Pythagorean and Fibonacci tuning

Summary: The relationship between mathematics and music led to several tuning systems.

A musical scale is a sequence of ordered notes used to construct music compositions. Scales can be classified according to their starting point, the intervals between their notes, or the number of notes they contain. Instruments may be tuned according to many possible systems. There are close mathematical connections between musical scales, tuning systems, and number theory, as well as dynamical systems. Mathematics also plays a critical role in designing playable and efficient keyboards for instruments that will be tuned to something other than the standard eight-note Western scale.

Most Western music uses an eight-note “octave” scale (do, re, mi, fa, sol, la, ti, do), where the two “do” notes have the same tone but different pitches. The piano keyboard is set up in the C major key, where the white keys starting with C correspond to the eight notes in the octave.

There are also tones between some of the notes on the scale, represented on the piano by the black keys. Counting from C to B, there are 12 equal semitones in the chromatic scale of Western music.

To tune an instrument with strings, the lengths of the strings are adjusted to produce the correct pitch. Pythagoras of Samos (570–495 b.c.e.) is credited with realizing two things that allowed him to calculate the string lengths for the 12 semitones of the chromatic scale:

• 1. A string that is half as long produces the tone that is one octave higher. A string that is twice as long produces a tone that is one octave lower.
• 2. A string that is two-thirds as long produces a tone that is up five notes (called a fifth, or do-sol interval), seven semitones higher in the 12-tone chromatic scale.

Pythagoras saw that seven and 12 share no common factors and that he could use this fact to generate the lengths of all 12 strings in the chromatic scale.

• 1. Start with a string that sounds like a C note.
• 2. Cut a string that is two-thirds of the C string to give G.
• 3. Cut a string that is twice as long as G, yielding the same tone down an octave.
• 4. Cut a string two-thirds of this new lower G to give D.
• 5. Cut a string two-thirds as long as D to give A.
• 6. Cut a string twice as long as A, yielding A down an octave.
• 7. Cut a string two-thirds of the lower A to give E.
• 8. Cut a string two-thirds of E to give B.
• 9. Cut a string twice as long as B, yielding B down an octave.

Continue in this pattern, shortening a string to two-thirds to produce new higher notes and doubling the string when needed to avoid going past the top of the octave. After 19 steps, all of the strings of the C to C octave are determined, as well as a few extra notes below C (see Figure 1).

Called the “circle of fifths,” this method of tuning by shortening the string to move up seven semitones (and back 12 when needed) would not work if the two numbers involved shared a common factor, such as four and 12. Not all of the semitones would be “hit” in that case.

Equal Tuning

Pythagoras was a little off when he assumed that a string two-thirds as long would produce the seventh semitone. In actuality, using irrational numbers (something Pythagoras did not believe in), the lengths of string needed to produce all of the semitones can be found more precisely. Starting with a string of length two, one can factor two into 12 equal parts or “twelfth roots.” This method of tuning, used in the twenty-first century for most music, is called “equal tuning” (see Figure 2). The values of these irrational numbers to three decimal places show that the fifth note (or seventh semitone) string, G, is actually slightly more than two-thirds of the C string: two-thirds of a string of length 2 would yield a G string of length 1.333 rather than the equal tuning length of approximately 1.335. This little bit of difference is magnified when the circle of fifths technique is used to tune the strings, yielding notes that sound flat.

Other Tuning Systems

Between Pythagoras’s time and the twenty-first century, a number of other tuning strategies were developed as music and mathematics knowledge grew. Popular in the medieval age, for example, was “just” tuning, which differs from both Pythagorean and equal tuning. To use equal tuning in the twenty-first century, one does not have to physically measure strings precisely; equipment can be used to measure the fundamental frequency (related to the pitch) of the sound wave generated by the string in order to tighten the string to the correct length.

There is also a method of tuning based on the Fibonacci series of Leonardo Pisano Fibonacci, which has been analyzed by English mathematician Sir James Jeans. The numbers in the musical Fibonacci series (2, 5, 7, 12, 19,…) can be generated by increasingly long series of musical fourths and fifths from the octave scale. An interval of two tones that are a fifth apart, such as F and C, have a frequency ratio of three-halves. The next fifth is a G, which is musically very close to the original F, but an octave higher, so the two-tone scale is left as F and C. Extending the fifths to a five-tone scale gives F, C, G, D, and A. This would be followed by E, which is again almost the initial F. A slight modification made by slightly raising all the tones (after the initial F) would create a five-note equal tuning scale. Increasingly larger scales can be made by continuing this pattern.

Bibliography

Ashton, Anthony. Harmonograph: A Visual Guide to the Mathematics of Music. New York: Walker & Co., 2003.

Hall, Rachel W., and Kresimir Josic. “The Mathematics of Musical Instruments.” American Mathematical Monthly 108, no. 4 (2001).

Jeans, James. Science and Music. New York: Dover Publications, 1968.