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Encyclopedia of Microtonal Music Theory

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3mu / triamu

[Joe Monzo, Tonalsoft Encyclopedia of Microtonal Music Theory]

A term coined in July 2003 by a group of tuning theorists (including Aaron Hunt, Gene Ward Smith, and Joe Monzo), to describe one of a family of terms referring to units of resolution in MIDI tuning, used in electronic music software and computer music software. 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". In this work the numerical figure is used in preference to the verbal prefix.

At the setting for 3mu pitch-bend resolution, a semitone is divided into 23 = 8 pitch-bend units. Thus there are 8 * 12 = 96 3mus in an "octave", so the 3mu measurement system may be thought of as 96-edo tuning, with a 3mu being one degree in 96-edo.

A 3mu is calculated as the 96th root of 2 -- 96√2, or 2(1/96) -- with a ratio of approximately 1:1.007246412. It is an irrational number, but is very close to the ratio 139:138 : the difference is ~1/16,000 of a cent, which for all intents and purposes makes the 3mu identical to that ratio. The formula for calculating the 3mu-value of any ratio is: 3mus = log10r * [ (23 * 12) / log10(2)] or 3mus = log2r * (23 * 12) , where r is the ratio.

A 3mu is:

The 3mu is also known as a 1/16-tone, a tuning which was used and advocated by Julian Carillo.

The internal data structure of the 3mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or -) showing the direction of the pitch-bend up or down, and three other bits which are not used, as follows:

  let "d" designate the bits that cannot be used
  because it is reserved for the SysEx flag, to
  indicate that this is a byte of pitch-bend data.

  let "s" designate the bit that represents the
  sign of the pitch-bend data, + or - .

  let "x" designate unused bits

  the 3mu spec thus uses a total of 2+3 = 5 bits.

  thus, the maximum possible value is:

      ds11 1xxx  [binary]

    =  +/-    38  [hex]

    =  +/-    56  [decimal]

    note that the first nibble can only indicate the sign + or -
    and the data-values 0, 16, 32, or 48 [decimal].

Below is an illustration of exactly how this works.

The "x" represents the status flag at the beginning of the byte,
and is not recognized as part of the tuning resolution.

The bit which represents 64 [decimal] is the sign bit.

The actual tuning data begins with the bit representing 32 [decimal].

x 64 32 16   8  4  2  1  --  decimal value
x  1  0  0   0  x  x  x  --  bits

= 64 decimal = 40 hex = the plain MIDI-note, 0 cents deviation from 12edo.

x 64 32 16   8  4  2  1  --  decimal value
x  1  0  0   1  x  x  x  --  bits

= 72 decimal = 48 hex = one unit (12.5 cents) above the 12edo MIDI-note.

x 64 32 16   8  4  2  1  --  decimal value
x  0  1  1   1  x  x  x  --  bits

= 56 decimal = 38 hex = one unit (12.5 cents) below the 12edo MIDI-note.

Therefore the 3mu gives a range of possible values +/- as follows:

decimal  hex    cents
    0     00     0
    8     08    12.5
   16     10    25
   24     18    37.5
   32     20    50
   40     28    62.5
   48     30    75
   56     38    87.5
  (64     40   100)

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, as A-1, Bb2, etc.). Then the remainder or deficit is converted into 3mus plus or minus, respectively.

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