Yale University Department of Music

The Number Seven in the Theory of IntonationAuthor(s): Eric RegenerSource: Journal of Music Theory, Vol. 19, No. 1 (Spring, 1975), pp. 140-153Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843753Accessed: 15/03/2010 07:45

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140

THE NUMBER SEVEN

IN

1 Introduction

Since the acceptance of the senario of Zarlino, the number 7 has been the stepchild of intonation theory. In theory textbooks the seventh partial of the fundamental C is usually indicated as Bb, with the disapproving footnote "out of tune" or "slightly flat." The best-known theorists are not sure what names to use for ratios involving the number 7. Hindemith [1940, p. 105] suggests that the augmented fourth can be either of the ratios 7:5 or 10:7, implying that the former ratio could be either an augmented fourth or a diminished fifth. Barbour [1953, p. 129] speaks, similarly, of "the augmented sixth (that is, the minor seventh, 7:4)."

141

THE THEORY OF

INTONATION

ERIC REGENER

One source of the theorists' difficulties is a failure to dissociate pitch notation from its now standard interpretation in twelve- tone tempered intonation. The enharmonic equivalence of the diminished second and the perfect prime is a unique feature of this particular intonation system and is not inherent in the no- tation. In talking about intonation systems in general, the as- sumption that enharmonically equivalent notes are the same is therefore bound to lead to confusion. Such difficulties are easily overcome by means of a consistent and objective mathematical framework such as that of Regener [1973], in which the product of two frequency-ratios always corresponds to the sum of the corresponding intervals without enharmonic "cheating," and in which the intervals of the notation are considered as purely

142

abstract entities without any a priori assumption as to their pitch representation. Such a framework imparts a new useful- ness to the notation, since it frees the notes from an invidious identification with any particular intonation system.

In a consistent system of just intonation, it is easy to determine what notated interval corresponds to a given frequency-ratio, provided the ratio contains only powers of 2, 3, and 5 as fac- tors. For example, the ratio

9 3 3 4 5 2 ' 2 ' 5

corresponds to an interval of two fifths up, followed by a major third down, or an upward minor seventh (see Figure la). The ratio

16 2 2 -~-

2 2 ? - 3

corresponds to the same interval (Figure lb); but the ratio

125 2 2 2 5 5 5 72 3 3 *4' 44

corresponds to an augmented sixth and to no other interval (Figure Ic).

Even in a consistent system, the number 7 poses areal dilem- ma. Given any ratio including 7 as a prime factor (and no other prime factors other than 2, 3, and 5), the corresponding interval will be uniquely determined, provided we can assign an interval with certainty to any one ratio involving the number 7, say the ratio 7/4. The hesitancy of the theorists is based here on a real uncertainty, not necessarily due to the confusion engendered by enharmonic equivalence. It is not clear, as it is with the more fundamental ratios 3/2 and 5/4, what interval is "meant" by the ratio 7/4.

We propose to show that this dilemma is susceptible to a con- vincing abstract solution using only the simplest assumptions as to the nature of just intonation. The ratio 7/4 will be seen to correspond best, in fact, to the interval of an augmented sixth. Before assembling the evidence, we spend a section developing the necessary mathematical background. This sec- tion is a digest of the relevant technical portions of Regener [1973] with slight differences in symbolism and, in the author's opinion, a considerable gain in clarity.

143

FIGURE

1

a :b c

iiT +t-i +L iF I ^

, -

^ | ^ "

^ | , ^ (~~~~~~~~~momumoo

144

2 Background

Any notated interval can be described as a sum of k upward octaves and q upwardperfect fifths, where k and q are integers (which may be negative). For example, a downward major third is the sum of 2 octaves and -4 upward (i.e., 4 downward) fifths, and anupward major third is the sum of -2 octaves and 4 fifths. It will be convenient to write these two numbers as an ordered pair (k, q) representing the interval. For example, an upward octave is (1, 0), an upward perfect fifth (0, 1), an upward major third (-2, 4), and so forth. It is clear, con- versely, that every orderedpair of integers determines a unique interval: thus, we can use the two designations interchangeably.

