Note | Understanding Western Music Theory

Note | Understanding Western Music Theory: This article is about the musical term. In music, the term note has two primary meanings:

1. A sign used in musical notation to represent the relative duration and pitch of a sound (♪, ♫);
2. A pitched sound itself.

Notes are the “atoms” of much Western music: discretizations of musical phenomena that facilitate performance, comprehension, and analysis.

The term note can be used in both generic and specific senses: one might say either “the piece ‘Happy Birthday to You’ begins with two notes having the same pitch,” or “the piece begins with two repetitions of the same note.” In the former case, one uses a note to refer to a specific musical event;

in the latter, one uses the term to refer to a class of events sharing the same pitch. (See also: Key signature names and translations).

Two notes with fundamental frequencies in a ratio equal to any power of two (e.g. half, twice, or four times) are perceived as very similar. Because of that, all notes with these kinds of relations can be grouped under the same pitch class.

In traditional music theory, most countries in the world use the naming convention Do-Re-Mi-Fa-Sol-La-Si, including for instance Italy, Spain, France, Romania, most Latin American countries, Greece, Bulgaria, Turkey,

Note | Understanding Western Music Theory

Names of some notes without accidentals

Russia, and all the Arabic-speaking or Persian-speaking countries. However, within the English-speaking and Dutch-speaking world, pitch classes are typically represented by the first seven letters of the Latin alphabet (A, B, C, D, E, F, and G). A few European countries, including Germany, adopt an almost identical notation, in which H is substituted for B (see below for details).

The eighth note, or octave, is given the same name as the first but has double its frequency. The name octave is also used to indicate the span between a note and another with double frequency. To differentiate two notes that have the same pitch class but fall into different octaves,

the system of scientific pitch notation combines a letter name with an Arabic numeral designating a specific octave. For example, the now-standard tuning pitch for most Western music, 440 Hz, is named a′ or A4.

There are two formal systems to define each note and octave, the Helmholtz pitch notation and the Scientific pitch notation.

Accidentals

Letter names are modified by the accidentals. A sharp ♯ raises a note by a semitone or half-step, and a flat ♭ lowers it by the same amount. In modern tuning, a half step has a frequency ratio of ¹²√2 , approximately 1.059. The accidentals are written after the note name: so, for example, F♯ represents F-sharp, and B♭ is B-flat.

Additional accidentals are the double-sharp, raising the frequency by two semitones, and double-flat, lowering it by that amount.

Frequency vs Position on Treble Clef. Each note shown has a

frequency of the previous note multiplied by ¹²√2

In musical notation, accidentals are placed before the note symbols. Systematic alterations to the seven lettered pitches in the scale can be indicated by placing the symbols in the key signature, which then apply implicitly to all occurrences of corresponding notes.

Explicitly noted accidentals can be used to override this effect for the remainder of a bar. A special accidental, the natural symbol ♮, is used to indicate an unmodified pitch.

Effects of key signature and local accidentals do not cumulate. If the key signature indicates G-sharp, a local flat before a G makes it G-flat (not G natural), though often this type of rare accidental is expressed as a natural, followed by a flat (♮♭) to make this clear. Likewise (and more commonly), a double sharp sign on a key signature with a single sharp ♯ indicates only a double sharp, not a triple sharp.

Assuming enharmonicity, many accidentals will create equivalences between pitches that are written differently. For instance, raising the note B to B♯ is equal to the note C. Assuming all such equivalences, the complete chromatic scale adds five additional pitch classes to the original seven lettered notes for a total of 12 (the 13th note completing the octave), each separated by a half step.

Notes that belong to the diatonic scale relevant in the context are sometimes called diatonic notes; notes that do not meet that criterion are then sometimes called chromatic notes.

Another style of notation, rarely used in English, uses the suffix “is” to indicate a sharp and “es” (only “s” after A and E) for a flat, e.g. Fis for F♯, Ges for G♭, Es for E♭. This system first arose in Germany and is used in almost all European countries whose main language is not English, Greek, or a Romance language.

In most countries using these suffixes, the letter H is used to represent what is B natural in English, the letter B is used instead of B♭, and Heses (i.e., H ) is used instead CHAPTER 6. NOTE

of B  (not Bes, which would also have fit into the system). Dutch speakers in Belgium and the Netherlands use the same suffixes, but applied throughout to the notes A to G, so that B, B♭ and B  have the same meaning as in English, although they are called B, Bes, and Beses instead of B, B flat and B double flat. Denmark also uses H, but uses Bes instead of Heses for B .

12-tone chromatic scale

The following chart lists the names used in different countries for the 12 notes of a chromatic scale built on C. The corresponding symbols are shown within parenthesis. Differences between German and English notation are highlighted in bold typeface. Although the English and Dutch names are different, the corresponding symbols are identical.

 Note designation in accordance with octave name

The table of each octave and the frequencies for every note of pitch class A is shown below. The traditional (Helmholtz) system centers on the great octave (with capital letters) and small octave (with lower case letters). Lower octaves are named “contra” (with primes before), higher ones “lined” (with primes after).

