# 6mu / hexamu / 768-edo

[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 6mu pitch-bend resolution, a semitone is divided into 26 = 64 pitch-bend units. Thus there are 64 * 12 = 768 6mus in an "octave", so the 6mu measurement system may be thought of as 768-edo tuning, with a 6mu being one degree in 768-edo.

A 6mu is calculated as the 768th root of 2 -- 768√2, or 2(1/768) -- with a ratio of approximately 1:1.000902943. It is an irrational number, but is extremely close to the ratio 2217:2215 ( 31 5-1 443-1 7391 ):: the difference is ~ 1/70,000 of a cent, which for all intents and purposes makes the 6mu identical to that ratio. The formula for calculating the 6mu-value of any ratio is: 6mus = log10r * [ (26 * 12) / log10(2)] or 6mus = log2r * (26 * 12) , where r is the ratio.

A 6mu is:

• exactly 25/64 (= 0.390625 = ~ 2/5) of a 300-edo savart
• exactly 51/64 (= 0.796875 = ~ 4/5) of a 612-edo schisma
• exactly 1 29/96 (= 1.30208333... = ~ 1 1/3) millioctaves
• exactly 1 1/3 (= 1.3333... ) yamaha-units
• exactly 1 9/16 (= 1.5625 = ~ 1 1/2) cents
• exactly 3 13/64 (= 3.203125 = ~ 3 1/5) minas
• exactly 11 91/768 (= 11.11848958,3... = ~ 11 1/9) tinas
• exactly 13 77/96 (= 13.80208333... = ~ 13 4/5) türk-sents
• exactly 39 151/768 (= 39.19661458333... = ~ 39 1/5) jots
• approximately 48 (= ~ 47.95394297 = ~ 47 62/65) tuning-units
• exactly 64 12mus
• exactly 256 14mus

The internal data structure of the 6mu 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 all six of the remaining bits used for the tuning data, 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 6mu spec thus uses a total of 2+6 = 8 bits.

thus, the maximum possible value is:

ds11 1111  [binary]

=  +/-   3F  [hex]

=  +/-   63  [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  0  --  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  0  1  --  bits

= 65 decimal = 41 hex = one unit (1.5625 cents) above the 12edo MIDI-note.

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

= 63 decimal = 3F hex = one unit (1.5625 cents) below the 12edo MIDI-note.
```

Therefore the 6mu gives a range of possible values +/- as follows:

```-------- bits ---------                  ------- cents ------
x 64 32 16   8  4  2  1  decimal  hex     decimal    fraction

(128     80    +100.0000   +100)

x  1  1  1   1  1  1  1    127     7F    + 98.4375   + 98 7/16
x  1  1  1   1  1  1  0    126     7E    + 96.8750   + 96 14/16
x  1  1  1   1  1  0  1    125     7D    + 95.3125   + 95 5/16
x  1  1  1   1  1  0  0    124     7C    + 93.7500   + 93 12/16
x  1  1  1   1  0  1  1    123     7B    + 92.1875   + 92 3/16
x  1  1  1   1  0  1  0    122     7A    + 90.6250   + 90 10/16
x  1  1  1   1  0  0  1    121     79    + 89.0625   + 89 1/16
x  1  1  1   1  0  0  0    120     78    + 87.5000   + 87 8/16
x  1  1  1   0  1  1  1    119     77    + 85.9375   + 85 15/16
x  1  1  1   0  1  1  0    118     76    + 84.3750   + 84 6/16
x  1  1  1   0  1  0  1    117     75    + 82.8125   + 82 13/16
x  1  1  1   0  1  0  0    116     74    + 81.2500   + 81 4/16
x  1  1  1   0  0  1  1    115     73    + 79.6875   + 79 11/16
x  1  1  1   0  0  1  0    114     72    + 78.1250   + 78 2/16
x  1  1  1   0  0  0  1    113     71    + 76.5625   + 76 9/16
x  1  1  1   0  0  0  0    112     70    + 75.0000   + 75

x  1  1  0   1  1  1  1    111     6F    + 73.4375   + 73 7/16
x  1  1  0   1  1  1  0    110     6E    + 71.8750   + 71 14/16
x  1  1  0   1  1  0  1    109     6D    + 70.3125   + 70 5/16
x  1  1  0   1  1  0  0    108     6C    + 68.7500   + 68 12/16
x  1  1  0   1  0  1  1    107     6B    + 67.1875   + 67 3/16
x  1  1  0   1  0  1  0    106     6A    + 65.6250   + 65 10/16
x  1  1  0   1  0  0  1    105     69    + 64.0625   + 64 1/16
x  1  1  0   1  0  0  0    104     68    + 62.5000   + 62 8/16
x  1  1  0   0  1  1  1    103     67    + 60.9375   + 60 15/16
x  1  1  0   0  1  1  0    102     66    + 59.3750   + 59 6/16
x  1  1  0   0  1  0  1    101     65    + 57.8125   + 57 13/16
x  1  1  0   0  1  0  0    100     64    + 56.2500   + 56 4/16
x  1  1  0   0  0  1  1     99     63    + 54.6875   + 54 11/16
x  1  1  0   0  0  1  0     98     62    + 53.1250   + 53 2/16
x  1  1  0   0  0  0  1     97     61    + 51.5625   + 51 9/16
x  1  1  0   0  0  0  0     96     60    + 50.0000   + 50

