale in meantone tuning. Introduction ; The meantone perfect fifth ; The diatonic meantone intervals expressed as int (i) ; The major scale and the names of the diatonic meantone intervals ; Diatonic seventh chords and the major scale -- IX. Extended meantone tuning. Introduction ; Names of notes and numerical data ; Chromatic intervals in meantone tuning ; The meantone chromatic scale ; Major keys, minor keys, triads, and wolves ; Procedure for putting a harpsichord or an organ in meantone tuning ; Musical examples in meantone tuning ; Werckmeister's tuning -- X. The general family of recognizable diatonic tunings. Introduction ; Change in size of the diatonic intervals relative to the amount of tempering applied to the perfect fifth ; The range of recognizability and its relation to the size of the perfect fifth ; Expressions for intervals and for the range of recognizability in terms of the major second and the minor second ; 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 ; Expressions for intervals and definitions of notes in terms of R ; Silbermann's one-sixth comma temperament -- XI. Equal tunings and closed circles of fifths. Introduction ; Necessary and sufficient conditions for a particular extended diatonic tuning eventually to produce a closed circle of fifths ; Diatonic tunings and irrational numbers ; The equal tuning theorem ; Representation of certain equal tunings by the conventional musical notation ; The circle of fifths theorem ; Diatonic and chromatic intervals in 12-note equal tuning ; Recommended procedure for putting a piano in 12-note equal tuning ; Application of the conventional musical notation to the diatonic tunings where R = 3/2 and R = 3 ; Equal tunings that do not contain recognizable diatonic scales ; Behavior of the circle of fifths when R is a fraction not in its lowest terms -- XII. The diatonic equal tunings. Introduction ; The diatonic equal tunings regarded as temperaments ; Representation of certain of the commas of just tuning in the diatonic equal tunings ; Numerical data pertaining to the diatonic equal tunings of fewer than thirty-six notes ; Distribution of various versions of recognizable and unrecognizable diatonic scales within the equal tunings ; Equal tunings that are successively closer approximations to Pythagorean tuning, meantone tuning, and Silbermann's tuning ; Equal tunings that contain approximations to just tuning ; Enharmonic modulations and modulating sequences. |
| Abstract |
In a comprehensive work with important implications for tuning theory and musicology, Easley Blackwood, a distinguished-composer, establishes a mathematical basis for the family of diatonic tunings generated by combinations of perfect fifths and octaves. |
| General note | Includes index. |
| LCCN | 85042972 |
| ISBN | 0691091293 (alk. paper) : |
