  # 4mu / tetramu

[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 4mu pitch-bend resolution, a semitone is divided into 24 = 16 pitch-bend units. Thus there are 16 * 12 = 192 4mus in an "octave", so the 4mu measurement system may be thought of as 192-edo tuning, with a 4mu being one degree in 192-edo.

A 4mu is calculated as the 192nd root of 2 -- 192√2, or 2(1/192) -- with a ratio of approximately 1:1.003616666. It is an irrational number, but is extremely close to the ratio 555:553 (31 51 7-1 371 79-1): a 4mu is only a little more than 1/120 of a 14mu (~ 1/20,000 of a cent) larger than 555:553, which for all intents and purposes makes the 4mu identical to that ratio. The formula for calculating the 4mu-value of any ratio is: 4mus = log10r * [ (24 * 12) / log10(2)] or 4mus = log2r * (24 * 12) , where r is the ratio.

A 4mu is:

• approximately 1/4 of a 53-edo, syntonic-, or pythagorean-comma
• exactly 3/8 (0.375, ~1/3) of a 72-edo morion
• exactly 1 9/16 (1.5625, ~ 11/2) 300-edo savarts
• exactly 3 3/16 (3.1875, ~ 31/5) 612-edo schismas
• exactly 5 5/24 (5.208333..., ~ 51/5) 1000-edo milli8ves
• exactly 6 1/4 (6.25) 1200-edo cents
• exactly 55 5/24 (55.208333..., ~ 551/5) 10600-edo türk-sents

The internal data structure of the 4mu 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 two 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 4mu spec thus uses a total of 2+4 = 6 bits.

thus, the maximum possible value is:

ds11 11xx  [binary]

=  +/-    F  [hex]

=  +/-   15  [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  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   0  1  x  x  --  bits

= 68 decimal = 44 hex = one unit (6.25 cents) above the 12edo MIDI-note.

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

= 60 decimal = 3C hex = one unit (6.25 cents) below the 12edo MIDI-note.
```

Therefore the 4mu gives a range of possible values +/- as follows:

```                 ----- cents -----
decimal  hex     decimal   fraction
0     00      0          0
4     04      6.25       6 1/4
8     08     12.5       12 2/4
12     0C     18.75      18 3/4
16     10     25         25
20     14     31.25      31 1/4
24     18     37.5       37 2/4
28     1C     43.75      43 3/4
32     20     50         50
36     24     56.25      56 1/4
40     28     62.5       62 2/4
44     2C     68.75      68 3/4
48     30     75         75
52     34     81.25      81 1/4
56     38     87.5       87 2/4
60     3C     93.75      93 3/4
(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 4mus plus or minus, respectively.

. . . . . . . . .

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