  # 2mu / doamu (also 48-ed2 / 48-edo / 48-ET / 48-tone equal-temperament)

[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 2mu pitch-bend resolution, a semitone is divided into 22 = 4 pitch-bend units. Thus there are 4 * 12 = 48 2mus in an "octave", so the 2mu measurement system may be thought of as 48-edo tuning, with a 2mu being one degree in 48-edo.

A 2mu is calculated as the 48th root of 2 -- 48√2, or 2(1/48) -- with a ratio of approximately 1:1.014545335. It is an irrational number, but is very close to the ratio 70:69 : the difference is ~1/11 of a cent, which in most cases would be hard to distinguish. The formula for calculating the 2mu-value of any ratio is: 2mus = log10r * [ (22 * 12) / log10(2)] or 2mus = log2r * (22 * 12) , where r is the ratio.

A 2mu is:

• exactly 43/48 (0.8958333..., ~ 8/9) of a meride
• exactly 1 1/2 ( 1.5 ) 72-edo moria
• exactly 6 1/4 ( 6.25 ) 300-edo savarts
• exactly 12 3/4 ( 12.75 ) 612-edo schismas
• exactly 20 5/6 ( 20.8333... ) 1000-edo millioctaves
• exactly 25 cents

The 2mu is also known as an "eighth-tone" or 1/8-tone.

The internal data structure of the 2mu 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 four 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 2mu spec thus uses a total of 2+2 = 4 bits.

thus, the maximum possible value is:

ds11 xxxx  [binary]

=  +/-    30 [hex]

=  +/-    48 [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   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  0  1   x  x  x  x  --  bits

= 80 decimal = 50 hex = one unit (25 cents) above the 12edo MIDI-note.

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

= 48 decimal = 30 hex = one unit (25 cents) below the 12edo MIDI-note.
```

Therefore the 2mu gives a range of possible values +/- as follows:

```decimal  hex    cents
0     00     0
16     10    25
32     20    50
48     30    75
(64     40   100)
```

The total possible set of 2mu values is thus:

```x 64 32 16   8  4  2  1
x  1  1  1   x  x  x  x   =  64 + 48 = 112  =  12-edo MIDI-note + 75 cents
x  1  1  0   x  x  x  x   =  64 + 32 =  96  =  12-edo MIDI-note + 50 cents
x  1  0  1   x  x  x  x   =  64 + 16 =  80  =  12-edo MIDI-note + 25 cents
x  1  0  0   x  x  x  x   =  64             =  12-edo MIDI-note
x  0  1  1   x  x  x  x   =  64 - 16 =  48  =  12-edo MIDI-note - 25 cents
x  0  1  0   x  x  x  x   =  64 - 32 =  32  =  12-edo MIDI-note - 50 cents
x  0  0  1   x  x  x  x   =  64 - 48 =  16  =  12-edo MIDI-note - 75 cents
```

All mus operate in this fashion.

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 2mus plus or minus, respectively.

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by selecting the highest level of financial support that you can afford. Thank you.

 support level patron: \$50 USD savior: \$100 USD angel of tuning: \$500 USD microtonal god: \$1000 USD friend: \$25 USD donor: \$5 USD 