Semitone

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semitone
Inverse major seventh (for minor second); diminished octave (for augmented unison); augmented octave (for diminished unison)
Name
Other names minor second
or diatonic semitone;
augmented unison and diminished unison
or 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
Equal temperament 100 [1]
24 equal temperament 100
Just intonation 112, [1] 105, 133, 92 [1] , 71 [1] , 90
Minor second   Play  

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 12-tone scale. For example, C is adjacent to C; the interval between them is a semitone.

In music theory, an interval is the difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

The chromatic scale is a musical scale with twelve pitches, each a semitone above or below its adjacent pitches. As a result, in 12-tone equal temperament, the chromatic scale covers all 12 of the available pitches. Thus, there is only one chromatic scale.

Contents

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.

Major third musical interval

In classical music from Western culture, a third is a musical interval encompassing three staff positions, and the major third is a third spanning four semitones. Along with the minor third, the major third is one of two commonly occurring thirds. It is qualified as major because it is the larger of the two: the major third spans four semitones, the minor third three. For example, the interval from C to E is a major third, as the note E lies four semitones above C, and there are three staff positions from C to E. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones.

Perfect fifth musical interval

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.

In music theory, a distinction is made [6] between a diatonic semitone, or minor second (an interval encompassing two different 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.

Music theory considers the practices and possibilities of music

Music theory is the study of the practices and possibilities of music. The Oxford Companion to Music describes three interrelated uses of the term "music theory":

The first is what is otherwise called 'rudiments', currently taught as the elements of notation, of key signatures, of time signatures, of rhythmic notation, and so on. [...] The second is the study of writings about music from ancient times onwards. [...] The third is an area of current musicological study that seeks to define processes and general principles in music — a sphere of research that can be distinguished from analysis in that it takes as its starting-point not the individual work or performance but the fundamental materials from which it is built.

Enharmonic (in modern musical notation and tuning) note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently

In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. For example, in twelve-tone equal temperament, the notes C and D are enharmonic notes. Namely, they are the same key on a keyboard, and thus they are identical in pitch, although they have different names and different roles in harmony and chord progressions. Arbitrary amounts of accidentals can produce further enharmonic equivalents, such as B, although these are much rarer and have less practical use.

Equal temperament The musical tuning system where the ratio between successive notes is constant

An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In other words, the ratios of the frequencies of any adjacent pair of notes is the same, and, as pitch is perceived roughly as the logarithm of frequency, equal perceived "distance" from every note to its nearest neighbor.

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.

Pythagorean tuning

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ~702 cents wide.

Pythagorean comma small interval in Pythagorean tuning between two enharmonically equivalent notes such as C and B♯

In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B (Play ), or D and C. It is equal to the frequency ratio (1.5)12/128 = about 1.01364, or about 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73). The comma which musical temperaments often refer to tempering is the Pythagorean comma.

Quarter-comma meantone, or 14-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80), with respect to its just intonation used in Pythagorean tuning ; the result is . The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

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.

Anhemitonic scale

Musicology commonly classifies scales as either hemitonic or anhemitonic. Hemitonic scales contain one or more semitones, while anhemitonic scales do not contain semitones. For example, in traditional Japanese music, the anhemitonic yo scale is contrasted with the hemitonic in scale. The simplest and most commonly used scale in the world is the atritonic anhemitonic "major" pentatonic scale. The whole tone scale is also anhemitonic.

In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Some scales contain different pitches when ascending than when descending, for example, the melodic minor scale.

Chord (music) harmonic set of three or more notes

A chord, in music, is any harmonic set of pitches consisting of two or more notes that are heard as if sounding simultaneously.

Minor second

The melodic minor second is an integral part of most cadences of the Common practice period.   Play  
minor second
Inverse major seventh
Name
Other names semitone, half step
Abbreviation m2
Size
Semitones 1
Interval class 1
Cents
Equal temperament 100.0

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).

Major scale describes a type of music of acoustic tones

The major scale is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at double its frequency so that it is called a higher octave of the same note.

In western music theory, a diatonic scale is a heptatonic scale that includes five whole steps and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, depending on their position in the scale. This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other.

Major seventh musical interval

In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions, and the major seventh is one of two commonly occurring sevenths. It is qualified as major because it is the larger of the two. The major seventh spans eleven semitones, its smaller counterpart being the minor seventh, spanning ten semitones. For example, the interval from C to B is a major seventh, as the note B lies eleven semitones above C, and there are seven staff positions from C to B. Diminished and augmented sevenths span the same number of staff positions, but consist of a different number of semitones.

  Listen to a minor second in equal temperament  . 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.

