  # 11mu / endekamu

[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 11mu pitch-bend resolution, a semitone is divided into 211 = 2048 pitch-bend units. Thus there are 2048 * 12 = 24576 11mus in an "octave", so the 11mu measurement system may be thought of as 24576-edo tuning, with an 11mu being one degree in 24576-edo.

An 11mu is calculated as the 24576th root of 2 -- 24576√2, or 2(1/24576) -- with a ratio of approximately 1:1.000028205. It is an irrational number, but is extremely close to the ratio 35456:35455 : the difference is less than 1/4,000,000 of a cent, which for all intents and purposes makes the 11mu identical to that ratio. The formula for calculating the 11mu-value of any ratio is: 11mus = log10(ratio) * [ (211 * 12) / log10(2)] or 11mus = log2r * (211 * 12) , where r is the ratio.

An 11mu is:

• exactly 125/3072 (= 0.0406901041666... ~= 1/25) of a millioctave
• exactly 25/512 (= 0.048828125 ~= 1/20 ) of a cent
• exactly 1325/3072 (= 0.4313151041666... ~= 1/2 ) of a türk-sent
• exactly 1 5527/24576 (= 1.2248942057291666... ~= 1 1/4 ) jots
• approximately 1 1/2 (~= 1.498560718 ) temperament-units

The internal data structure of the _mu requires two bytes, with the first bits of each byte reserved as a flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or -) showing the direction of the pitch-bend up or down, and 2 other bits which are not used. 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.

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 11mus plus or minus, respectively.

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