endekamu, 11mu

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

An endekamu is calculated as the 24576th root of 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, making them for all intents and purposes identical.


An endekamu is

• exactly ¹²⁵/₃₀₇₂ (= 0.0406901041666... = ~ ¹/₂₅ ) of a milli8ve,

• 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) tuning units.


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

endekamus = log₁₀(ratio) * [ (2¹¹ * 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 endekamus 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
enneamu
dekamu
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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