Reply To: An error in "The Circle of Fifths!"

[Jonathan MarshallParticipant]

Hey everyone,

Just thought I'd chime in from my experience writing a software DX7 and in particular calculating sine wave frequencies from MIDI note on events. Lisa has mentioned J.S. Bach and equal temperament where the scale is divided into 12 notes with equal frequency ratios. This is the basis of all western music and the reason that you can switch from one key to another on a 'modern' (post-Bach) keyboard.

Each octave the frequency doubles - so an octave higher from A (440Hz) is A (880Hz). The frequency difference between consecutive notes on a well tempered clavier and other modern keyboards is the 12th root of 2. So on an equally tempered scale the perfect 5th interval is actually:

(12th root of 2) to the power of 7 = 2^(7/12) = 1.4983... which is a little short of 1.5

It's to the power of 7 because there are 7 notes between the fundamental and the 5th - in the key of C: C=0 C#=1 D=2 D#=3 E=4 F=5 F#=6 G=7. This makes the fifth a little less than perfect. However the circle of fifths does work on a modern keyboard because:

7 octaves = 2^7 = 128.

12 fifths = (2^(7/12)) ^ 12 = 2^(7/12*12) = 2^7 = 128.

The problem with a fretless instrument such as violin is that the musician must pick a pitch and they will naturally play the perfect fifth as 1.5 times - which is of course what it should be.

Cheers,
Jonathan