Pythagoras - Life and Findings
Pythagoras did one of the first mathematical investigations into music. The Ancient Greek mathematician is most known for his research on geometry but it is a little known fact that he discovered the basic geometry of music. The story goes that as Pythagoras passed a blacksmith he was drawn to the harmonious tones that were being made by the metalworkers as they struck their hammers to their anvils. Curious of how such tones were being made Pythagoras investigated further into how harmonious tones are produced. At the end of his investigation Pythagoras theorized that harmonious tones are based upon simple ratios.
He found that the ratio between two octaves is 2:1. In the following sections you will learn about Pythagorean principles applied to music notation.
Principles of Strings
A string instrument is, in essence, a hollow wooden box that keeps a number of strings under tension. String instruments can be played either by bowing, plucking or striking the strings. Pythagoras discovered that if you cut a string in half, it would be an octave above the note produced by the whole string. On certain string instruments, such as the cello, this would also produce a harmonic, or overtone.
There are three ways to change the pitch on a string. You can change the length of the string, tension, and density (thickness of strings). For tuning string instruments, adjust the tension because changing length or density would be impractical.
The length of a string is inversely proportional to the pitch, meaning a longer string results in a lower pitch. For tension, a lower tension results in a lower pitch. You can imagine this by stretching a rubber band; the more it is tightened, the higher the pitch is. The density of a string is inversely proportional to the pitch. The more mass in the string, the more the pitch goes down. In the following formulas represent the relationships between pitch (f) against a corresponding string length, tension, or density:
A string length is inversely proportional to the pitch it produces. In this formula, l represents the length of the string.
A string tension is directly proportional to the pitch; the more tension, the higher the pitch. In this formula, T represents the length of the string.
A string density is inversely proportional to the pitch; the thicker the string, the lower would be its corresponding pitch. In this formula, p represents the density of the string.
In the Pythagorean scale, the notes consist of perfect fifths stacked up on each other. To create this scale, we start with a note, deeming it with a frequency of 1 for simplicity. From there, we escalate in the interval of perfect fifths. It would look like this:
From there, we convert the notes to fit into one octave. We do this by multiplying the ratios of each note to the reciprocal of an octave, or 1/2 for each octave. For example, the note in our example D5 is one octave higher than our starting note C4. Therefore, we multiply the ratio of D5, which is 9/4, by 1/2. For E6, we take that ratio of 81/16 and multiply it by 1/4, since E6 is two octaves above C4. In essence, it would look like this:
Which is simplified subsequently as:
Which is simplified subsequently as:
We then put them in ascending order:
As some people might have noticed, we are missing one note from this sequence. Remember, the 2 is the same note as the 1, just an octave higher. Following our example based on C4, we are missing F, which is a perfect fourth from the starting note of C.
However, the missing note would have the same interval between the starting notes no matter what, the C4 is just an example that helps clarify things further. To obtain this missing note, we go down a perfect fifth, meaning that we divide by 3/2, or multiply by the reciprocal, 2/3, and go up one octave. In numbers, it would look like this: 2/3 * 2/1 = 4/3. The full scale is now complete:
Looking at the intervals between each step, we can see that this scale represents a major scale:
where 9/8 is the whole step and 256/243 is the half step. The exact ratios that we have are the perfect fifth and perfect fourth, or 3/2 and 4/3 respectively.
This scale fits the white keys on the piano and is called the Pythagorean diatonic scale. The larger interval 9:8 is the Pythagorean whole tone. The smaller interval 256:243 is the diatonic semitone. We all know the piano also has black keys between the whole tone spaces. What would happen if we add these chromatic notes to the corresponding spots in the scale we have found? The note F# is a fourth below B, which makes (3/4) * (243/128) = 729/512. The 3/4 comes from the reciprocal of a fourth, since we are going down. The 243/128 is the reciprocal of the diatonic semitone, 256:243, multiplied by two. The interval F# to G would be (3/2) / (729/512) = 256/243, the diatonic semitone. On the other hand, the interval F to F# is (729/512) / (4/3) = 2187/ 2048. This interval is the Pythagorean chromatic semitone and is a little larger than the diatonic semitone. Therefore, this scale has two semitones of different sizes.
There is also another flaw in this scale. If we keep adding all the chromatic notes in the scale, we would find that the B#, which is the enharmonic equivalent to C, have different relative frequencies. We see this by going up from C in whole tone steps: 9/8, (9/8)2, (9/8)3 and so on until we reach B#, which has the relative frequency (9/8)6. The C above the starting C, or an octave higher, would have a relative frequency of 2. The interval B# to C is (9/8)6: 2, which elaborated shows the ratio 531441: 524288. This means that the B# is higher than its enharmonic equivalent C. This same ratio comes up if we find the interval between the chromatic semitone and the diatonic semitone, which would be (2187/2048) / (256/243). This interval has been named the Pythagorean comma.