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. Music theory is the field of study that deals with the Mechanics of music and how Music works Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio. In Music, a pitch class is a set of all pitches that are a whole number of Octaves apart e The chromatic scale is a Musical scale with twelve pitches each a Semitone or Half step apart The circle of fifths was first described by Johann David Heinichen, in his 1728 treatise Der Generalbass in der Composition. Johann David Heinichen ( 17 April 1683 - 16 July 1729) was a German Baroque Composer and Music theorist
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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. The Circle of fifths text table shows the number of flats or sharps in each of the Diatonic musical scales/keys In Music theory, the term interval describes the relationship between the pitches of two Notes Intervals may be described as vertical The perfect fifth ( is the Musical interval between a note and the note seven Semitones above it on the musical scale In Music, a pitch class is a set of all pitches that are a whole number of Octaves apart e
Since the space is circular, it is also possible to descend by fourths. 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. In Music theory, a diatonic scale (from the Greek διατονικος, meaning " through tones" also known as the heptatonia prima and 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. In Musical notation, a key signature is a series of sharp or flat symbols placed on the staff, designating notes that are to be consistently 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. Minor Scale was a test conducted by the United States Defense Nuclear Agency (now part of the Defense Threat Reduction Agency) involving the detonation g. A minor has 0 sharps or flats and E minor has 1 sharp. (See relative minor/major for details. In Music, the relative minor of a particular major key (or the Relative major of a minor key is the key which has the same Key signature but )
Tonal music often modulates by moving between adjacent scales on the circle of fifths. In Music, modulation is most commonly the act or process of changing from one key ( tonic, or tonal center) to another 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. The chromatic circle is a geometrical space that shows relationships among the 12 equal-tempered Pitch classes making up the familiar Chromatic scale. 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. The chromatic circle is a geometrical space that shows relationships among the 12 equal-tempered Pitch classes making up the familiar Chromatic scale. In Music, a pitch class is a set of all pitches that are a whole number of Octaves apart e 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, 
. In Music, a pitch class is a set of all pitches that are a whole number of Octaves apart e In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers 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. The chromatic circle is a geometrical space that shows relationships among the 12 equal-tempered Pitch classes making up the familiar Chromatic scale.
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, counting seven keys to the right (both black and white) to get to the next note on the circle above — which is a perfect fifth. A musical keyboard is the set of adjacent depressible levers or keys on a Musical instrument, particularly the piano Seven half steps, 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 mathematics describes.
The frequencies of two notes that are a perfect fifth apart differ by a ratio of approximately 3:2. A ratio of exactly 3:2 would sound best, but for mathematical reasons it is not possible to get the circle of fifths to 'join up' (that is, to return to the original pitch after going round the circle). Therefore the 3:2 ratio is slightly detuned so that perfect fifths do cycle. This slight detuning is part of musical temperament. The primary tuning system used for Western instruments today is called twelve-tone equal temperament. Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio.
In coordinate system
In order to understand and memorize the relations between pitch classes easier, it is possible to put all the keys into a Cartesian coordinate system according to their signatures. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane
In the resulting graph, the X coordinate represents the number of minor key signatures, treating the number of flats as a negative (as it makes a note negative). The Y coordinate represents the major key signature count the same way. C is therefore in (-3;0) as C minor has 3 flats and C major a 0 (zero).
When minor and major keys are paired into single dots, various relations become visible more clearly from the graph:
- Minor and major key signature amounts are always separated by 3 units.
- Examples: if we know that C major has 0, we get that C minor would have 0 + 3 = 3 flats. If G minor has 2 flats (-2), G major would have -2 + 3 = 1 sharp.
- The amount of signatures in a sharp/flat key is always 7 more than the same note key without sharp/flat tonic.
- Examples: if C major has 0, C sharp major would have 7 sharps. If C minor has -3, C sharp minor would be -3 + 7 = 4 sharps.
- The relative key is also easy to find. If a relative is needed of the C major, which stands on y = 0, we look for a key that stands on 0 of the minor axis. In another example, the relative of B minor (x = 2) is D major (y = 2).
- The order of signature placement is the same as the key order on the axes. If the key goes down becoming flat, flats are placed from B downwards. Respectively, sharps are placed from F upwards.
- It is also clear that all of the keys tend to recur every 12 units because of the chromatic scale consisting of 12 notes. The chromatic scale is a Musical scale with twelve pitches each a Semitone or Half step apart Therefore a major key with 120 sharp signatures would remain C major.
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. In Diatonic set theory structure implies multiplicity is a quality of a collection or scale.
Relation with chromatic scale
The circle of fifths, or fourths, may be mapped from the chromatic scale by multiplication, and vice versa. The chromatic scale is a Musical scale with twelve pitches each a Semitone or Half step apart 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). In integer notation, or the Integer model of pitch all Pitch classes and intervals between pitch classes are designated using the numbers 0 through 11 Twelve-tone technique (also dodecaphony, especially in British usage twelve-note composition) is a method of musical composition devised by Arnold
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♯. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying In Music, a tone row or note row ( German: Reihe or Tonreihe) also series and set, refers to a non-repetitive 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. The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure 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. In modern Music and notation, an enharmonic equivalent is a Note ( enharmonic tone) interval ( enharmonic interval) or For example, the key of B, with five sharps, is enharmonically equivalent to the key of C♭, with 7 flats. But the circle of fifths 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, coincidentally, 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
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. In Acoustics and Telecommunication, the harmonic of a Wave is a component Frequency of the signal that is an Integer A pentatonic scale is a musical scale with five pitches per Octave in contrast to an heptatonic (seven note scale such as the Major scale In Music theory, the major scale or Ionian scale is one of the diatonic scales It is made up of seven distinct Notes plus an eighth In Music theory, the term key is used in many different and sometimes contradictory ways This allows a very easy fingerpicking technique without picking false notes, if the right key is chosen. Fingerstyle guitar is the technique of playing the Guitar by plucking the strings directly with the fingertips fingernails or picks attached to fingers as opposed to
Accordions also commonly use the stradella bass system for the left hand buttons. The accordion is a portable box-shaped Musical instrument of the hand-held Bellows -driven free-reed aerophone family sometimes referred to as a Squeezebox It follows the circle of fifths.
See also
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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