Editing Mathematical coincidence (section)

==== Concerning musical intervals ====
{{See also|Musical temperament}}
In music, the distances between notes (intervals) are measured as ratios of their frequencies, with near-rational ratios often sounding harmonious. In western [[twelve-tone equal temperament]], the ratio between consecutive note frequencies is <math>\sqrt[12]{2}</math>. 
* The coincidence <math>2^{19} \approx 3^{12}</math>, from <math>\frac{\log3}{\log2} = 1.5849\ldots \approx \frac{19}{12}</math>, closely relates the interval of 7 [[semitone]]s in [[equal temperament]] to a [[perfect fifth]] of [[just intonation]]:  <math>2^{7/12}\approx 3/2</math>, correct to about 0.1%. The just fifth is the basis of [[Pythagorean tuning]]; the difference between [[circle of fifths|twelve just fifths]] and seven octaves is the [[Pythagorean comma]].<ref name="schroeder">
{{cite book
 | title = Number theory in science and communication
 | author = Manfred Robert Schroeder
 | publisher = Springer
 | edition = 2nd
 | year = 2008
 | isbn = 978-3-540-85297-1
 | pages = 26–28
 | url = https://books.google.com/books?id=2KV2rfP0yWEC&q=coincidence+circle-of-fifths+1024+7-octaves+%22one+part+in+a+thousand%22&pg=PA27
}}</ref>
* The coincidence <math>{(3/2)}^{4} = (81/16) \approx 5</math> permitted the development of [[meantone temperament]], in which just perfect fifths (ratio <math>3/2</math>) and [[major third]]s (<math>5/4</math>) are "tempered" so that four <math>3/2</math>'s is approximately equal to <math>5/1</math>, or a <math>5/4</math> major third up two octaves. The difference (<math>81/80</math>) between these stacks of intervals is the [[syntonic comma]].{{cn|date=March 2022}}
* The coincidence <math>\sqrt[12]{2}\sqrt[7]{5} = 1.33333319\ldots \approx \frac43</math> leads to the [[Schisma|rational version]] of [[12-TET]], as noted by [[Johann Kirnberger]].{{Citation needed|date=September 2010}}
* The coincidence <math>\sqrt[8]{5}\sqrt[3]{35} = 4.00000559\ldots \approx 4</math> leads to the rational version of [[quarter-comma meantone]] temperament.{{Citation needed|date=September 2010}}
* The coincidence of powers of 2, above, leads to the approximation that three major thirds concatenate to an octave, <math>{(5/4)}^{3} \approx {2/1}</math>.  This and similar approximations in music are called [[Diesis|dieses]].