Chromatic scale

{{Short description|Musical scale set of twelve pitches}} thumb|Chromatic scale: every key of one octave on the piano keyboard

In Western music, a '''chromatic scale''' (or '''twelve-tone scale''') is a set of twelve pitches within an octave, where the interval between any two adjacent notes is a semitone.

If the scale is tuned such that the interval between any two adjacent notes may function both as a diatonic and chromatic semitone (as in the modern 12-tone equal temperament), it provides a practical approximation of acoustically pure intervals in every key, and serves as a superset containing subsets like diatonic scales.

Chromatic instruments, such as the piano, are made to produce the chromatic scale. 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.

==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.{{efn|As every chromatic scale is identical under transposition, inversion, and retrograde to every other.}} 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>{{cite book |last=Jeans |first=James |author-link=James Jeans |date=1923 |title=Science and Music |url=https://archive.org/details/in.ernet.dli.2015.459051/page/n33/mode/1up |publisher=Cambridge University Press |pages=24–25 |via=Internet Archive}}</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.

{{Quote|The ''chromatic scale''...is a series of half steps which comprises all the pitches of our [12-tone] equal-tempered system.|Allen Forte (1979)<ref name="Forte">Forte, Allen, ''Tonal Harmony'', third edition (S.l.: Holt, Rinehart, and Wilson, 1979): pp. 4–5. {{ISBN|0-03-020756-8}}.</ref>}}

{{Quote|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)<ref>Piston, Walter (1987/1941). ''Harmony'', p. 5. 5th ed. revised by DeVoto, Mark. W. W. Norton, New York/London. {{ISBN|0-393-95480-3}}.</ref>}}

{{Quote|A ''chromatic scale'' is a nondiatonic scale consisting entirely of half-step intervals. Since each tone of the scale is equidistant from the next {{bracket|symmetry}} it has no tonic {{bracket|key}}.<ref name="B&S">{{cite book|last1=Benward|first1=Bruce|last2=Saker|first2=Marilyn Nadine|year=2003|title=Music in Theory and Practice|volume=I|page=37|publisher=McGraw-Hill |edition=7th|isbn=978-0-07-294262-0}}</ref> ...<br/> 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.<ref>Benward & Saker (2003). "Glossary", p. 359.</ref>|Benward & Saker (2003)}}

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|thumb|Chromatic scale drawn as a circle]] thumb|300px|The diatonic scale notes (above) and the non-scale chromatic notes (below)<ref name="Forte"/>

{{Quote|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)<ref>Kamien, Roger (1990). ''Music: An Appreciation'', p. 44. Brief edition. McGraw-Hill. {{ISBN|0-07-033568-0}}.</ref>}}

==Notation== [[File:Pitch class space star.svg|thumb|The circle of fifths drawn within the chromatic circle as a star dodecagram.<ref>{{cite journal|last=McCartin|first=Brian J.|date=November 1998|title=Prelude to Musical Geometry|journal=The College Mathematics Journal|volume=29|issue=5|pages=354–370 (364)|doi=10.1080/07468342.1998.11973971 |jstor=2687250}}</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, G{{music|flat}} – G{{music|natural}} – G{{music|sharp}}).

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== {{anchor|Tuning|Tunings}} ===Pythagorean=== {{Main|Pythagorean tuning}}

The most common conception of the chromatic scale before the 13th century was the Pythagorean chromatic scale ({{audio|Shí èr lǜ on C.mid|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{{music|b}} to A{{music|#}} centered on D (in bold) (G{{music|b}}–D{{music|b}}–A{{music|b}}–E{{music|b}}–B{{music|b}}–F–C–G–'''D'''–A–E–B–F{{music|#}}–C{{music|#}}–G{{music|#}}–D{{music|#}}–A{{music|#}}), with sharps ''higher'' than their enharmonic flats (cents rounded to one decimal):

