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{{Wikipedia}}
{{Infobox interval region|Name=Semitone, minor second, augmented unison|Cents lower=75|Cents lower wide=60|Cents upper=125|Cents upper wide=140|JI intervals=16/15, 25/24|MOSes=1L 8s, 9L 1s, 1L 9s|Complement=[[Major seventh]]|Lower region=[[Comma and diesis]]|Higher region=[[Neutral second]]}}{{Wikipedia|Semitone}}
A '''semitone''', as a concrete [[interval region]], is typically near 100{{cent}} in size, distinct from [[commas and dieses]] (less than 60{{c}}), and from [[neutral second]]s (about 150{{c}}). A rough tuning range for the semitone is about 60{{c}} to 125{{c}} according to [[Margo Schulter]]'s theory of interval regions.
Functionally, a semitone is an interval that makes up part of a [[tone]], often as one step of a 12-tone chromatic scale, which is a possible criterion for the classification of an interval as a semitone in [[just intonation]].


A '''semitone''' is an interval that is near 100 [[Cent|cents]] in size, distinct from [[Comma and diesis|commas and dieses]] (less than 60 cents), and from [[Major second|major seconds]] (about 200 cents). A rough tuning range for the semitone is about 50 cents to 140 cents, though this is extremely wide; some might prefer to restrict it to around 70 cents to 130 cents.  
Semitones come in two functional categories based on their number of steps in the [[5L 2s|diatonic]] scale:
*[[Diatonic semitone]]s, minor seconds (m2), or limmas,
*[[Chromatic semitone]]s, augmented unisons (A1), or chromas.


"Semitone" also refers to an interval measure of exactly 100 cents, which is not the subject of this article.
The intervals covered in this article range from 50{{c}} to 140{{c}}.  
 
Semitones tend to fall into one of two functional categories, based on the system being used: '''diatonic semitones''' (or minor seconds), and '''chromatic semitones''' (chromas, or augmented unisons). This page covers both categories of intervals, as the distinction between them is largely a matter of the [[diatonic]] MOS, and is also not the subject of this article.  


== In just intonation ==
== In just intonation ==
=== By prime limit ===
In the low prime limits, up to the 5-limit, in which the West has developed a formal system of diatonic harmony, the distinction between diatonic and chromatic semitones is the clearest, so a pair of 2 semitones will be provided for each. However, higher than the 5-limit, function as diatonic vs. chromatic tends to become less clear, and larger intervals can be seen as belonging to neither category.
In the low prime limits, up to the 5-limit, in which the West has developed a formal system of diatonic harmony, the distinction between diatonic and chromatic semitones is the clearest, so a pair of 2 semitones will be provided for each. However, higher than the 5-limit, function as diatonic vs. chromatic tends to become less clear, and larger intervals can be seen as belonging to neither category.


