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A Pythagorean tuning of the diatonic scale

A tuning system based on the line of fifths.

by REGINALD BAIN

 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

It is said that the Greek philosopher and religious teacher Pythagoras (c. 550 BC) created a seven-tone scale from a series of consecutive 3:2 perfect fifths. The Pythagorean cult's preference for proportions involving whole numbers is evident in this scale's construction, as all of its tones may be derived from interval frequency ratios based on the first three counting numbers: 1, 2, and 3. This scale has historically been referred to as the Pythagorean scale, however, from the point of view of modern tuning theory, it is perhaps convenient to think of it as an alternative tuning system for our modern diatonic scale.

The white keys on a piano form a diatonic scale. One of the most important characteristics of this diatonic scale is that the octave is partitioned into adjacent intervals of the following type and quantities: five whole-steps and two half-steps arranged in the assymetrical pattern shown below in Fig. 1: Fig. 1. The diatonic scale on C4.

In Fig. 1, the octave, or interval whose frequency ratio is 2:1, is the basic interval. A basic interval defines where a scale repeats its pattern. A generating interval is required to generate the steps of a scale. In the case of a Pythagorean tuning, the generating interval is a 3:2 fifth. Notice that a sequence of five consecutive upper 3:2 fifths based on C4, and one lower 3:2 fifth, produces a seven-tone scale, as shown in Fig. 2. Playback:

 Performing the calculations: To simply the calculations, the frequency of C4 is assigned the variable x . (In a 12TET system based on 440 Hz., x = 261.62 Hz.) Up a 3:2 fifth from C4 is G4. So the frequency of G4 may be calculated as follows: Frequency of G4 = x * 3/2 = 3/2 x Up a 3:2 fifth from G4 is D5. The frequency of D5 may be calculated as follows: Frequency of D5 = x * 3/2 * 3/2 = 9/4 x Notice that D5 exceeds the octave-related pitch C5. So we need to take D5 down an octave to get D4, a tone just above C4. We take a pitch down an octave as follows: Frequency of D4 = 9/4 x divided by 2/1 = 9/4 x * 1/2 = 9/8 x Note that dividing by a fraction is equivalent to multiplying by its reciprocal. This process may be used to calcuate the frequency of each of the remaining scale degrees, except F4. F4 is obtained by going down a 3:2 fifth from C4 to F3 (x * 2/3 = 2/3 x), and then take F3 up an octave to F4. Frequency of F4 = 2/3 x * 2/1 = 4/3 x

It should be notated that in theory, a sequence of 3:2-fifth-related pitches can produce any number of tones within an octave. Stoping at the number seven is completely arbitrary, and was perhaps a consequence of the fact that in the time of Pythagoras there were seven known heavenly bodies: the Sun, the Moon, and five planets (Venus, Mars, Jupiter, Saturn and Mercury).

In Fig. 2, the frequency of C4 is represented by the variable x so that frequency of each pitch in the tuning can be expressed as a ratio. We begin the scale creation process by dividing up the musical space spanned by the basic interval. We do this one 3:2 fifth at a time. When pitches are generated outside the octave, we take them down or up an octave (i.e., divide by 2) as necessary as indicated by the colored lines in Fig.2. See the sidebar (right) for a detailed explanation of all of the required calcuations.

Properties of the Pythagorean Tuning System

Notice that a Pythagorean tuning system has a number of interesting features. For example, the system may be built using only two intervals:

1. The basic interval: 2:1 octave
2. The generating inteval: 3:2 fifth

Additionally, the Pythagorean fourth may be derived as the difference between the octave and fifth. For example,

2/1 divided by 3/2 = 2/1 * 2/3 = 4/3

The Pythagorean whole tone may be derived as the difference between the fifth and fourth. For example,

3/2 divided by 4/3 = 3/2 * 3/4 = 9/8

The Pythagorean semitone may be derived as the difference between the Pythagorean fourth (4/3) and two whole tones (9/8 * 9/8). For example,

4/3 divided by (9/8 * 9/8) = 4/3 divided by 81/64 = 4/3 * 64/81 = 256/243 = 90 cents

Notice that the above ratios are equivalent to the adjacent intervals produced by the sequence of 3:2 fifths shown in Fig. 2. Finally, it should be noted that any tone in the Pythagorean system can be expressed as a power of 3:2.

However, also notice that two Pythagorean semitones (256/243 * 256/243) do not add up to a Pythagorean whole tone (9/8). The Greeks called the difference between the Pythagorean whole tone and semitone aptome, meaning cut off, and its size may be expressed as:

9/8 divided by 256/243 = 9/8 * 243/256 = 2187/2048 = 114 cents

Comparing a Pythagorean Tuning with a Modern Piano Tuning

Fig. 3a introduces a convenient way to chart frequency relationships created by a particular tuning. The top row lists US Standard pitch names. The octave-related pitch is shown in parentheses. The next two rows show the relative frequency based on the first scale degree: first as a frequency ratio, then that ratio's equivalent in cents. The last two rows compare the size of each scale step. Again, first as a frequency ratio ,and then that ratio's equivalent in cents. Notice that a Pythagorean tuning of the diatonic scale creates only two different sized steps: a major tone (9:8 or 204 cents) and the Pythagorean semitone (256:243 or 90 cents)--hence the name diatonic.

 Pitch Name C4 D4 E4 F4 G4 A4 B4 (C5) Frequency ratio above C4 1:1 9:8 81:64 4:3 3:2 27:16 243:128 2:1 No. of cents above C4 0 204 408 498 702 906 1,110 1,200 Step-size (as ratio) 9:8 9:8 256:243 9:8 9:8 9:8 256:243 Step-size (in cents) 204 204 90 204 204 204 90 2 step sizes: 1. major tone = 9:8 or 204 cents; 2. Pythagorean semitone = 256:243 or 90 cents

Click here to view an animation that comparing a Pythagorean tuning of the diatonic scale with a 12TET version. Please note that all of the 12TET frequency values were calculated based on A4 440 Hz, while the Pythagorean frequency values were calculated based on C4 261.62. The first scale degree is sustained as a pedal tone so you may hear the beating of partials.

Extending a Pythagorean tuning to include all 12 tones of the chromatic scale

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. 4. Playback:

To see where the ratios come from, it may be helpful to write out a line of 12 fifths in US Standard pitch notation that is symmetric about about C4 - G4 as shown in Fig. 5:

 Db1  Ab1  Eb2  Bb2  F3  C4 G4 D5  A5  E6  B6  F#7

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

Updated: September 24, 2002

Reginald Bain | University of South Carolina | School of Music | Disclaimer
http://www.music.sc.edu/fs/bain/atmi02/
A Web-based Multimedia Approach to the Harmonic Series: A Pythagorean tuning of the diatonic scale