Twelfth Root of Two - The Equal-tempered Chromatic Scale

The Equal-tempered Chromatic Scale

Since a musical interval is a ratio of frequencies, the equal-tempered chromatic scale divides the octave (which has a ratio of 2:1) into twelve equal parts.

Applying this value successively to the tones of a chromatic scale, starting from A above middle C with a frequency of 440 Hz, produces the following sequence of pitches:

Note
Frequency
Hz
Multiplier
Coefficient
(to six places)
A 440.00 20/12 1.000000
A♯ B♭ 466.16 21/12 1.059463
B 493.88 22/12 1.122462
C 523.25 23/12 1.189207
C♯ D♭ 554.37 24/12 1.259921
D 587.33 25/12 1.334839
D♯ E♭ 622.25 26/12 1.414213
E 659.26 27/12 1.498307
F 698.46 28/12 1.587401
F♯ G♭ 739.99 29/12 1.681792
G 783.99 210/12 1.781797
G♯ A♭ 830.61 211/12 1.887748
A 880.00 212/12 2.000000

The final A (880 Hz) is twice the frequency of the lower A (440 Hz), that is, one octave higher.

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