Semitone
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Inverse | major seventh (for minor second); diminished octave (for augmented unison); augmented octave (for diminished unison) |
---|---|
Name | |
Other names | minor second, diatonic semitone, augmented unison, diminished unison, chromatic semitone |
Abbreviation | m2; A1 |
Size | |
Semitones | 1 |
Interval class | 1 |
Just interval | 16:15,[1] 17:16,[2] 27:25, 135:128,[1] 25:24,[1] 256:243 |
Cents | |
12-Tone equal temperament | 100[1] |
Just intonation | 112,[1] 105, 133, 92,[1] 71,[1] 90 |
A semitone, also called a half step or a half tone,[3] is the smallest musical interval commonly used in Western tonal music,[4] and it is considered the most dissonant[5] when sounded harmonically. It is defined as the interval between two adjacent notes in a
In a 12-note approximately equally divided scale, any interval can be defined in terms of an appropriate number of semitones (e.g. a
In
In
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
Inverse | major seventh |
---|---|
Name | |
Other names | semitone, half step |
Abbreviation | m2 |
Size | |
Semitones | 1 |
Interval class | 1 |
Cents | |
12-Tone equal temperament | 100.0 |
The minor second occurs in the
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.
In other temperaments
In
Augmented unison
The augmented unison, the interval produced by the
In addition to this kind of usage, harmonic augmented unisons are frequently written in modern works involving
History
The semitone appeared in the music theory of Greek antiquity as part of a diatonic or chromatic
Though it would later become an integral part of the musical
"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 ." 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."[13]
However, beginning in the 13th century
By the
In the 20th century, however, composers such as
Semitones in different tunings
The exact size of a semitone depends on the tuning system used. Meantone temperaments have two distinct types of semitones, but in the exceptional case of equal temperament, there is only one. The unevenly distributed well temperaments 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 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.
Chromatic semitone | 76.0 | 76.0 | 76.0 | 76.0 | 76.0 | |||||||||||||||||||||
Pitch | C | C♯ | D | E♭ | E | F | F♯ | G | G♯ | A | B♭ | B | C | |||||||||||||
Cents | 0.0 | 76.0 | 193.2 | 310.3 | 386.3 | 503.4 | 579.5 | 696.6 | 772.6 | 889.7 | 1006.8 | 1082.9 | 1200.0 | |||||||||||||
Diatonic semitone | 117.1 | 117.1 | 117.1 | 117.1 | 117.1 | 117.1 | 117.1 |
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.
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 21/12 (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:
suggested by Vincenzo Galilei and used by luthiers of the Renaissance,
suggested by Marin Mersenne as a constructible and more accurate alternative,
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
Pythagorean tuning
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 (ⓘ), and is often called the Pythagorean limma. It is also sometimes called the Pythagorean minor semitone. It is about 90.2 cents.
It can be thought of as the difference between three
.The Pythagorean chromatic semitone has a ratio of 2187/2048 (ⓘ). It is about 113.7 cents. It may also be called the Pythagorean apotome[14][15][16] or the Pythagorean major semitone. (See Pythagorean interval.)
It can be thought of as the difference between four perfect
The Pythagorean limma and Pythagorean apotome are
Just 5-limit 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.[17] 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) and a
- perfect octave(2:1)
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".[8]
An "augmented unison" (sharp) in just intonation is a different, smaller semitone, with frequency ratio 25:24 (ⓘ) or 1.0416... (approximately 70.7 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 used a sharp (♯) to indicate a note is raised 70.7 cents, or a flat (♭) to indicate a note is lowered 70.7 cents.[18] (This is the standard practice for just intonation, but not for all other microtunings.)
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, spanning 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 chromatic semitone
- chromatic semitone, or smaller, or minor chromatic semitone between harmonically related flats and sharps e.g. between E♭ and E (6:5 and 5:4):
- Larger chromatic semitone
- or major chromatic semitone, or larger limma, or major chroma,[18] e.g. between C and an accute C♯ (C♯ raised by a syntonic comma) (1:1 and 135:128):
- Just diatonic semitone
- or smaller, or minor diatonic semitone, e.g. between E and F (5:4 to 4:3):
- Larger diatonic semitone
- or greater or major diatonic semitone, e.g. between A and B♭ (5:3 to 9:5), or C and chromatic D♭ (27:25), or F♯ and G (25:18 and 3:2):
The most frequently occurring semitones are the just ones (S3, 16:15, and S1, 25:24): S3 occurs at 6 short intervals out of 12, S1 3 times, S2 twice, and S4 at only one interval (if diatonic D♭ replaces chromatic D♭ 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 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).
Extended just intonations
In
Under 11 limit tuning, there is a fairly common undecimal
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 (ⓘ), and the minor diatonic semitone is 17:16 or 105.0 cents,[19] 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 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-tone equal temperament distinguishes between the chromatic and diatonic semitones; in this tuning, the chromatic semitone is one step of the scale (ⓘ), and the diatonic semitone is two (ⓘ). 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 (ⓘ) and 7 (ⓘ) 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).
See also
- List of meantone intervals
- List of musical intervals
- List of pitch intervals
- Approach chord
- Major second
- Neutral second
- Pythagorean interval
- Regular temperament
References
- ^ ISBN 978-0-393-33420-3. Retrieved 28 June 2017.
- ISBN 0-8247-4714-3. Overtone semitone.
- ^ Semitone, half step, half tone, halftone, and half-tone are all variously used in sources.[1][2][3][4][5]
Aaron Copland, Leonard Bernstein, and others use "half tone".[6] [7][8][9]
One source says that step is "chiefly US",[10] and that half-tone is "chiefly N. Amer."[11] - ISBN 1-59257-437-8. p. 19.
- ^ Capstick, John Walton (1913). Sound: An Elementary Text-book for Schools and Colleges. Cambridge University Press.
- ^ "musictheory.net". www.musictheory.net. Retrieved 2024-01-04.
- ISBN 978-1-55440-283-0.
- ^ a b Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction, p. 165. Theodore Baker, trans. G. Schirmer.
- ISBN 978-0-07-294262-0. Specific example of an A1 not given but general example of perfect intervals described.
- ISBN 0-07-285260-7. "There is no such thing as a diminished unison."
- ISBN 0-7645-7838-3. "There is no such thing as a diminished unison, because no matter how you change the unisons with accidentals, you are adding half steps to the total interval."
- ISBN 978-0-7390-3635-8. Since lowering either note of a perfect unison would actually increase its size, the perfect unison cannot be diminished, only augmented.
- ^ ISBN 0-691-09135-8.
- ISBN 0-415-12411-5.
- ^ Hermann von Helmholtz (1885). On the Sensations of Tone as a Physiological Basis for the Theory of Music, p. 454.
- ISBN 0-521-85387-7.
- Proceedings of the Royal Society of London. 30. Great Britain: Royal Society: 531. 1880.
digitized 26 Feb 2008; Harvard University
- ^ a b Fonville, J. (Summer 1991). "Ben Johnston's extended just intonation – a guide for interpreters". Perspectives of New Music. 29 (2): 106–137.
... the 25/24 ratio is the sharp (♯) ratio ... this raises a note approximately 70.6 cents.(p109)
- ISBN 1-4102-1920-8.
Further reading
- ISBN 0-393-97527-4.
- ISBN 0-393-09090-6.