The chromatic circle is a
clock
A clock or a timepiece is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month and t ...
diagram for displaying relationships among the 12
equal-tempered pitch class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...
es making up the familiar
chromatic scale
The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce th ...
on a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
.
Explanation
If one starts on any equal-tempered pitch and repeatedly ascends by the
musical interval of a
semitone
A semitone, also called a 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 two adjacent no ...
, one will eventually land on a pitch with the same pitch class as the initial one, having passed through all the other
equal-tempered chromatic pitch classes in between. Since the space is circular, it is also possible to descend by semitone.
The chromatic circle is useful because it represents melodic distance, which is often correlated with physical distance on musical instruments. For instance, to move from any C on a piano keyboard to the nearest E, one must move up four semitones, corresponding to four clockwise steps on the chromatic circle. One can also move ''down'' by eight semitones, corresponding to eight counterclockwise steps on the pitch class circle.
Larger motions on the piano (or in
pitch space
In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apa ...
) can be represented in pitch class space by paths that "wrap around" the chromatic circle one or more times.
One can represent the twelve equal-tempered pitch classes by the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order twelve, or equivalently, the
residue classes modulo twelve, Z/12Z. 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
In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval of ...
.
Comparison with circle of fifths
A key difference between the chromatic circle and the
circle of fifths
In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval of ...
is that the former is truly a continuous space: every point on the circle corresponds to a conceivable
pitch class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...
, 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.
Pitch constellation
![pitch constellation chromatic](https://upload.wikimedia.org/wikipedia/commons/9/90/Pitch_constellation_chromatic.svg)
A pitch constellation is a graphical representation of
pitches used to describe
musical scale
In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale.
Often, especially in the ...
s,
modes
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
,
chords
Chord may refer to:
* Chord (music), an aggregate of musical pitches sounded simultaneously
** Guitar chord a chord played on a guitar, which has a particular tuning
* Chord (geometry), a line segment joining two points on a curve
* Chord ( ...
or other groupings of
pitches within an octave range.
[.][.] It consists of a circle with markings along the circumference or lines from the center which indicate pitches. Most pitch constellations use of a subset of pitches chosen from the twelve pitch
chromatic scale
The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce th ...
. In this case the points on the circle are spaced like the twelve hour markings on an analog clock where each tick mark represents a semitone.
Scales and modes
The pitch constellation provides an easy way to identify certain patterns and similarities between
harmonic structures.
For example.
* A
major scale
The major scale (or Ionian mode) is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at double ...
consists of a circle with markings at 0 (or 12), 2, 4, 5, 7, 9 and 11 o'clock.
* A
minor scale
In music theory, the minor scale is three scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending) – rather than just two as with the major scale, which ...
consists of a circle with markings at 0 (or 12), 2, 3, 5, 7, 8 and 10 o'clock.
![Pitch constellation degrees](https://upload.wikimedia.org/wikipedia/commons/4/4d/Pitch_constellation_degrees.svg)
The diagrams above show the two scales marked with
"scale degrees". It can be observed that the
tonic, second, fourth and fifth are shared, while the minor scale
flattens the third, sixth and seventh notes relative to the major scale.
[.] Another observation is that the minor scale's constellation is the same as the major scale, but rotated +90 degrees.
In the following drawing all of the major/minor scales are drawn. Note that the constellation for all the major scales or all the minor scales are identical. The different scales are generated by rotating the note overlay. The notes that need to be
sharpened/flattened can be easily identified.
Moreover, if we draw all seven
diatonic modes
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
we can see them all as rotations of the
Ionian mode
Ionian mode is a musical mode or, in modern usage, a diatonic scale also called the major scale.
It is the name assigned by Heinrich Glarean in 1547 to his new authentic mode on C (mode 11 in his numbering scheme), which uses the diatonic octave ...
.
Note also the significance of the 6 o'clock point. This corresponds to a
tritone
In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three adj ...
. The modes including pitches a tritone from the tonic (
Locrian and
Lydian) are least used. The 5 o'clock and 7 o'clock pitches are also important points corresponding to a
perfect fourth
A fourth is a 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 interval from C to ...
and
perfect fifth respectively. The most used scales/modes - major (
Ionian mode
Ionian mode is a musical mode or, in modern usage, a diatonic scale also called the major scale.
It is the name assigned by Heinrich Glarean in 1547 to his new authentic mode on C (mode 11 in his numbering scheme), which uses the diatonic octave ...
), minor (
Aeolian mode
The Aeolian mode is a musical mode or, in modern usage, a diatonic scale also called the natural minor scale. On the white piano keys, it is the scale that starts with A. Its ascending interval form consists of a ''key note, whole step, half ste ...
