Pythagorean Tuning
Pythagoras devised a system of tuning based upon the interval of a fifth, the most consonant interval after the unison and the octave. It’s generally believed that by constructing a series of both descending and ascending fifths from the same starting point he came across a basic diatonic scale. Basically, Pythagoras took the series of fifths from a starting point 1 / 1, and produced them by successively multiplying by 3 / 2 to give 3 / 2, 9 / 8, 27/ 16, 81/64, and 243/128. To get the interval for a 4th he multiplied down a fourth or 2/3. Look below at how a Major Diatonic Scale is constructed in the Key of C Major.
|
Pitch |
Ratio |
|
F |
2/3 |
|
C |
1 / 1 |
|
G |
3/2 |
|
D |
9/8 |
|
A |
27/16 |
|
E |
81/64 |
|
B |
243/128 |
To get the Diatonic Scale the next step is then to arrange these in order by pitch sequence.
|
Sorted fifth |
Note name |
|
1 / 1 |
C |
|
9/8 |
D |
|
81/64 |
E |
|
4/3 |
F |
|
3/2 |
G |
|
27/16 |
A |
|
243/128 |
B |
|
2/1 |
D |
One anomaly of this tuning is it creates the “Pythagorean Comma”. This is defined as the slight difference that results after tuning each successive note from the circle of fifths starting from the first note. For example, if you start with note C and tune a series of 12 perfect fifths, C D G A B F# C# G# D# A# E# B#(C) the resulting B# will be slightly higher in pitch (23.5 cents) than the original note C that you started tuning from 7 octaves higher. In other words, twelve perfect fifths are not exactly equal to seven perfect octaves, and the Pythagorean comma is the amount of the discrepancy.
Note: Scientists and Music Theorists have devised a standard unit for measuring the size of intervals resulting from two frequencies vibrating at a given ratio. This unit is called a cent because it equals 1/100th of a half step.
Here is the formula for figuring out how many cents large an interval is:
Divide 1200 by the logarithm of 2. If you use base 10 logarithms (any base is permitted), 1200/log 2 = 3986.3137… For any ratio n/p, the number of cents in the interval is log (n/p) x 1200/log 2, If you’re using log 10, then cents = log (n/p) x 3986.3137…