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The Structure of Recognizable Diatonic Tunings

The Structure of Recognizable Diatonic Tunings

Copyright Date: 1985
Pages: 328
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  • Book Info
    The Structure of Recognizable Diatonic Tunings
    Book Description:

    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.

    Originally published in 1986.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-5407-3
    Subjects: Music

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-viii)
  3. Preface
    (pp. ix-2)
  4. I Fundamental Properties of Musical Intervals
    (pp. 3-21)

    As we shall eventually see (Chs. III, IV, VIII, IX, and X), the structure of recognizable diatonic tunings is basically an array of intricate interconnections among acceptable approximations to intervals, certain of which may be tuned individually so as to be free of beats produced by interacting harmonics. These interconnections, which are the very foundation of what is perceived as tonal harmonic motion, are shaped by the short-term span of human memory, the tolerance range of the human ear (Ch. X), and the peculiar manner in which intervals are perceived. This chapter is concerned primarily with the latter subject. Section...

  5. II The Diatonic Scale in Pythagorean Tuning
    (pp. 22-30)

    The diatonic scale, with only slight variations in tuning, has served as the essential organizing element of the notes of multi-part music for over eight hundred years. There exists an infinite number of tunings for the diatonic scale in which conventional tonal and modal harmonic motion is recognizable (Ch. X), and we shall ultimately explore the properties of the entire family of recognizable diatonic tunings. But the nature of the organization is quite complex, and is best explored initially within the confines of one particular tuning. The tuning most readily comprehensible at the beginning, and presenting relatively few computational difficulties,...

  6. III Names and Distributional Patterns of the Diatonic Intervals
    (pp. 31-48)

    In this chapter, we take up a systematic investigation of those intricate interconnections among intervals that will eventually be shown to apply to the entire family of recognizable diatonic tunings. The subject is best described by the algebra of congruences and residues, and since this branch of number theory is generally not included in standard texts and the reader is likely to be unfamiliar with it, we give in Section 1 such elementary aspects of the subject as are needed in the theory of diatonic scales. In Section 2 we derive a pair of formulas which relate the scale to...

  7. IV Extended Pythagorean Tuning
    (pp. 49-66)

    We now extend the basic concepts of Pythagorean tuning beyond the confines of notes 0 through 6. In Section 1 we present a precise description of the conventional musical symbols ♯ and ♭, demonstrating that a flat is an inverse sharp, and Section 2 shows how to compute the frequency of any pitch in Pythagorean tuning within the octave starting at middle C, and how to determine its distance from middle C in cents. Section 3 treats extensively the names of certain chromatic intervals, and shows how to derive the name of the interval, given the names of the notes...

  8. V The Diatonic Major Scale in Just Tuning
    (pp. 67-90)

    Just tuning of the major scale results from an attempt to remove the impurity of the major and minor thirds and sixths produced by Pythagorean tuning (Ch. III, Sec. 7). As in the discussion of Pythagorean tuning, we develop the theory for one diatonic system—C major—and then extend the tuning to include other keys. In Section 1 we extend the conventional nomenclature of notes so as to reveal at a glance differences between pure and Pythagorean intervals. In Section 2 it is shown that pure tuning of the primary triads of C major does not give a tuning...

  9. VI Extended Just Tuning
    (pp. 91-128)

    The question of extended just tuning involves the tuning of the entire family of major and minor keys, the just tuning of chromatic intervals and chords where such exists, and a detailed study of the resulting maze of commas.

    In Section 1 we explore the properties of the just tuning of the family of major keys, using the techniques developed during the investigation of extended Pythagorean tuning (Ch. IV, Sec. 6) in conjunction with the subscript notation used to describe the just tuning of C major (Ch. V, Secs. 1–3). Section 2 deals formally with linear relations connecting five...

  10. VII Musical Examples in Just Tuning
    (pp. 129-153)

    If just tuning is taken to mean the most euphonious possible arrangement of all simultaneous combinations of notes, then it follows that any musical passage has a just tuning. On the other hand, as we have seen, certain chords, or combinations of notes, cannot be tuned in such a manner as to consist solely of pure intervals (Ch. V, Sec. 6; Ch. VI, Secs. 6 and 7).

    The problem of determining the just tuning for a given musical fragment is generally solvable by the application of principles developed in Chapters V and VI, and involves attaching the appropriate subscripts to...

  11. VIII The Diatonic Scale in Meantone Tuning
    (pp. 154-164)

    Meantone tuning, like just tuning, is a device whose purpose is to correct the unmusical impurities associated with Pythagorean triads. Unlike just tuning, in which the triads are pure, meantone tuning allows a slight impurity, which makes possible the elimination of the disturbing melodic discontinuities associated with the syntonic and septimal commas. In Section 1 we see how a slight but noticeable distortion of perfect fifths CG, GD, DA, and AE produces pure tuning of major third CE, and also removes the distinction of the major and minor tones of just tuning. Section 2 defines the diatonic scale in a...

  12. IX Extended Meantone Tuning
    (pp. 165-192)

    The theory of extended meantone tuning is a matter of great practical importance, for there is abundant historical evidence that meantone tuning was the preferred tuning for keyboard instruments during the period 1550 to 1650. As in the case of the diatonic intervals, we develop the theory of the meantone chromatic intervals and chromatic scale from what was discovered with regard to extended Pythagorean tuning. Before continuing, the reader should review the contents of Chapter IV.

    In Section 1 we extend the notion of the unending line of pure perfect fifths to meantone fifths, and show how to calculate the...

  13. X The General Family of Recognizable Diatonic Tunings
    (pp. 193-220)

    In this chapter, we determine the range over which the generating interval may vary to produce a diatonic tuning that exhibits the same subjective character and structural organization as Pythagorean tuning and meantone tuning. This also serves to frame precise definitions of the names and symbols associated with notes and intervals. In Section 1 it is shown how a small change in the size of the generating interval affects the triads and the scale, and suggests that an improvement in the former causes a deterioration in the latter, and vice versa. Section 2 determines that the range over which the...

  14. XI Equal Tunings and Closed Circles of Fifths
    (pp. 221-260)

    In this chapter, we investigate a special class of recognizable diatonic tunings—those in which the perfect fifths ultimately form a closed circle. In Section 1 it is shown that this property depends solely on whetherRandvare rational numbers. In Section 2 we demonstrate thatRand the ratio of the perfect fifth are incommensurable, and consequently irrational numbers are associated with all recognizable diatonic tunings in some respect. In Section 3 we show that if the notes forming any closed circle of intervals are reproduced in all registers by octave transpositions, the result is an equal...

  15. XII The Diatonic Equal Tunings
    (pp. 261-314)

    The subject of this final chapter is the distribution and behavior of certain equal tunings that contain diatonic scales or approximations to just tuning. The discussion makes extensive use of the circle of fifths theorem (Theorem 40; Ch. IX, Sec. 5), with which the reader should be thoroughly familiar.

    Section 1 shows how all the diatonic equal tunings may be regarded as temperaments, and Section 2 describes the behavior of the principal commas of just tuning (syntonic comma, Pythagorean comma, schisma, diaschisma, and diesis) within the various temperaments with respect toR. Before studying Section 2, the reader will find...

  16. Index
    (pp. 315-318)
  17. Back Matter
    (pp. 319-319)