An important function of an interval i = (k, q) is its "diatonic span" ds(i), defined as

ds(k, q) = 7k + 4q.

It is easily verified that ds(i) is the number of lines and spaces (diatonic steps) traversed between the two notes of any repre- sentation of the interval on the staff; it is positive if the second note is above the first, negative if below, and zero if the inter- val is a perfect prime (0, 0) or a sum of augmented primes (-4, 7). The sum of two arbitrary intervals (k, q) and (1, r) is defined in the obvious way:

(k, q) + (1, r) = (k + 1, q + r).

Note that the representation of an interval in Regener [1973] is not (k, q) but (d, q), where d = ds(k, q) is its diatonic span. The former representation is chosen here since it simplifies the mathematics.

An "intonation system" is a function assigning a unique fre-

quency-ratio to every interval, in such a way that the sum of two intervals always corresponds to the product of their fre- quency-ratios. An intonation system in which the octave has the frequency-ratio 2 is called "regular". It is convenient to take logarithms to the base 2: an intonation system is then a function, say "lf", such that the sum of two intervals always corresponds to the sum of their log frequency-ratios.

(1) lf(il + i2) = lf(il) + lf(i2),

and such that the octave has log frequency-ratio 1:

145

(2) lf(l, 0) = 1.

Regener [1973, pp. 90-93] proves that a regular system is completely determined by specifying the log frequency-ratio of the upward perfect fifth: this log frequency-ratio is called the "determining constant" c. For example, in ordinary twelve- tone intonation the fifth comprises 7 out of 12 equal steps whose sum is the octave (log frequency-ratio 1), and the determining constant of this system is therefore 7/12. To derive the ex- pression for the log frequency-ratio of any interval (k, q), we split it into its octave- and fifth-components:

lf(k, q) = lf(k, 0) + lf(O, q) = k lf(l, 0) + q- lf(, 1),

both by (1); then by (2) and the definition of c, we have

(3) lf(k, q) = k + qc.

The system of just intonation, as ordinarily defined (without the number 7), consists of a set of rational ratios (quotients of integers) together with a rule specifying what notated interval corresponds to a given ratio. The ratios ordinarily used are all those which involve only the numbers 2, 3, and 5 as factors, that is, all ratios of the form

2u 3v 5W,

where u, v, and w are integers (which may be positive, nega- tive, or zero). We shall designate the set of all ratios of this form by the symbolQ5. The rule, based on the harmonic ser- ies, specifies that the ratio 2/1 (u = 1, v = w = 0) corres- ponds to the octave, the ratio 3/2 = 2-1 * 3 to the perfect fifth, and the ratio 5/4 = 2-2 5 to the major third, and that the product of two ratios corresponds to the sum of the two corres- ponding intervals. Defining "int" to be the function which as- signs the interval (k, q) to a frequency-ratio belonging toQ5, we can write the given rules in mathematical terms as follows:

int(2) = (1, 0), the upward octave; int(3/2) = (0, 1), the upward perfect fifth; int(5/4) = (-2, 4), the upward major third;

and the "rule of consistency", int(r ? s) = int(r) + int(s).

146

We derive the general expression for the notated interval corresponding to a frequency-ratio in Q5 by finding out what interval corresponds to the ratios 3 and 5 (that for 2 is given), using the rule of consistency:

int(3) = int(3/2) + int(2) = (1, 1); int(5) = int(5/4) + int(2) + int(2) = (0, 4).

Therefore the general expression, again by the rule of consis- tency, is

int(2u 3V 5W) = int(2u) + int (3V) + int(5W) = u ? int(2) + v int(3) + w int(5) = (u, 0) + (v, v) + (0, 4w) or

(4) int(2u 3v 5w) = (u + v, v + 4w).

