Calculating tempered intervals
Calcualating 'pure (meantone) intervals
SNDAN Home page
Note frequencies for eight octaves in
standard Equal Temperament, based on A=440.C4= 'Middle C'.
|Note||Octave 0||Octave 1||Octave 2||Octave 3||Octave 4||Octave 5||Octave 6||Octave 7|
The two values required to find the single-note interval (S) in a tempered scale system are the bounding interval (I), and the number of notes (n).These values are calculated using the formula:
S = I1/nFor standard Equal Temperament, the bounding interval is the octave, a doubling of frequency, hence I = 2, and for the chromatic scale, n = 12. Thus the semitone is the "twelfth root of 2", or 21/12. This works out at approximately 1.0594631. Thus, given A = 440, one semitone up = 440 * 1.0594631 = 466.163764. Rounded to three decimal places, this is 466.164, as shown in the table above.
This principle can extend to any bounding interval, and to any number of notes. For example, to define a equal tempered scale of 20 notes over two octaves, the interval will be 41/20, which is 1.0717735. This calculation can be done using any standard scientific calulator. Windows includes just such a calculator. The sequence for standard equal temperament is as follows:
The Cent, introduced in the 19th Century by A.J. Ellis, divides each equal-tempered semitone into 100, and thus the octave into 1200. The calculation can be made exactly as given above. One Cent is approximately 1.00057779.
To convert a numeric ratio into Cents, the logarithm of the ratio must be multiplied by the constant 3986.3137. In practice, the error when the more convenient value 4000 is used, is of little consequence in most cases.
Meantone intervals derive directly from the harmonic series; thus the intervals are described as 'pure'. For any harmonic H, the frequency is H*F, where F is the fundamental frequency.
Thus with F= 100, the third harmonic is 100*3 = 300, and the fourth harmonic is 100*4 = 400. This gives the interval of a 'perfect fourth', for which the ratio is simply 4/3, or 1.333... . For the descending interval, the ratio will be 3/4, or 0.75. Some intervals, such as the perfect fifth (3/2), are almost identical to their equal-tempered equivalents (the difference is around 2 Cents in this case). Others, notably the major third, are very different - the 'pure' third (5/4, = 1.25) is much narrower than the equal-tempered third, = 1.256. The difference in Cents is 14 (400 - 386).
[SNDAN Home page]