Editing Semitone (section)

==Semitones in different tunings==

The exact size of a semitone depends on the [[Musical tuning|tuning]] system used. [[Meantone temperament]]s have two distinct types of semitones, but in the exceptional case of [[equal temperament]], there is only one. The unevenly distributed [[well temperament]]s contain many different semitones. [[Pythagorean tuning]], similar to meantone tuning, has two, but in other systems of just intonation there are many more possibilities.

===Meantone temperament===

In [[meantone temperament|meantone]] systems, there are two different semitones. This results because of the break in the [[circle of fifths]] that occurs in the tuning system: diatonic semitones derive from a chain of five fifths that does not cross the break, and chromatic semitones come from one that does.

The chromatic semitone is usually smaller than the diatonic. In the common [[quarter-comma meantone]], tuned as a cycle of [[Musical temperament|tempered]] [[Perfect fifth|fifths]] from E{{music|flat}} to G{{music|sharp}}, the chromatic and diatonic semitones are 76.0 and 117.1 cents wide respectively.

{| class="wikitable" style="text-align:center" align="center"
| bgcolor="#ffeeee" | '''Chromatic semitone'''
|
| colspan="2" bgcolor="#ffeeee" | <small>76.0</small>
| colspan="2" |
| colspan="2" |
| colspan="2" bgcolor="#ffeeee" | <small>76.0</small>
| colspan="2" |
| colspan="2" bgcolor="#ffeeee" | <small>76.0</small>
| colspan="2" |
| colspan="2" bgcolor="#ffeeee" | <small>76.0</small>
| colspan="2" |
| colspan="2" |
| colspan="2" bgcolor="#ffeeee" | <small>76.0</small>
| colspan="2" |
|-
| bgcolor="#fffbee" | '''Pitch'''
| colspan="2" bgcolor="#fffbee" | C
| colspan="2" bgcolor="#fffbee" | C{{music|sharp}}
| colspan="2" bgcolor="#fffbee" | D
| colspan="2" bgcolor="#fffbee" | E{{music|flat}}
| colspan="2" bgcolor="#fffbee" | E
| colspan="2" bgcolor="#fffbee" | F
| colspan="2" bgcolor="#fffbee" | F{{music|sharp}}
| colspan="2" bgcolor="#fffbee" | G
| colspan="2" bgcolor="#fffbee" | G{{music|sharp}}
| colspan="2" bgcolor="#fffbee" | A
| colspan="2" bgcolor="#fffbee" | B{{music|flat}}
| colspan="2" bgcolor="#fffbee" | B
| colspan="2" bgcolor="#fffbee" | C
|-
| bgcolor="#fffbee" | '''Cents'''
| colspan="2" bgcolor="#fffbee" | <small>0.0</small>
| colspan="2" bgcolor="#fffbee" | <small>76.0</small>
| colspan="2" bgcolor="#fffbee" | <small>193.2</small>
| colspan="2" bgcolor="#fffbee" | <small>310.3</small>
| colspan="2" bgcolor="#fffbee" | <small>386.3</small>
| colspan="2" bgcolor="#fffbee" | <small>503.4</small>
| colspan="2" bgcolor="#fffbee" | <small>579.5</small>
| colspan="2" bgcolor="#fffbee" | <small>696.6</small>
| colspan="2" bgcolor="#fffbee" | <small>772.6</small>
| colspan="2" bgcolor="#fffbee" | <small>889.7</small>
| colspan="2" bgcolor="#fffbee" | <small>1006.8</small>
| colspan="2" bgcolor="#fffbee" | <small>1082.9</small>
| colspan="2" bgcolor="#fffbee" | <small>1200.0</small>
|- <!--- this row inserted to avoid bug in Chrome causing misalignment of columns -->
| || || || || || || || || || || || || || || || || || || || || || || || || || ||
|-
| bgcolor="#eeeeff" | '''Diatonic semitone'''
| colspan="3" |
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" |
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" |
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" |
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" |
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
|
|}

Extended meantone temperaments with more than 12 notes still retain the same two semitone sizes, but there is more flexibility for the musician about whether to use an augmented unison or minor second. [[31-TET|31-tone equal temperament]] is the most flexible of these, which makes an unbroken circle of 31 fifths, allowing the choice of semitone to be made for any pitch.

