music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first is the "Elements of music, ...

, the circle of fifths (sometimes also cycle of fifths) is a way of organizing pitches as a sequence of

perfect fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval f ...

s. Starting on a C, and using the standard system of tuning for Western music (

12-tone equal temperament 12 equal temperament (12-ET) is the musical system that divides the octave into 12 parts, all of which are Equal temperament, equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the Twelfth root of two, 12th root of 2 ...

), the sequence is: C, G, D, A, E, B, F/G, C/D, G/A, D/E, A/B, F, and C. This order places the most closely related

key signature In Western musical notation, a key signature is a set of sharp (), flat (), or rarely, natural () symbols placed on the staff at the beginning of a section of music. The initial key signature in a piece is placed immediately after the cl ...

s adjacent to one another. Twelve-tone equal temperament tuning divides each octave into twelve equivalent semitones, and the circle of fifths leads to a C seven octaves above the starting point. If the fifths are tuned with an exact frequency ratio of 3:2 (the system of tuning known as

just intonation In music, just intonation or pure intonation is a musical tuning, tuning system in which the space between notes' frequency, frequencies (called interval (music), intervals) is a natural number, whole number ratio, ratio. Intervals spaced in thi ...

), this is not the case (the circle does not "close").

Definition

The circle of fifths organizes pitches in a sequence of

s, generally shown as a circle with the pitches (and their corresponding keys) in clockwise order. It can be viewed in a counterclockwise direction as a circle of fourths. Harmonic progressions in Western music commonly use adjacent keys in this system, making it a useful reference for musical composition and harmony. The top of the circle shows the key of C Major, with no sharps or flats. Proceeding clockwise, the pitches ascend by fifths. The key signatures associated with those pitches change accordingly: the key of G has one sharp, the key of D has 2 sharps, and so on. Proceeding counterclockwise from the top of the circle, the notes change by descending fifths and the key signatures change accordingly: the key of F has one flat, the key of B has 2 flats, and so on. Some keys (at the bottom of the circle) can be notated either in sharps or in flats. Starting at any pitch and ascending by a fifth generates all tones before returning to the beginning pitch class (a pitch class consists of all of the notes indicated by a given letter regardless of octave—all "C"s, for example, belong to the same pitch class). Moving counterclockwise, the pitches descend by a fifth, but ascending by a

perfect fourth A fourth is a interval (music), musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending int ...

will lead to the same note an octave higher (therefore in the same pitch class). Moving counter-clockwise from C could be thought of as descending by a fifth to F, or ascending by a fourth to F.

Structure and use

Diatonic key signatures

Each pitch can serve as the tonic of a major or minor key, and each of these keys will have a

diatonic scale In music theory a diatonic scale is a heptatonic scale, heptatonic (seven-note) scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by eith ...

associated with it. The circle diagram shows the number of sharps or flats in each

, with the major key indicated by a capital letter and the minor key indicated by a lower-case letter. Major and minor keys that have the same key signature are referred to as ''relative major'' and ''relative minor'' of one another.

Modulation and chord progression

Tonal music often modulates to a new tonal center whose key signature differs from the original by only one flat or sharp. These closely-related keys are a fifth apart from each other and are therefore adjacent in the circle of fifths.

Chord progression In a musical composition, a chord progression or harmonic progression (informally chord changes, used as a plural, or simply changes) is a succession of chords. Chord progressions are the foundation of harmony in Western musical tradition from ...

s also often move between chords whose roots are related by perfect fifth, making the circle of fifths useful in illustrating the "harmonic distance" between chords. The circle of fifths on umbilic torus surface

The circle of fifths on umbilic torus surface

The circle of fifths is used to organize and describe the harmonic or tonal function of chords. Chords can progress in a pattern of ascending perfect fourths (alternately viewed as descending perfect fifths) in "functional succession". This can be shown "...by the circle of fifths (in which, therefore,

scale degree In music theory, the scale degree is the position of a particular note on a scale relative to the tonic—the first and main note of the scale from which each octave is assumed to begin. Degrees are useful for indicating the size of intervals ...

