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Semitone (interval region): Difference between revisions

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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 half of a [[tone]]. It is most commonly used as an [[interval size measure|unit of interval size]] with a value equal to [[12edo|1\12]], exactly 100{{cent}}. An exact semitone in [[just intonation]] would be expressed as sqrt(9/8), with a value of about 102{{cent}}.  
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.


Besides the unmodified ''semitone'', used as an interval size measure, we also speak of the [[diatonic semitone]], or minor second, and the [[chromatic semitone]], or augmented unison, of a [[diatonic scale]]. These are equated ([[enharmonic equivalence|enharmonically equivalent]]) if and only if 12edo is used; for example, they are not the same thing in [[meantone]], where the diatonic semitone is larger than the chromatic semitone, or in [[Pythagorean tuning]], where the diatonic semitone is smaller instead.  
The intervals covered in this article range from 50{{c}} to 140{{c}}.
 
== 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 3-limit:
** 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{{c}}.
* In the 5-limit:
** 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{{c}}.
*** There is also a ''ptolemaic chromatic semitone'', which is a ratio of [[135/128]], and is about 92{{c}}.
* In higher limits:
** 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{{c}}.
** 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{{c}}.
** 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{{c}}.
** 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{{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 ==
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"
|-
! EDO
! 16/15
! 25/24
! Other semitones
|-
| 12
| colspan="2" | 100{{c}}
|
|-
| 15
| colspan="2" | 80{{c}}
|
|-
| 16
| colspan="2" | 75{{c}}
|
|-
| 17
| 141{{c}}
| 71{{c}}
|
|-
| 19
| 126{{c}}
| 63{{c}}
|
|-
| 22
| 109{{c}}
| 55{{c}}
|
|-
| 24
| 100{{c}}
| 50{{c}}
|
|-
| 25
| 96{{c}}
| *
|
|-
| 26
| colspan="2" | 92{{c}}
|
|-
| 27
| 133{{c}}
| 89{{c}}
|
|-
| 29
| 124{{c}}
| 83{{c}}
|
|-
| 31
| 116{{c}}
| 77{{c}}
|
|-
| 34
| 106{{c}}
| 71{{c}}
|
|-
| 41
| 117{{c}}
| 59{{c}}
| {{nowrap|88{{c}} ≈ 256/243}}
|-
| 53
| 113{{c}}
| 68{{c}}
| {{nowrap|91{{c}} ≈ 256/243}}
|}
 
== In regular temperaments ==
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 ===
* [[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]].
* [[Chlorine]], based on [[17edo]], stacking seventeen 25/24s to make an octave.
 
=== 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.
 
[[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 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}}


[[Category:Interval size measures]]
[[Category:12edo]]
[[Category:12edo]]
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