x  1  0  1   1  1  1  1     95     5F    + 48.4375   + 48 7/16
x  1  0  1   1  1  1  0     94     5E    + 46.8750   + 46 14/16
x  1  0  1   1  1  0  1     93     5D    + 45.3125   + 45 5/16
x  1  0  1   1  1  0  0     92     5C    + 43.7500   + 43 12/16
x  1  0  1   1  0  1  1     91     5B    + 42.1875   + 42 3/16
x  1  0  1   1  0  1  0     90     5A    + 40.6250   + 40 10/16
x  1  0  1   1  0  0  1     89     59    + 39.0625   + 39 1/16
x  1  0  1   1  0  0  0     88     58    + 37.5000   + 37 8/16
x  1  0  1   0  1  1  1     87     57    + 35.9375   + 35 15/16
x  1  0  1   0  1  1  0     86     56    + 34.3750   + 34 6/16
x  1  0  1   0  1  0  1     85     55    + 32.8125   + 32 13/16
x  1  0  1   0  1  0  0     84     54    + 31.2500   + 31 4/16
x  1  0  1   0  0  1  1     83     53    + 29.6875   + 29 11/16
x  1  0  1   0  0  1  0     82     52    + 28.1250   + 28 2/16
x  1  0  1   0  0  0  1     81     51    + 26.5625   + 26 9/16
x  1  0  1   0  0  0  0     80     50    + 25.0000   + 25

x  1  0  0   1  1  1  1     79     4F    + 23.4375   + 23 7/16
x  1  0  0   1  1  1  0     78     4E    + 21.8750   + 21 14/16
x  1  0  0   1  1  0  1     77     4D    + 20.3125   + 20 5/16
x  1  0  0   1  1  0  0     76     4C    + 18.7500   + 18 12/16
x  1  0  0   1  0  1  1     75     4B    + 17.1875   + 17 3/16
x  1  0  0   1  0  1  0     74     4A    + 15.6250   + 15 10/16
x  1  0  0   1  0  0  1     73     49    + 14.0625   + 14 1/16
x  1  0  0   1  0  0  0     72     48    + 12.5000   + 12 8/16
x  1  0  0   0  1  1  1     71     47    + 10.9375   + 10 15/16
x  1  0  0   0  1  1  0     70     46    +  9.3750   +  9 6/16
x  1  0  0   0  1  0  1     69     45    +  7.8125   +  7 13/16
x  1  0  0   0  1  0  0     68     44    +  6.2500   +  6 4/16
x  1  0  0   0  0  1  1     67     43    +  4.6875   +  4 11/16
x  1  0  0   0  0  1  0     66     42    +  3.1250   +  3 2/16
x  1  0  0   0  0  0  1     65     41    +  1.5625   +  1 9/16
x  1  0  0   0  0  0  0     64     40       0.0000      0

x  0  1  1   1  1  1  1     63     3F    -  1.5625   -  1 9/16
x  0  1  1   1  1  1  0     62     3E    -  3.1250   -  3 2/16
x  0  1  1   1  1  0  1     61     3D    -  4.6875   -  4 11/16
x  0  1  1   1  1  0  0     60     3C    -  6.2500   -  6 4/16
x  0  1  1   1  0  1  1     59     3B    -  7.8125   -  7 13/16
x  0  1  1   1  0  1  0     58     3A    -  9.3750   -  9 6/16
x  0  1  1   1  0  0  1     57     39    - 10.9375   - 10 15/16
x  0  1  1   1  0  0  0     56     38    - 12.5000   - 12 8/16
x  0  1  1   0  1  1  1     55     37    - 14.0625   - 14 1/16
x  0  1  1   0  1  1  0     54     36    - 15.6250   - 15 10/16
x  0  1  1   0  1  0  1     53     35    - 17.1875   - 17 3/16
x  0  1  1   0  1  0  0     52     34    - 18.7500   - 18 12/16
x  0  1  1   0  0  1  1     51     33    - 20.3125   - 20 5/16
x  0  1  1   0  0  1  0     50     32    - 21.8750   - 21 14/16
x  0  1  1   0  0  0  1     49     31    - 23.4375   - 23 7/16
x  0  1  1   0  0  0  0     48     30    - 25.0000   - 25

