# 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.

## Frequency values

If f is a frequency in hertz, then the corresponding MIDI note number N$_{MIDI}$ is given by the formula
    $N_{\mathsf {MIDI}}=69+12\cdot \log _{2}\left({\frac {f}{\ 440\ \mathrm {Hz} \ }}\right)=69+{\frac {12}{\ \log 2\ }}\log \left({\frac {f}{\ 440\ \mathrm {Hz} \ }}\right)\ ,$
where "log" in the second expression is any logarithm (e.g. either the common logarithm log$_{10}$ , the natural logarithm ln ≡ log$_e$ , or any other).
The quantity log$_2$( ⁠ f / 440 Hz ⁠ ) is the number of octaves above the 440 Hz concert A, or A$_4$, or a′ . Multiplying it by 12 gives the number of semitones above 440 Hz (the value is negative if the frequency f is lower in pitch than 440 Hz). Adding 69 (decimal, or 0x45 hexadecimal) gives the number of semitones above the C five octaves below middle C.
Not only is 440 Hz the standard central pitch for MIDI, it is also widely used as the concert A standard pitch (A$_4$ e.g. USA, UK), and since that is represented in MIDI signals by the integer 69 (nine semitones above middle C (C$_4$, c′), which is 60 decimal or 0x3C hexadecimal), this gives a real number which expresses pitch in a manner consistent with midiand integer notation, known as the MIDI note number, N$_{MIDI}$ .
Converting from MIDI note number (N$_{MIDI}$) to frequency (f) is given by the following formula:
    $\ f=440\ {\mathsf {Hz}}\cdot 2^{\left[\ \left(\ N_{\mathsf {MIDI}}\ -\ 69\ \right)/12\ \right]}\ =440\ {\mathsf {Hz}}\cdot \exp \left[\ {\frac {\ \ln 2\ }{12}}\left(\ N_{\mathsf {MIDI}}\ -\ 69\ \right)\ \right]~.$

## Frequency Data Format

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 = 12543.854 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 / 2$^{14}$ = .0061 cents."
This higher resolution allows a logarithmic representation of pitch in which the semitone is divided into 128$^2$ = 2$^{14}$ = 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.

## Applications

The precision pitch values may be used in microtonal music, just intonation, meantone temperament, or other alternative tunings.
Software which supports MTS includes Scala, TiMidity++, ZynAddSubFX and FluidSynth.
Software plugin instruments which support MTS include Native Instruments FM8, Synthogy Ivory, and Xen-Arts' various xenharmonic VSTi plugins, including the FMTS FM synthesizer, Ivor virtual analog synthesizer, and XenFont SoundFont sample player.
Hardware instruments in current production which support MTS include: Dave Smith Instruments (DSI) Rev-2, Prophet-12, Prophet-6, Oberheim OB-6, Moog Sub37, Minitaur, Novation Bass Station II, Peak, Sonoclast Plastic Pitch MIDI Microtuner, and the Waldorf Kyra.