Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited

dodekamu, 12mu


    (NOTE: This term supercedes cawapu.)


    A term I and a few other tuning theorists coined in July 2003, based on an idea by Aaron Hunt, to describe one a family of terms referring to units of resolution in MIDI tuning, in which the prefix specifies the exponent of 2 which describes the number of MIDI tuning units per semitone and the final "mu" is an acronym for "MIDI unit".

    At the setting for dodekamu pitch-bend resolution, a semitone is divided into 212 = 4096 pitch-bend units. Thus there are 4096 * 12 = 49152 dodekamus in an "octave", so the dodekamu measurement system may be thought of as 49152-EDO tuning, with a dodekamu being one degree in 49152-EDO.

    A dodekamu is calculated as the 49152th root of 2, or 2(1/49152), with a ratio of approximately 1:1.000014102. It is an irrational number, but is extremely close to the ratio 70913:70912 : the difference is less than 1/2,500,000 of a cent, making them for all intents and purposes identical.


    A dodekamu is

    • exactly 125/6144 (= 0.02034505208333... = ~ 1/49 ) of a milli8ve,

    • exactly 25/1024 (= 0.0244140625 = ~ 1/41 ) of a cent,

    • exactly 1325/6144 (= 0.21565755208333... = ~ 1/5 ) of a türk-sent,

    • exactly 30103/49152 (= 0.61244710286458333... = ~ 3/5 ) of a jot,

    • approximately 3/4 (= ~ 0.749280359 ) of a tuning unit.


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

    dodekamus = log10(ratio) * [ (212 * 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 dodekamus 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:

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

    [from Joe Monzo, JustMusic: A New Harmony]

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