# 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 {Interval} (p) & = && \ log _ {2} (p) \; \ mathrm {octave} \\ & = 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.