doamu, 2mu

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 doamu pitch-bend resolution, a semitone is divided into 2² = 4 pitch-bend units. Thus there are 4 * 12 = 48 doamus in an "octave", so the doamu measurement system may be thought of as 48-EDO tuning, with a doamu being one degree of 48-EDO.


A doamu is calculated as the 48th root of 2, or 2^(1/48), with a ratio of approximately 1:1.014545335. It is an irrational number, but is close to the ratio 70:69 : the difference is ~ ¹/₁₁ of a cent, which under most circumstances would be hard to distinguish.


A doamu is

• exactly ⁴³/₄₈ ( 0.8958333..., ~ ⁸/₉ ) of a meride,

• exactly 1 ¹/₂ ( 1.5 ) 72edo-moria,

• exactly 6 ¹/₄ ( 6.25 ) 300edo-savarts,

• exactly 12 ³/₄ ( 12.75 ) 612edo-schismas,

• exactly 20 ⁵/₆ ( 20.8333... ) millioctaves,

• exactly 25 cents,


The doamu is also known as an "eighth-tone".


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

doamus = log₁₀(ratio) * [ (2² * 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 doamus 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
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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