Chromatic scale

Last updated
Chromatic scale: every key of one octave on the piano keyboard PianoKeyboard.svg
Chromatic scale: every key of one octave on the piano keyboard

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. Almost all western musical 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.

Contents

Most music uses subsets of the chromatic scale such as diatonic scales. While the chromatic scale is fundamental in western music theory, it is seldom directly used in its entirety in musical compositions or improvisation.

Definition

The chromatic scale is a musical scale with twelve pitches, each a semitone, also known as a half-step, above or below its adjacent pitches. As a result, in 12-tone equal temperament (the most common tuning in Western music), the chromatic scale covers all 12 of the available pitches. Thus, there is only one chromatic scale. [lower-alpha 1]

In equal temperament, all the semitones have the same size (100 cents), and there are twelve semitones in an octave (1200 cents). As a result, the notes of an equal-tempered chromatic scale are equally-spaced.

The chromatic scale...is a series of half steps which comprises all the pitches of our [12-tone] equal-tempered system.

Allen Forte (1979) [1]

All of the pitches in common use, considered together, constitute the chromatic scale. It is made up entirely of successive half steps, the smallest interval in Western music....Counting by half steps, an octave includes twelve different pitches, white and black keys together. The chromatic scale, then, is a collection of all the available pitches in order upward or downward, one octave's worth after another.

Walter Piston (1987) [2]

A chromatic scale is a nondiatonic scale consisting entirely of half-step intervals. Since each tone of the scale is equidistant from the next [ symmetry ] it has no tonic [ key ]. [3] ...
Chromaticism [is t]he introduction of some pitches of the chromatic scale into music that is basically diatonic in orientation, or music that is based on the chromatic scale instead of the diatonic scales. [4]

Benward & Saker (2003)

The ascending and descending chromatic scale is shown below. [3]

Chromatic scale
Chromatic scale drawn as a circle Pitch class space.svg
Chromatic scale drawn as a circle
The diatonic scale notes (above) and the non-scale chromatic notes (below) Chromatic notes diagram.png
The diatonic scale notes (above) and the non-scale chromatic notes (below)

The twelve notes of the octave—all the black and white keys in one octave on the piano—form the chromatic scale. The tones of the chromatic scale (unlike those of the major or minor scale) are all the same distance apart, one half step. The word chromatic comes from the Greek chroma, color; and the traditional function of the chromatic scale is to color or embellish the tones of the major and minor scales. It does not define a key, but it gives a sense of motion and tension. It has long been used to evoke grief, loss, or sorrow. In the twentieth century it has also become independent of major and minor scales and is used as the basis for entire compositions.

Roger Kamien (1976) [5]

Notation

The circle of fifths drawn within the chromatic circle as a star dodecagram. Pitch class space star.svg
The circle of fifths drawn within the chromatic circle as a star dodecagram.

The chromatic scale has no set enharmonic spelling that is always used. Its spelling is, however, often dependent upon major or minor key signatures and whether the scale is ascending or descending. In general, the chromatic scale is usually notated with sharp signs when ascending and flat signs when descending. It is also notated so that no scale degree is used more than twice in succession (for instance, G – G – G).

Similarly, some notes of the chromatic scale have enharmonic equivalents in solfege. The rising scale is Do, Di, Re, Ri, Mi, Fa, Fi, Sol, Si, La, Li, Ti and the descending is Ti, Te/Ta, La, Le/Lo, Sol, Se, Fa, Mi, Me/Ma, Re, Ra, Do, However, once 0 is given to a note, due to octave equivalence, the chromatic scale may be indicated unambiguously by the numbers 0-11 mod twelve. Thus two perfect fifths are 0-7-2. Tone rows, orderings used in the twelve-tone technique, are often considered this way due to the increased ease of comparing inverse intervals and forms (inversional equivalence).

Pitch-rational tunings

Pythagorean

The most common conception of the chromatic scale before the 13th century was the Pythagorean chromatic scale ( Loudspeaker.svg Play  ). Due to a different tuning technique, the twelve semitones in this scale have two slightly different sizes. Thus, the scale is not perfectly symmetric. Many other tuning systems, developed in the ensuing centuries, share a similar asymmetry.

In Pythagorean tuning (i.e. 3-limit just intonation) the chromatic scale is tuned as follows, in perfect fifths from G to A centered on D (in bold) (G–D–A–E–B–F–C–G–D–A–E–B–F–C–G–D–A), with sharps higher than their enharmonic flats (cents rounded to one decimal):

CDCDEDEFGFGAGABABC
Pitch
ratio
12562432187204898322719683163848164431024729729512321288165614096271616959049327682431282
Cents090.2113.7203.9294.1317.6407.8498588.3611.7702792.2815.6905.9996.11019.61109.81200

where 256243 is a diatonic semitone (Pythagorean limma) and 21872048 is a chromatic semitone (Pythagorean apotome).

The chromatic scale in Pythagorean tuning can be tempered to the 17-EDO tuning (P5 = 10 steps = 705.88 cents).

