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Encyclopedia of Microtonal Music Theory

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1mu / enamu

[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".

At the setting for enamu pitch-bend resolution, a semitone is divided into 21 = 2 pitch-bend units. Thus there are 2 * 12 = 24 enamus in an "octave", so the enamu measurement system may be thought of as 24-EDO tuning, with a enamu being one degree in 24-EDO.

An enamu is calculated as the 24th root of 2 -- 24√2, or 2(1/24) -- with a ratio of approximately 1:1.029302237. It is an irrational number, but is very close to the ratio 35:34 ( 2-1 51 71 17-1 ): the difference is ~ -0.184210833552446 (~ 1/5) of a cent, which in most cases would be hard to distinguish. The formula for calculating the 1mu-value of any ratio is: 1mus = log10r * [ (21 * 12) / log10(2)] or 1mus = log2r * (21 * 12) , where r is the ratio.

An enamu is exactly 50 cents, and is identical to the quarter-tone. Since "quarter-tone" is already such a well-established term, "enamu" is not likely to gain much currency. (More information can thus be found under the "quarter-tone" entry.)

The internal data structure of the 1mu 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 five 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 1mu spec thus uses a total of 2+1 = 3 bits.

  thus, the maximum possible value is:

      ds1x xxxx  [binary]

    =  +/-    20 [hex]

    =  +/-    32 [decimal]

    note that the first nibble can only indicate the sign + or -
    and the data-values 0 or 32 [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  x   x  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  1  x   x  x  x  x  --  bits

= 96 decimal = 60 hex = one unit (50 cents) above the 12edo MIDI-note.

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

= 32 decimal = 20 hex = one unit (50 cents) below the 12edo MIDI-note.

Therefore the 1mu gives a range of possible values of 0 or +/- 32 [decimal] = 0 or +/- 20 [hex]. Even for humans (at least, those familiar with hexadecimal numbering) this provides a convenient measurement as it only requires one hexadecimal digit to show deviation from the 12edo MIDI-note.

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 1mus plus or minus, respectively.

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