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


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 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  

Augmented unison

Augmented unison on C.
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.   Play  

The augmented unison, the interval produced by the augmentation, or widening by one-half step, of the perfect unison, [8] does not occur between diatonic scale steps, but instead between a scale step and a chromatic alteration of the same step. It is also called a chromatic semitone. The augmented unison is abbreviated A1, or aug 1. Its inversion is the diminished octave (d8, or dim 8). The augmented unison is also the inversion of the augmented octave, because the interval of the diminished unison does not exist. [9] This is because a unison is always made larger when one note of the interval is changed with an accidental. [10] [11]

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 ).

Franz Liszt'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

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 [ 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.” [12]

A dramatic chromatic scale in the opening measures of Luca Marenzio's Solo e pensoso, ca. 1580. (   Play  

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. [12] These cadences would become a fundamental part of the musical language, even to the point where the usual accidental accompanying the minor second in a cadence was often omitted from the written score (a practice known as musica ficta). By the 16th century, the semitone had become a more versatile interval, sometimes even appearing as an augmented unison in very chromatic passages. Semantically, in the 16th century the repeated melodic semitone became associated with weeping, see: passus duriusculus, lament bass, and pianto.

By the Baroque era (1600 to 1750), the tonal harmonic framework was fully formed, and the various musical functions of the semitone were rigorously understood. Later in this period the adoption of well temperaments for instrumental tuning and the more frequent use of enharmonic equivalences increased the ease with which a semitone could be applied. Its function remained similar through the Classical period, and though it was used more frequently as the language of tonality became more chromatic in the Romantic period, the musical function of the semitone did not change.

In the 20th century, however, composers such as Arnold Schoenberg, Béla Bartók, and Igor Stravinsky sought alternatives or extensions of tonal harmony, and found other uses for the semitone. Often the semitone was exploited harmonically as a caustic dissonance, having no resolution. Some composers would even use large collections of harmonic semitones (tone clusters) as a source of cacophony in their music (e.g. the early piano works of Henry Cowell). By now, enharmonic equivalence was a commonplace property of equal temperament, and instrumental use of the semitone was not at all problematic for the performer. The composer was free to write semitones wherever he wished.

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 semitone76.076.076.076.076.0
PitchCCDEEFFGGABBC
Cents0.076.0193.2310.3386.3503.4579.5696.6772.6889.71006.81082.91200.0
Diatonic semitone117.1117.1117.1117.1117.1117.1117.1

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. 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 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 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 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 tuning

Pythagorean limma on C.   Play  .
Pythagorean apotome on C.   Play  .

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.

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 .)

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

16:15 diatonic semitone.
16:15 diatonic semitone   Play  .
'Larger' or major limma on C   Play  .

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 ().

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 an accidental 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:

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 (   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 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. [19]

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

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).

See also

Related Research Articles

Major second musical interval

In Western music theory, a major second is a second spanning two semitones. A second is a musical interval encompassing two adjacent staff positions. For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff positions. Diminished, minor and augmented seconds are notated on adjacent staff positions as well, but consist of a different number of semitones.

Minor third musical interval

In the music theory of Western culture, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions. The minor third is one of two commonly occurring thirds. It is called minor because it is the smaller of the two: the major third spans an additional semitone. For example, the interval from A to C is a minor third, as the note C lies three semitones above A, and (coincidentally) there are three staff positions from A to C. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones. The minor third is a skip melodically.

Major sixth musical interval

In music from Western culture, a sixth is a musical interval encompassing six note letter names or staff positions, and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the larger of the two. The major sixth spans nine semitones. Its smaller counterpart, the minor sixth, spans eight semitones. For example, the interval from C up to the nearest A is a major sixth. It is a sixth because it encompasses six note letter names and six staff positions. It is a major sixth, not a minor sixth, because the note A lies nine semitones above C. Diminished and augmented sixths span the same number of note letter names and staff positions, but consist of a different number of semitones.

Minor sixth musical interval

In classical music from Western culture, a sixth is a musical interval encompassing six staff positions, and the minor sixth is one of two commonly occurring sixths. It is qualified as minor because it is the smaller of the two: the minor sixth spans eight semitones, the major sixth nine. For example, the interval from A to F is a minor sixth, as the note F lies eight semitones above A, and there are six staff positions from A to F. Diminished and augmented sixths span the same number of staff positions, but consist of a different number of semitones.

Comma (music) small musical interval, the difference between two tunings of the same note

In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning system, and another F tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81/80 are considered the same note because the 12 note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B and A are both approximated by the same interval although they are a septimal kleisma apart.

Augmented second musical interval

In classical music from Western culture, an augmented second is an interval that, in equal temperament, is sonically equivalent to a minor third, spanning three semitones, and is created by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and the interval from C to D is an augmented second, spanning three semitones.

31 equal temperament musical tuning system with 31 pitches equally-spaced on a logarithmic scale

In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO, also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps. Play  Each step represents a frequency ratio of 312, or 38.71 cents.