Just intonation

In 5-limit just intonation the chromatic scale, Ptolemy's intense chromatic scale[ citation needed ], is as follows, with flats higher than their enharmonic sharps, and new notes between E–F and B–C (cents rounded to one decimal):

CCDDDEEE/FFFGGGAAABBB/CC
Pitch ratio1252416159875646554322543251836253225168553125729515848252
Cents070.7111.7203.9274.6315.6386.3427.4498568.7631.3702772.6813.7884.49551017.61088.31129.31200

The fractions 98 and 109, 65 and 3227, 54 and 8164, 43 and 2720, and many other pairs are interchangeable, as 8180 (the syntonic comma) is tempered out.[ clarification needed ]

Just intonation tuning can be approximated by 19-EDO tuning (P5 = 11 steps = 694.74 cents).

Non-Western cultures

The ancient Chinese chromatic scale is called Shí-èr-lǜ . However, "it should not be imagined that this gamut ever functioned as a scale, and it is erroneous to refer to the 'Chinese chromatic scale', as some Western writers have done. The series of twelve notes known as the twelve were simply a series of fundamental notes from which scales could be constructed." [7] However, "from the standpoint of tonal music [the chromatic scale] is not an independent scale, but derives from the diatonic scale," [1] making the Western chromatic scale a gamut of fundamental notes from which scales could be constructed as well.

See also

Notes

  1. As every chromatic scale is identical under transposition, inversion, and retrograde to every other.

Sources

  1. 1 2 3 Forte, Allen, Tonal Harmony, third edition (S.l.: Holt, Rinehart, and Wilson, 1979): pp. 4–5. ISBN   0-03-020756-8.
  2. Piston, Walter (1987/1941). Harmony, p. 5. 5th ed. revised by DeVoto, Mark. W. W. Norton, New York/London. ISBN   0-393-95480-3.
  3. 1 2 Benward, Bruce; Saker, Marilyn Nadine (2003). Music in Theory and Practice. Vol. I (7th ed.). p. 37. ISBN   978-0-07-294262-0.
  4. Benward & Saker (2003). "Glossary", p. 359.
  5. Kamien, Roger (1990). Music: An Appreciation, p. 44. Brief edition. McGraw-Hill. ISBN   0-07-033568-0.
  6. McCartin, Brian J. (November 1998). "Prelude to Musical Geometry". The College Mathematics Journal . 29 (5): 354–370 (364). doi:10.1080/07468342.1998.11973971. JSTOR   2687250.
  7. Needham, Joseph (1962/2004). Science and Civilization in China, Vol. IV: Physics and Physical Technology, pp. 170–171. ISBN   978-0-521-05802-5.

Further reading

Related Research Articles

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.

Equal temperament Musical tuning system where the ratio between successive notes is constant

An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, which gives an equal perceived step size as pitch is perceived roughly as the logarithm of frequency.

Just intonation Musical tuning based on pure intervals

In music, just intonation or pure intonation is the attempt to tune all musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and may be called a just interval; when it is sounded, no beating is heard. Just intervals consist of members of a single harmonic series of an implied fundamental. For example, in the diagram, the notes G3 and C4 may be tuned as members of the harmonic series of the lowest C, in which case their frequencies will be 3 and 4 times, respectively, the fundamental frequency and their interval ratio equal to 4:3; they may also be tuned differently.

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 Method of tuning a musical instrument

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.

In music theory, the tritone is defined as a musical interval composed of 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. According to this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale. A tritone is also commonly defined as an interval spanning six semitones. According to this definition, a diatonic scale contains two tritones for each octave. For instance, the above-mentioned C major scale contains the tritones F–B and B–F. In twelve-equal temperament, the tritone divides the octave exactly in half as 6 of 12 semitones or 600 of 1,200 cents.

In music theory, a tetrachord is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion —but in modern use it means any four-note segment of a scale or tone row, not necessarily related to a particular tuning system.

Enharmonic

In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. So, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. The term is derived from Latin enharmonicus, from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος (enarmónios), from ἐν (en) and ἁρμονία (harmonía).

Perfect fifth Musical interval

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.

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 C and B (Play ), or D and C. It is equal to the frequency ratio (1.5)1227 = 531441524288 ≈ 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 refer to tempering is the Pythagorean comma.

Circle of fifths Relationship among tones of the chromatic scale

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.

Semitone Musical interval

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 the musical system of ancient Greece, genus is a term used to describe certain classes of intonations of the two movable notes within a tetrachord. The tetrachordal system was inherited by the Latin medieval theory of scales and by the modal theory of Byzantine music; it may have been one source of the later theory of the jins of Arabic music. In addition, Aristoxenus calls some patterns of rhythm "genera".

Comma (music)

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 14-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 × 14 = 45 ≈ 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.

Diminished second Musical interval

In modern Western tonal music theory, a diminished second is the interval produced by narrowing a minor second by one chromatic semitone. It is enharmonically equivalent to a perfect unison. Thus, it is the interval between notes on two adjacent staff positions, or having adjacent note letters, altered in such a way that they have no pitch difference in twelve-tone equal temperament. An example is the interval from a B to the C immediately above; another is the interval from a B to the C immediately above.

Diatonic and chromatic Terms in music theory to characterize scales

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.

Regular diatonic tuning

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

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.