# Calculating intervals: the Pitch Table

A basic scientific calculator is very useful when calculating scales and pitch ratios. Since not eveyone has one to hand,  this page provides a simple table giving the frequencies for notes of the musical scale.

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 C 16.351 32.703 65.406 130.813 261.626 523.251 1046.502 2093.005 C#,Db 17.324 34.646 69.296 138.591 277.183 554.365 1100.731 2217.461 D 18.354 36.708 73.416 146.832 293.665 587.330 1174.659 2349.318 D#,Eb 19.445 38.891 77.782 155.563 311.127 622.254 1244.508 2489.016 E 20.061 41.203 82.407 164.814 329.626 659.255 1318.510 2367.021 F 21.827 43.654 87.307 174.614 349.228 698.456 1396.913 2637.021 F#,Gb 23.124 46.249 92.449 184.997 369.994 739.989 1474.978 2959.955 G 24.499 48.999 97.999 195.998 391.995 783.991 1567.982 3135.964 G#,Ab 25.956 51.913 103.826 207.652 415.305 830.609 1661.219 3322.438 A 27.500 55.000 110.000 220.000 440.000 880.000 1760.000 3520.000 A#,Bb 29.135 58.270 116.541 233.082 466.164 932.326 1664.655 3729.310 B 30.868 61.735 123.471 246.942 493.883 987.767 1975.533 3951.066

### Calculating tempered intervals.

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/n
For 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:

• enter 2
• click xy
• click  (left bracket)
• enter 12
• click  1/x
• click ) (right bracket)
• click =
Most electronic calulators include a button for x1/y, making the calculation even quicker.

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### The Cent

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.

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### Calculating 'pure' (meantone) intervals.

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).