Circle of fifths
music theory, the circle of fifths is a sequence encompassing all of the notes in the equally tempered chromatic scale. Starting on any note and repeatedly ascending by the musical interval of a perfect fifth, one will eventually land on the same note, after reaching all of the other notes:
-
The numbers on the inside of the circle also show how many sharps or flats would be in the key signature for a major scale built on that note. Thus a major scale built on A will have three sharps in its key signature. To figure the key signatures of minor keys see: relative minor/major. The circle of fifths can also be used to determine which order sharps or flats are added to key signatures. The first sharp added is F#, the next is C# and so on. The first flat added is Bb, the next Eb, and so on.
Descending by fifths, and ascending or descending by fourths also works, since motion in one direction by a fourth is equivalent to motion in the opposite direction by a fifth. For this reason the circle of fifths is also known as the circle of fourths.
The circle of fifths, or fourths, may be mapped from the chromatic scale by multiplication, and vice versus. To map between the circle of fifths and the chromatic scale (in integer notation) multiply by 7 (M7), and for the circle of fourths multiply by 5 (M5).
Here is a demonstration of this procedure. Start of with an ordered 12-tuple (tone row) of integers
- (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
representing the notes of the chromatic scale: 0 = C, 2 = D, 4 = E, 5 = F, 7 = G, 9 = A, 11 = B, 1 = C#, 3 = D#, 6 = F#, 8 = G#, 10 = A#. Now multiply the entire 12-tuple by 7:
- (0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77)
and then apply a modulo 12 reduction to each of the numbers (subtract 12 from each number as many times as necessary until the number becomes smaller than 12):
- (0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5)
which is equivalent to
- (C, G, D, A, E, B, F#, C#, G#, D#, A#, F),
which is the circle of fifths.
Moving around the circle of fifths is a common way to modulate.
This was supposedly invented in the sixth century B.C. by Pythagoras. It is said that Pythagoras also had the idea of tuning an instrument by fifths and thus discovered the Pythagorean comma.
See also: enharmonic, cadence (music), sonata form