* In the 3-limit:
* In the 3-limit:
** The '''limma,''' or Pythagorean diatonic semitone, is a ratio of 256/243, and is about 90 cents.
** The ''limma'', or ''Pythagorean diatonic semitone'', is a ratio of [[256/243]], and is about 90{{c}}.
** The '''apotome,''' or Pythagorean chromatic semitone, is a ratio of 2187/2048, and is about 114 cents.
** The ''apotome'', or ''Pythagorean chromatic semitone'', is a ratio of [[2187/2048]], and is about 114{{c}}.
* In the 5-limit:
* In the 5-limit:
** The '''classical diatonic semitone''' is a ratio of 16/15, and is about 112 cents.
** The ''classical diatonic semitone'' is a ratio of [[16/15]], and is about 112{{c}}.
** The '''classical chromatic semitone''' is a ratio of 25/24, and is about 71 cents.
** The ''classical chromatic semitone'' is a ratio of [[25/24]], and is about 71{{c}}.
*** There is also a '''ptolemaic chromatic semitone,''' which is a ratio of 135/128, and is about 92 cents.
*** There is also a ''ptolemaic chromatic semitone'', which is a ratio of [[135/128]], and is about 92{{c}}.
* In higher limits:
* In higher limits:
** The 7-limit '''third-tone''' is a ratio of 28/27, and is about 63 cents.
** The 7-limit ''third-tone'' is a ratio of [[28/27]], and is about 63{{c}}.
** The 7-limit '''minor semitone''' is a ratio of 21/20, and is about 84 cents.
** The 7-limit ''minor semitone'' is a ratio of [[21/20]], and is about 84{{c}}.
** The 7-limit '''major semitone''' is a ratio of 15/14, and is about 119 cents.
** The 7-limit ''major semitone'' is a ratio of [[15/14]], and is about 119{{c}}.
** The 11-limit '''minor semitone''' is a ratio of 22/21, and is about 81 cents.
** The 11-limit ''minor semitone'' is a ratio of [[22/21]], and is about 81{{c}}.
** The 13-limit '''sinaic''' is a ratio of 14/13, and is about 128 cents.
** The 13-limit ''sinaic'' is a ratio of [[14/13]], and is about 128{{c}}.
** The 13-limit '''greater 2/3 tone''' is a ratio of 13/12, and is about 139 cents.
** The 13-limit ''greater 2/3-tone'' is a ratio of [[13/12]], and is about 139{{c}}.
** The 17-limit '''large semitone''' is a ratio of 17/16, and is about 104 cents.
** The 17-limit ''large semitone'' is a ratio of [[17/16]], and is about 104{{c}}.
** The 17-limit '''small semitone''' is a ratio of 18/17, and is about 99 cents.
** The 17-limit ''small semitone'' is a ratio of [[18/17]], and is about 99{{c}}.
 
=== By delta ===
This table lists just semitones by [[Delta-N|delta]]; simple semitone ratios tend to be [[Superparticular ratio|superparticular]].
{| class="wikitable"
|-
! Delta 1 (Superparticular)
! Cents
|-
| [[13/12]]
| 139{{c}}
|-
| [[14/13]]
| 128{{c}}
|-
| [[15/14]]
| 119{{c}}
|-
| [[16/15]]
| 112{{c}}
|-
| [[17/16]]
| 104{{c}}
|-
| [[18/17]]
| 99{{c}}
|-
| [[19/18]]
| 94{{c}}
|-
| [[20/19]]
| 89{{c}}
|-
| [[21/20]]
| 85{{c}}
|-
| [[22/21]]
| 81{{c}}
|-
| [[23/22]]
| 77{{c}}
|-
| [[24/23]]
| 74{{c}}
|-
| [[25/24]]
| 71{{c}}
|-
| [[26/25]]
| 68{{c}}
|-
| [[27/26]]
| 65{{c}}
|-
| [[28/27]]
| 63{{c}}
|-
| [[29/28]]
| 61{{c}}
|-
| [[30/29]]
| 59{{c}}
|-
| [[31/30]]
| 57{{c}}
|-
| [[32/31]]
| 55{{c}}
|-
| [[33/32]]
| 53{{c}}
|-
| [[34/33]]
| 52{{c}}
|-
| [[35/34]]
| 50{{c}}
|}