x  0  1  0   1  1  1  1     47     2F    - 26.5625   - 26 9/16
x  0  1  0   1  1  1  0     46     2E    - 28.1250   - 28 2/16
x  0  1  0   1  1  0  1     45     2D    - 29.6875   - 29 11/16
x  0  1  0   1  1  0  0     44     2C    - 31.2500   - 31 4/16
x  0  1  0   1  0  1  1     43     2B    - 32.8125   - 32 13/16
x  0  1  0   1  0  1  0     42     2A    - 34.3750   - 34 6/16
x  0  1  0   1  0  0  1     41     29    - 35.9375   - 35 15/16
x  0  1  0   1  0  0  0     40     28    - 37.5000   - 37 8/16
x  0  1  0   0  1  1  1     39     27    - 39.0625   - 39 1/16
x  0  1  0   0  1  1  0     38     26    - 40.6250   - 40 10/16
x  0  1  0   0  1  0  1     37     25    - 42.1875   - 42 3/16
x  0  1  0   0  1  0  0     36     24    - 43.7500   - 43 12/16
x  0  1  0   0  0  1  1     35     23    - 45.3125   - 45 5/16
x  0  1  0   0  0  1  0     34     22    - 46.8750   - 46 14/16
x  0  1  0   0  0  0  1     33     21    - 48.4375   - 48 7/16
x  0  1  0   0  0  0  0     32     20    - 50.0000   - 50

x  0  0  1   1  1  1  1     31     1F    - 51.5625   - 51 9/16
x  0  0  1   1  1  1  0     30     1E    - 53.1250   - 53 2/16
x  0  0  1   1  1  0  1     29     1D    - 54.6875   - 54 11/16
x  0  0  1   1  1  0  0     28     1C    - 56.2500   - 56 4/16
x  0  0  1   1  0  1  1     27     1B    - 57.8125   - 57 13/16
x  0  0  1   1  0  1  0     26     1A    - 59.3750   - 59 6/16
x  0  0  1   1  0  0  1     25     19    - 60.9375   - 60 15/16
x  0  0  1   1  0  0  0     24     18    - 62.5000   - 62 8/16
x  0  0  1   0  1  1  1     23     17    - 64.0625   - 64 1/16
x  0  0  1   0  1  1  0     22     16    - 65.6250   - 65 10/16
x  0  0  1   0  1  0  1     21     15    - 67.1875   - 67 3/16
x  0  0  1   0  1  0  0     20     14    - 68.7500   - 68 12/16
x  0  0  1   0  0  1  1     19     13    - 70.3125   - 70 5/16
x  0  0  1   0  0  1  0     18     12    - 71.8750   - 71 14/16
x  0  0  1   0  0  0  1     17     11    - 73.4375   - 73 7/16
x  0  0  1   0  0  0  0     16     10    - 75.0000   - 75

x  0  0  0   1  1  1  1     15     0F    - 76.5625   - 76 9/16
x  0  0  0   1  1  1  0     14     0E    - 78.1250   - 78 2/16
x  0  0  0   1  1  0  1     13     0D    - 79.6875   - 79 11/16
x  0  0  0   1  1  0  0     12     0C    - 81.2500   - 81 4/16
x  0  0  0   1  0  1  1     11     0B    - 82.8125   - 82 13/16
x  0  0  0   1  0  1  0     10     0A    - 84.3750   - 84 6/16
x  0  0  0   1  0  0  1      9     09    - 85.9375   - 85 15/16
x  0  0  0   1  0  0  0      8     08    - 87.5000   - 87 8/16
x  0  0  0   0  1  1  1      7     07    - 89.0625   - 89 1/16
x  0  0  0   0  1  1  0      6     06    - 90.6250   - 90 10/16
x  0  0  0   0  1  0  1      5     05    - 92.1875   - 92 3/16
x  0  0  0   0  1  0  0      4     04    - 93.7500   - 93 12/16
x  0  0  0   0  0  1  1      3     03    - 95.3125   - 95 5/16
x  0  0  0   0  0  1  0      2     02    - 96.8750   - 96 14/16
x  0  0  0   0  0  0  1      1     01    - 98.4375   - 98 7/16
x  0  0  0   0  0  0  0      0     00    -100.0000   -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 6mus plus or minus, respectively.

. . . . . . . . .

### 6mus calculator

Ratio may be entered as fraction or floating-point decimal number.
(value must be greater than 1)

For EDOs (equal-temperaments), type: "a/b" (without quotes)
where "a" = EDO degree and "b" = EDO cardinality.
(value must be less than 1)

Enter ratio: = 6mus

. . . . . . . . .

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