Semitone (interval region): Difference between revisions

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 {{Wikipedia}}
-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}}.
+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.
-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.
+"Semitone" also refers to an interval measure of exactly 100 cents, which is not the subject of this article.
-[[Category:Interval size measures]]
+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 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 cents.
+** The '''apotome,''' or Pythagorean chromatic semitone, is a ratio of 2187/2048, and is about 114 cents.
+* In the 5-limit:
+** The '''classical diatonic semitone''' is a ratio of 16/15, and is about 112 cents.
+** The '''classical chromatic semitone''' is a ratio of 25/24, and is about 71 cents.
+*** There is also a '''ptolemaic chromatic semitone,''' which is a ratio of 135/128, and is about 92 cents.
+* In higher limits:
+** The 7-limit '''third-tone''' is a ratio of 28/27, and is about 63 cents.
+** The 7-limit '''minor semitone''' is a ratio of 21/20, and is about 84 cents.
+** The 7-limit '''major semitone''' is a ratio of 15/14, and is about 119 cents.
+** The 11-limit '''minor semitone''' is a ratio of 22/21, and is about 81 cents.
+** The 13-limit '''sinaic''' is a ratio of 14/13, and is about 128 cents.
+** The 13-limit '''greater 2/3 tone''' is a ratio of 13/12, and is about 139 cents.
+** The 17-limit '''large semitone''' is a ratio of 17/16, and is about 104 cents.
+** The 17-limit '''small semitone''' is a ratio of 18/17, and is about 99 cents.
+== In EDOs ==
+The following table lists the best tuning of 16/15, 25/24, and other semitones if present, in various significant [[EDOs]].
+{| class="wikitable"
+|+
+!EDO
+!16/15
+!25/24
+!Other semitones
+|-
+|12
+| colspan="2" |100c
+|
+|-
+|15
+| colspan="2" |80c
+|
+|-
+|16
+| colspan="2" |75c
+|
+|-
+|17
+|141c
+|71c
+|
+|-
+|19
+|126c
+|63c
+|
+|-
+|22
+|109c
+|55c
+|
+|-
+|24
+|100c
+|50c
+|
+|-
+|25
+|96c
+|*
+|
+|-
+|26
+| colspan="2" |92c
+|
+|-
+|27
+|133c
+|89c
+|
+|-
+|29
+|124c
+|83c
+|
+|-
+|31
+|116c
+|77c
+|
+|-
+|34
+|106c
+|71c
+|
+|-
+|41
+|117c
+|59c
+|88c ≈ 256/243
+|-
+|53
+|113c
+|68c
+|91c ≈ 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 ===
+* 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
+=== Temperaments that use 16/15 as a generator ===
+* TBD
+When 25/24 is tempered out, it leads to [[dicot]] temperament.
+When 16/15 is tempered out, it leads to [[father]] temperament.
+{{Navbox intervals}}
 [[Category:12edo]]