Chromatic scale

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. 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.

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.

DefinitionEdit

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.Template:Efn The ratio of the frequency of one note in the scale to that of the preceding note is given by <math>\sqrt[12]{2} \approxeq 1.06</math>.<ref>Template:Cite book</ref>

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.

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The ascending and descending chromatic scale is shown below.<ref name="B&S"/>

<score sound="1"> {

\override Score.TimeSignature #'stencil = ##f \relative c' {

 \clef treble \time 12/4
 c4^\markup { Ascending } cis d dis e f fis g gis a ais b
 c^\markup { Descending } b bes a aes g ges f e es d des c
 }

} </score>

File:Pitch class space.svg

Chromatic scale drawn as a circle

File:Chromatic notes diagram.png

The diatonic scale notes (above) and the non-scale chromatic notes (below)<ref name="Forte"/>

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NotationEdit

File:Pitch class space star.svg

The circle of fifths drawn within the chromatic circle as a star dodecagram.<ref>Template:Cite journal</ref>

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, GTemplate:Music – GTemplate:Music – GTemplate:Music).

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 tuningsEdit

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PythagoreanEdit

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The most common conception of the chromatic scale before the 13th century was the Pythagorean chromatic scale ({{#if:Shí èr lǜ on C.mid|{{#ifexist:Media:Shí èr lǜ on C.mid|<phonos file="Shí èr lǜ on C.mid">Play</phonos>|{{errorTemplate:Main other|Audio file "Shí èr lǜ on C.mid" not found}}Template:Category handler}}}}). 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 GTemplate:Music to ATemplate:Music centered on D (in bold) (GTemplate:Music–DTemplate:Music–ATemplate:Music–ETemplate:Music–BTemplate:Music–F–C–G–D–A–E–B–FTemplate:Music–CTemplate:Music–GTemplate:Music–DTemplate:Music–ATemplate:Music), with sharps higher than their enharmonic flats (cents rounded to one decimal):

	C	DTemplate:Music	CTemplate:Music	D	ETemplate:Music	DTemplate:Music	E	F	GTemplate:Music	FTemplate:Music	G	ATemplate:Music	GTemplate:Music	A	BTemplate:Music	ATemplate:Music	B	C
Pitch ratio	1	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	2
Cents	0	90.2	113.7	203.9	294.1	317.6	407.8	498	588.3	611.7	702	792.2	815.6	905.9	996.1	1019.6	1109.8	1200

where Template:Frac is a diatonic semitone (Pythagorean limma) and Template:Frac 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 intonationEdit

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

	C	CTemplate:Music	DTemplate:Music	D	DTemplate:Music	ETemplate:Music	E	ETemplate:Music/FTemplate:Music	F	FTemplate:Music	GTemplate:Music	G	GTemplate:Music	ATemplate:Music	A	ATemplate:Music	BTemplate:Music	B	BTemplate:Music/CTemplate:Music	C
Pitch ratio	1	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	Template:Frac	2
Cents	0	70.7	111.7	203.9	274.6	315.6	386.3	427.4	498	568.7	631.3	702	772.6	813.7	884.4	955	1017.6	1088.3	1129.3	1200

The fractions Template:Frac and Template:Frac, Template:Frac and Template:Frac, Template:Frac and Template:Frac, Template:Frac and Template:Frac, and many other pairs are interchangeable, as Template:Frac (the syntonic comma) is tempered out.Template:Clarify

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

Non-Western culturesEdit

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 lü were simply a series of fundamental notes from which scales could be constructed."<ref>Needham, Joseph (1962/2004). Science and Civilization in China, Vol. IV: Physics and Physical Technology, pp. 170–171. Template:ISBN.</ref> However, "from the standpoint of tonal music [the chromatic scale] is not an independent scale, but derives from the diatonic scale,"<ref name="Forte"/> making the Western chromatic scale a gamut of fundamental notes from which scales could be constructed as well.