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Facts of the Matter Richard Brill |
Math of melodies
reverberates
across the agesThe melodic and harmonic structure of music is important to anyone who enjoys music, trained or otherwise.
It is amazing that certain combinations of musical notes strike an aesthetic chord in the brain that crosses spans of time and boundaries of culture.
Bone flutes made 30,000 years ago have holes that are spaced to produce sounds that are tuned to four-note scales bearing unmistakable mathematical relationships to modern instruments. Lyres and harps dating from 2800 B.C. are tuned with the same four note scale. Both employ the same ratios -- 1, 4/3, 3/2, 2/1, known as "perfect intervals" -- that are used in the music of virtually every culture on the planet.
The most basic harmony is the octave, represented by the 2/1 ratio.
The mathematical relationships underlying harmony fascinated a young boy in ancient Greece around 500 B.C. His name was Pythagoras and he would grow up to be a great philosopher and mathematician whose far-reaching influence resonates today in music and in science.
He is the fellow for whom the Pythagorean theorem is named, the one that relates the sides of a right triangle. He founded and was the leader of a communal cult whose members sacrificed all rights to possessions including ideas, so we don't know who really came up with the "Pythagorean" theorem or the other things that have become attributed to Pythagoras through the haze of history.
What is known is that he was brilliant. As a young man, Pythagoras discovered that strings and pipes produce harmonious sounds only if their lengths are in small, whole- number (integer) ratios. Out of this fascination with harmony grew a conceptual scheme of integers and shapes associated with them as mystical, mathematical models for understanding the universe.
Pythagoras devised a musical scale that is the earliest known form of a diatonic scale, which divides the octave into eight notes (hence the name "octave"). It is the "do re mi" scale, consisting of the musical notes C, D, E, F, G, A, B, C, which correspond to the white keys on a piano. To create the eight-note diatonic scale, Pythagoras first divided the octave into five notes, a pentatonic scale, choosing a ratio of 9/8 for each musical step, that being the closest small integer ratio (equivalent to the decimal 1.125) approximating the 5th root of 2 (1.149), the ratio of the octave.
To complete the diatonic scale, Pythagoras further subdivided the pentatonic intervals with hemitones, separated by intervals in the ratio 256/243 (which is a close integer approximation to the square root of 9/8 -- 1.053 compared to 1.060). Then he sequenced the tones and hemitones: 2 tones, 1 hemitone, 3 tones, 1 hemitone. This is the same sequence of intervals used in our modern scale, but the slight discrepancy in the size of the hemitone intervals of the Pythagorean scale gives it a slightly "mistuned" sound that evokes images of ancient, pagan rituals.
There were numerous attempts to find more appropriate intervals in the 600 years that followed, which may not seem like much of a challenge in today's world of gigahertz number crunching. But until well into the first millennium A.D. there was no such thing as numbers, let alone decimals and calculators. Calculations were done with fractions using alphabetic letters and truth tables rather than numbers and the rote algorithms that we learn in school.
Even using modern decimal notation, something as simple as calculating a square root is not really simple at all without the electronic silicon brains that are programmed using thousands of years of accumulated mathematical knowledge. Just try calculating the square root of a fraction like 9/8 without using a calculator or decimals and you'll see what I mean. If you can do that, then try to find a ratio of integers that approximates it. Then try calculating the fifth root of 2.
There are several alternative ways to produce the diatonic intervals. Claudius Ptolemy, the great philosopher, mathematician and astronomer of the ancient city of Alexandria, published a dozen or so such algorithms in the second century A.D. One method that he particularly liked was to use three triads (groups of three notes), the first note of which is one of the tones of a perfect interval, with each triad in turn consisting of two other notes at "perfect intervals" above the lowest note of the triad. "Triads of a triad" as it were.
An example in modern terminology would be major triads based on the notes C, F, G (the first, fourth, and fifth notes of the diatonic scale). The triads would then consist of the notes (C, E, G); (F, A, C); (G, B, D), which comprises all of the notes of the C major scale, including the octave.
Ptolemy called this particular tuning the "syntonic diatonic." It was rediscovered in the 15th century and has remained the favored scale of musicians and composers in the Western world ever since, being used in everything from Gregorian chants to hymns, classical music to rock and roll, folk, spirituals and jazz. The perfect intervals that we know as the first, fourth and fifth notes of the diatonic scale are still the basis of melodic and harmonic structure in virtually all music today.
