Editing Syntonic comma (section)

==Notation==
{{multiple image
| direction = vertical
| width     = 200
| image1    = Major chord on C.png
| caption1  = Just major chord on C in Ben Johnston's notation. {{audio|Major chord on C in just intonation.mid|Play}} Pythagorean major chord on C in Helmholtz-Ellis notation. {{audio|Pythagorean major chord on C.mid|Play}}
| image2    = Pythagorean major chord on C.png
| caption2  = Pythagorean major chord, Ben Johnston's notation.
| image3    = Just major chord on C HE notation.png
| caption3  = Just major chord, in Helmholtz-Ellis notation.
}}

[[Moritz Hauptmann]] developed a method of notation used by [[Hermann von Helmholtz]]. Based on Pythagorean tuning, subscript numbers are then added to indicate the number of syntonic commas to lower a note by. Thus a Pythagorean scale is C D E F G A B, while a just scale is C D E<sub>1</sub> F G A<sub>1</sub> B<sub>1</sub>. [[Carl Eitz]] developed a similar system used by [[J. Murray Barbour]]. Superscript positive and negative numbers are added, indicating the number of syntonic commas to raise or lower from Pythagorean tuning. Thus a Pythagorean scale is C D E F G A B, while the 5-limit Ptolemaic scale is C D E<sup>−1</sup> F G A<sup>−1</sup> B<sup>−1</sup>.

In [[Helmholtz-Ellis notation]], a syntonic comma is indicated with up and down arrows added to the traditional accidentals. Thus a Pythagorean scale is C D E F G A B, while the 5-limit Ptolemaic scale is C D E[[File:HE syntonic comma - natural down.png|9px]] F G A [[File:HE syntonic comma - natural down.png|9px]] B[[File:HE syntonic comma - natural down.png|9px]].

Composer [[Ben Johnston (composer)|Ben Johnston]] uses a "−" as an accidental to indicate a note is lowered by a syntonic comma, or a "+" to indicate a note is raised by a syntonic comma.<ref name="Fonville"/> Thus a Pythagorean scale is C D E+ F G A+ B+, while the 5-limit Ptolemaic scale is C D E F G A B.

{| class="wikitable"
|-
!
! 5-limit just
! Pythagorean
|-
| HE
| C D E[[File:HE syntonic comma - natural down.png|9px]] F G A[[File:HE syntonic comma - natural down.png|9px]] B[[File:HE syntonic comma - natural down.png|9px]]
| C D E F G A B
|-
| Johnston
| C D E F G A B
| C D E+ F G A+ B+
|}