Another system (scientific) suffixes a number (starting with 0, or sometimes −1). In this system A4 is nowadays standardised to 440 Hz, lying in the octave containing notes from C4 (middle C) to B4. The lowest note on most pianos is A0, the highest C8. The MIDI system for electronic musical instruments and computers uses a straight count starting with note 0 for C-1 at 8.1758 Hz up to note 127 for G9 at 12,544 Hz.

Written notes

A written note can also have a note value, a code that determines the note’s relative duration. In order of halving duration, we have: double note (breve); whole note (semibreve); half note (minim); quarter note (crotchet); eighth note (quaver); sixteenth note (semiquaver). Smaller still are the thirty-second note (demisemiquaver), sixty-fourth note (hemidemisemiquaver), and hundred twenty-eighth note (semihemidemisemiquaver) or 1

2 note, 1

4 note, 1

8 note, 1

16 note, 1

32 note, 1

64 note, and 1 128 note.

6.6. HISTORY OF NOTE NAMES

When notes are written out in a score, each note is assigned a specific vertical position on a staff position (a line or a space) on the staff, as determined by the clef. Each line or space is assigned a note name. These names are memorized by musicians and allow them to know at a glance the proper pitch to play on their instruments for each note-head marked on the page.

The staff above shows the notes C, D, E, F, G, A, B, C listen and then in reverse order, with no key signature or accidentals.

Note frequency (hertz)

Main article: Mathematics of musical scales

In all technicality, music can be composed of notes at any arbitrary physical frequency. Since the physical causes of music are vibrations of mechanical systems, they are often measured in hertz (Hz), with 1 Hz = one vibration per second. For historical and other reasons, especially in Western music, only twelve notes of fixed frequencies are used.

These fixed frequencies are mathematically related to each other, and are defined around the central note, A4. The current “standard pitch” or modern “concert pitch” for this note is 440 Hz, although this varies in actual practice (see History of pitch standards).

The note-naming convention specifies a letter, any accidentals, and an octave number. Any note is an integer of half-steps away from middle A (A4). Let this distance be denoted n. If the note is above A4, then n is positive; if it is below A4, then n is negative. The frequency of the note (f) (assuming equal temperament) is then:

f = 2^n/12 × 440 Hz

For example, one can find the frequency of C5, the first C above A4. There are 3 half-steps between A4 and C5 (A4 → A♯4 → B4 → C5), and the note is above A4, so n = +3. The note’s frequency is:

f = 2^3/12 × 440 Hz ≈ 523.2 Hz

To find the frequency of a note below A4, the value of n is negative. For example, the F below A4 is F4. There are 4 half-steps (A4 → A♭4 → G4 → G♭4 → F4), and the note is below A4, so n = −4. The note’s frequency is:

f = 2^−4/12 × 440 Hz ≈ 349.2 Hz

Finally, it can be seen from this formula that octaves automatically yield powers of two times the original frequency, since n is therefore a multiple of 12 (12k, where k is the number of octaves up or down), and so the formula reduces to:

f = 2^12k/12 × 440 Hz = 2^k × 440 Hz

yielding a factor of 2. In fact, this is the means by which this formula is derived, combined with the notion of equally-spaced intervals.

The distance of an equally tempered semitone is divided into 100 cents. So 1200 cents are equal to one octave — a frequency ratio of 2:1. This means that a cent is precisely equal to the 1200th root of 2, which is approximately 1.000578.

For use with the MIDI (Musical Instrument Digital Interface) standard, a frequency mapping is defined by:

Hz

Where p is the MIDI note number. And in the opposite direction, to obtain the frequency from a MIDI note p, the formula is defined as:

f = 2(p−69)/12 × 440 Hz

For notes in an A440 equal temperament, this formula delivers the standard MIDI note number (p). Any other frequencies fill the space between the whole numbers evenly. This allows MIDI instruments to be tuned very accurately in any microtuning scale, including non-western traditional tunings.

History of note names

Music notation systems have used letters of the alphabet for centuries. The 6th-century philosopher Boethius is known to have used the first fourteen letters of the classical Latin alphabet,

A-B-C-D-E-F-G-H-I-K-L-M-N-O (the letter J didn’t exist until the 16th century)

to signify the notes of the two-octave range that was in use at the time, and which in modern scientific pitch notation is represented as

A₂-B₂-C₃-D₃-E₃-F₃-G₃-A₃-B₃-C₄-D₄-E₄F₄-G₄.

Though it is not known whether this was his devising or common usage at the time, this is nonetheless called Boethian notation. Although Boethius is the first author which is known to have used this nomenclature in the literature, the above-mentioned two-octave range was already known five centuries before by Ptolemy, who called it the “perfect system” or “complete system”,

as opposed to other systems of notes of smaller range, which did not contain all the possible species of an octave (i.e., the seven octaves starting from A, B, C, D, E, F, and G).

Following this, the range (or compass) of used notes was extended to three octaves, and the system of repeating letters A-G in each octave was introduced, these being written as a lower case for the second octave (a-g) and double lowercase letters for the third (aa-gg).

When the range was extended down by one note, to a G, that note was denoted using the Greek G (Γ), gamma. (It is from this that the French word for scale, gamme is derived, and the English word gamut, from “Gamma-Ut”, the lowest note in Medieval music notation.)

