Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited

pentamu, 5mu


    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 number of bits resolution (i.e., 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 pentamu pitch-bend resolution, a semitone is divided into 25 = 32 pitch-bend units. Thus there are 32 * 12 = 384 pentamus in an "octave", so the pentamu measurement system may be thought of as 384-EDO tuning, with a pentamu being one degree of 384-EDO.

    A pentamu is calculated as the 384th root of 2, or 2(1/384), with a ratio of approximately 1:1.001806701. It is an irrational number, but is extremely close to the ratio 1109:1107 (= 3-3 41-1 11091): the difference is only ~ 1/36,000 of a cent, which makes them for all intents and purposes identical.


    A pentamu is


    • approximately 1/7 of a 53edo, syntonic, or Pythagorean comma,

    • exactly 3/16 (= 0.1875 = ~ 1/5 ) of a 72edo-morion,

    • exactly 25/32 (= 0.78125 = ~ 7/9 ) of a 300edo-savart,

    • exactly 1 19/32 (= 1.59375 = ~ 1 3/5 ) 612edo-schismas,

    • exactly 2 29/48 (= 2.6041666... = ~ 2 3/5 ) milli8ves,

    • exactly 3 1/8 (= 3.125 ) cents.

    • exactly 27 29/48 ( 27.6041666..., ~ 27 3/5 ) türk-sents.


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

    pentamus = log10(ratio) * [ (25 * 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 pentamus 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
    hexamu
    heptamu
    oktamu
    enneamu
    dekamu
    endekamu
    dodekamu
    tridekamu
    tetradekamu
    cawapu
    midipu
    my Gentle Introduction to the MIDI Tuning Specification
    the Official MIDI Tuning Specification.

    [from Joe Monzo, JustMusic: A New Harmony]

updated:

    2003.07.04 -- page created

(to download a zip file of the entire Dictionary, click here)

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