enamu, 1mu

A term I and a few other tuning theorists coined in July 2003, based on an idea by Aaron Hunt, to describe one of 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 enamu pitch-bend resolution, a semitone is divided into 2¹ = 2 pitch-bend units. Thus there are 2 * 12 = 24 enamus in an "octave", so the enamu measurement system may be thought of as 24-EDO tuning, with a enamu being one degree in 24-EDO.


An enamu is calculated as the 24th root of 2, or 2^(1/24), with a ratio of approximately 1:1.029302237. It is an irrational number, but is very close to the ratio 35:34 ( 2^-1 5¹ 7¹ 17^-1 ): the difference is ~ -0.184210833552446 (~ ¹/₅ ) of a cent, which in most cases would be hard to distinguish.


An enamu is exactly 50 cents, and is identical to the quarter-tone. Since "quarter-tone" is already such a well-established term, "enamu" is not likely to gain much currency. (More information can thus be found under the "quarter-tone" entry.)


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

enamus = log₁₀(ratio) * [ (2¹ * 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 enamus plus or minus, respectively.

See also:

MIDI tuning units
doamu
triamu
tetramu
pentamu
hexamu
heptamu
oktamu
enneamu
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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