II is closer to the dominant than scale degree IV)". In this view the tonic or tonal center is considered the end point of a

chord progression In a musical composition, a chord progression or harmonic progression (informally chord changes, used as a plural, or simply changes) is a succession of chords. Chord progressions are the foundation of harmony in Western musical tradition from ...

derived from the circle of fifths. Ii-V-I turnaround in C

According to Richard Franko Goldman's ''Harmony in Western Music'', "the IV chord is, in the simplest mechanisms of diatonic relationships, at the greatest distance from I. In terms of the escendingcircle of fifths, it leads away from I, rather than toward it." He states that the progression I–ii–V–I (an

authentic cadence In Western musical theory, a cadence () is the end of a phrase in which the melody or harmony creates a sense of full or partial resolution, especially in music of the 16th century onwards.Don Michael Randel (1999). ''The Harvard Concise Dict ...

) would feel more final or resolved than I–IV–I (a plagal cadence). Goldman concurs with Nattiez, who argues that "the chord on the fourth degree appears long before the chord on II, and the subsequent final I, in the progression I–IV–vii^o–iii–vi–ii–V–I", and is farther from the tonic there as well. (In this and related articles, upper-case Roman numerals indicate major triads while lower-case Roman numerals indicate minor triads.)

Circle closure in non-equal tuning systems

Using the exact 3:2 ratio of frequencies to define a perfect fifth (

) does not quite result in a return to the pitch class of the starting note after going around the circle of fifths. Twelve-tone equal temperament tuning produces fifths that return to a tone exactly seven

octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...

s above the initial tone and makes the frequency ratio of the chromatic semitone the same as that of the diatonic semitone. The standard tempered fifth has a frequency ratio of 2^7/12:1 (or about 1.498307077:1), approximately two cents narrower than a justly tuned fifth. Ascending by twelve justly tuned fifths fails to close the circle by an excess of approximately 23.46 cents, roughly a quarter of a

semitone A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between ...

, an interval known as the Pythagorean comma. If limited to twelve pitches per octave, Pythagorean tuning markedly shortens the

width Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Intern ...

of one of the twelve fifths, which makes it severely dissonant. This anomalous fifth is called the wolf fifth – a humorous reference to a wolf howling an off-pitch note. Non-extended quarter-comma meantone uses eleven fifths slightly narrower than the equally tempered fifth, and requires a much wider and even more dissonant wolf fifth to close the circle. More complex tuning systems based on just intonation, such as 5-limit tuning, use at most eight justly tuned fifths and at least three non-just fifths (some slightly narrower, and some slightly wider than the just fifth) to close the circle.

Equal-tempered tunings with more than twelve notes

Nowadays, with the advent of electronic isomorphic keyboards, equal temperament tunings with more than twelve notes per octave can be used to close the circle of fifths for other tunings. For example, 31-tone equal temperament closely approximates quarter-comma meantone, and 53-tone equal temperament closely approximates Pythagorean tuning.

History

The circle of fifths developed in the late 1600s and early 1700s to theorize the modulation of the Baroque era (see ). The first circle of fifths diagram appears in the ''Grammatika'' (1677) of the composer and theorist Nikolay Diletsky, who intended to present

as a tool for composition. It was "the first of its kind, aimed at teaching a Russian audience how to write Western-style polyphonic compositions." A circle of fifths diagram was independently created by German composer and theorist Johann David Heinichen in his ''Neu erfundene und gründliche Anweisung'' (1711), which he called the "Musical Circle" (German: ''Musicalischer Circul''). This was also published in his ''Der General-Bass in der Composition'' (1728). Heinichen placed the relative minor key next to the major key, which did not reflect the actual proximity of keys.

