oktamu, octamu, 8mu

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 oktamu pitch-bend resolution, a semitone is divided into 2⁸ = 256 pitch-bend units. Thus there are 256 * 12 = 3072 oktamus in an "octave", so the oktamu measurement system may be thought of as 3072-EDO tuning, with a oktamu being one degree of 3072-EDO.

An oktamu is calculated as the 3072nd root of 2, or 2^(1/3072), with a ratio of approximately 1:1.000225659. It is an irrational number, but is extremely close to the ratio 8865:8863 : the difference is only ~ ¹/_278,000 of a cent, making them for all intents and purposes identical.


An oktamu is

• exactly ¹²⁵/₃₈₄ (= 0.3255208333... = ~ ¹/₃ ) of a milli8ve,

• exactly ²⁵/₆₄ (= 0.390625 = ~ ²/₅ ) of a cent,

• exactly 3 ¹⁷³/₃₈₄ (= 3.4505208333... = ~ 3 ¹/₂ ) türk-sents,

• exactly 9 ²⁴⁵⁵/₃₀₇₂ (= 9.7991536458333... = ~ 9 ⁴/₅ ) jots,

• approximately 12 (= ~ 11.98848574 ) tuning units.


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

oktamus = log₁₀(ratio) * [ (2⁸ * 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 oktamus 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, 1mu
doamu, 2mu
triamu, 3mu
tetramu, 4mu
pentamu, 5mu
hexamu, 6mu
heptamu, 7mu
enneamu, 9mu
dekamu, 10mu
endekamu, 11mu
dodekamu, 12mu
tridekamu, 13mu
tetradekamu, 14mu
cawapu
midipu
my Gentle Introduction to the MIDI Tuning Specification
the Official MIDI Tuning Specification.

[from Joe Monzo, JustMusic: A New Harmony]

---

(to download a zip file of the entire Dictionary, click here)

For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here

I welcome feedback about this webpage: corrections, improvements, good links. Let me know if you don't understand something. return to the Microtonal Dictionary index return to my home page return to the Sonic Arts home page