# Circle of fifths

In music theory, the circle of fifths is a model of pitch space and is the series encompassing all of the notes in the equally tempered chromatic scale. Starting on any note and repeatedly ascending by the musical interval of a perfect fifth, one will eventually land on the same note, after reaching all of the other notes:

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Fifths.png
C - G - D - A - E - B - F# - C# - G# (Ab) - Eb - Bb - F - C

The numbers on the inside of the circle also show how many sharps or flats would be in the key signature for a major scale built on that note. Thus a major scale built on A will have three sharps in its key signature. To figure the key signatures of minor keys see: relative minor/major. The circle of fifths can also be used to determine which order sharps or flats are added to key signatures. The first sharp added is F#, the next is C# and so on. The first flat added is Bb, the next Eb, and so on.

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.

Descending by fifths, and ascending or descending by fourths also works, since 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 of fifths, or fourths, may be mapped from the chromatic scale by multiplication, and vice versus. 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 of 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.

Moving around the circle of fifths is a common way to modulate.

This was supposedly invented in the sixth century B.C. by Pythagoras. It is said that Pythagoras also had the idea of tuning an instrument by fifths and thus discovered the Pythagorean comma.

One theory regarding harmonic functionality is that "functional succession is explained by the circle of fifths (in which, therefore, scale degree II is closer to the dominant than scale degree IV)." According to Goldman's Harmony in Western Music, "the IV chord is actually, in the simplest mechanisms of diatonic relationships, at the greatest distance from I. In terms of the circle of fifths, it leads away from I, rather than toward it." (1965, p.68) Thus the progression I-ii-V-I would comply more with tonal logic. However, Goldman (ibid., chapter 3), as well as Jean-Jacques Nattiez, points out that "the chord on the fourth degree appears long before the chord on II, and the subsequent final I, in the progression I-IV-viio-iii-vi-ii-V-I." (Nattiez 1990, p. 226) Goldman also points out that, "historically the use of the IV chord in harmonic design, and especially in cadences, exhibits some curious features. By and large, one can say that the use of IV in final cadences becomes more common in the nineteenth century than it was in the eighteenth, but that it may also be understood as a substitute for the ii chord when it proceeds V. It may also be quite logically construed as an incomplete ii7 chord (lacking root)." (1968, p.68) However, Nattiez calls this, "a narrow escape: only the theory of a ii chord without a root allows Goldman to maintain that the circle of fifths is completely valid from Bach to Wagner." (1990, p.226)

For information on the circle of fifths in popular music harmony see Winkler, P. (1978). Profane Culture. London.

## Source

• Nattiez, Jean-Jacques (1990). Music and Discourse: Toward a Semiology of Music (Musicologie générale et sémiologue, 1987). Translated by Carolyn Abbate (1990). ISBN 0691027145.
• D'Indy (1903).
• Goldman (1965). Harmony in Western Music.

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