Editing Chromatic scale (section)

==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 tuning|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 system]]s, 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]]).

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

===Just intonation===
<!--Ptolemy's intense chromatic scale]] redirects directly here.-->
{{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, ...)?}}

Just intonation tuning can be approximated by [[19 equal temperament|19-EDO tuning]] (P5 = 11 steps = 694.74 cents).