Editing Pythagorean comma (section)

==Circle of fifths and enharmonic change==
{{Image frame|width=250
| content = <score raw="1">p = \markup { \lower #1 "+" } pps = \markup { \concat { \lower #1 "++" \sharp }} ppps = \markup { \concat { \lower #1 "+++" \sharp }} \new PianoStaff \with { \override Accidental.stencil = #ly:text-interface::print \override StaffGrouper.staff-staff-spacing.basic-distance = #15 \omit TimeSignature }
<< \new Staff \with{ \magnifyStaff #3/2 } {\relative c'
\tweak AccidentalPlacement.positioning-done ##f <\tweak Accidental.text \pps \tweak Accidental.X-offset #-10.75 fis
\tweak Accidental.text \pps \tweak Accidental.X-offset #-6 cis'
\tweak Accidental.text \pps \tweak Accidental.X-offset #-10.75 gis'
\tweak Accidental.text \pps \tweak Accidental.X-offset #-6  dis'
\tweak Accidental.text \ppps \tweak Accidental.X-offset #-14.75 ais'
\tweak Accidental.text \ppps \tweak Accidental.X-offset #-8  eis'>1 }

\new Staff \with{ \magnifyStaff #3/2 } {\relative c,, {\clef bass <c g' d'
\tweak Accidental.text \p ais'
\tweak Accidental.text \p eis'
\tweak Accidental.text \p bis'>1 } }
>> \paper {tagline=##f}
</score>
| caption = Pythagorean comma as twelve justly tuned perfect fifths in Ben Johnston notation[[File:Just perfect fifth on C.mid]]
}}

The Pythagorean comma can also be thought of as the discrepancy between 12 [[just intonation|justly tuned]] [[perfect fifth]]s (ratio 3:2) and seven octaves (ratio 2:1):

:<math>\frac{\hbox{twelve fifths}}{\hbox{seven octaves}}
=\left(\tfrac32\right)^{12} \!\!\Big/\, 2^{7}
= \frac{3^{12}}{2^{19}}
= \frac{531441}{524288}
= 1.0136432647705078125
\!</math>

{|
|-----
| valign="top" |
{| class="wikitable"
|+Ascending by perfect fifths
|-----
! Note
! [[Perfect fifth|Fifth]]
! Frequency ratio
! Decimal ratio
|-
! C
| align="center"| 0 || align="center"| 1 ''':''' 1 || align="center"| &nbsp; 1
|-
! G
| align="center"| 1 || align="center"| 3 ''':''' 2 || align="center"| &nbsp; 1.5
|-
! D
| align="center"| 2 || align="center"| 9 ''':''' 4 || align="center"| &nbsp; 2.25
|-
! A
| align="center"| 3 || align="center"| 27 ''':''' 8 || align="center"| &nbsp; 3.375
|-
! E
| align="center"| 4 || align="center"| 81 ''':''' 16 || align="center"| &nbsp; 5.0625
|-
! B
| align="center"| 5 || align="center"| 243 ''':''' 32 || align="center"| &nbsp; 7.59375
|-
! F{{music|sharp}}
| align="center"| 6 || align="center"| 729 ''':''' 64 || align="center"| &nbsp; 11.390625
|-
! C{{music|sharp}}
| align="center"| 7 || align="center"| 2187 ''':''' 128 || align="center"| &nbsp; 17.0859375
|-
! G{{music|sharp}}
| align="center"| 8 || align="center"| 6561 ''':''' 256 || align="center"| &nbsp; 25.62890625
|-
! D{{music|sharp}}
| align="center"| 9 || align="center"| 19683 ''':''' 512 || align="center"| &nbsp; 38.443359375
|-
! A{{music|sharp}}
| align="center"|10 || align="center"| 59049 ''':''' 1024 || align="center"| &nbsp; 57.6650390625
|-
! E{{music|sharp}}
| align="center"|11 || align="center"| 177147 ''':''' 2048 || align="center"| &nbsp; 86.49755859375
|-
! '''B{{music|sharp}}''' (≈ C)
| align="center"|12 || align="center"|531441 ''':''' 4096  || align="center"| &nbsp; 129.746337890625
|}
| valign="top" |
{| class="wikitable"
|+Ascending by octaves
|-----
! Note
! [[Octave]]
! Frequency ratio
|-----
| align="center"|'''C''' || align="center"|0 || align="center"|1 ''':''' 1
|-
| align="center"|'''C''' || align="center"|1 || align="center"|2 ''':''' 1
|-
| align="center"|'''C''' || align="center"|2 || align="center"|4 ''':''' 1
|-----
| align="center"|'''C''' || align="center"|3 || align="center"|8 ''':''' 1
|-----
| align="center"|'''C''' || align="center"|4 || align="center"|16 ''':''' 1
|-----
| align="center"|'''C''' || align="center"|5 || align="center"|32 ''':''' 1
|-----
| align="center"|'''C''' || align="center"|6 || align="center"|64 ''':''' 1
|-----
| align="center"|'''C''' || align="center"|7 || align="center"|128 ''':''' 1
|}
|}

In the following table of [[musical scale]]s in the [[circle of fifths]], the Pythagorean comma is visible as the small interval between, e.g., F{{music|sharp}} and G{{music|flat}}. Going around the circle of fifths with just intervals results in a [[comma pump]] by the Pythagorean comma.

The 6{{music|flat}} and the 6{{music|sharp}} scales{{efn|group=lower-roman|1=The 7{{music|flat}} and 5{{music|sharp}}, respectively 5{{music|flat}} and 7{{music|sharp}} scales differ in the same way by one Pythagorean comma. Scales with seven accidentals are seldom used,<ref>[http://www.cisdur.de/e_index.html "Complete Overview of Compositions with Seven Accidentals"], Ulrich Reinhardt</ref> because the enharmonic scales with five accidentals are treated as equivalent.}} are not identical—even though they are on the [[piano keyboard]]—but the {{music|flat}} scales are one Pythagorean comma lower. Disregarding this difference leads to [[enharmonic|enharmonic change]].

[[File:Circle of fifths unrolled, pythagorean comma.svg|1000px]]
{{notelist|group=lower-roman}}
{{clear}}

This interval has serious implications for the various [[musical tuning|tuning]] schemes of the [[chromatic scale]], because in Western music, [[circle of fifths|12 perfect fifths]] and seven octaves are treated as the same interval. [[Equal temperament]], today the most common tuning system in the West, reconciled this by flattening each fifth by a twelfth of a Pythagorean comma (approximately 2 cents), thus producing perfect octaves.

Another way to express this is that the just fifth has a frequency ratio (compared to the tonic) of 3:2 or 1.5 to 1, whereas the seventh semitone (based on 12 equal logarithmic divisions of an octave) is the seventh power of the [[twelfth root of two]] or 1.4983... to 1, which is not quite the same (a difference of about 0.1%). Take the just fifth to the 12th power, then subtract seven octaves, and you get the Pythagorean comma (about a 1.4% difference).