Frequencies of Musical Notes

Equal Temperament

The equal temperament is a way of tuning musical instruments so they can play any music regardless of its key signature. All notes are equally spaced on a logarithmic scale. This allows the existence of such instruments as the piano, the pipe organ, or the accordeon, where a tune is emitted by pressing a key, as well as fretted string instruments, such as the guitar, banjo, or mandolin. It is also commonly used with electronic instruments and music synthesized by computers.

In modern days, the frequency of each equal tempered note is calculated by equal logarithmic scale based on the arbitrarily chosen frequency of exactly 440 Hz for the A4 note. Results are in this table:

N.B. The frequency of each note is 1.059463 times greater than the note that precedes it. Why? Because the frequency of each note is exactly twice the frequency of the note one octave lower. And since there are 12 notes in the scale, if the scale is to be equal tempered, each note is a 12^th root of two greater than the one before it. And 1.059463 is the 12^th root of 2.

Cents

To add a cent to a note frequency, multiply it by 1.0005777895. To subtract a cent, multiply it by 0.9994225441 instead. To add (or subtract) more than one cent, you need to multiply the frequency as many times as the number of cents you want to add or subtract. This can involve too many multiplications, so it is simpler to just multiply the frequency according to this table:

Diatonic Temperament

In classical times, musical instruments were tuned to the diatonic temperament, also called the just scale, or Pythagorean scale, and other names. Ancient Greeks considered it the absolute tuning, perfectly natural, pleasant to the ear.

Its main disadvantage, and the main reason most modern instruments do not use it, is that it is different for each key signature. Nevertheless, some instruments, such as the meditative sounding fujara, use it (the fujara, for example, is typically tuned to the G3 key signature). Also, non-fretted string instruments, such as the violin, are often played diatonically.

It is rare for synthesized instruments to be tuned according to the diatonic temperament, for the same reasons the piano used the equal temperament: So they can play any music regardless of its key signature. If, however, you are synthesizing a specific musical piece from scratch, played at a specific key signature (as most pieces are), you may want to consider synthesizing it according to the rules of the diatonic temperament. The result is most likely going to sound much better than the typical synthesized music.

To do that, first find the fundamental frequency, which is the frequency of the key note (e.g., G for a G Major piece) for each octave used in the piece. To find it, simply look it up in the above table of equal temperament. Then calculate the rest of the notes depending on their interval from the fundamental by multiplying the fundamental frequency by the corresponding number in the second column of the following table and then dividing it by the number in the third column:

N.B. There is no octave in the above table, as that is the fundamental note of the next octave, so you just look it up in the first table above.

Copyright © 2007 G. Adam Stanislav
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