Semitone

In Western music theory, a semitone (also called a half step or half tone) is one of two parts into which a whole tone is divided. Semitones appear as the smallest steps in chromatic scales (which divide the octave into twelve semitones), arising on keyboards between the pitches of two adjacent keys. For example, C is adjacent to D; the interval between them is a semitone. Semitones are among the most dissonant intervals when sounded harmonically.

In musical notation, two types of semitones are distinguished. A diatonic semitone, or minor second, arises between two notes encompassing different staff positions, e.g. from C to D. A chromatic semitone, or augmented unison, arises between two notes at the same staff position, e.g. from D to D. The sum of these two intervals, the interval from C to D, is a whole tone, or major second.

The modern tuning system of 12-tone equal temperament divides the octave into twelve equal semitones, each with a frequency ratio of the twelfth root of two (100 cents). In this system, a given tonal interval can be represented by a specific number of semitones from any starting note. For example, two semitones may represent a major second, four semitones a major third, and seven semitones a perfect fifth .

In tuning systems derived from stacking perfect fifths, i.e. Pythagorean tuning and the related meantone temperaments, diatonic semitones all have the same size, as do chromatic semitones, but the two sizes are distinct. For example, they are not the same size in Pythagorean tuning, where the diatonic semitone is distinguished from the larger chromatic semitone, or in quarter-comma meantone temperament, where the diatonic semitone is larger instead.

The condition of having semitones is called hemitonia; that of having no semitones is anhemitonia. A musical scale or chord containing semitones is called hemitonic; one without semitones is anhemitonic.

Minor second

The minor second (m2 or -2) 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. Its inversion is the major seventh (M7 or Ma7).

Melodically, this interval is very frequently used, and is of particular importance in cadences. In the authentic 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.

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" from Pictures at an Exhibition. More recently, the music to the movie Jaws exemplifies the minor second.

Augmented unison

The augmented unison, augmented prime Historically, this interval, like the tritone, is described as being and was associated with the .

thumb|left|Augmented unisons often appear as a consequence of [[secondary dominants, such as those in the soprano voice of this sequence from Felix Mendelssohn's Song Without Words Op. 102 No. 3, mm. 47–49.

File:Mendelssohn dominants.mid]]

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).

thumb|right|[[Franz Liszt's second Transcendental Étude, measure 63]]

Harmonically, augmented unisons are quite rare in tonal repertoire. In the example to the right, Liszt wrote 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 solo piano.

Diminished unison

The term diminished unison or diminished prime is also found occasionally. It is found once in Rameau's writings, for example,

Other sources reject the possibility or utility of the diminished unison on the grounds that any alteration to the unison increases its size, thus augmenting rather than diminishing it. The term is sometimes justified as a negative-numbered interval, and also in terms of violin double-stopping technique on analogy to parallel intervals found on other strings. Some theoreticians make a distinction for this diminished form of the unison, stating it is only valid as a melodic interval, not a harmonic one.

History

The semitone appeared in the music theory of Greek antiquity as part of a diatonic or chromatic tetrachord, and it has always had a place in the diatonic scales of Western music since. The various modal scales of medieval music theory were all based upon this diatonic pattern of tones and semitones.

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 remainder between the perfect fourth and the ditone <math>\left(\begin{matrix} \frac{4}{3} \end{matrix} / ^2} = \begin{matrix} \frac{256}{243} \end{matrix}\right)</math>." 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."

thumb|Dramatic chromatic scale in the opening measures of [[Luca Marenzio's Solo e pensoso, ca. 1580 File:Marenzio solo e pensoso opening.MID|450x450px]]

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.)

:<math>\frac{2187}{2048} = \frac{3^7}{2^{11 = \frac{(3/2)^7}{2^4} \approx 113.7\text{ cents}</math>

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 only a Pythagorean comma apart, and may be considered enharmonic equivalents, in contrast to the diatonic and chromatic semitones in meantone temperament and 5-limit just intonation, which are farther apart.

Meantone temperament

Meantone temperaments are generated by a sequence of tempered perfect fifths of the same size, similarly to Pythagorean tuning, where the perfect fifths are pure instead. Likewise, the diatonic and chromatic semitones depend on the size of the fifth, where the diatonic semitone is derived by descending 5 fifths, and the chromatic semitone is derived by ascending 7 fifths. These are typically of different sizes, with the exception of 12 equal temperament, discussed below. Unlike in Pythagorean tuning, the chromatic semitone is usually smaller than the diatonic.

In the common quarter-comma meantone, the chromatic and diatonic semitones are 76.0 and 117.1 cents wide respectively. They differ by the lesser diesis of ratio 128:125 or 41.1 cents.

Quarter-comma meantone is largely used to create a 12-tone tuning, usually via a sequence of fifths from E to G. There is a break in the circle of fifths in this tuning; the interval between G and E does not represent a perfect fifth, but rather a diminished sixth. Such intervals are referred to as wolf intervals. Ascending by a semitone from an arbirary pitch, the resulting interval depends on this break: diatonic semitones derive from a descending chain of 5 fifths that does not cross the break, and chromatic semitones come from one that does, where the note is instead derived by an ascending chain of 7 fifths.

{| class="wikitable" style="text-align:center" align="center"

| bgcolor="#ffeeee" | Chromatic semitone

| colspan="2" bgcolor="#ffeeee" | 76.0

| colspan="2" |

| colspan="2" bgcolor="#ffeeee" | 76.0

| colspan="2" |

| colspan="2" bgcolor="#ffeeee" | 76.0

| colspan="2" |

| colspan="2" bgcolor="#ffeeee" | 76.0

| colspan="2" |

| colspan="2" bgcolor="#ffeeee" | 76.0

| colspan="2" |

| bgcolor="#fffbee" | Pitch

| colspan="2" bgcolor="#fffbee" | C

| colspan="2" bgcolor="#fffbee" | D

| colspan="2" bgcolor="#fffbee" | E

| colspan="2" bgcolor="#fffbee" | F

| colspan="2" bgcolor="#fffbee" | G

| colspan="2" bgcolor="#fffbee" | A

| colspan="2" bgcolor="#fffbee" | B

| colspan="2" bgcolor="#fffbee" | C

| bgcolor="#fffbee" | Cents

| colspan="2" bgcolor="#fffbee" | 0.0

| colspan="2" bgcolor="#fffbee" | 76.0

| colspan="2" bgcolor="#fffbee" | 193.2

| colspan="2" bgcolor="#fffbee" | 310.3

| colspan="2" bgcolor="#fffbee" | 386.3

| colspan="2" bgcolor="#fffbee" | 503.4

| colspan="2" bgcolor="#fffbee" | 579.5

| colspan="2" bgcolor="#fffbee" | 696.6

| colspan="2" bgcolor="#fffbee" | 772.6

| colspan="2" bgcolor="#fffbee" | 889.7

| colspan="2" bgcolor="#fffbee" | 1006.8

| colspan="2" bgcolor="#fffbee" | 1082.9

| colspan="2" bgcolor="#fffbee" | 1200.0

|-

| || || || || || || || || || || || || || || || || || || || || || || || || || ||

| bgcolor="#eeeeff" | Diatonic semitone

| colspan="3" |