tridekamu, tridecamu, 13mu

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 tridekamu pitch-bend resolution, a semitone is divided into 2¹³ = 8192 pitch-bend units. Thus there are 8192 * 12 = 98304 tridekamus in an "octave", so the tridekamu measurement system may be thought of as 98304-EDO tuning, with a tridekamu being one degree in 98304-EDO.

A tridekamu is calculated as the 98304th root of 2, or 2^(1/98304), with a ratio of approximately 1:1.000007051. It is an irrational number, but is extremely close to the ratio 141825:141824 : the difference is only ~ ¹/_6,000,000 of a cent, making them for all intents and purposes identical.


A tridekamu is

• exactly ¹²⁵/₁₂₂₈₈ (= 0.010172526041666... = ~ ¹/₉₈ ) of a milli8ve,

• exactly ²⁵/₂₀₄₈ (= 0.01220703125 = ~ ¹/₈₂ ) of a cent,

• exactly ¹³²⁵/₁₂₂₈₈ (= ~ 0.107828776041666... = ~ ¹/₉ ) of a türk-sent,

• exactly ³⁰¹⁰³/₉₈₃₀₄ (= 0.306223551432291666... = ~ ¹/₃ ) of a jot,

• approximately ³/₈ (= ~ 0.374640179 ) of a tuning unit.


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

tridekamus = log₁₀(ratio) * [ (2¹³ * 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 tridekamus plus or minus, respectively.

See also:

MIDI tuning units
enamu
doamu
triamu
tetramu
pentamu
hexamu
heptamu
oktamu
enneamu
dekamu
endekamu
dodekamu
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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