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Originally published in 1986.
The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover 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.
Product Details
| ISBN-13: | 9780691638935 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 04/19/2016 |
| Series: | Princeton Legacy Library , #327 |
| Pages: | 330 |
| Product dimensions: | 6.20(w) x 9.30(h) x 1.00(d) |
Read an Excerpt
The Structure of Recognizable Diatonic Tunings
By Easley Blackwood
PRINCETON UNIVERSITY PRESS
Copyright © 1985 Princeton University PressAll rights reserved.
ISBN: 978-0-691-09129-7
CHAPTER 1
Fundamental Properties of Musical Intervals
Introduction. 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 1 deals with the ear's logarithmic perception of pitch increments; in Section 2 we investigate the elementary aspects of the most euphonious intervals — the pure intervals — with particular attention to the unique property possessed by the octave. In Section 3 we derive a method for computing the beat frequency associated with intervals that depart slightly from the pure tuning. In Section 4 it is revealed that pure intervals cannot necessarily be expressed simply in terms of each other, as might have been supposed, or hoped. In Section 5 we define a new class of intervals in terms of which the pure intervals may be expressed rationally, and we investigate and describe the fundamentally important interval size convention. Section 6 describes the conventional method of measuring intervals in terms of a fixed unit, the cent. Section 7 shows how to express certain commonly occurring intervals in terms of the first few basic intervals — a technique that has extensive application in the theory of just tuning (Chs. V, VI, and VII). And finally, Section 8 gives several methods for determining the size in cents of an interval from its ratio, and vice versa. Also included is a short logarithmic table to facilitate computation if electronic means are unavailable.
1. The unique determination of musical intervals by frequency ratios.
Before we begin the systematic investigation of our elusive subject, it will be helpful to recall certain elementary facts concerning the human perception of musical sounds. We need not concern ourselves, however, with such questions as why a periodic disturbance in the atmosphere may be perceived as a tone, or why a higher frequency corresponds to a higher pitch. The purpose of our study is not to tackle problems of a metaphysical or biopsychological nature; rather it is to depict in mathematical terms, insofar as possible, the behavior of certain musical elements that we take as given.
The idea of a relation between two tones gives rise to the concept of an interval; this word may be defined as the distance separating two pitches. Thus an interval may be thought of as two pitches heard in succession, or simultaneously. In order for two intervals to have the same sound other than a difference in register, it is both necessary and sufficient that the ratios of the frequencies forming them should be equal. There is no known reason or proof for this statement; it is simply a fact which may be observed, and is to be regarded as an axiom. Now let there be two intervals, I1 and I2, the first composed of frequencies f1 and f2, and the second of frequencies f3 and f4. Let r1 and r2 be numbers such that f2/f1 = r1, and f4/f3 = r2. The axiom may now be stated as follows: intervals I1 and I2 are identical except for a difference in register if, and only if, r1 = r2.
Let us now consider the slightly different situation in which I1 and I2 are contiguous — that is, the higher pitch of I1 coincides with the lower pitch of I2 — and call the three pitches involved f1, f2, and f3, as in Figure 1. Clearly there exists another interval I3 formed by pitches f1 and f3, and it is in accord with intuition to write I1 + I2 = I3. In the present case, the frequency ratios are given by f2/f1 = r1, and f3/f2 = r2, or f2 = r1f1, and f3 = r2f2; and hence f3 = r2(r1f1) = r1r2f1. Let us now denote the ratio of interval I3 by r3 — that is, f3/f1 = r3, or f3 = r3f1. If in this equation we now substitute r1r2f1 for f3, it follows at once that r1r2f1 = r3f1, and also (since f1 ≠ 0) that r1r2 = r3. We may now state a fundamental principle proceeding directly from the axiom previously stated: The arithmetic process corresponding to the intuitive notion of adding two intervals is the multiplication of their frequency ratios. Now if r1r2 = r3, then log r1 + log r2 = log r3; thus I1 = log r1, I2 = log r2, and I3 = log r3. Hence I1, I2, and I3 are numbers, and we may now state that the number corresponding to the intuitive notion of the size of an interval is the logarithm of its frequency ratio.
