Semitone

A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale (e.g. from C to C). This implies that its size is exactly or approximately equal to 100 cents, a twelfth of an octave.

In a 12-note approximately equally divided scale, any interval can be defined in terms of an appropriate number of semitones (e.g. a whole tone or major second is 2 semitones wide, a major third 4 semitones, and a perfect fifth 7 semitones.

In music theory, a distinction is made between a diatonic semitone, or minor second (an interval encompassing two staff positions, e.g. from C to D) and a chromatic semitone or augmented unison (an interval between two notes at the same staff position, e.g. from C to C). These are enharmonically equivalent when twelve-tone equal temperament is used, but are not the same thing in meantone temperament, where the diatonic semitone is distinguished from and larger than the chromatic semitone (augmented unison.) See Interval (music)#Number for more details about this terminology.

In twelve-tone equal temperament all semitones are equal in size (100 cents). In other tuning systems, "semitone" refers to a family of intervals that may vary both in size and name. In Pythagorean tuning, seven semitones out of twelve are diatonic, with ratio 256:243 or 90.2 cents (Pythagorean limma), and the other five are chromatic, with ratio 2187:2048 or 113.7 cents (Pythagorean apotome); they differ by the Pythagorean comma of ratio 531441:524288 or 23.5 cents. In quarter-comma meantone, seven of them are diatonic, and 117.1 cents wide, while the other five are chromatic, and 76.0 cents wide; they differ by the lesser diesis of ratio 128:125 or 41.1 cents. 12-tone scales tuned in just intonation typically define three or four kinds of semitones. For instance, Asymmetric five-limit tuning yields chromatic semitones with ratios 25:24 (70.7 cents) and 135:128 (92.2 cents), and diatonic semitones with ratios 16:15 (111.7 cents) and 27:25 (133.2 cents). For further details, see below.

Minor second

The minor second occurs in the major scale, between the third and fourth degree, (mi (E) and fa (F) in C major), and between the seventh and eighth degree (ti (B) and do (C) in C major). It is also called the diatonic semitone because it occurs between steps in the diatonic scale. The minor second is abbreviated m2 (or −2). Its inversion is the major seventh (M7, or +7).

. Here, middle C is followed by D, which is a tone 100 cents sharper than C, and then by both tones together.

Melodically, this interval is very frequently used, and is of particular importance in cadences. In the perfect and deceptive cadences it appears as a resolution of the leading-tone to the tonic. In the plagal cadence, it appears as the falling of the subdominant to the mediant. It also occurs in many forms of the imperfect cadence, wherever the tonic falls to the leading-tone.

Harmonically, the interval usually occurs as some form of dissonance or a nonchord tone that is not part of the functional harmony. It may also appear in inversions of a major seventh chord, and in many added tone chords.

's Prelude in C major from the WTC book 1, mm. 7-9. The minor second may be viewed as a suspension of the B resolving into the following A minor seventh chord.]]

's "wrong note" Étude.]] In unusual situations, the minor second can add a great deal of character to the music. For instance, Frédéric Chopin's Étude Op. 25, No. 5 opens with a melody accompanied by a line that plays fleeting minor seconds. These are used to humorous and whimsical effect, which contrasts with its more lyrical middle section. This eccentric dissonance has earned the piece its nickname: the "wrong note" étude. This kind of usage of the minor second appears in many other works of the Romantic period, such as Modest Mussorgsky's Ballet of the Unhatched Chicks. More recently, the music to the movie Jaws exemplifies the minor second.

Augmented unison

Melodically, an augmented unison very frequently occurs when proceeding to a chromatic chord, such as a secondary dominant, a diminished seventh chord, or an augmented sixth chord. Its use is also often the consequence of a melody proceeding in semitones, regardless of harmonic underpinning, e.g. D, D, E, F, F. (Restricting the notation to only minor seconds is impractical, as the same example would have a rapidly increasing number of accidentals, written enharmonically as D, E, F, G, A).

's second Transcendental Etude, measure 63.]] Harmonically, augmented unisons are quite rare in tonal repertoire. In the example to the right, Liszt had written an E against an E in the bass. Here E was preferred to a D to make the tone's function clear as part of an F dominant seventh chord, and the augmented unison is the result of superimposing this harmony upon an E pedal point.

In addition to this kind of usage, harmonic augmented unisons are frequently written in modern works involving tone clusters, such as Iannis Xenakis' Evryali for piano solo.

History

Though it would later become an integral part of the musical cadence, in the early polyphony of the 11th century this was not the case. Guido of Arezzo suggested instead in his Micrologus other alternatives: either proceeding by whole tone from a major second to a unison, or an occursus having two notes at a major third move by contrary motion toward a unison, each having moved a whole tone.

“As late as the 13th century the half step was experienced as a problematic interval not easily understood, as the irrational [sic] remainder between the perfect fourth and the ditone $\backslash left(\backslash begin\{matrix\}\; \backslash frac\{4\}\{3\}\; \backslash end\{matrix\}\; /^2\}\; =\; \backslash begin\{matrix\}\; \backslash frac\{256\}\{243\}\; \backslash end\{matrix\}\backslash right)$.” In a melodic half step, no “tendency was perceived of the lower tone toward the upper, or of the upper toward the lower. The second tone was not taken to be the ‘goal’ of the first. Instead, the half step was avoided in clausulae because it lacked clarity as an interval.”

's Solo e pensoso, ca. 1580. ( ]] However, beginning in the 13th century cadences begin to require motion in one voice by half step and the other a whole step in contrary motion. or the Pythagorean major semitone. (See Pythagorean interval.) :$\backslash frac\{2187\}\{2048\}\; =\; \backslash frac\{3^7\}\{2^\{11\}\}\; \backslash approx\; 113.7\backslash text\{\; cents\}$

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.

Just intonation

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

An augmented unison in just intonation is another semitone of 25:24 () 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 an accidental to indicate a note is raised 70.7 cents, or a flat to indicate a note is lowered 70.7 cents.

Two other kinds of semitones are produced by 5-limt 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: : $S\_1\; =\; \{25\; \backslash over\; 24\}\; \backslash approx\; 70.7\; \backslash \; \backslash hbox\{cents\}$ Larger, or major, chromatic semitone, or larger limma, or major chroma, e.g. between D and D: : $S\_2\; =\; \{135\; \backslash over\; 128\}\; \backslash approx\; 92.2\; \backslash \; \backslash hbox\{cents\}$ Just, or smaller, or minor, diatonic semitone, e.g. between C and D: : $S\_3\; =\; \{16\; \backslash over\; 15\}\; \backslash approx\; 111.7\; \backslash \; \backslash hbox\{cents\}$ Larger, or major, diatonic semitone, e.g. between A and B: : $S\_4\; =\; \{27\; \backslash over\; 25\}\; \backslash approx\; 133.2\; \backslash \; \backslash hbox\{cents\}$

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).

Other ratios may function as a minor second. In 7-limit there is the septimal diatonic semitone of 15:14 () 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 () 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) (), 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 17-limit just intonation, the major diatonic semitone is 15:14 or 119.4 cents (), and the minor diatonic semitone is 17:16 or 105.0 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).

Other equal temperaments

In general, because the two semitones can be viewed as the difference between major and minor thirds, and the difference between major thirds and perfect fourths, tuning systems that match these just intervals closely will also distinguish between the two types of semitones and match their just intervals closely.

See also

References

Further reading

Category:Intervals