For example, the interval corresponding to the ratio 9/8 is

int(9/8) = int(2-3 32 50) = (-3 + 2, 2) = (-1, 2),

an upward major second; corresponding to the ratio 10/9 we have

int(10/9) = int(21 3-2 51) = (1 - 2, -2 + 4) = (-1, 2),

the same interval. The quotient of these two intervals is the ratio

9 9 81 2-4 34 5-1 . 1-0 2-

the "syntonic comma". Any power of this ratio corresponds to the notated interval of a perfect prime:

(5) int(81r/80r) = int(2-4 34r 5r) = (-4r + 4r, 4r - 4r) = (0, 0),

for any integer r.

Clearly, any two frequency-ratios in Q5 whose quotient is a power of the syntonic comma will correspond to the same in- terval. The ordinary theory of just intonation specifies, rough- ly, that the more "musically useful" of two such ratios is that

147

involving the lowest numbers when in lowest terms. For ex- ample, the ratios 81/64, 5/4, and 100/81 all correspond to the major third, but 5/4 is the most useful; of the ratios 729/640, 9/8, 10/9, and 800/729, all of which correspond to the major second, 9/8 and 10/9 are the most useful.

Regener [1974] shows that, if considerations of musical context are excluded, it is possible to define which is the most musi- cally useful, in this sense, of all the frequency-ratios corres- ponding to the same interval (k, q). If the ratios are close to 1, the most useful, roughly speaking again, is one of the two ratios whose value is closest to

2k 5q/4

This is the frequency-ratio for the interval (k, q) in strict meantone intonation; it can be verified that the frequency-ratio for the major third (-2, 4) is precisely 5/4. Meantone intona- tion is a regular intonation system in which the determining constant c = log2 5/4 = (log2 5)/4.

Let us defineQ7 to be the set of all rational ratios having no prime factor greater than 7. Our task will be to investigate the possible ways of extending the explicit correspondence be- tween rational ratios and notated intervals fromQ5 to Q7: that is, to ratios involving powers of the prime number 7 as well as of 2, 3, and 5. As our standard of ambiguity in distinguish- ing different ratios we shall use the syntonic comma: this de- cision deliberately excludes perceptual and psychological fac- tors from what is meant to be a purely abstract discussion.

3 Intervals corresponding to the ratio 7

We want to choose a notated interval (k, q) which best corres- ponds to the ratio 7:

int(7) = (k, q).

The rule of consistency implies that this choice will determine the interval corresponding to any ratio inQ7 by the obvious ex- tension of (4):

(6) int(2u 3v 5w 7x) = (u + v + kx, v + 4w + qx)

for any integers u, v, w, and x.

148

But the interval (k, q) already corresponds to a frequency-ratio in Q5. In fact, it corresponds to a class of such ratios whose quotients are powers of the syntonic comma. Suppose that s is one of these ratios: that is, s is a ratio in Q5 such that

int(s) = (k, q).

This implies that

(7) int(s/7) = (0, 0),

the perfect prime, in analogy to the relation (5) for the syntonic comma.

We shall use the name "seven-comma" for any ratio of the form s/7, where s is a rational ratio involving only the num- bers 2, 3, and 5 as prime factors. Making a seven-comma s/7 correspond to the interval of the perfect prime, as in (7), automatically implies that

int(7) = int(s).

We shall now see that a useful choice of the interval int(7) de- pends on the properties of the "class" of seven-commas s/7 for which int(7) = int(s), rather than on any particular ratio.

For a minimum of ambiguity in the recognition of intervals, we want s/7 to be relatively close to 1 (that is, zero cents): by our earlier assumption, it should not be much further from 1 than 81/80, the syntonic comma. Musical usefulness requires in addition that the ratio itself involve numbers as low as pos- sible.

The latter condition implies that s must be among the most useful ratios of its class. For example, the seven-comma

14348907/14336000 = 2-14 315 5-3/7 = 1.558 cents

is much closer to 1 than the syntonic comma of 21. 506 cents. The corresponding interval is (-14 + 15, 15 - (4 ? 3)) = (1, 3), a major sixth plus two octaves, for which the ratio involving the lowest numbers is 4 - 5/3 = 20/3: thus the most useful seven-comma corresponding to this interval is 20/21. This ratio, though it involves smallnumbers, is about -84.467 cents, almost a semitone. It will turn out that this interval is not one of the best choices for int(7).

