fifth

In music theory, the circle of fifths (or cycle of fifths) is an imaginary geometrical space that depicts relationships among the 12 equal-tempered pitch classes comprising the familiar chromatic scale. The circle of fifths was first described by Johann David Heinichen, in his 1728 treatise Der Generalbass in der Composition.

Structure and Use

If one starts on any equal-tempered pitch and repeatedly ascends by the musical interval of a perfect fifth, one will eventually land on a pitch with the same pitch class as the initial one, passing through all the other equal-tempered chromatic pitch classes in between.

Since the space is circular, it is also possible to descend by fifths. In pitch class space, motion in one direction by a fourth is equivalent to motion in the opposite direction by a fifth. For this reason the circle of fifths is also known as the circle of fourths.

The circle is commonly used to represent the relations between diatonic scales. Here, the letters on the circle are taken to represent the major scale with that note as tonic. The numbers on the inside of the circle show how many sharps or flats the key signature for this scale would have. Thus a major scale built on A will have three sharps in its key signature. The major scale built on F would have one flat. For minor scales, rotate the letters counter-clockwise by 3, so that e.g. A minor has 0 accidentals and E minor has 1 sharp. (See relative minor/major for details.)

Tonal music often modulates by moving between adjacent scales on the circle of fifths. This is because diatonic scales contain seven pitch classes that are contiguous on the circle of fifths. It follows that diatonic scales a perfect fifth apart share six of their seven notes. Furthermore, the notes not held in common differ by only a semitone. Thus modulation by perfect fifth can be accomplished in an exceptionally smooth fashion. For example, to move from the C major scale F - C - G - D - A - E - B to the G major scale C - G - D - A - E - B - F♯, one need only move the C major scale's "F" to "F♯."

In Western tonal music, one also finds chord progressions between chords whose roots are related by perfect fifth. For instance, root progressions such as D-G-C are common. For this reason, the circle of fifths can often be used to represent "harmonic distance" between chords.

The circle of fifths is closely related to the chromatic circle, which also arranges the twelve equal-tempered pitch classes 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, 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, the twelve equal-tempered pitch classes can be represented by the cyclic group of order twelve, or equivalently, the residue classes modulo twelve, . The group 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 fifth gives rise to the circle of fifths shown here.

In layman's terms

A simple way to see the relationship between these notes is by looking at a piano keyboard, and starting at any key and counting 7 keys to the right (both black and white) to get to the next note on the circle above - which is a perfect fifth. 7 half steps or the distance from the 1st to the 8th key on a piano is a perfect fifth.

A simple way to hear the relationship between these notes is by playing them on a piano keyboard. If you traverse the circle of fifths backwards, the notes will feel as though they fall into each other. This aural relationship is what the math describes.

The frequencies of two notes that are a perfect fifth apart differ by a ratio of approximately 3:2 = 1.5. A ratio of exactly 1.5 sounds best, and this explains why a perfect fifth sounds consonant, though for mathematical reasons it is not possible to get the circle of fifths to 'join up' (i.e. return to the original pitch after going round the circle) unless a close approximation to a perfect fifth is used, namely 2 to the power 7/12 = 1.498. This is the basis of the tuning nowadays used for Western instruments, called Equal temperament, which enables music to be played in any key by having all intervals tuned the same way (i.e. using the same ratios) in all keys.

Related concepts

Diatonic circle of fifths

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

Relation with chromatic scale

The circle of fifths, or fourths, may be mapped from the chromatic scale by multiplication, 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 (M5).

Here is a demonstration of this procedure. Start off with an ordered 12-tuple (tone row) 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 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. Note that this is enharmonically identical to:

(C, G, D, A, E, B, G♭, D♭, A♭, E♭, B♭, F)

Infinite Series

The “bottom keys” of the circle of fifths are often written in flats and sharps, as they are easily interchanged using enharmonics. For example, the key of B, with five sharps, is enharmonically equivalent to the key of C♭, with 7 flats. But the circle of sharps doesn’t stop at 7 sharps (C♯) nor 7 flats (C♭). Following the same pattern, one can construct a circle of fifths with all sharp keys, or all flat keys.

After C♯ comes the key of G♯ (following the pattern of being a fifth higher, and, coincidently, enharmonically equivalent to the key of A♭). The “8th sharp” is placed on the F♯, to make it F. The key of D♯, with 9 sharps, has another sharp placed on the C♯, making it C. The same for key signatures with flats is true; The key of E (four sharps) is equivalent to the key of F♭ (again, one fifth below the key of C♭, following the pattern of flat key signatures. The double-flat is placed on the B♭, making it B.)

The circle used in instrument building

Piano keys translated to the tuning of Landmans Moodswinger

Experimental luthier Yuri Landman created a twelve string overtone zither and tuned this in a circle of fourths:

E-A-D-G-C-F-A♯-D♯-G♯-C♯-F♯-B, arranged in 3 clusters of 4 strings to make the field of strings more readable.

Because of this tuning all five neighbouring strings form a harmonic pentatonic scale and all seven neighbouring strings form a major scale, available in every key. This allows a very easy fingerpicking technique without picking false notes, if the right key is chosen.

See also

• enharmonic
• cadence (music)
• sonata form
• chromatic circle
• Well temperament

Diatonic Scales and Keys
	Flats Sharps Major minor Major minor 0 C (Major), a (minor) 1 F d G e 2 B♭ g D b 3 E♭ c A f♯ 4 A♭ f E c♯ 5 D♭ b♭ B g♯ 6 G♭ e♭ F♯ d♯ 7 C♭ a♭ C♯ a♯
lower case letters are minor                         [redisplay] the table indicates the number of sharps or flats in each scale

External links

• Interactive Circle of Fifths
• Bach's Tuning by Bradley Lehman
• mnemonics
• Circle of Fifths - Memory Technique
• Circle of Fifths - Diagram
• Circle of Fifths - Complete Detailed Diagram
• Circle of Fifths - In Bass Clef
• Free learning tool for the Circle of Fifths

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