Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited

tetradekamu, tetradecamu, 14mu



    (NOTE: This term supercedes midipu.)


    I had previously suggested the name 'midipu' for the 14-bit resolution of the MTS (MIDI tuning standard). In July 2003, Aaron Hunt suggested the general name 'Mu' for 'MIDI unit', and suggested that Greek prefixes be used to specify the number of bits resulting in the unit. Prefixes were offered and agreed upon by Aaron, myself, Gene Ward Smith and John Chalmers.

    The tetradekamu is specified in the MIDI spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999).

    At the setting for finest pitch-bend resolution, a semitone is divided into 214 = 16384 pitch-bend units, or tetradekamus. Thus there are 16384 * 12 = 196608 tetradekamus in an "octave", so the tetradekamus measurement system may be thought of as 196608-EDO tuning, with a tetradekamu being one degree of 196608-EDO.

    A tetradekamu is calculated as the 196608th root of 2, or 2(1/196608), with a ratio of approximately 1:1.000003526. It is an irrational number, but is extremely close to the ratio 283608:283607 : the difference is only ~ 1/1,000,000 of a cent, making them for all intents and purposes identical.


    A tetradekamu is

    • exactly 125/24576 (= 0.0050862630208333... = ~ 1/197 ) of a milli8ve,

    • exactly 25/4096 (= 0.006103515625 = ~ 1/164 ) of a cent,

    • exactly 1325/24576 (= 0.0539143880208333... = ~ 1/19 ) of a türk-sent,

    • exactly 30103/196608 (= 0.1531117757161458333... = ~ 1/7 ) of a jot,

    • approximately 1/5 (= ~ 0.18732009 ) of a tuning unit.


    The formula for calculating the tetradekamu-value of any ratio is:

    midipus = log10(ratio) * [ (214 * 12) / log10(2) ]

    For practical use in tuning MIDI-files, an interval's semitone value must first be calculated. The nearest integer semitone is translated into a MIDI note-number (which can generally also be described by letter-name plus optional accidental: A, Bb, C#, etc., followed by an "octave" register-number). Then the remainder or deficit is converted into tetradekamus plus or minus, respectively. These give the correct tuning to a tolerance that is far better than anything that the human auditory system can detect, or indeed far better than what is available in the output of any electronic instruments.

    See also:

    my Gentle Introduction to the MIDI Tuning Specification
    the Official MIDI Tuning Specification.
    enamu
    doamu
    triamu
    tetramu
    pentamu
    hexamu
    heptamu
    oktamu
    enneamu
    dekamu
    endekamu
    dodekamu
    tridekamu
    cawapu
    midipu

    [from Joe Monzo, JustMusic: A New Harmony]

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