Obviously the frequency of any pitch is a positive number, hence the case in which r1< 0 has no musical meaning. Now if r1 > 1, then 0 < f1< f2, and the interval is depicted as ascending. And if 0 < r1< 1, then o < f2< f1, and the interval is depicted as descending. Thus for an ascending interval, log r1 is positive, and for a descending interval, log r1 is negative. Of course it is necessary to observe strictly this convention regarding sign.
Example: Determine the ratio of the interval formed by adding the two intervals whose ratios are 5/4 and 6/5. We have (5/4)(6/5) = 3/2, and so the sum is the interval whose ratio is 3/2.
Example: Find the ratio of the interval formed by subtracting the interval whose ratio is 5/4 from the one whose ratio is 5/3. This difference may be seen to be the sum of the two intervals if we construe the latter as ascending and the former as descending. Hence we have (5/3)(4/5) = 4/3, and the resulting difference has the ratio 4/3.
2. Pure tuning, notes, and pitches. It has long been noted that the common harmonious, consonant intervals are more or less accurately represented by frequency ratios involving small integers. Thus the ratios 2/1, 3/2, 4/3, 5/3, 5/4, and 6/5 correspond to those intervals known in musical terms as an octave, perfect fifth, perfect fourth, major sixth, major third, and minor third, respectively. Here we should pause for a moment, however. It is not precise to use musical terms until exact definitions have been framed. For example, a "fifth" would seem to contain five units, but it is not yet apparent what these units are. So for the moment, instead of "perfect fifth," we shall use the more cumbersome phrase "the interval whose ratio is 3/2," and similarly for the other terms above. We make an exception in the case of the octave, this word being universally understood to mean the interval whose ratio is 2/1.
We now come to another phenomenon for which there is no known reason or explanation, and which must be simply observed and taken as self-evident — the octave possesses unique musical characteristics. Specifically, two pitches separated by an octave (or any number of octaves) are perceived as the same note, but in different registers. This gives rise to a distinction between pitch and note which we shall observe scrupulously. A pitch is a tone of a specific frequency, say f, but a note may be any of the several pitches whose frequencies are given by f(2n), where n is an integer or zero. It is important to observe that musical terminology generally names notes rather than pitches. When it is desired to specify a pitch, both the name of the note and the register must be given. Thus "middle C" is a pitch but "c" is a note — it may be anyone of the eight or so pitches within the commonly used musical range of roughly 30–4300 Hz. That is, if we take middle C to be 264 Hz, we apply the name C also to the pitches of 33, 66, 132, 528, 1,056, 2,112, and 4,224 Hz. All these numbers are obtained by multiplying or dividing the frequency of middle C by integral powers of 2.
When an interval whose ratio is composed of small integers deviates slightly from the exact value, it maintains its recognizable character, but takes on a rougher sound when the pitches forming it are played simultaneously. This is partly due to the presence of harmonics in the pitches themselves (it is assumed that the reader is familiar with this phenomenon), and partly to nonlinearities in the middle ear which produce both harmonic and intermodulation distortion. As an example, consider the interval formed by pitches of 200 Hz and 300 Hz. This interval has the sound of a smooth, exactly tuned "perfect fifth." The interval formed by pitches of 200 Hz and 302 Hz also sounds like a "perfect fifth," but trained musicians find it to be slightly out of tune. Inparticular, the third harmonic of the lower pitch (600 Hz) interacts with the second harmonic of the higher pitch (604 Hz) to produce the subjective effect known as beats. The beat frequency is the difference between the interacting frequencies, in this case 4 Hz. The response characteristics of the ear cause the beat frequency to be perceived as a faint wavering sound which seems to have no point of origin. The presence of beating intervals has generally been regarded as undesirable, especially in music in which major triads are treated as consonances. Intervals tunes so as to be free of beats are said to be tuned purely.
The prominence of beats produced by an impure interval varies greatly from situation to situation, and is much influenced by instrumental timbre. The beat frequency is important theoretically, for it provides an accurate indicator of how subjectively out of tune an impure interval is, even in situations where the beats themselves are virtually inaudible. For this purpose, it is necessary to fix the lower pitch in every case at an arbitrary reference point, and we shall use middle C as the standard.
