Editing Diatonic scale (section)

===Just intonation===
{{main|Just intonation}}
Just intonation often is represented using [[Leonhard Euler]]'s [[Tonnetz]], with the horizontal axis showing the perfect fifths and the vertical axis the perfect major thirds. In the Tonnetz, the diatonic scale in just intonation appears as follows:

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;text-align:center"
| A || E || B ||
|-
| style="width:2em;" | F
| style="width:2em;" | C
| style="width:2em;" | G
| style="width:2em;" | D
|}

F–A, C–E and G–B, aligned vertically, are perfect major thirds; A–E–B and F–C–G–D are two series of perfect fifths. The notes of the top line, A, E and B, are lowered by the [[syntonic comma]], {{frac|81|80}}, and the "wolf" fifth D–A is too narrow by the same amount. The tritone F–B is {{frac|45|32}} ≈ 1.40625.

This tuning has been first described by [[Ptolemy]] and is known as [[Ptolemy's intense diatonic scale]]. It was also mentioned by Zarlino in the 16th century and has been described by theorists in the 17th and 18th centuries as the "natural" scale.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;text-align:center" cellspacing="0" border="1"
! style="text-align:left" | notes
| style="width: 60px;" | C
| style="width: 60px;" | D
| style="width: 60px;" | E
| style="width: 60px;" | F
| style="width: 60px;" | G
| style="width: 60px;" | A
| style="width: 60px;" | B
| style="width: 60px;" | C'
|-
! style="text-align:left" | pitch
| {{frac|1|1|}} || {{frac|9|8}} || {{frac|5|4}} || {{frac|4|3}} || {{frac|3|2}} || {{frac|5|3}} || {{frac|15|8}} || {{frac|2|1|}}
|-
! style="text-align:left" | interval between notes
| || {{frac|9|8}} || {{frac|10|9}} || {{frac|16|15}} || {{frac|9|8}} || {{frac|10|9}} || {{frac|9|8}} || {{frac|16|15}}
|}
Since the frequency ratios are based on simple powers of the [[prime number]]s 2, 3, and 5, this is also known as [[five-limit tuning]].