149

In order to see just what the desirable choices are, we spread our net relatively wide. We ask for all the seven-commas smaller than a quarter-tone (+50 cents) and involving numbers of less than six figures. A computer search reveals the possi- bilities listed in Table 1. The seven-commas r = s/7 are given in fractional form, in cents (1,200 times the\base-2 log- arithms; u, v, and w are the powers of 2, 3, and 5 appearing in s = 2u 3v 5w; k and q are the octave - and fifth-components of the interval int(s) (k, q). The columns headed d and int display the diatonic span ds(int)s/4)) and the name of the interval int(s/4) obtained by subtracting two octaves from int(s); if s/ 7 is equated to a perfect prime as in (6), this interval is int(7/4). The ratios corresponding to the same interval are grouped together.

The last column, headed "meantone," presents the values t/7 in cents for each interval, where

t = 2k 5q/4

is the frequency-ratio in meantone intonation corresponding to the interval (k, q). Recalling that the most useful ratios s corresponding to this interval are roughly those whose values are nearest t, we see that the seven-commas r = s/7 involving the lowest numbers are those near t/7. The quantity t/7 there- fore shows where these lowest-numbered seven-commas are to be found, even if they exceed a quarter-tone and therefore do not appear in the table. It shows, for example, that the ratio 65536/65625 is the most useful of the seven-commas s/7 for which int(s/4) is a triply diminished octave, while 2187/ 2240 is far from being the most useful of those corresponding to a major sixth (cf. the example above). We are using the word "useful" here in its defined sense of "involving the lowest numbers," of course: the most "useful" is not necessarily the most useful for all purposes.

4 Superiority of the augmented sixth

Table 1 shows seven intervals which, within rather wide limits, could be taken to correspond to the ratio 7/4. Considering this ratio rather than 7 simplifies naming the intervals without changing the relationships. To be precise, if a given seven- comma r = s/7 is taken to correspond to the perfect prime, then the interval i on the same line corresponds both to 7/4 and to s/4, where the latter ratio is in Q5. The best intervals i have ratios s/4 involving low numbers and close to 7/4. The column headed "r (cents)" in fact displays the difference in cents between s/4 and 7/4, since

150

s s/4 7 7/4'

We now examine the intervals in the table on their merits.

Though the ratio for the triply diminished octave is quite close to 7/4 and involves the lowest numbers of its class, it involves the highest numbers of all the ratios in the table. The other three intervals possessing one ratio apiece combine undesirably large numbers with considerable wideness of the mark.

The diminished and the minor seventh are both more reasonable candidates. While their most useful seven-commas (512/525 and 36/35, respectively)are nearly as large as a quarter-tone, each possesses another seven-comma (864/875 and 64/63, respectively) which involves numbers of comparable size and which is closer to zero cents (though still at least slightly larger than the syntonic comma). Of these, the minor seventh seems preferable because the numbers involved in its most useful seven-commas (64/63 and 36/35) are even smaller than those of the syntonic comma.

Of the two most useful seven-commas for the augmented sixth, one (125/126) is less than 1 and one (225/224) is greater. This means that the corresponding ratios, 125/72 and 225/128, bracket the value 7/4. In fact, as the last column shows, the meantone value for the augmented sixth is different from 7/4 by only just over three cents. The numbers involved are quite small as well, considering that no other seven-comma com- parably close to 1 involves numbers less than 4,000. It seems reasonable, therefore, to consider the augmented sixth as the best interval to represent the ratio 7/4 in an extended system of just intonation.

