The chromatic circle is a clock diagram for displaying relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale on a circle.
The chromatic circle is a clock diagram for displaying relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale on a circle.
If one starts on any equal-tempered pitch and repeatedly ascends by the musical interval of a semitone, 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) 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 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.
A key difference between the chromatic circle and the circle of fifths is that the former is truly a continuous space: 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.
A pitch constellation is a graphical representation of pitches used to describe musical scales, modes, chords or other groupings of pitches within an octave range. [2] [3] [4] It consists of a circle with markings along the circumference or lines from the center which indicate pitches. Most pitch constellations use a subset of pitches chosen from the twelve pitch chromatic scale. 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.
The pitch constellation provides an easy way to identify certain patterns and similarities between harmonic structures.
For example.
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. [5] 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.
Major and minor scales   |
Moreover, if we draw all seven diatonic modes we can see them all as rotations of the Ionian mode. [2] [6] Note also the significance of the 6 o'clock point. This corresponds to a tritone. 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 and perfect fifth respectively. The most used scales/modes - major (Ionian mode), minor (Aeolian mode) and Mixolydian - include these pitches.
Symmetric scales have simple representations in this scheme.
More exotic scales - such as the pentatonic, blues and octatonic - can also be drawn and related to the common scales.
A more complete list of musical scales and modes
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:
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.
Similarities between chords can also be observed as well as the significance of augmented/diminished notes. [3] [5]
For triads we have the following:
And for seventh chords:
Some chords are related to symmetrical (augmented or diminished) chords by a change in one note, which is related to otonality and utonality:
Beginning with a pitch constellation of a chromatic scale, the notes of a circle of fifths 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 ...).
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), resulting in a "broken" circle of fifths.
The ratio of the frequencies between two pitches in the constellation can be determined as follows. [7] 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:
In music theory, a diatonic scale is any heptatonic scale that includes five whole steps and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, depending on their position in the scale. This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other.
In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.
The major scale 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 its frequency so that it is called a higher octave of the same note.
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.
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈ 702 cents wide.
In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.
The chromatic scale is a set of twelve pitches used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the chromatic scale, while other instruments capable of continuously variable pitch, such as the trombone and violin, can also produce microtones, or notes between those available on a piano.
In music theory, the tritone is defined as a musical interval spanning three adjacent whole tones. For instance, the interval from F up to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B.
In music, especially regarding musical notation and tuning, two pitch classes have enharmonic equivalence when they are considered to produce the same pitch but are "spelled" differently within the naming system being used. This relation naturally extends from pitch classes to notes, chords, intervals, and key signatures. Thus, an enharmonic spelling is an alternative representation of a given pitch class. The term derives from Latin enharmonicus, in turn from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος, from ἐν ('in') and ἁρμονία ('harmony').
In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.
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 C and B♯, or D♭ and C♯. It is equal to the frequency ratio (1.5)12⁄27 = 531441⁄524288 ≈ 1.01364, or about 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73). The comma that musical temperaments often "temper" is the Pythagorean comma.
In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths.. If C is chosen as a starting point, the sequence is: C, G, D, A, E, B, F♯, C♯, A♭, E♭, B♭, F. Continuing the pattern from F returns the sequence to its starting point of C. This order places the most closely related key signatures adjacent to one another. It is usually illustrated in the form of a circle.
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 notes in a 12-tone scale. For example, C is adjacent to C♯; the interval between them is a semitone.
In Western classical music, an augmented fifth is an interval produced by widening a perfect fifth by a chromatic semitone. For instance, the interval from C to G is a perfect fifth, seven semitones wide, and both the intervals from C♭ to G, and from C to G♯ are augmented fifths, spanning eight semitones. Being augmented, it is considered a dissonant interval.
In Western music, the adjectives major and minor may describe an interval, chord, scale, or key. A composition, movement, section, or phrase may also be referred to by its key, including whether that key is major or minor.
In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.
Quarter-comma meantone, or 1⁄4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81 : 80), with respect to its just intonation used in Pythagorean tuning ; the result is 3/2 × 1⁄4 = 4√5 ≈ 1.49535, or a fifth of 696.578 cents. This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.
Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.
A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.
Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.
