Semitone

In just intonation a 16:15 minor second arises in the C major scale between B & C and E & F, and is, "the sharpest dissonance found in the [major] scale."[7] Play B & C 

In 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 tempered fifths from E♭ to G♯, the chromatic and diatonic semitones are 76.0 and 117.1 cents wide respectively.

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-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.

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 2^1/12 (approximately 1.05946), or 100 cents, and is 11.7 cents narrower than the 16:15 ratio (its most common form in just intonation, discussed below).

All diatonic intervals can be expressed as an equivalent number of semitones. For instance a whole tone equals two semitones.

There are many approximations, rational or otherwise, to the equal-tempered semitone. To cite a few:

• ${\displaystyle 18/17\approx 99.0{\text{ cents,}}}$
  suggested by Vincenzo Galilei and used by luthiers of the Renaissance,

• ${\displaystyle {\sqrt[{4}]{\frac {2}{3-{\sqrt {2}}}}}\approx 100.4{\text{ cents,}}}$
  suggested by Marin Mersenne as a constructible and more accurate alternative,

• ${\displaystyle (139/138)^{8}\approx 99.9995{\text{ cents,}}}$
  used by Julián Carrillo as part of a sixteenth-tone system.

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

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 had a slightly different sonic color or character, beyond the limitations of conventional notation.

Pythagorean limma on C

Pythagorean apotome on C

Pythagorean limma as five descending just perfect fifths from C (the inverse is B+)

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 (play ), and is often called the Pythagorean limma. It is also sometimes called the Pythagorean minor semitone. It is about 90.2 cents.

${\displaystyle {\frac {256}{243}}={\frac {2^{8}}{3^{5}}}\approx 90.2{\text{ cents}}}$

It can be thought of as the difference between three octaves and five just fifths, and functions as a diatonic semitone in a Pythagorean tuning.

The Pythagorean chromatic semitone has a ratio of 2187/2048 (play ). It is about 113.7 cents. It may also be called the Pythagorean apotome[13][14][15] or the Pythagorean major semitone. (See Pythagorean interval.)

${\displaystyle {\frac {2187}{2048}}={\frac {3^{7}}{2^{11}}}\approx 113.7{\text{ cents}}}$

It can be thought of as the difference between four perfect octaves and seven 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.

16:15 diatonic semitone

16:15 diatonic semitone

'Larger' or major limma on C

Relationship between the 4 common 5-limit semitones

A minor second in just intonation typically corresponds to a pitch ratio of 16:15 (play ) or 1.0666... (approximately 111.7 cents), called the just diatonic semitone.[16] This is a practical just semitone, since it is the difference between a perfect fourth and major third (${\displaystyle {\tfrac {4}{3}}\div {\tfrac {5}{4}}={\tfrac {16}{15}}}$).

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."[7]

An augmented unison in just intonation is another semitone of 25:24 (play ) or 1.0416... (approximately 70.7 cents). It is the difference between a 5:4 major third and a 6:5 minor third. Composer Ben Johnston uses a sharp (♯) to indicate a note is raised 70.7 cents, or a flat (♭) to indicate a note is lowered 70.7 cents.[17]

Two other kinds of semitones are produced by 5-limit tuning. A chromatic scale defines 12 semitones as the 12 intervals between the 13 adjacent notes forming a full octave (e.g. from C4 to C5). The 12 semitones produced by a commonly used version of 5-limit tuning have four different sizes, and can be classified as follows:

• Just, or smaller, or minor, chromatic semitone, e.g. between E♭ and E (6/5 and 5/4):

${\displaystyle S_{1}={25 \over 24}\approx 70.7\ {\hbox{cents}}}$

• Larger, or major, chromatic semitone, or larger limma, or major chroma,[17] e.g. between C and C♯ (1/1 and 135/128):

${\displaystyle S_{2}={135 \over 128}\approx 92.2\ {\hbox{cents}}}$

• Just, or smaller, or minor, diatonic semitone, e.g. between C and D♭ (1/1 and 16/15):

${\displaystyle S_{3}={16 \over 15}\approx 111.7\ {\hbox{cents}}}$

• Larger, or major, diatonic semitone, e.g. between A and B♭ (5/3 and 16/9):

${\displaystyle S_{4}={27 \over 25}\approx 133.2\ {\hbox{cents}}}$

The most frequently occurring semitones are the just ones (S₃ and S₁): S₃ occurs six times out of 12, S₁ three times, S₂ twice, and S₄ only once.

The smaller chromatic and diatonic semitones differ from the larger by the syntonic comma (81:80 or 21.5 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 cents), the outer differ by the greater diesis (648:625 or 62.6 cents).

In 7-limit there is the septimal diatonic semitone of 15:14 (play ) available between the 5-limit major seventh (15:8) and the 7-limit minor seventh (7:4). There is also a smaller septimal chromatic semitone of 21:20 (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-limit neighbours, although the former was often implemented by theorist Henry Cowell, while Harry Partch used the latter as part of his 43-tone scale.

Under 11-limit tuning, there is a fairly common undecimal neutral second (12:11) (play ), but it lies on the boundary between the minor and major second (150.6 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-limit tuning, there is a tridecimal 2/3 tone (13:12 or 138.57 cents) and tridecimal 1/3 tone (27:26 or 65.34 cents).

In 17-limit just intonation, the major diatonic semitone is 15:14 or 119.4 cents (Play ), and the minor diatonic semitone is 17:16 or 105.0 cents,[18] and septendecimal limma is 18:17 or 98.95 cents.

Though the names diatonic and chromatic are often used for these intervals, their musical function is not the same as the two 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 usage (diatonic semitone function is more prevalent).

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