One way to tune a 7-note diatonic scale is to use only the basic intervals whose frequency ratios are 2:1 and 3:2--the first two intervals of the harmonic series. Today we refer to these intervals as the perfect octave and fifth respectively. As this method of tuning the diatonic scale is commonly attributed to Pythagoras (c. 500 B.C.) it is called a Pythagorean tuning. It is assumed that the Pythagoreans regarded this particular method of constructing a 7-note scale to be an elegant mathematical solution to the problem of dividing up the octave into discrete steps, as it utilizes only the first three counting numbers (1, 2, and 3).
On the Origin of Scales
"Although in theory any number of pitches may be used as a basis for musical expression, in practice different cultures have adopted patterns made up of relatively few discrete pitches which they consistently employ."Murray Campbell and Clive Greated,
The Musician's Guide to Acoustics
Fig. 1: A Pythagorean tuning of the diatonic scale: a scale whose steps are
tuned using only the basic intervals 2:1 and 3:2. Click here to go to a page where you can hear this scale.
If one assumes as Pythagoreans did, that pitches related by a frequency ratio of 2:1 may be considered equivalent, then the process of creating a scale might best be described as filling in the musical space spanned by the octave with smaller basic intervals. Figure 1 shows one way to go about calculating the frequency of each scale step in a diatonic scale. Beginning on the pitch C4 (c. 264 Hz.), the next note in the series, G4, is calculated by multiplying the frequency of C4 by 3/2. To show that C4 can be any frequency we use the variable x to represent any frequency.
Fig. 2: A chart showing the size of each adjacent scale step in a Pythagorean diatonic scale on C4 in cents.
Pitch C4 D4 E4 F4 G4 A4 B4 C5 Frequency ratio
above C41 9:8 81:64 4:3 3:2 27:16 243:128 2 No. of cents above C4 0 204 408 498 702 906 1,110 1,200 Size of step (in cents) 204 204 90 204 204 204 90 2 step sizes: 1. major tone = 9/8 or 204 cents, 2. diesis (difference) = 256/243 or 90 cents It is interesting to note that a sequence of ever higher 3:2 "fifths" never produces a frequency ratio that can be reduced to the ratio 2:1.{1} For example,
Calculating from F3, 12 fifths (3/2)12 produces the pitch E#10which is 129.75 times higher than the original starting point. In comparison, 7 octaves above F3,(2/1)7, produces a tone 128 times higher. From a musical point of view, it would be nice if these two numbers were the same. Of course, they are not. This type of discrepancy is called a comma. A large number of such discrepancies arise in tuning theory. Each comma is assigned a unique name, and the discrepancy discussed above which arose from this tuning of the Pythagorean scale is called the Pythagorean comma. Its size may approximately be expressed as the ratio 129.75:128, 23 cents, about one-fourth of a semitone.
Fig. 3: Calculating the Pythagorean comma: 12 3:2 fifths vs. 7 2:1 octaves, 129.75:128
2:1-related with F3? Pitch F3 C4 G4 D5 A4 E6 B7 F#7 C#8 G#8 D#9 A#9 (E#10) No. of 3:2s 1 2 3 4 5 6 7 8 9 10 11 12
Obviously, the Pythagorean approach to scale creation may be continued to produce scales containing more than 7-notes. For example, a Pythagorean tuning of the 12-note chromatic scale on C is shown in Fig. 5. To see where the ratios come from, it may be helpful to write out a line of 12 fifths that is symmetric about about C4 - G4 as shown in Fig. 4:
Fig. 4: A line of 12 fifths symmetric about C4 - G4 Db1 Ab1 Eb2 Bb2 F3 C4 G4 D5 A5 E6 B6 F#7
Fig. 5: A screen shot from Soundscape Productions' JICalc showing
a Pythagorean tuning of the 12-note chromatic scale.
1The "unique prime factorization theorem" can be used to prove the validity of this conjecture. For a clear and concise summary of its mathematical proof consult Blackwood, Easley, The Structure of Recognizable Diatonic Tunings, (Princeton, NJ: Princeton University Press, 1985), 8-11.
2The Pythagorean semitone was also called the limma (left over), as it was calculated by the Greeks to be the difference (or amount left over) between a fourth and two whole tones. See Willi Apel, ed., Harvard Dictionary of Music (Cambridge: Harvard University Press, 1969), pp. 709-10.Further Study
See Bibliography On-line Sources
Campbell/Greated, 170-71.
Partch, 398-406.
BAIN The Overtone Series
Soundscape Production's JICalc, a Hypercard stack that can be used to explore a variety of tunings and temperaments.
Margo Schulter's FAQ on Pythagorean Tuning
Reginald Bain |
University of South Carolina |
School of Music |
Disclaimer
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A Pythagorean Tuning of the Diatonic Scale v2.01b
Copyright © 1997-2001 by Reginald Bain. All rights reserved.
These materials may not be distributed in any form without the express written permission of the author.