Johann Mattheson Johann Mattheson (28 September 1681 – 17 April 1764) was a German composer, critic, lexicographer and music theorist. His writings on the late Baroque and early Classical period were highly influential, specifically, "his biographical and the ...

(1735) and others attempted to improve this— David Kellner (1737) proposed having the major keys on one circle, and the relative minor keys on a second, inner circle. This was later developed into chordal space, incorporating the parallel minor as well. Some sources imply that the circle of fifths was known in antiquity, by

Pythagoras Pythagoras of Samos (; BC) was an ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of P ...

. This is a misunderstanding and an anachronism. Tuning by fifths (so-called Pythagorean tuning) dates to Ancient Mesopotamia; see , though they did not extend this to a twelve-note scale, stopping at seven. The Pythagorean comma was calculated by

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

and by Chinese mathematicians (in the ''

Huainanzi The ''Huainanzi'' is an ancient Chinese text made up of essays from scholarly debates held at the court of Liu An, Prince of Huainan, before 139 BCE. Compiled as a handbook for an enlightened sovereign and his court, the work attempts to defi ...

''); see . Thus, it was known in antiquity that a cycle of twelve fifths was almost exactly seven octaves (more practically, alternating ascending fifths and descending fourths was almost exactly an octave). However, this was theoretical knowledge, and was not used to construct a repeating twelve-tone scale, nor to modulate. This was done later in

meantone temperament Meantone temperaments are musical temperaments; that is, a variety of Musical tuning#Tuning systems, tuning systems constructed, similarly to Pythagorean tuning, as a sequence of equal fifths, both rising and descending, scaled to remain within th ...

and twelve-tone equal temperament, which allowed modulation while still being in tune, but did not develop in Europe until about 1500. Although popularized as the circle of fifths, its Anglo-Saxon etymological origins trace back to the name "wheel of fifths."

Use

In musical pieces from the

Baroque music Baroque music ( or ) refers to the period or dominant style of Classical music, Western classical music composed from about 1600 to 1750. The Baroque style followed the Renaissance music, Renaissance period, and was followed in turn by the Class ...

era and the Classical era of music and in Western

popular music Popular music is music with wide appeal that is typically distributed to large audiences through the music industry. These forms and styles can be enjoyed and performed by people with little or no musical training.Popular Music. (2015). ''Fun ...

traditional music Folk music is a music genre that includes traditional folk music and the contemporary genre that evolved from the former during the 20th-century folk revival. Some types of folk music may be called world music. Traditional folk music has b ...

and

folk music Folk music is a music genre that includes #Traditional folk music, traditional folk music and the Contemporary folk music, contemporary genre that evolved from the former during the 20th-century folk revival. Some types of folk music may be ca ...

, when pieces or songs modulate to a new key, these modulations are often associated with the circle of fifths. In practice, compositions rarely make use of the entire circle of fifths. More commonly, composers make use of "the compositional idea of the 'cycle' of 5ths, when music moves consistently through a smaller or larger segment of the tonal structural resources which the circle abstractly represents." The usual practice is to derive the circle of fifths progression from the seven tones of the diatonic scale, rather than from the full range of twelve tones present in the chromatic scale. In this diatonic version of the circle, one of the fifths is not a true fifth: it is a tritone (or a diminished fifth), e.g. between F and B in the "natural" diatonic scale (i.e. without sharps or flats). Here is how the circle of fifths derives, through

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

from the diatonic major scale: And from the (natural) minor scale: The following is the basic sequence of chords that can be built over the major bass-line: And over the minor: Adding sevenths to the chords creates a greater sense of forward momentum to the harmony:

Baroque era

According to Richard Taruskin,

Arcangelo Corelli Arcangelo Corelli (, also , ; ; 17 February 1653 – 8 January 1713) was an List of Italian composers, Italian composer and violinist of the middle Baroque music, Baroque era. His music was key in the development of the modern genres of Sonata a ...