===Equal temperament===

[[12-tone equal temperament]] is a form of meantone tuning in which the diatonic and chromatic semitones are exactly the same, because its circle of fifths has no break. Each semitone is equal to one twelfth of an octave. This is a ratio of [[Twelfth root of two|2<sup>1/12</sup>]] (approximately 1.05946), or 100 cents, and is 11.7 cents narrower than the 16:15 ratio (its most common form in [[just intonation]], [[#Just intonation|discussed below]]).

All diatonic intervals can be expressed as an equivalent number of semitones. For instance a [[major sixth]] equals nine semitones.

There are many approximations, [[Rational number|rational]] or otherwise, to the equal-tempered semitone. To cite a few:
:*<math>18 / 17 \approx 99.0 \text{ cents,}</math><br />suggested by [[Vincenzo Galilei]] and used by [[luthier]]s of the [[Renaissance music|Renaissance]],

:*<math>\sqrt[4]{\frac{2}{3-\sqrt{2}}} \approx 100.4 \text{ cents,}</math><br />suggested by [[Marin Mersenne]] as a [[Constructible number|constructible]] and more accurate alternative,

:*<math>(139 / 138 )^8  \approx 99.9995 \text{ cents,}</math><br />used by [[Julián Carrillo]] as part of a sixteenth-tone system.

For more examples, see Pythagorean and Just systems of tuning below.

===Well temperament===
There are many forms of [[well temperament]], but the characteristic they all share is that their semitones are of an uneven size. Every semitone in a well temperament has its own interval (usually close to the equal-tempered version of 100 cents), and there is no clear distinction between a ''diatonic'' and ''chromatic'' semitone in the tuning. Well temperament was constructed so that [[enharmonic]] equivalence could be assumed between all of these semitones, and whether they were written as a minor second or augmented unison did not effect a different sound. Instead, in these systems, each [[Key (music)|key]] had a slightly different sonic color or character, beyond the limitations of conventional notation.

===Pythagorean tuning===
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| image1 = Pythagorean limma on C.png
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| caption1 = Pythagorean limma on C[[File:Pythagorean minor semitone on C.mid|90px]]
| image2 = Pythagorean apotome on C.png
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| caption2 = Pythagorean apotome on C[[File:Pythagorean apotome on C.mid|90px]]
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{{multiple image
   | width1    = 200
   | image1    = Pythagorean limma.png
   | caption1  = Pythagorean limma as five descending just perfect fifths from C (the inverse is B+)
   | width2    = 200
   | image2    = Pythagorean apotome.png
   | caption2  = Pythagorean apotome as seven just perfect fifths
}}

Like meantone temperament, [[Pythagorean tuning]] is a broken [[circle of fifths]]. This creates two distinct semitones, but because Pythagorean tuning is also a form of 3-limit [[just intonation]], these semitones are rational. Also, unlike most meantone temperaments, the chromatic semitone is larger than the diatonic.

The '''Pythagorean diatonic semitone''' has a ratio of 256/243 ({{Audio|Pythagorean minor semitone on C.mid|play}}), and is often called the '''Pythagorean limma'''. It is also sometimes called the ''Pythagorean minor semitone''. It is about 90.2 cents.
:<math>\frac{256}{243} = \frac{2^8}{3^5} \approx 90.2 \text{ cents}</math>

It can be thought of as the difference between three [[octaves]] and five [[perfect fifth|just fifths]], and functions as a [[#Minor second|diatonic semitone]] in a [[Pythagorean tuning]].

The '''Pythagorean chromatic semitone''' has a ratio of 2187/2048 ({{Audio|Pythagorean apotome on C.mid|play}}). It is about 113.7 [[Cent (music)|cents]]. It may also be called the '''Pythagorean apotome'''<ref name="Rashed">Rashed, Roshdi (ed.) (1996). ''Encyclopedia of the History of Arabic Science, Volume 2'', pp. 588, 608. Routledge. {{ISBN|0-415-12411-5}}.</ref><ref>[[Hermann von Helmholtz]] (1885). ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', p. 454.</ref><ref>Benson, Dave (2006). ''Music: A Mathematical Offering'', p. 369. {{ISBN|0-521-85387-7}}.</ref> or the ''Pythagorean major semitone''. (''See [[Pythagorean interval]]''.)
:<math>\frac{2187}{2048} = \frac{3^7}{2^{11}} \approx 113.7\text{ cents}</math>

It can be thought of as the difference between four perfect [[octave]]s and seven [[perfect fifth|just fifths]], and functions as a [[chromatic semitone]] in a [[Pythagorean tuning]].

