  Encyclopedia of Microtonal Music Theory

[Joe Monzo, Tonalsoft Encyclopedia of Microtonal Music Theory]

NOTE: This page contained serious mathematical errors until it was corrected on 2013.0131 (31 January 2013). If this page was accessed before that date, the previous version should be replaced with this one, and any work based on the previous version should be corrected.

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.

The 14mu is specified in the MIDI spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999). (Note: this term supercedes "midipu", the term previously coined by Monzo to designate this unit.) The 14mu is the smallest unit of tuning resolution which has ever been put into common musical use, and provides extremely accurate tuning in microtonal electronic music. Note that it differs from all other mus in that it does not specify the 12-edo MIDI-note as a center frequency with pitch-bend to be applied as either positive or negative values, but rather specifies the 12-edo MIDI-note as the bottom frequency with all pitch-bend values positive.

At the setting for 14mu pitch-bend resolution, a semitone is divided into 212 = 16384 pitch-bend units. Thus there are 16384 * 12 = 196608 14mus in an "octave", so the 14mu measurement system may be thought of as 196608-edo tuning, with a 14mu being one degree in 196608-edo.

A 14mu is calculated as the 196608th root of 2 -- 196608√2, or 2(1/196608) -- with a ratio of approximately 1:1.000003526. It is an irrational number, but is extremely close to the ratio 283608:283607 : the difference is ~ 1/1,000,000 of a cent, which for all intents and purposes makes the 14mu identical to that ratio. The formula for calculating the 14mu-value of any ratio is: 14mus = log10(ratio) * [ (214 * 12) / log10(2)] or 14mus = log2r * (214 * 12) , where r is the ratio.

A 14mu is:

• exactly 125/24576 (= 0.0050862630208333... ~= 1/197) of a millioctave
• exactly 25/4096 (= 0.006103515625 ~= 1/164) of a cent
• exactly 1325/24576 (= 0.0539143880208333... ~= 1/19) of a türk-sent
• exactly 30103/196608 (= 0.1531117757161458333... ~= 1/7) of a jot
• approximately 1/5 (~= 0.18732009 ) of a temperament-unit

The internal data structure of the 14mu requires two bytes, with the first bits of each byte reserved as a flags to indicate the byte's status as data. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.

```  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.

the 14mu spec thus uses a total of 7+7 = 14 bits.

thus, the maximum possible value is:

d111 1111  d111 1111  [binary]

=  +/-   7F 7F  [hex]

=  +/-   _  [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.

x 64 32 16   8  4  2  1     x 8192 4096 2048  1024 512 256 128 --  decimal value
x  0  0  0   0  0  0  0     x   0    0    0     0   0   0   0

= 00 00 hex = 0 decimal = the plain MIDI-note, 0 cents deviation from 12edo.

x 64 32 16   8  4  2  1     x 8192 4096 2048  1024 512 256 128 --  decimal value
x  0  0  0   0  0  0  1     x   0    0    0     0   0   0   0

= 00 01 hex = 1 decimal = one unit  (exactly 0.006103515625 = 25/4096 cents) above the 12-edo MIDI-note

x 64 32 16   8  4  2  1     x 8192 4096 2048  1024 512 256 128 --  decimal value
x  1  1  1   1  1  1  1     x   1    1    1     1   1   1   1
= 7F 7F hex = 16383 decimal = 16383 units (exactly 99.993896484375 = 99 4071/4096 cents) above the 12-edo MIDI-note
= one unit (0.006103516 = 25/4096 cents) below the next higher 12-edo MIDI-note
```

Therefore the 14mu gives a range of possible values +/- as follows -- the top end, middle, and bottom end of the range are shown for illustration. (Also shown is one section where the cents values become offset -- i.e., because one increment is 25/4096 of a cent, and 4096 does not divide evenly by 25, there will be a value of 4100/4096 cents.):

```Numbers under "LSB" and "MSB" are the exponent of 2 which that bit represents.
Cents values given are exact, in both decimal and fraction formats.

(The top value, 2*2^13, is obviously not possible in binary arithmetic, and is
provided only to show how the final increment would work. In actuality, the byte
representing the MIDI-note would increase by one and both of these bytes return to zero.)

---------------------- bits -----------------------
-------- LSB ----------     -------- MSB ----------    M  L           ---------- cents ------------
x  6  5  4   3  2  1  0     x 13 12 11  10  9  8  7     hex  decimal     decimal          fraction

x  0  0  0   0  0  0  0     x  2  0  0   0  0  0  0    80 00  16384   +100                +100
x  1  1  1   1  1  1  1     x  1  1  1   1  1  1  1    7F 7F  16383   + 99.993896484375   + 99 4071/4096
x  1  1  1   1  1  1  0     x  1  1  1   1  1  1  1    7F 7E  16382   + 99.98779296875    + 99 4046/4096
x  1  1  1   1  1  0  1     x  1  1  1   1  1  1  1    7F 7D  16381   + 99.981689453125   + 99 4021/4096
.
.
.
x  0  1  0   0  1  0  0     x  0  0  0   0  0  0  1    01 24    164   +  1.000976563      +  1    4/4096 = 4100/4096
x  0  1  0   0  0  1  1     x  0  0  0   0  0  0  1    01 23    163   +  0.994873047      +    4075/4096
.
.
.
x  0  0  0   0  1  0  0     x  0  0  0   0  0  0  0    00 04      4   +  0.0244140625     +     100/4096
x  0  0  0   0  0  1  1     x  0  0  0   0  0  0  0    00 03      3   +  0.018310546875   +      75/4096
x  0  0  0   0  0  1  0     x  0  0  0   0  0  0  0    00 02      2   +  0.01220703125    +      50/4096
x  0  0  0   0  0  0  1     x  0  0  0   0  0  0  0    00 01      1   +  0.006103515625   +      25/4096
x  0  0  0   0  0  0  0     x  0  0  0   0  0  0  0    00 00      0      0                0  (i.e., the plain 12-edo MIDI-note)
```

For practical use in tuning MIDI-files, an interval's semitone value must first be calculated. The nearest integer semitone below the target frequency 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 is converted into 14mus plus (above) the 12-edo MIDI-note.

. . . . . . . . .

14mus 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: = 14mus

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

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