was the most influential composer to establish the pattern as a standard harmonic "trope": "It was precisely in Corelli's time, the late seventeenth century, that the circle of fifths was being 'theorized' as the main propellor of harmonic motion, and it was Corelli more than any one composer who put that new idea into telling practice." The circle of fifths progression occurs frequently in the music of J. S. Bach. In the following, from ''Jauchzet Gott in allen Landen'', BWV 51, even when the solo bass line implies rather than states the chords involved: Handel uses a circle of fifths progression as the basis for the Passacaglia movement from his Harpsichord suite No. 6 in G minor.

Baroque The Baroque ( , , ) is a Western Style (visual arts), style of Baroque architecture, architecture, Baroque music, music, Baroque dance, dance, Baroque painting, painting, Baroque sculpture, sculpture, poetry, and other arts that flourished from ...

composers learnt to enhance the "propulsive force" of the harmony engendered by the circle of fifths "by adding sevenths to most of the constituent chords." "These sevenths, being dissonances, create the need for resolution, thus turning each progression of the circle into a simultaneous reliever and re-stimulator of harmonic tension... Hence harnessed for expressive purposes." Striking passages that illustrate the use of sevenths occur in the aria "Pena tiranna" in Handel's 1715 opera '' Amadigi di Gaula'': – and in Bach's keyboard arrangement of Alessandro Marcello's Concerto for Oboe and Strings.

Nineteenth century

Franz Schubert Franz Peter Schubert (; ; 31 January 179719 November 1828) was an Austrian composer of the late Classical period (music), Classical and early Romantic music, Romantic eras. Despite his short life, Schubert left behind a List of compositions ...

's Impromptu in E-flat major, D 899, contains harmonies that move in a modified circle of fifths: The Intermezzo movement from

Mendelssohn Jakob Ludwig Felix Mendelssohn Bartholdy (3 February 18094 November 1847), widely known as Felix Mendelssohn, was a German composer, pianist, organist and conductor of the early Romantic period. Mendelssohn's compositions include symphonie ...

's String Quartet No.2 has a short segment with circle-of-fifths motion (the ii° is substituted by iv): Robert Schumann's "Child falling asleep" from his '' Kinderszenen'' uses the progression, changing it at the end—the piece ends on an A minor chord, instead of the expected tonic E minor. In Wagner's opera, ''

Götterdämmerung ' (; ''Twilight of the Gods''), Wagner-Werk-Verzeichnis, WWV 86D, is the last of the four epic poetry, epic music dramas that constitute Richard Wagner's Literary cycle, cycle ''Der Ring des Nibelungen'' (English: ''The Ring of the Nibelung''). I ...

'', a cycle of fifths progression occurs in the music which transitions from the end of the prologue into the first scene of Act 1, set in the imposing hall of the wealthy Gibichungs. "Status and reputation are written all over the motifs assigned to Gunther", chief of the Gibichung clan:

Jazz and popular music

The enduring popularity of the circle of fifths as both a form-building device and as an expressive musical trope is evident in the number of " standard" popular songs composed during the twentieth century. It is also favored as a vehicle for improvisation by jazz musicians, as the circle of fifths helps songwriters understand intervals, chord-relationships and progressions. * Bart Howard, " Fly Me to the Moon" *

Jerome Kern Jerome David Kern (January 27, 1885 – November 11, 1945) was an American composer of musical theatre and popular music. One of the most important American theatre composers of the early 20th century, he wrote more than 700 songs, used in over ...

, "

All the Things You Are "All the Things You Are" is a song composed by Jerome Kern with lyrics written by Oscar Hammerstein II. The song was written for the musical '' Very Warm for May'' (1939)Ray Noble, "

Cherokee The Cherokee (; , or ) people are one of the Indigenous peoples of the Southeastern Woodlands of the United States. Prior to the 18th century, they were concentrated in their homelands, in towns along river valleys of what is now southwestern ...

