MIDI Tuning Standard
MIDI Tuning Standard (MTS) is a specification of precise musical pitch agreed to by the MIDI Manufacturers Association in the MIDI protocol. MTS allows for both a bulk tuning dump message, giving a tuning for each of 128 notes, and a tuning message for individual notes as they are played.
If ƒ is a frequency, then the corresponding frequency data value d may be computed by
The quantity log2 (ƒ / 440 Hz) is the number of octaves above the 440-Hz concert A (it is negative if the frequency is below that pitch). Multiplying it by 12 gives the number of semitones above that frequency. Adding 69 gives the number of semitones above the C five octaves below middle C.
Since 440 Hz is a widely used standard concert A (e.g. USA, UK), and since that is represented in MIDI terms by the integer 69 (nine semitones above middle C, which is 60), this gives a real number which expresses pitch in a manner consistent with MIDI and integer notation, known as the midi note number.
Converting from midi note number (d) to frequency (f) is given by the following formula:
The frequency data format allows for the precise notation of frequencies that differ from equal temperament.
"Frequency data shall be defined in [units] which are fractions of a semitone. The frequency range starts at MIDI note 0, C = 8.1758 Hz, and extends above MIDI note 127, G = 12543.875 Hz. The first byte of the frequency data word specifies the highest equal-tempered semitone not exceeding the frequency. The next two bytes (14 bits) specify the fraction of 100 cents above the semitone at which the frequency lies. Effective resolution = 100 cents / 214 = .0061 cents."
This higher resolution allows a logarithmic representation of pitch in which the semitone is divided into 1282 = 214 = 16384 parts, which means the octave is divided into 196608 (logarithmically) equal parts. These parts are exactly 100/16384 cents (approximately 0.0061 cents) in size, which is far below the threshold of human pitch perception and which therefore allows a very accurate representation of pitch.
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