Pythagorean interval

In musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 (equivalent to 31/21) and the perfect fourth with ratio 4/3 (equivalent to 22/31) are Pythagorean intervals.

Diminished second musical interval

In modern Western tonal music theory, a diminished second is the interval produced by narrowing a minor second by one chromatic semitone. It is enharmonically equivalent to a perfect unison. Thus, it is the interval between notes on two adjacent staff positions, or having adjacent note letters, altered in such a way that they have no pitch difference in twelve-tone equal temperament. An example is the interval from a B to the C immediately above; another is the interval from a B to the C immediately above.

Diminished third musical interval

In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A to C, and from A to C are diminished thirds, two semitones wide. Being diminished, it is considered a dissonant interval.

Music and mathematics

Music theory has no axiomatic foundation in modern mathematics, yet the basis of musical sound can be described mathematically and exhibits "a remarkable array of number properties". Elements of music such as its form, rhythm and metre, the pitches of its notes and the tempo of its pulse can be related to the measurement of time and frequency, offering ready analogies in geometry.

Regular diatonic tuning

A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.

Augmented seventh musical interval

In classical music from Western culture, an augmented seventh is an interval produced by widening a major seventh by a chromatic semitone. For instance, the interval from C to B is a major seventh, eleven semitones wide, and both the intervals from C to B, and from C to B are augmented sevenths, spanning twelve semitones. Being augmented, it is classified as a dissonant interval. However, it is enharmonically equivalent to the perfect octave.

Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.

References

  1. 1 2 3 4 5 6 7 Duffin, Ross W. (2008). How equal temperament ruined harmony : (and why you should care) (First published as a Norton paperback. ed.). New York: W. W. Norton. p. 163. ISBN   978-0-393-33420-3 . Retrieved 28 June 2017.
  2. Haluska, Jan (2003). The Mathematical Theory of Tone Systems, p.xxiv. ISBN   0-8247-4714-3. Overtone semitone.
  3. Semitone, half step, half tone, halftone, and half-tone are all variously used in sources.
    Aaron Copland, Leonard Bernstein, and others use "half tone".
    One source says that step is "chiefly US",
    and that half-tone is "chiefly N. Amer."
  4. Miller, Michael. The Complete Idiot's Guide to Music Theory, 2nd ed. [Indianapolis, IN]: Alpha, 2005. ISBN   1-59257-437-8. p. 19.
  5. Capstick, John Walton (1913). Sound: An Elementary Text-book for Schools and Colleges. Cambridge University Press.
  6. Wharram, Barbara (2010). Elementary Rudiments of Music (2nd ed.). Mississauga, ON: Frederick Harris Music. p. 17. ISBN   978-1-55440-283-0.
  7. 1 2 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.
  8. Benward & Saker (2003). Music: In Theory and Practice, Vol. I, p.54. ISBN   978-0-07-294262-0. Specific example of an A1 not given but general example of perfect intervals described.
  9. Kostka and Payne (2003). Tonal Harmony, p.21. ISBN   0-07-285260-7. "There is no such thing as a diminished unison."
  10. Day and Pilhofer (2007). Music Theory for Dummies, p.113. 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."
  11. Surmani, Andrew; Karen Farnum Surmani; Morton Manus (2009). Alfred's Essentials of Music Theory: A Complete Self-Study Course for All Musicians. p. 135: Alfred Music Publishing. p. 153. ISBN   0-7390-3635-1. "Since lowering either note of a perfect unison would actually increase its size, the perfect unison cannot be diminished, only augmented.".
  12. 1 2 Dahlhaus, Carl, trans. Gjerdingen, Robert O. Studies in the Origin of Harmonic Tonality. Princeton University Press: Princeton, 1990. ISBN   0-691-09135-8.
  13. Rashed, Roshdi (ed.) (1996). Encyclopedia of the History of Arabic Science, Volume 2, p.588 and 608. Routledge. ISBN   0-415-12411-5.
  14. Hermann von Helmholtz (1885). On the Sensations of Tone as a Physiological Basis for the Theory of Music, p.454.
  15. Benson, Dave (2006). Music: A Mathematical Offering, p.369. ISBN   0-521-85387-7.
  16. Royal Society (Great Britain) (1880, digitized Feb 26, 2008). Proceedings of the Royal Society of London, Volume 30, p.531. Harvard University.
  17. John Fonville. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p.109, Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991), pp. 106–137. "...the 25/24 ratio is the sharp (#) ratio...this raises a note approximately 70.6 cents."
  18. John Fonville. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p.109, Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991), pp. 106–137.
  19. Prout, Ebenezer (2004). Harmony, p.325. ISBN   1-4102-1920-8.

Further reading