  # 5mu / pentamu

[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. Te 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 5mu pitch-bend resolution, a semitone is divided into 25 = 32 pitch-bend units. Thus there are 32 * 12 = 384 5mus in an "octave", so the 5mu measurement system may be thought of as 384-edo tuning, with a 5mu being one degree in 384-edo.

A 5mu is calculated as the 384th root of 2 -- 384√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 for all intents and purposes makes the 5mu identical to that ratio. The formula for calculating the 5mu-value of any ratio is: 5mus = log10r * [ (25 * 12) / log10(2)] or 5mus = log2r * (25 * 12) , where r is the ratio.

A 5mu is:

• approximately 1/7 of a 53-edo, pythagorean-, or syntonic-comma
• exactly 3/16 (0.1875..., ~ 1/5) of a 72-edo morion
• exactly 25/32 (0.78125..., ~ 3/4) of a 300-edo savart
• exactly 1 19/32 (1.59375..., ~ 1 3/5) 612-edo schismas
• exactly 2 29/48 (2.6041666......, ~ 2 3/5) 1000-edo milli8ves
• exactly 3 1/8 (3.125) 1200-edo cents
• exactly 27 29/48 (27.6041666..., ~ 27 3/5) 10600-edo türk-sents

The internal data structure of the 5mu 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 one other bit which is 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 5mu spec thus uses a total of 2+5 = 7 bits.

thus, the maximum possible value is:

ds11 111x  [binary]

=  +/-   3E  [hex]

=  +/-   62  [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  0  0  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   0  0  1  x  --  bits

= 66 decimal = 42 hex = one unit (3.125 cents) above the 12edo MIDI-note.

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

= 62 decimal = 3E hex = one unit (3.125 cents) below the 12edo MIDI-note.
```

Therefore the 5mu gives a range of possible values +/- as follows:

```                ----- cents -----
decimal  hex    decimal  fraction

0     00     0
2     02     3.125    3 1/8
4     04     6.25     6 2/8
6     06     9.375    9 3/8
8     08    12.5     12 4/8
10     0A    15.625   15 5/8
12     0C    18.75    18 6/8
14     0E    21.875   21 7/8
16     10    25       25
18     12    28.125   28 1/8
20     14    31.25    31 2/8
22     16    34.375   34 3/8
24     18    37.5     37 4/8
26     1A    40.625   40 5/8
28     1C    43.75    43 6/8
30     1E    46.875   46 7/8
32     20    50       50
34     22    53.125   53 1/8
36     24    56.25    56 2/8
38     26    59.375   59 3/8
40     28    62.5     62 4/8
42     2A    65.625   65 5/8
44     2C    68.75    68 6/8
46     2E    71.875   71 7/8
48     30    75       75
50     32    78.125   78 1/8
52     34    81.25    81 2/8
54     36    84.375   84 3/8
56     38    87.5     87 4/8
58     3A    90.625   90 5/8
60     3C    93.75    93 6/8
62     3E    96.875   96 7/8
(64     40   100)     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 5mus plus or minus, respectively.

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