== In EDOs ==
== In EDOs ==
The following table lists the best tuning of 16/15, 25/24, and other semitones if present, in various significant [[EDOs]].
The following table lists the best tuning of 16/15, 25/24, and other semitones if present, in various significant [[edo|EDO]]s.
{| class="wikitable"
{| class="wikitable"
|+
!EDO
!16/15
!25/24
!Other semitones
|-
|-
|12
! EDO
| colspan="2" |100c
! 16/15
|
! 25/24
! Other semitones
|-
| 12
| colspan="2" | 100{{c}}
|  
|-
|-
|15
| 15
| colspan="2" |80c
| colspan="2" | 80{{c}}
|
|  
|-
|-
|16
| 16
| colspan="2" |75c
| colspan="2" | 75{{c}}
|
|  
|-
|-
|17
| 17
|141c
| 141{{c}}
|71c
| 71{{c}}
|
|  
|-
|-
|19
| 19
|126c
| 126{{c}}
|63c
| 63{{c}}
|
|  
|-
|-
|22
| 22
|109c
| 109{{c}}
|55c
| 55{{c}}
|
|  
|-
|-
|24
| 24
|100c
| 100{{c}}
|50c
| 50{{c}}
|
|  
|-
|-
|25
| 25
|96c
| 96{{c}}
|*
| *
|
|  
|-
|-
|26
| 26
| colspan="2" |92c
| colspan="2" | 92{{c}}
|
|  
|-
|-
|27
| 27
|133c
| 133{{c}}
|89c
| 89{{c}}
|
|  
|-
|-
|29
| 29
|124c
| 124{{c}}
|83c
| 83{{c}}
|
|  
|-
|-
|31
| 31
|116c
| 116{{c}}
|77c
| 77{{c}}
|
|  
|-
|-
|34
| 34
|106c
| 106{{c}}
|71c
| 71{{c}}
|
|  
|-
|-
|41
| 41
|117c
| 117{{c}}
|59c
| 59{{c}}
|88c ≈ 256/243
| {{nowrap|88{{c}} ≈ 256/243}}
|-
|-
|53
| 53
|113c
| 113{{c}}
|68c
| 68{{c}}
|91c ≈ 256/243
| {{nowrap|91{{c}} ≈ 256/243}}
|}
|}


== In regular temperaments ==
== In regular temperaments ==
Two important, simple semitone ratios are 16/15 and 25/24. The following notable temperaments are generated by them:
Two important, simple semitone ratios are 16/15 and 25/24. The following notable temperaments are generated by them:


=== Temperaments that use 25/24 as a generator ===
=== Temperaments that use 25/24 as a generator ===
 
* [[Valentine]], which divides [[3/2]] into nine small semitones, five of which make [[5/4]]. See also the related [[Carlos Alpha]].
* Vishnu, which stacks seven 25/24s to make a just [[perfect fourth]] of [[4/3]]
* [[Vishnu]], which stacks seven 25/24s to make a just [[perfect fourth]] of [[4/3]].
* Chlorine, equivalent to [[17edo]], stacking seventeen 25/24s to make an octave
* [[Chlorine]], based on [[17edo]], stacking seventeen 25/24s to make an octave.


=== Temperaments that use 16/15 as a generator ===
=== Temperaments that use 16/15 as a generator ===
* [[Miracle]], which splits [[3/2]] into six semitones, each representing both [[15/14]] and [[16/15]].
* [[Negri]], which splits [[4/3]] into four semitones, such that three of them represent [[5/4]].
* [[Diaschismic]], which is usually described as having a fifth as its second generator, but can alternatively be generated by a half-octave and a semitone.


* TBD
[[Compton]] has one step of [[12edo]] as its first generator, representing [[256/243]].
 


When 25/24 is tempered out, it leads to [[dicot]] temperament.
When 25/24 is tempered out, it leads to [[dicot]] temperament.


When 16/15 is tempered out, it leads to [[father]] temperament.
When 16/15 is tempered out, it leads to [[father]] temperament.
== In moment-of-symmetry scales ==
Intervals between 100 and 133{{c}} generate the following [[MOS]] scales:
These tables start from the last monolarge [[MOS]] generated by the interval range.
MOSes with more than 12 notes are not included.
{| class="wikitable"
|-
! Range
! colspan="2" | MOS
|-
| 100–109{{c}}
| [[1L 10s]]
| [[11L 1s]]
|-
| 109–120{{c}}
| [[1L 9s]]
| [[10L 1s]]
|-
| 120–133{{c}}
| [[1L 8s]]
| [[9L 1s]]
|}
== See also ==
* [[Semitone]] (disambiguation page)
{{Navbox intervals}}
{{Navbox intervals}}
[[Category:12edo]]
[[Category:12edo]]
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