The present writer is convinced that many earlier theorists have attached far too much importance to the desirability of tuning the most consonant intervals purely. Even so, it is almost certainly true that traditional tonal and pretonal polyphonic music is hardly enhanced by the presence of impure intervals. All in all, it is not unreasonable to proceed to the construction of scales and tuning systems for the existing polyphonic repertoire upon the assumption that impure intervals are undesirable. However, as will be seen (Ch. V, Sec. 4), it is impossible to construct scales that contain no impure intervals at all.
3. To determine the beat frequency of an impure interval. In the previous section, it was observed that the interval formed by frequencies of 200 Hz and 302 Hz produces beats at a rate that is the difference between the frequency of the third harmonic of the lower pitch (600 Hz) and the second harmonic of the higher pitch (604 Hz). More generally, an impure interval whose ratio is close to 3/2 produces beats through an interaction of these same two harmonics, and the difference of their frequencies is the beat frequency. Let the frequency of the lower pitch be f; then the frequency of its third harmonic is 3f. If the ratio of the interval is r, then the frequency of the higher pitch is rf, and the frequency of its second harmonic is 2rf. The beat frequency is thus 2rf - 3f = f(2r - 3). Note that this quantity is positive if r > 3/2, and negative if r< 3/2. It will be convenient to adopt the convention that the beat frequency of an interval smaller than pure is given by a negative number.
Example: Determine the beat frequency of the interval whose ratio is 40/27 and whose lower pitch is middle C (264 Hz). Since 40/27 is only a little less than 3/2 (the value of six decimals is 1.481481), the beat frequency is 264[(2)(1.481481) - 3] = (264)(-.037037) = -9.778 per second.
Now let the ratio of any slightly impure interval be r where r > 1, and let r be nearly equal to p/q, the ratio of the pure version. The beat frequency of such an interval generally is the difference between the qth harmonic of the higher pitch and the pth harmonic of the lower pitch. If f is the frequency of the lower pitch, then rf is the frequency of the higher pitch. The qth harmonic of the higher pitch is then qrf, the pth harmonic of the lower pitch is pf, and the difference is f(qr - p). This number is positive if r >p/q, and negative if r< p/q. To summarize, the beat frequency of a slightly impure interval whose ratio is r may be either f(r - 2), f(2r - 3), f(3r - 4), f(3r - 5), f(4r - 5), or f(5r - 6), depending on whether r is nearly 2, 3/2, 4/3, 5/3, or 6/5, respectively, and where f is the frequency of the lower pitch.
Example: Determine the beat frequency of the interval whose ratio is 81/64 and whose lower pitch is an octave below middle C (132 Hz). We find that 81/64 = 1.265625, which is very near 5/4. Hence the beat frequency is 132[(4)(1.265625) - 5] = (132)(5.0625 - 5) = 8.25 per second.
Example: Determine the beat frequency of the interval whose ratio is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and whose lower pitch is 132 Hz. In the next section, we shall prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not a fraction; but the beat frequency does not depend on whether or not r is a rational number. We find that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], only a little larger than 5/3 (= 1.666667), and so the beat frequency is 132[(3)(1.681793) – 5] = (132)(5.045378 – 5) = (132)(.045378) = 5.990 per second.
(Continues...)
Excerpted from The Structure of Recognizable Diatonic Tunings by Easley Blackwood. Copyright © 1985 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
- FrontMatter, pg. i
- Contents, pg. v
- Preface, pg. ix
- Chapter I. Fundamental Properties of Musical Intervals, pg. 3
- Chapter II. The Diatonic Scale in Pythagorean Tuning, pg. 22
- Chapter III. Names and Distributional Patterns of the Diatonic Intervals, pg. 31
- Chapter IV. Extended Pythagorean Tuning, pg. 49
- Chapter V. The Diatonic Scale in Just Tuning, pg. 67
- Chapter VI. Extended Just Tuning, pg. 91
- Chapter VII. Musical Examples in Just Tuning, pg. 129
- Chapter VIII. The Diatonic Scale in Meantone Tuning, pg. 154
- Chapter IX. Extended Meantone Tuning, pg. 165
- Chapter X. The General Family of Recognizable Diatonic Tunings, pg. 193
- Chapter XI. Equal Tunings and Closed Circles of Fifths, pg. 221
- Chapter XII. The Diatonic Equal Tunings, pg. 261
- Index, pg. 315