Millioctave

Physical unit
Unit name	Millioctave
Unit symbol	${\displaystyle \mathrm {mO} }$
Physical quantity (s)	musical interval
Formula symbol	${\displaystyle \Delta }$
dimension	${\displaystyle {\mathsf {{\frac {T^{-1}}{T^{-1}}}=1}}}$
In SI units	${\displaystyle 1}$
Named after	ancient Greek ὄκτω óktō = "eight"
Derived from	octave
See also: Cent , Neper , Savart

The millioctave (mO) is an auxiliary unit of measurement for the size of musical intervals . 1000 mO correspond to an octave or 1200  cents or an interval with a frequency ratio (proportion ) of 2: 1. ${\displaystyle p}$

The millioctave never prevailed against the cent measure. However, it is still used occasionally by authors who want to avoid the obvious association of cents with equal intervals.

definition

Diatonic intervals
Prime second third fourth fifth sixth seventh octave none decime undezime duodecime tredezime semitone / whole tone
Special intervals
Microinterval Comma Diësis Limma Apotome Ditone Tritone Wolf fifth Natural septime
units
Cent Millioctave Octave Savart

The following applies (see interval ):

${\displaystyle {\begin{alignedat}{2}\Delta =\mathrm {Intervall} (p)&=&&\log _{2}(p)\;\mathrm {Oktave} \\&=1000\cdot &&\log _{2}(p)\;\mathrm {mO} \end{alignedat}}}$

${\displaystyle {\begin{aligned}\Leftrightarrow p=\mathrm {Proportion} (\Delta )&=2^{\frac {\Delta }{1000\;\mathrm {mO} }}\\&=({\sqrt[{1000}]{2}})^{\frac {\Delta }{\mathrm {mO} }}\\&\approx 1{,}00069339\ ^{\frac {\Delta }{\mathrm {mO} }}\end{aligned}}}$

Like the more common cent measure , the millioctave is a logarithmic measure for intervals. Therefore, one can add interval sizes in millioctaves instead of having to multiply them as with frequency ratios.

With

${\displaystyle {\begin{aligned}\log _{10}(p)&=\log _{2}(p)\cdot \log _{10}(2)\\\Leftrightarrow {\frac {\log _{10}(p)}{\log _{10}(2)}}&=\log _{2}(p)\end{aligned}}}$

the result is the calculator-friendly equation:

${\displaystyle {\begin{aligned}\Rightarrow \Delta &=1000\cdot {\frac {\log _{10}(p)}{\log _{10}(2)}}\;\mathrm {mO} \\&=1000\cdot {\frac {\log _{10}(p)}{0{,}30103}}\;\mathrm {mO} \\&\approx 3321{,}93\cdot \log _{10}(p)\;\mathrm {mO} \end{aligned}}}$

Example:

interval	Frequency ratio	in millioctaves	In cents
1 octave	2: 1	1000	1200
2 octaves	4: 1	2000	2400
3 octaves	8: 1	3000	3600
Fifth	3: 2	585	702
Fourth	4: 3	415	498
major third	5: 4	322	386

Conversions

1 mO = 1.2 cents = log ₁₀ (2)  Savart ≈ 0.301 Savart

history

The millioctave was introduced in 1903 by the German physicist Arthur von Oettingen in his essay The dual system of harmony . As early as 1871, in On Sound and Atmospheric Vibrations with the Mathematical Elements of Music , George Biddell Airy discussed John Frederick William Herschel's proposal to divide the octave into 1000 parts.

See also

• Mood

Individual evidence

1. ↑ Arthur von Oettingen: The dual system of harmony . ( Memento of the original from August 24, 2011 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. In: Annalen der Naturphilosophie 1 (1902), pp. 62–75; 2 (1903/4), pp. 375-403; 3 (1904), pp. 241-269; 4 (1905), pp. 116-152 and 301-338; 5 (1906), pp. 449-503. "The millioctave is the 83rd part of a semitone and an interval so small that it can no longer be distinguished as the difference between two tones." P. 388 f. 
2. ↑ George Biddell Airy: On sound and atmospheric vibrations: with the mathematical elements of music . 2nd Edition. London 1871, p. 222. “We are permitted by Sir John Herschel to explain a system proposed by him which possesses that advantage. It consists in using such a modulus that the logarithm of 2 is 1000. "