Editing Semitone (section)

===Just 5-limit intonation {{anchor|Just intonation}}===
<!--[[Just diatonic semitone]], [[Just chromatic semitone]], and [[Semitone maximus]] redirect directly here.-->
[[File:Just diatonic semitone.png|thumb|right|16:15 [[#Minor second|diatonic semitone]]]]
[[File:Just diatonic semitone on C.png|thumb|right|16:15 diatonic semitone[[File:Just diatonic semitone on C.mid]]]]
[[File:Major limma on C.png|thumb|right|'Larger' or major limma on C[[File:Greater chromatic semitone on C.mid]]]]
[[File:Semitone 5-limit diamond.png|thumb|right|300px|Relationship between the 4&nbsp;common 5&nbsp;limit semitones]]

A minor second in [[just intonation]] typically corresponds to a pitch [[ratio]] of 16:15 ({{Audio|Just diatonic semitone on C.mid|play}}) or 1.0666... (approximately 111.7&nbsp;[[cent (music)|cent]]s), called the '''just diatonic semitone'''.<ref>{{cite journal |title={{grey|[no title cited]}} |publisher=Royal Society |place=Great Britain |year=1880 |quote=digitized 26 Feb 2008; Harvard University |journal=[[Proceedings of the Royal Society of London]] |volume=30 |page=531}}</ref> This is a practical just semitone, since it is the interval that occurs twice within the diatonic scale between a:
: [[major third]] (5:4) and [[perfect fourth]] (4:3) <math>\ \left(\ \tfrac{4}{3} \div \tfrac{5}{4} = \tfrac{16}{15}\ \right)\ ,</math> and a
: [[major seventh]] (15:8) and the [[perfect octave]] (2:1) <math>\ \left(\ \tfrac{2}{1} \div \tfrac{15}{8} = \tfrac{16}{15}\ \right) ~.</math>

The 16:15 just minor second arises in the C major scale between B & C and E & F, and is, "the sharpest dissonance found in the scale".<ref name="books.google.com"/>

An "augmented unison" (sharp) in just intonation is a different, smaller semitone, with frequency ratio 25:24 ({{Audio|Just chromatic semitone on C.mid|play}}) or 1.0416... (approximately 70.7&nbsp;cents). It is the interval between a [[major third]] (5:4) and a minor third (6:5). In fact, it is the spacing between the minor and major thirds, sixths, and sevenths (but not necessarily the major and minor second). Composer [[Ben Johnston (composer)|Ben Johnston]] used a sharp ({{music|#}}) to indicate a note is raised 70.7&nbsp;cents, or a flat ({{Music|b}}) to indicate a note is lowered 70.7&nbsp;cents.<ref name=Fonville>{{cite journal |first=J. |last=Fonville |author-link=John Fonville |date=Summer 1991 |title=[[Ben Johnston (composer)|Ben Johnston]]'s extended just intonation – a guide for interpreters |journal=[[Perspectives of New Music]] |volume=29 |issue=2 |pages=106–137 |doi=10.2307/833435 |jstor=833435 |quote=...&nbsp;the {{sfrac|25|24}} ratio is the sharp ({{music|#}}) ratio ... this raises a note approximately 70.6&nbsp;cents.{{rp|style=ama|p=109}} }}</ref> (This is the standard practice for just intonation, but not for all other microtunings.)

Two other kinds of semitones are produced by 5&nbsp;limit tuning. A [[chromatic scale]] defines 12&nbsp;semitones as the 12&nbsp;intervals between the 13&nbsp;adjacent notes, spanning a full octave (e.g. from C{{sub|4}} to C{{sub|5}}). The 12&nbsp;semitones produced by a [[Five-limit tuning#Size of intervals|commonly used version]] of 5&nbsp;limit tuning have four different sizes, and can be classified as follows:

; Just chromatic semitone : ''chromatic semitone'', or ''smaller'', or ''minor chromatic semitone'' between harmonically related flats and sharps e.g. between E{{Music|b}} and E (6:5 and 5:4):
: <math> S_1 = \tfrac{5}{4} \div \tfrac{6}{5} = \tfrac{25}{24} \approx 70.7 \ \hbox{cents}</math>
; Larger chromatic semitone : or ''major chromatic semitone'', or ''larger limma'', or ''major chroma'',<ref name=Fonville/> e.g. between C and an accute&nbsp;C{{music|#}} (C{{music|#}} raised by a [[syntonic comma]]) (1:1 and 135:128):
: <math>S_2 = \tfrac{25}{24} \times \tfrac{81}{80} = \tfrac{135}{128} \approx 92.2 \ \hbox{cents}</math>
; Just diatonic semitone: or ''smaller'', or ''minor diatonic semitone'', e.g. between E and F (5:4 to 4:3):
: <math>S_3 = \tfrac{4}{3} \div \tfrac{5}{4} = \tfrac{16}{15} \approx 111.7 \ \hbox{cents}</math>
; Larger diatonic semitone: or ''greater'' or ''major diatonic semitone'', e.g. between A and B{{music|b}} (5:3 to 9:5), or C and chromatic D{{music|b}} (27:25), or F{{music|#}} and G (25:18 and 3:2):
: <math>S_4 = \tfrac{9}{5} \div \tfrac{5}{3} = \tfrac{27}{25} \approx 133.2 \ \hbox{cents}</math>

The most frequently occurring semitones are the just ones ({{mvar|S}}{{sub|3}}, 16:15, and {{mvar|S}}{{sub|1}}, 25:24): S{{sub|3}} occurs at 6&nbsp;short intervals out of 12, {{mvar|S}}{{sub|1}} 3&nbsp;times, {{mvar|S}}{{sub|2}} twice, and {{mvar|S}}{{sub|4}} at only one interval (if diatonic D{{music|b}} replaces chromatic D{{music|b}} and sharp notes are not used).

The smaller chromatic and diatonic semitones differ from the larger by the [[syntonic comma]] (81:80 or 21.5&nbsp;cents). The smaller and larger chromatic semitones differ from the respective diatonic semitones by the same 128:125 diesis as the above meantone semitones. Finally, while the inner semitones differ by the [[diaschisma]] (2048:2025 or 19.6&nbsp;cents), the outer differ by the greater diesis (648:625 or 62.6&nbsp;cents).