." Many jazz musicians have found this particularly challenging as the middle eight progresses so rapidly through the circle, "creating a series of II–V–I progressions that temporarily pass through several tonalities." * Kosma, Prévert and Mercer, " Autumn Leaves" *

The Beatles The Beatles were an English Rock music, rock band formed in Liverpool in 1960. The core lineup of the band comprised John Lennon, Paul McCartney, George Harrison and Ringo Starr. They are widely regarded as the Cultural impact of the Beatle ...

, " You Never Give Me Your Money" *

Mike Oldfield Michael Gordon Oldfield (born 15 May 1953) is an English retired musician, songwriter and producer best known for his debut studio album ''Tubular Bells'' (1973), which became an unexpected critical and commercial success. Though primarily a gu ...

, " Incantations" *

Carlos Santana Carlos Humberto Santana Barragán (; born July 20, 1947) is an American guitarist, best known as a founding member of the Rock music, rock band Santana (band), Santana. Born and raised in Mexico where he developed his musical background, he r ...

, " Europa (Earth's Cry Heaven's Smile)" *

Gloria Gaynor Gloria Fowles (born September 7, 1943), known professionally as Gloria Gaynor, is an American singer, best known for the disco era hits "I Will Survive" (1978), "I Have a Right, Let Me Know (I Have a Right)" (1979), "I Am What I Am (Broadway mus ...

, "

I Will Survive "I Will Survive" is a song recorded by American singer Gloria Gaynor, released in October 1978 by Polydor Records as the second single from her sixth album, ''Love Tracks (Gloria Gaynor album), Love Tracks'' (1978). It was written by Freddie Pe ...

" *

Pet Shop Boys Pet Shop Boys are an English synth-pop duo formed in London in 1981. Consisting of vocalist Neil Tennant and keyboardist Chris Lowe, they have sold more than 100 million records worldwide and were listed as the most successful duo in UK music h ...

, " It's a Sin" *

Donna Summer Donna Adrian Gaines (December 31, 1948May 17, 2012), known professionally as Donna Summer, was an American singer and songwriter. She gained prominence during the disco era of the 1970s and became known as the "Queen of Disco", while her music ...

, " Love to Love you, Baby"

Related concepts

Diatonic circle of fifths

The diatonic circle of fifths is the circle of fifths encompassing only members of the diatonic scale. Therefore, it contains a diminished fifth, in C major between B and F. See structure implies multiplicity. The

circle progression A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting ...

is commonly a circle of fifths through the diatonic chords, including one diminished chord. A circle progression in C major with chords I–IV–vii^o–iii–vi–ii–V–I is shown below. :

Chromatic circle

The circle of fifths is closely related to the chromatic circle, which also arranges the equal-tempered pitch classes of a particular tuning in a circular ordering. A key difference between the two circles is that the chromatic circle can be understood as a continuous space where every point on the circle corresponds to a conceivable pitch class, and every conceivable pitch class corresponds to a point on the circle. By contrast, the circle of fifths is fundamentally a ''discrete'' structure arranged through distinct intervals, and there is no obvious way to assign pitch classes to each of its points. In this sense, the two circles are mathematically quite different. However, for any positive integer ''N'', the pitch classes in ''N''-tone equal temperament can be represented by the

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

of order ''N'', or equivalently, the residue classes modulo equal to ''N'',

\mathbb/N\mathbb

. In twelve-tone equal temperament, the group

\mathbb_

has four generators, which can be identified with the ascending and descending semitones and the ascending and descending perfect fifths. The semitonal generator gives rise to the chromatic circle while the perfect fourth and perfect fifth give rise to the circle of fifths. In most other tunings, such as in 31 equal temperament, many more intervals can be used as the generator, and many more circles are possible as a result.

Relation with chromatic scale

The circle of fifths, or fourths, may be mapped from the chromatic scale by

multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...