The remaining five notes of the chromatic scale (the black keys on a piano keyboard) were added gradually; the first being B♭, since B was flattened in certain modes to avoid the dissonant tritone interval. This change was not always shown in notation,

but when written, B♭ (B-flat) was written as a Latin, round “b”, and B♮ (B-natural) a Gothic or “hard-edged” b. These evolved into the modern flat (♭) and natural (♮) symbols respectively. The sharp symbol arose from a barred b, called the “canceled b”.

In parts of Europe, including Germany, the Czech Republic, Slovakia, Poland, Hungary, Norway, Denmark, Serbia, Croatia, Finland, Iceland and Sweden, the Gothic b transformed into the letter H (possibly for hart, German for hard, or just because the Gothic b resembled an H). Therefore, in German music notation, H is used in lieu of B♮ (B-natural), and B in lieu of B♭ (B-flat). Occasionally, music written in German for international use will use H for B-natural and B^b for B-flat (with a modern script lowercase b instead of a flat sign). Since a Bes or B♭ in Northern Europe (i.e. a B  elsewhere) is both rare and unorthodox (more likely to be expressed as Heses), it is generally clear what this notation means.

In  Italian,  Portuguese,   Spanish,   French,  Romanian,

Greek, Russian, Mongolian, Flemish, Persian, Arabic, Hebrew, Bulgarian and Turkish notation the notes of scales are given in terms of Do-Re-Mi-Fa-Sol-La-Si rather than C-D-E-F-G-A-B. These names follow the original names reputedly given by Guido d’Arezzo, who had taken them from the first syllables of the first six musical phrases of a Gregorian Chant melody Ut queant laxis, which began on the appropriate scale degrees. These became the basis of the solfege system.

“Do” later replaced the original “Ut” for ease of singing (most likely from the beginning of Dominus, Lord), though “Ut” is still used in some places. “Si” or “Ti” was added as the seventh degree (from Sancte Johannes, St. John, to whom the hymn is dedicated). The use of ‘Si’ versus ‘Ti’ varies regionally.

NOTE

The two notation systems most commonly used nowadays are the Helmholtz pitch notation system and the Scientific pitch notation system. As shown in the table above, they both include several octaves, each starting from C rather than A. The reason is that the most commonly used scale in Western music is the major scale,

and the sequence C-D-E-F-G-A-B (the C-major scale) is the simplest example of a major scale. Indeed, it is the only major scale that can be obtained using natural notes (the white keys on the piano keyboard), and is typically the first musical scale taught in music schools.

In a newly developed system, primarily in use in the United States, notes of scales become independent of the music notation. In this system the natural symbols C-DE-F-G-A-B refer to the absolute notes, while the names Do-Re-Mi-Fa-So-La-Ti are relativized and show only the relationship between pitches,

where Do is the name of the base pitch of the scale, Re is the name of the second pitch, etc. The idea of so-called movable-do, originally suggested by John Curwen in the 19th century, was fully developed and involved into a whole educational system by Zoltán Kodály in the middle of the 20th century, which system is known as the Kodály Method or Kodály Concept.

Semitone | Note | Understanding Western Music Theory

This article is about musical intervals. For the printing method, see halftone.

A semitone also called a half step or a half tone, is

Minor second Play.

the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonic. 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 a ratio of 256:243 or 90.2 cents (Pythagorean limma), and the other five are chromatic, with a ratio of 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 of 25:24 (70.7 cents) and 135:128 (92.2 cents),

and diatonic semitones with ratios of 16:15 (111.7 cents) and 27:25 (133.2 cents). For further details, see below. Main article: Anhemitonic scale

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

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).

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.

AUGMENTED UNISON

The melodic minor second is an integral part of most cadences of the Common practice period.

A harmonic minor second in J.S. Bach’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.

The opening measures of Frédéric Chopin’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

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.

The augmented unison, the interval produced by the augmentation, or widening by one-half step, of the perfect unison, 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. This is because a unison is always made larger when one note of the interval is changed with an

accidental.

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      ).

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.

Franz Liszt’s second Transcendental Etude, measure 63.

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.”

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. 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.

Extended meantone temperaments with more than 12

SEMITONES IN DIFFERENT TUNINGS

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 a 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 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:

• 18/17 ≈ 99.0cents, suggested by Vincenzo Galilei and used by luthiers of the Renaissance,

cents,

suggested by Marin Mersenne as a constructible and more accurate alternative,

• (139/138)⁸ ≈ 99.9995cents, 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 uneven size. Every semitone in a good 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 affect 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 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.

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 or the Pythagorean major semitone. (See Pythagorean interval.)

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 5limit 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. 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.”

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 and 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-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:

cents

• Larger, or major, chromatic semitone, or larger limma, or major chroma, g. between D♭ and D:

cents

• Just, or smaller, or minor, diatonic semitone,

e.g. between C and D♭:

cents

• Larger, or major, diatonic semitone, e.g. between A and B♭:

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 the 7-limit there is the septimal diatonic semitone of 15:14 (  play

REFERENCES

) 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.

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 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.