) and
Mixolydian
Mixolydian mode may refer to one of three things: the name applied to one of the ancient Greek ''harmoniai'' or ''tonoi'', based on a particular octave species or scale; one of the medieval church modes; or a modern musical mode or diatonic scal ...
- include these pitches.
Symmetric scale
In music, a symmetric scale is a music scale which equally divides the octave. The concept and term appears to have been introduced by Joseph Schillinger and further developed by Nicolas Slonimsky as part of his famous ''Thesaurus of Scales and M ...
s have simple representations in this scheme.
![Symmetric](https://upload.wikimedia.org/wikipedia/commons/c/c5/Symmetric.svg)
More exotic scales - such as the
pentatonic
A pentatonic scale is a musical scale (music), scale with five Musical note, notes per octave, in contrast to the heptatonic scale, which has seven notes per octave (such as the major scale and minor scale).
Pentatonic scales were developed ...
,
blues and
octatonic - can also be drawn and related to the common scales.
![Exotic](https://upload.wikimedia.org/wikipedia/commons/a/aa/Exotic.svg)
A more complete
list of musical scales and modes
The following is a list of musical scales and modes. Degrees are relative to the major scale.
See also
* Bebop scale
* Chord-scale system
*Heptatonic scale
* Jazz scale
*List of chord progressions
* List of chords
* List of musical interva ...
Other overlays
In previous sections we saw how various overlays (scale degrees, semi-tone numbering, notes) can be used to notate the circumference of the constellation. Various other overlays can be laid around the constellation. For example:
*
Intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets
* A statistical level of measurement
* Interval e ...
.
*
Solfège
In music, solfège (, ) or solfeggio (; ), also called sol-fa, solfa, solfeo, among many names, is a music education method used to teach aural skills, pitch and sight-reading of Western music. Solfège is a form of solmization, though the tw ...
.
*
Pitch ratios (ratios of pitch frequencies).
![Overlays](https://upload.wikimedia.org/wikipedia/commons/3/34/Overlays.svg)
Note that once a pitch constellation has been determined, any number of overlays (notes, solfège, intervals, etc.) may be placed on top for analysis/comparison. Often generating one harmonic relationship from another is simply a matter of rotating the overlay or constellation or shifting one or two pitch locations.
Chords
Similarities between
chords
Chord may refer to:
* Chord (music), an aggregate of musical pitches sounded simultaneously
** Guitar chord a chord played on a guitar, which has a particular tuning
* Chord (geometry), a line segment joining two points on a curve
* Chord ( ...
can also be observed as well as the significance of
augmented/diminished notes.
For
triads we have the following:
![Pitch constellation triads](https://upload.wikimedia.org/wikipedia/commons/a/a9/Pitch_constellation_triads.svg)
And for
seventh chord
A seventh chord is a chord consisting of a triad plus a note forming an interval of a seventh above the chord's root. When not otherwise specified, a "seventh chord" usually means a dominant seventh chord: a major triad together with a mi ...
s:
Circle of fifths
Beginning with a pitch constellation of a chromatic scale, the notes of a
circle of fifths
In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval of ...
can be easily generated. Starting at C and moving across the circle and then one tick clockwise a line is drawn with an arrow indicating the direction moved. Continuing from that point (across the circle and one tick clockwise) all points are connected. Moving through this pattern the notes of the circle of fifths
can be determined (C, G, D, A ...).
![PythagoreanTuningGeometric](https://upload.wikimedia.org/wikipedia/commons/a/a8/PythagoreanTuningGeometric.png)
One can also depict non-tempered intervals on a chromatic circle, which allows one to depict
commas (small intervals), particularly
comma pumps. For example, using a sequence of twelve just fifths (3:2 ratio) does not quite return to the starting point (the size of the gap is the
Pythagorean comma
In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as ...
), resulting in a "broken" circle of fifths.
Technical note
The ratio of the frequencies between two pitches in the constellation can be determined as follows.
[.] Take the length of the arc (measured clockwise) between the two points and divide by the circumference of the circle. The frequency ratio is two raised to this power. For example, for a fifth (P5, which is located at 7 o'clock relative to the tonic T) the frequency ratio is:
References
Further reading
*.
*.
*
External links
Notenscheibe web application- pitch constellations of scales, triads, intervals and the circle of fifths, with basic audio
On-line app illustrating pitch constellationsScaleTapper- IPhone app which utilizes pitch constellations.
PDF of musical scales
{{Pitch space
Pitch space
Post-tonal music theory