Since for this choice we have

int(7) = (-3, 10),

the interval corresponding to any ratio in Q7 is shown by (6) to be

(7) int(2u 3v 5w 7x) = (u + v - 3x, v + 4w + lOx)

for any integers u, v, w, and x. This function produces some interesting small-number ratios for intervals with a relatively wide span in fifths. For example,

151

TABLE

1

r(cents) log2 r

-2.34948 -.00195789 16 -1 -5

-43.40834 -.03617361 9 -1 -2

-21.90205 -.01825170 5 3 -3

-.39576 -.00032979 1 7 -4

5.75780 .00479817 10 -6 1

27.26409 .02272008 6 -2 0

48.77038 .04064198 2 2 -1

-41.45461 -.03454551 -6 7 -1

-35.30106 -.02941755 3 -6 4

-13. 79477 -.01149564 -1 -2 3

7.71152 .00642627 -5 2 2

29.21781 .02434817 -9 6 1

35.37137 .02947614 0 -7 6

-33.34734 -.02778945 -12 2 5

u v w k q d int meantone (cents)

15 -21 7 ddd8 3.02709

8 -9 6 d7 -38.03177

4 -2 6 m7 38.01723

1 3 5 M6 -79.09062

-3 10 5 A6 -3.04163

-7 17 5 AA6 73.00737

-10 22 4AAA5 -44.10048

Ratio r

65536/65625

512/525

864/875

4374/4375

5120/5103

64/63

36/35

2187/2240

5000/5103

125/126

225/224

3645/3584

15625/15309

28125/28672

152

int(12/7) = (6, -9), a diminished seventh; int(14/9) = (-4, 8), an augmented fifth; int(7/5) = (-3, 6), an augmented fourth.

Svensson [1950] gives a diminished fifth for the last ratio.

It is easily verified that the class of small-number ratios in Q7 corresponding to a given interval has a much narrower range under equation (6) than under the analogous equations derived from any of the other possibilities in Table 1. In fact, an examination of all the small-number ratios in Q7 (up to a certain limit) and their corresponding intervals turns up an interesting relationship with 31-tone intonation, which willbe pursued in a later article. We may remark, at least, that the augmented sixth has a ratio in 31-tone intonation remarkly close to'7/4: its logarithm is 25/31, or roughly 0.80645161 (967.74194 cents), while log2 (7/4)is about 0.80735492 (968.82591 cents): the difference is only 0.00090331 (1.08397 cents), very nearly too small to be noticeable.

ACKNOWLEDGMENT

The author gratefully acknowledges the assistance of Grant #S72-0087 and #S73-1210 from the Canada Council for the Arts,

and the stimulation of conversations with Bengt Hambraeus (who kindly provided the reference to Svensson) and John Rea, both of McGill University.

153

BIBLIOGRAPHY

Barbour, J. Murray. TUNING AND TEMPERAMENT: A HISTORICAL SURVEY. East Lansing, Michigan: Michigan State College Press.

Hindemith, Paul. UNTERWEISUNG IM TONSATZ. Mainz: B. Schott's Sohne, 1940.

Regener, Eric. PITCH NOTATION AND EQUAL TEMPERAMENT: A FORMAL STUDY. Occasional Papers, No. 6: Music. Berkeley: University of California Press, 1973.

Regener, Eric. "A 'Lucky' System of Intonation. " Unpublished article, 1974.

Svensson, Sven E. "Vart Tonssystem och dess Temperaturer 1." SVENSK TID- SKRIFT FOR MUSIKFORSKNING 32:150-186.

Article Contentsp. 140p. 141p. 142p. 143p. 144p. 145p. 146p. 147p. 148p. 149p. 150p. 151p. 152p. 153

Issue Table of ContentsJournal of Music Theory, Vol. 19, No. 1 (Spring, 1975), pp. 1-182Front Matter [pp. 1 - 139]Some Models of Unity in Musical Form [pp. 2 - 30]A Sixteenth-Century Hexachord Composition [pp. 32 - 45]Analysis Symposium: Webern, Orchestra Pieces (1913) Movement I ("Bewegt") [pp. 47 - 64]Some Comparisons between Intuitive and Scientific Descriptions of Music [pp. 66 - 110]Choron, Ftis, and the Theory of Tonality [pp. 112 - 138]The Number Seven in the Theory of Intonation [pp. 140 - 153]Book Reviewsuntitled [pp. 154 - 160]untitled [pp. 161 - 171]

Letters: The Structure of Atonal Music [pp. 172 - 173]Books and Articles [pp. 174 - 179]Corrections to Seay, Gilbert, Morris and Cogan [p. 180]Back Matter [pp. 181 - 182]