The Pythagorean limma and Pythagorean apotome are [[enharmonic]] equivalents (chromatic semitones) and only a [[Pythagorean comma]] apart, in contrast to diatonic and chromatic semitones in [[meantone temperament]] and 5-limit [[just intonation]].

===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).

===Extended just intonations===
In [[7-limit|7&nbsp;limit tuning]] there is the [[septimal diatonic semitone]] of 15:14 ({{Audio|Septimal diatonic semitone on C.mid|play}}) available in between the 5&nbsp;limit [[major seventh]] (15:8) and the [[septimal minor seventh|7&nbsp;limit minor seventh]] / [[harmonic seventh]] (7:4). There is also a smaller [[septimal chromatic semitone]] of 21:20 ({{Audio|Septimal chromatic semitone on C.mid|play}}) between a septimal minor seventh and a fifth (21:8) and an octave and a major third (5:2). Both are more rarely used than their 5&nbsp;limit neighbours, although the former was often implemented by theorist [[Henry Cowell|Cowell]], while [[Harry Partch|Partch]] used the latter as part of [[Harry Partch's 43-tone scale|his 43&nbsp;tone scale]].

Under 11&nbsp;limit tuning, there is a fairly common ''undecimal [[neutral second]]'' (12:11) ({{Audio|Neutral second on C.mid|play}}), but it lies on the boundary between the minor and [[major second]] (150.6&nbsp;cents). In just intonation there are infinitely many possibilities for intervals that fall within the range of the semitone (e.g. the Pythagorean semitones mentioned above), but most of them are impractical.

In 13&nbsp;limit tuning, there is a tridecimal {{sfrac|2|3}} tone (13:12 or 138.57&nbsp;cents) and tridecimal {{sfrac|1|3}} tone (27:26 or 65.34&nbsp;cents).

In 17&nbsp;limit just intonation, the major diatonic semitone is 15:14 or 119.4&nbsp;cents ({{audio|Major diatonic semitone on C.mid|Play}}), and the minor diatonic semitone is 17:16 or 105.0&nbsp;cents,<ref>{{cite book |author-link=Ebenezer Prout |last=Prout |first=E. |year=2004 |title=Harmony |page=325 |publisher=University Press of the Pacific |isbn=1-4102-1920-8}}</ref> and septendecimal limma is 18:17 or 98.95&nbsp;cents.

Though the names ''diatonic'' and ''chromatic'' are often used for these intervals, their musical function is not the same as the meantone semitones. For instance, 15:14 would usually be written as an augmented unison, functioning as the ''chromatic'' counterpart to a ''diatonic'' 16:15. These distinctions are highly dependent on the musical context, and just intonation is not particularly well suited to chromatic use (diatonic semitone function is more prevalent).

===Other equal temperaments===

[[19 equal temperament|19-tone equal temperament]] distinguishes between the chromatic and diatonic semitones; in this tuning, the chromatic semitone is one step of the scale ({{Audio|1 step in 19-et on C.mid|play 63.2 cents}}), and the diatonic semitone is two ({{Audio|2 steps in 19-et on C.mid|play 126.3 cents}}). [[31 equal temperament|31-tone equal temperament]] also distinguishes between these two intervals, which become 2 and 3 steps of the scale, respectively. [[53 equal temperament|53-ET]] has an even closer match to the two semitones with 3 and 5 steps of its scale while [[72 equal temperament|72-ET]] uses 4 ({{Audio|4 steps in 72-et on C.mid|play 66.7 cents}}) and 7 ({{Audio|7 steps in 72-et on C.mid|play 116.7 cents}}) steps of its scale.

In general, because the smaller semitone can be viewed as the difference between a minor third and a major third, and the larger as the difference between a major third and a perfect fourth, tuning systems that closely match those just intervals (6/5, 5/4, and 4/3) will also distinguish between the two types of semitones and closely match their just intervals (25/24 and 16/15).