Tuning

12-TET

12-tone equal temperament (12-TET) is the tuning system most commonly used today in Western Music. It refers to a tuning system that divides the octave into 12 divisions. The divisions are equal on a logarithmic scale and are perceived to be equally spaced.

In 12-TET, the distance between each division in an octave is called a semi-tone (or a half-step). Two semi-tones are called a whole-tone (or a whole-step).

A system for measuring distance between tones uses a unit called a cent. In 12-TET there are 100 cents between each semi-tone. Thus, an octave is comprised of 1200 cents, and a perfect fifth is 700 cents.

To change the pitch of an audio sample by x semi-tones, use the following formula:

frequency * (2^(x/12))

Where x is the number of semi-tones up or down (e.g., 1 is up a semi-tone, 0 is no change, and -1 is down a semi-tone).

Just-Intonation

Just-intonation refers to a variety of tuning systems that use whole number ratios for determining the divisions of the octave.

To change the pitch of an audio sample using just-intonation, use the following formula:

frequency * (x/y)

Where x/y is the just-intonation ratio (for example, multiplying a frequency by 3/2 is a near equivalent of a 12-TET perfect fifth up).

To determine how large a just-intonation ratio is in cents, use the following formula:

log (x/y) * (1200/log 2)

Where x/y is the just-intonation ratio (for example, multiplying a frequency by 3/2 is the equivalent of going up 702 cents, or, in other words, it is close to going up a perfect fifth in 12-TET).

Calculating Frequencies

This section describes how to calculate the frequency of a note.

To begin, the frequency of a fixed note must be established. A widely used standard is ISO 16, which specifies that the A above middle C is 440 Hz.

Note, it is useful to label notes by octave numbers. A common approach is to label middle C as C4. Thus, an octave higher is C5, an octave lower is C3, and the A above middle C is A4.

For example, the frequency of D♯2 (i.e., 30 semitones below A4) is:

440 Hz * (2^(-30/12)) = 77.78 Hz

As another example, the frequency of B♭6 (i.e., 25 semitones above A4) is:

440 Hz * (2^(25/12)) = 1864.66 Hz