Editing Binary logarithm (section)

===Music theory===
In [[music theory]], the [[Interval (music)|interval]] or perceptual difference between two tones is determined by the ratio of their frequencies. Intervals coming from [[rational number]] ratios with small numerators and denominators are perceived as particularly euphonious. The simplest and most important of these intervals is the [[octave]], a frequency ratio of {{math|2:1}}. The number of octaves by which two tones differ is the binary logarithm of their frequency ratio.<ref name="mga">{{citation|title=The Musician's Guide to Acoustics|first1=Murray|last1=Campbell|first2=Clive|last2=Greated|publisher=Oxford University Press|year=1994|isbn=978-0-19-159167-9|page=78|url=https://books.google.com/books?id=iiCZwwFG0x0C&pg=PA78}}.</ref>

To study [[tuning system]]s and other aspects of music theory that require finer distinctions between tones, it is helpful to have a measure of the size of an interval that is finer than an octave and is additive (as logarithms are) rather than multiplicative (as frequency ratios are). That is, if tones {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} form a rising sequence of tones, then the measure of the interval from {{mvar|x}} to {{mvar|y}} plus the measure of the interval from {{mvar|y}} to {{mvar|z}} should equal the measure of the interval from {{mvar|x}} to {{mvar|z}}. Such a measure is given by the [[Cent (music)|cent]], which divides the octave into {{math|1200}} equal intervals ({{math|12}} [[semitone]]s of {{math|100}} cents each). Mathematically, given tones with frequencies {{math|''f''<sub>1</sub>}} and {{math|''f''<sub>2</sub>}}, the number of cents in the interval from {{math|''f''<sub>1</sub>}} to {{math|''f''<sub>2</sub>}} is<ref name="mga"/>
:<math>\left|1200\log_2\frac{f_1}{f_2}\right|.</math>
The [[millioctave]] is defined in the same way, but with a multiplier of {{math|1000}} instead of {{math|1200}}.<ref>{{citation|title=The Harvard Dictionary of Music|edition=4th|editor-first=Don Michael|editor-last=Randel|editor-link=Don Michael Randel|publisher=The Belknap Press of Harvard University Press|year=2003|isbn=978-0-674-01163-2|page=416|url=https://books.google.com/books?id=02rFSecPhEsC&pg=PA416}}.</ref>