Another way to produce the intervals of a diatonic scale uses the perfect intervals in a different way than Ptolemy's syntonic diatonic, but also generates the 12-note chromatic scale that divides the octave into twelve tones, including the eight diatonic tones within it.
In principle, the octave can be divided into any number of intervals. But to produce harmony, the intervals must be in small whole number ratios, as Pythagoras discovered. Twelve is the largest number of tones that can maintain those ratios, and is also the smallest common multiple of the integers 1, 2, 3, 4 that appear in the ratios of the perfect intervals.
To generate the chromatic scale, start with any note, then multiply by 3/2 to get a perfect fifth. Then multiply that result by 3/2 to get another perfect fifth, but drop it down an octave by dividing by 2. The remaining notes of the chromatic scale are generated similarly by multiplying each result by 3/2, dividing by 2 when necessary to keep the numbers within the octave.
As a numerical example, start with a frequency of 256 hertz, then multiply by 3/2 to get 384, corresponding to the note G above middle C. To get the next note multiply 384 by 3/4 (one-half of 3/2), which is 288. This corresponds to D, the next whole step above middle C.
To get the next note multiply 288 by 3/2 to get the frequency for the note A (one whole step above G) and so on dividing by 2 when necessary to keep the result between 256 and 512. This will generate: C, G, D, A, E, B, all white-key notes on the piano, all related by the ratio 3/2, the perfect fifth. Notice that the note "F," a perfect interval (4/3) above C, does not appear in this sequence.
Continuing the sequence will generate the remainder of the chromatic scale: F# (Gb), C# (Db), G# (Ab), D# (Eb), A# (Bb), which are the pentatonic notes of the black keys, and finally E#, which is the same as F, the remaining white key. Continuing from there will bring the sequence back to the original C.
The process actually produces a repeating sequence known as the circle of fifths, which contains all the notes of the chromatic scale. It can be started anywhere in the circle and will always produce the same sequence going around the circle in the same direction.
To compound the mystery, going "backwards" around the circle is a sequence of perfect fourths, which reflects the perfect interval of 4/3, from C to F to Bb, and so on, also generates the chromatic sequence (making it a "circle of fourths") using the same algorithm in reverse (the inverse) : Multiply by 2/3 (which is the inverse of 3/2), multiplying by 2 (the inverse operation of dividing by 2) where necessary to keep the values in the range of the octave (between 512 and 256).
The perfect intervals turn out to be inverses of each other precisely because of the numerical relationship of the octave. One perfect fourth up is a perfect fifth down and vice-versa: From C down to F is 7 half steps, but C up to F is only 5. From C up to G is also 7 half steps, but down to G is only 5. Seven plus 5 equals 12, the number of notes in the chromatic scale, not exactly mysterious, or higher math.
Consider now that the diatonic scale occupies one side of the circle, the pentatonic scale the other, as reflected on the piano keyboard, where the black keys (pentatonic scale) are one half pitch above the white keys (diatonic scale), and together constitute the entire 12 note chromatic scale.
Mathematically rich as the diatonic scale is, it does not favor transpositions from one key to another. This is due to the fact that the ratios are not the same between two notes that represent the same number of half step intervals on the chromatic scale.
Some instruments, such as the guitar, are tuned diatonically because it colors different fingerings of the chords differently, adding a richness and variety to the sound of the instrument. In contrast, concert instruments are tuned to an equal tempered scale, in which each chromatic note differs from the one below and above it by the same factor, numerically equal to the 12th root of 2, or 1.0594631. This equal distribution of intervals gives transpositions between keys a more well-tempered (well-tuned) sound.
It is not higher math, but it is mysterious that our brains are wired to respond to musical tones that produce a harmonically pleasant scale that has these numerical relationships.
The mysterious numerical patterns of music stimulated the growth of mathematics and science as much as did the movements of the planets. How brilliant Pythagoras and subsequent mathematicians were to eke out these relationships, and without the benefit of numbers and decimal notation to boot!
If these ratios and their harmonic relationships seem complicated, arcane, esoteric, or obscure to the reader today, imagine how magical, mystical, marvelous, and full of meaning they must have seemed to the Mediterranean and medieval mathematicians of yore.
Richard Brill picks up where your high school science teacher left off. He is a professor of science at Honolulu Community College, where he teaches earth and physical science and investigates life and the universe. He can be contacted by e-mail at rickb@hcc.hawaii.edu