:{| class="wikitable" style="text-align: center" |- !width=4%| !width=4%| C !width=4%| D{{music|flat}} !width=4%| C{{music|#}} !width=4%| D !width=4%| E{{music|flat}} !width=4%| D{{music|#}} !width=4%| E !width=4%| F !width=4%| G{{music|flat}} !width=4%| F{{music|#}} !width=4%| G !width=4%| A{{music|flat}} !width=4%| G{{music|#}} !width=4%| A !width=4%| B{{music|flat}} !width=4%| A{{music|#}} !width=4%| B !width=4%| C |- !Pitch<br />ratio | 1 || {{frac|256|243}} || {{frac|2187|2048}} || {{frac|9|8}} || {{frac|32|27}} || {{frac|19683|16384}} || {{frac|81|64}} || {{frac|4|3}} || {{frac|1024|729}} || {{frac|729|512}} || {{frac|3|2}} || {{frac|128|81}} || {{frac|6561|4096}} || {{frac|27|16}} || {{frac|16|9}} || {{frac|59049|32768}} || {{frac|243|128}} || 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 {{frac|256|243}} is a diatonic semitone (Pythagorean limma) and {{frac|2187|2048}} is a chromatic semitone (Pythagorean apotome).

===Just intonation===  {{Main|Just intonation#Twelve-tone scale}}

In 5-limit just intonation the chromatic scale, '''Ptolemy's intense chromatic scale'''{{fact|date=November 2019}}, is as follows, with flats ''higher'' than their enharmonic sharps, and new notes between E–F and B–C (cents rounded to one decimal):

:{| class="wikitable" style="text-align: center" |- ! ! C !! C{{music|#}} !! D{{music|flat}} !! D !! D{{music|#}} !! E{{music|flat}} !! E !! E{{music|#}}/F{{music|flat}} !! F !! F{{music|#}} !! G{{music|flat}} !! G !! G{{music|#}} !! A{{music|flat}} !! A !! A{{music|#}} !! B{{music|flat}} !! B !! B{{music|#}}/C{{music|flat}} !! C |- !Pitch ratio | 1 || {{frac|25|24}} || {{frac|16|15}} || {{frac|9|8}} || {{frac|75|64}} || {{frac|6|5}} || {{frac|5|4}} || {{frac|32|25}} || {{frac|4|3}} || {{frac|25|18}} || {{frac|36|25}} || {{frac|3|2}} || {{frac|25|16}} || {{frac|8|5}} || {{frac|5|3}} || {{frac|125|72}} || {{frac|9|5}} || {{frac|15|8}} || {{frac|48|25}} || 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 {{frac|9|8}} and {{frac|10|9}}, {{frac|6|5}} and {{frac|32|27}}, {{frac|5|4}} and {{frac|81|64}}, {{frac|4|3}} and {{frac|27|20}}, and many other pairs are interchangeable, as {{frac|81|80}} (the syntonic comma) is tempered out.{{clarify|date=October 2019|reason=Is this a process "as x is tempered out", a possibility "if x is tempered out", or a reason "since/because the x is tempered out"? What context or process includes or is defined by the syntonic comma being tempered out (equal temperament, 5-limit just, Pythagorean, ...)?}}

==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 ''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. {{ISBN|978-0-521-05802-5}}.</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.

==Notes== {{notelist}}

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==Sources== {{reflist}}

==Further reading== *Hewitt, Michael. 27 January 2013. ''Musical Scales of the World''. The Note Tree. {{ISBN|978-0957547001}}

==External links== *[http://robsilverguitars.blogspot.com/2010/05/chromatic-scales-for-guitar.html The Chromatic Scale arranged for guitar in several fingerings. (Formatted for easy printing)] * [http://www.skytopia.com/project/scale.html The 12 golden notes of music] * [https://ianring.com/scales/4095 Chromatic Scale – Analysis]

Category:Chromaticism Category:Musical scales Category:Post-tonal music theory Category:Musical symmetry Category:Hemitonic scales Category:Tritonic scales