The Pythagorean Scale: The Harmonic Model

The Pythagorean Scale a mathematical method that is used to create all the notes of the musical scale from the harmonic series; this scale is called the Pythagorean Scale. The naturally happening harmonics are whole number multiples. To get the other notes of the scale, we must use fractions.

The harmonic series gives us the ability to create periods of time of perfect 5ths, the ratio of 3/2f, "f" being the starting frequency. By using these 5ths, we can create the pythagorean scale. We can use the key of C, since the C scale only involves the white keys on the piano. If the starting note C is the frequency "f," then C an octave higher is 2f. The notes of the rising/increasing pythagorean C major scale are C=f, D=9/8f, E=81/64f, F=4/3f,
Any given note would have an (a) little different frequency in the different keys. If you started on C, or any other note, and traveled the circle of 5ths, using the perfect 5ths found in the harmonic series, when you arrived back at the starting note, it would not be the same pitch! Also, the half-step periods of time of a chromatic scale would not be equal. What was eventually done to correct these problems was that a little was added to some periods of time, and a little was shaved off others, until the period of time ratio between each of the half-step periods of time was the same. This change is called reducing, the result being the reduced scale, or the scale of equal personality. Equal Personality is usually said to have been invented by Andreas Werckmeister in about 1700, but events that prove of trying new things out/testing ideas scientifically with the idea much earlier. In the scale of equal personality, perfect 5ths, and other periods of time , are no longer mathematically perfect, according to the harmonic series, but they are so close that few people could hear the difference. Modern musicians that have grown up hearing the reduced scale do not even(see/her/become aware of) the small/short mistakes and flaws of its periods of time.This agreement where everyone meets in the middle of period of time, relationships make it possible to tune a piano to the reduced scale and play it in any one of the 12 major and 12 minor keys. Bach was so happy about this new scale that he wrote his now famous "The Well, Reduced Clavier" in 1722, which contains pieces of music in each of the 24 different keys, which now can be played without re-tuning the