, and vice versa. To map between the circle of fifths and the chromatic scale (in integer notation) multiply by 7 ( M7), and for the circle of fourths multiply by 5 (P5). In twelve-tone equal temperament, one can start off with an ordered 12-tuple (

tone row In music, a tone row or note row ( or '), also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometime ...

) of integers: : (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) representing the notes of the chromatic scale: 0 = C, 2 = D, 4 = E, 5 = F, 7 = G, 9 = A, 11 = B, 1 = C, 3 = D, 6 = F, 8 = G, 10 = A. Now multiply the entire 12-tuple by 7: : (0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77) and then apply a

modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...

12 reduction to each of the numbers (subtract 12 from each number as many times as necessary until the number becomes smaller than 12): : (0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5) which is equivalent to : (C, G, D, A, E, B, F, C, G, D, A, F) which is the circle of fifths. This is

enharmonic In music, two written notes have enharmonic equivalence if they produce the same pitch but are notated differently. Similarly, written intervals, chords, or key signatures are considered enharmonic if they represent identical pitches that ar ...

ally equivalent to: : (C, G, D, A, E, B, G, D, A, E, B, F).

Enharmonic equivalents, theoretical keys, and the spiral of fifths

Equal temperament An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system that approximates Just intonation, just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequency, frequencie ...

tunings do not use the exact 3:2 ratio of frequencies that defines a perfect fifth, whereas

uses this exact ratio. Ascending by fifths in equal temperament leads to a return to the starting pitch class—starting with a C and ascending by fifths leads to another C after a certain number of iterations. This does not occur if an exact 3:2 ratio is used (just intonation). The adjustment made in equal temperament tuning is called the Pythagorean comma. Because of this difference, pitches that are enharmonically equivalent in equal temperaments (such as C and D in 12-tone equal temperament, or C and D in

19 equal temperament In music, 19 equal temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), 19-ED2 ("Equal Division of 2:1) or 19 Equal temperament, ET, is the musical temperament, tempered scale derived by dividing the octave into 19 equal steps ...

) are not equivalent when using just intonation. In just intonation the sequence of fifths can therefore be visualized as a spiral, not a circle—a sequence of twelve fifths results in a " comma pump" by the Pythagorean comma, visualized as going up a level in the spiral. See also . Without enharmonic equivalences, continuing a sequence of fifths results in notes with double accidentals (double sharps or double flats), or even triple or quadruple accidentals. In most equal temperament tunings, these can be replaced by enharmonically equivalent notes. Keys with double or triple sharps and flats in key signatures are called theoretical keys; they are redundant in 12-tone equal temperament, and so their use is extremely rare, but if the number of notes per octave is not a multiple of 12, they are distinguished. Notation in these cases is not standardized. \relative c' The default behaviour of

LilyPond LilyPond is a computer program and file format for music engraving. One of LilyPond's major goals is to produce scores that are engraved with traditional layout rules, reflecting the era when scores were engraved by hand. LilyPond is cross-pla ...

(pictured above) writes single sharps or flats in the circle-of-fifths order, before proceeding to double sharps or flats. This is the format used in John Foulds' ''A World Requiem'', Op. 60, which ends with the key signature of G major, as displayed above. The sharps in the key signature of G major here proceed C, G, D, A, E, B, F. Single sharps or flats in the key signature are sometimes repeated as a courtesy, e.g. Max Reger's ''Supplement to the Theory of Modulation'', which contains D minor key signatures o
pp. 42–45
These have a B at the start and also a B at the end (with a double-flat symbol), going B, E, A, D, G, C, F, B. The convention of LilyPond and Foulds would suppress the initial B. Sometimes the double signs are written at the beginning of the key signature, followed by the single signs. For example, the F key signature is notated as B, E, A, D, G, C, F. This convention is used by Victor Ewald, by the program Finale, and by some theoretical works.

Notes

References

Sources

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