enneamu, 9mu

A term I and a few other tuning theorists coined in July 2003, based on an idea by Aaron Hunt, to describe one a family of terms referring to units of resolution in MIDI tuning, in which 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".

At the setting for enneamu pitch-bend resolution, a semitone is divided into 2⁹ = 512 pitch-bend units. Thus there are 512 * 12 = 6144 enneamus in an "octave", so the enneamu measurement system may be thought of as 6144-EDO tuning, with an enneamu being one degree in 6144-EDO.

An enneamu is calculated as the 6144th root of 2, or 2^(1/6144), with a ratio of approximately 1:1.000112823. It is an irrational number, but is extremely close to the ratio 8864:8863 : the difference is only ~ ¹/_108,000 of a cent, making them for all intents and purposes identical.


An enneamu is

• exactly ¹²⁵/₇₆₈ (= 0.16276041666... = ~ ¹/₆ ) of a milli8ve,

• exactly ²⁵/₁₂₈ (= 0.1953125 = ~ ¹/₅ ) of a cent,

• exactly 1 ⁵⁵⁷/₇₆₈ (= 1.72526041666... = ~ 1 ³/₄ ) türk-sents,

• exactly 4 ⁵⁵²⁷/₆₁₄₄ (= 4.89957682291666... = ~ 4 ⁸/₉ = ~ 5 ) jots,

• approximately 6 (= ~ 5.994242871 ) tuning units.


The formula for calculating the enneamu-value of any ratio is:

enneamus = log₁₀(ratio) * [ (2⁹ * 12) / log₁₀(2)]

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). Then the remainder or deficit is converted into enneamus plus or minus, respectively. These give the correct tuning to a tolerance that is far better than anything that the human auditory system can detect, or indeed far better than what is available in the output of any electronic instruments.

See also:

MIDI tuning units
enamu
doamu
triamu
tetramu
pentamu
hexamu
heptamu
oktamu
dekamu
endekamu
dodekamu
tridekamu
tetradekamu
cawapu
midipu
my Gentle Introduction to the MIDI Tuning Specification
the Official MIDI Tuning Specification.

[from Joe Monzo, JustMusic: A New Harmony]

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