Editing Syntonic comma (section)

==Relationships==
The prime factors of the just interval {{small|{{sfrac|81|80}}}} known as the syntonic comma can be separated out and reconstituted into various sequences of two or more intervals that arrive at the comma, such as {{nobr|{{math|{{small|{{sfrac|81|1}} × {{sfrac|1|80}} }}}} }} or (fully expanded and sorted by prime) {{nobr|{{math|{{small| {{sfrac| 3 × 3  × 3  × 3 | 2 × 2 × 2 × 2 × 5 }} }}}} .}} All sequences of notes that produce that fraction are mathematically valid, but some of the more musical sequences people use to remember and explain the comma's composition, occurrence, and usage are listed below:
* The ratio of the two kinds of [[major second]] which occur in [[5-limit tuning]]: the greater [[whole tone|tone]] (9:8, about 203.91 cents, or {{nobr|C {{small|↗}} D}} in [[just intonation|just]] C&nbsp;major) and lesser tone (10:9, about 182.40 cents, or {{nobr|D {{small|↗}} E}}). Namely, {{nobr| 9:8 ÷ 10:9 {{=}} 81:80 ,}} or equivalently, sharpening by a comma promotes a lesser [[major second]] to a greater second {{nobr| {{math|{{small| {{sfrac|10|9}} × {{sfrac|81|80}} {{=}} {{sfrac|9|8}} }}}} .}}<ref name=Lloyd/>
* The difference in [[Interval (music)#size|size]] between a Pythagorean [[ditone]] ([[interval ratio|frequency ratio]] 81:64, or about 407.82 [[Cent (music)|cents]]) and a just major third (5:4, or about 386.31 cents). Namely, {{nobr|{{small|{{math| {{sfrac|81|64}} ÷ {{sfrac|5|4}} {{=}} {{sfrac|81|80}} }}}} .}}
* The difference between four [[just intonation|justly]] tuned [[perfect fifth]]s, and two [[octave]]s plus a justly tuned [[major third]]. A just perfect fifth has a size of 3:2 (about 701.96 cents), and four of them are equal to 81:16 (about 2807.82 cents). A just major third has a size of [[sesquiquartum|5:4]] (about 386.31 cents), and one of them plus two octaves (4:1 or exactly 2400 cents) is equal to 5:1 (about 2786.31 cents). The difference between these is the syntonic comma. Namely, {{nobr| 81:16 ÷ 5:1 {{=}} 81:80 .}}
* The difference between one octave plus a justly tuned [[minor third]] (12:5, about 1515.64 cents), and three justly tuned [[perfect fourth]]s (64:27, about 1494.13 cents). Namely, 12:5&nbsp;÷&nbsp;64:27 =&nbsp;81:80.
* The difference between a [[Pythagorean tuning|Pythagorean]] [[major sixth]] (27:16, about 905.87 cents) and a [[5-limit tuning#The justest ratios|justly tuned]] or "pure" [[major sixth]] (5:3, about 884.36 cents). Namely, 27:16&nbsp;÷&nbsp;5:3 =&nbsp;81:80.<ref name=Lloyd/>

On a [[piano]] keyboard (typically tuned with [[Equal temperament#Twelve-tone equal temperament|12-tone equal temperament]]) a stack of four fifths (700&nbsp;×&nbsp;4 = 2800 cents) is exactly equal to two octaves (1200&nbsp;×&nbsp;2 = 2400 cents) plus a major third (400 cents). In other words, starting from a C, both combinations of intervals will end up at E. Using [[Just intonation|justly tuned]] octaves (2:1), fifths (3:2), and thirds (5:4), however, yields two slightly different notes. The ratio between their frequencies, as explained above, is a syntonic comma (81:80). [[Pythagorean tuning]] uses justly tuned fifths (3:2) as well, but uses the relatively complex ratio of 81:64 for major thirds. [[Quarter-comma meantone]] uses justly tuned major thirds (5:4), but flattens each of the fifths by a quarter of a syntonic comma, relative to their just size (3:2). Other systems use different compromises. This is one of the reasons why [[Equal temperament#Twelve-tone equal temperament|12-tone equal temperament]] is currently the preferred system for tuning most musical instruments{{clarify|date=July 2022}}.

Mathematically, by [[Størmer's theorem]], 81:80 is the closest [[Superparticular number|superparticular ratio]] possible with [[regular number]]s as numerator and denominator. A superparticular ratio is one whose numerator is 1 greater than its denominator, such as 5:4, and a regular number is one whose [[prime factor]]s are limited to 2, 3, and&nbsp;5. Thus, although smaller intervals can be described within 5-limit tunings, they cannot be described as superparticular ratios.