View source for MIDI tuning standard

{{Short description|Specification of precise musical pitch}}
{{One source|date=June 2023}}

'''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 [[Musical note|notes]], and a tuning message for individual notes as they are played.

==Frequency values==
If {{mvar|f}} is a [[frequency]] in [[Hertz (unit)|hertz]], then the corresponding MIDI note number {{mvar|N}}{{sub|{{sc|MIDI}}}} is given by the formula

:<math> 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)\ ,</math>

where "{{math|log}}" in the second expression is ''any'' logarithm (e.g. either the [[common logarithm]] {{math|log{{sub|10}}}}&nbsp;, the [[natural logarithm]] {{nobr|{{math|ln ≡ log{{sub|''e''}}}}}}&nbsp;, or any other).

The quantity {{nobr|{{math|log<sub>2</sub>{{big|(}} {{small|{{sfrac| {{big|''f''}} | 440 Hz }} }} {{big|)}} }} }} is the number of [[octaves]] above the 440&nbsp;Hz [[concert A|concert A]], or [[scientific pitch notation|A<sub>4</sub>]], or [[Helmholtz pitch notation|{{math|a}}&prime;]]&nbsp;. Multiplying it by&nbsp;12 gives the number of [[semitones]] above 440&nbsp;Hz (the value is negative if the frequency {{mvar|f}} is lower in pitch than 440&nbsp;Hz). Adding&nbsp;69 ([[decimal notation|decimal]], or 0x45&nbsp;[[hexadecimal]]) gives the number of semitones above the C five octaves below [[middle C]].

Not only is 440&nbsp;[[hertz|Hz]] the standard central pitch for MIDI, it is also widely used as the [[concert A]] standard pitch (A{{sub|4}} e.g. USA, UK), and since that is represented in MIDI signals by the integer&nbsp;69 (nine semitones above middle C (C{{sub|4}}, {{math|'''c&prime;'''}}), which is 60&nbsp;decimal or 0x3C&nbsp;[[hexadecimal]]), this gives a real number which expresses pitch in a manner consistent with midiand [[integer notation]], known as the MIDI note number, {{nobr|{{mvar|N}}{{sub|MIDI}} .}}

Converting from MIDI note number ({{mvar|N}}{{sub|MIDI}}) to frequency ({{mvar|f}}) is given by the following formula:

:<math>\ 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] ~.</math>

==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 [[Cent (music)|cents]] above the semitone at which the frequency lies. Effective resolution = 100 cents / 2<sup>14</sup> = .0061 cents."<ref name="refname1" />

This higher resolution allows a logarithmic representation of pitch in which the semitone is divided into 128<sup>2</sup> = 2<sup>14</sup> = 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 [[hearing range|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 (program)|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, [https://sonoclast.com/products/plastic_pitch_midi_microtuner/ Sonoclast Plastic Pitch MIDI Microtuner], and the Waldorf Kyra.

==See also==
*[[Microtonal music]]
*[[Microtuner]]
*[[Musical tuning]]

==References==
{{Reflist|refs=
<ref name="refname1">MIDI Manufacturers Association, MIDI Tuning Specification ( https://midi.org/midi-tuning-updated-specification/ )</ref>
}}

==External links==

{{Spoken Wikipedia|En-MIDI Tuning Standard-article.ogg|date=2016-04-25}}

===MTS specification===
* [https://midi.org/midi-tuning-updated-specification/ MTS specification]

===Microtonal music software supporting MTS===
* [http://www.huygens-fokker.org/scala/ Scala (Windows, Linux, Mac)]
* [http://hpi.zentral.zone Custom Scale Editor (Windows, Mac)]
* [https://hpi.zentral.zone/microsynth microsynth (Windows, Mac)]
* [http://www.tallkite.com/alt-tuner.html alt-tuner (Windows, Linux, Mac)]
* [http://timidity.sourceforge.net/ TiMidity++ (Windows, Linux, Mac)]
* [https://www.fluidsynth.org/ FluidSynth (Windows, Linux, Mac)]
* [http://www.robertinventor.com/software/tunesmithy/music.htm Tune Smithy (Windows)]
* [http://xen-arts.net/vsti/ Xen-Arts Microtonal MIDI Software (Windows VSTi)]
* [http://www.dynamictonality.com/relayer.htm Relayer (Windows, Mac)]
* [http://www.nonoctave.com/tuning/LilMissScaleOven/features.html L'il Miss' Scale Oven (Mac)] {{Webarchive|url=https://web.archive.org/web/20150102205116/http://www.nonoctave.com/tuning/LilMissScaleOven/features.html |date=2015-01-02 }}
* [https://archive.today/20130628023426/https://java.net/projects/gervill/pages/Home Gervill (Windows, Linux, Mac)]

[[Category:Computer file formats]]
[[Category:Electronic music]]
[[Category:MIDI standards]]
[[Category:Music notation file formats]]
[[Category:Musical tuning]]