Note - Note Frequency (hertz)

Note Frequency (hertz)

In all technicality, music can be composed of notes at any arbitrary physical frequency. Since the physical causes of music are vibrations of mechanical systems, they are often measured in hertz (Hz), with 1 Hz = 1 complete vibration per second. For historical and other reasons, especially in Western music, only twelve notes of fixed frequencies are used. These fixed frequencies are mathematically related to each other, and are defined around the central note, A4. The current "standard pitch" or modern "concert pitch" for this note is 440 Hz, although this varies in actual practice (see History of pitch standards).

The note-naming convention specifies a letter, any accidentals, and an octave number. Any note is an integer of half-steps away from middle A (A4). Let this distance be denoted n. If the note is above A4, then n is positive; if it is below A4, then n is negative. The frequency of the note (f) (assuming equal temperament) is then:

For example, one can find the frequency of C5, the first C above A4. There are 3 half-steps between A4 and C5 (A4 → A♯4 → B4 → C5), and the note is above A4, so n = +3. The note's frequency is:

To find the frequency of a note below A4, the value of n is negative. For example, the F below A4 is F4. There are 4 half-steps (A4 → A♭4 → G4 → G♭4 → F4), and the note is below A4, so n = −4. The note's frequency is:

Finally, it can be seen from this formula that octaves automatically yield powers of two times the original frequency, since n is therefore a multiple of 12 (12k, where k is the number of octaves up or down), and so the formula reduces to:

yielding a factor of 2. In fact, this is the means by which this formula is derived, combined with the notion of equally-spaced intervals.

The distance of an equally tempered semitone is divided into 100 cents. So 1200 cents are equal to one octave — a frequency ratio of 2:1. This means that a cent is precisely equal to the 1200th root of 2, which is approximately 1.000578.

For use with the MIDI (Musical Instrument Digital Interface) standard, a frequency mapping is defined by:

Where p is the MIDI note number. And in the opposite direction, to obtain the frequency from a MIDI note p, the formula is defined as:

For notes in an A440 equal temperament, this formula delivers the standard MIDI note number (p). Any other frequencies fill the space between the whole numbers evenly. This allows MIDI instruments to be tuned very accurately in any microtuning scale, including non-western traditional tunings.