Editing Diatonic scale (section)

==Tuning==
Diatonic scales can be tuned variously, either by iteration of a perfect or tempered fifth, or by a combination of perfect fifths and perfect thirds ([[Just intonation]]), or possibly by a combination of fifths and thirds of various sizes, as in [[well temperament]].

===Iteration of the fifth===
{{main|Pythagorean tuning}}
If the scale is produced by the iteration of six perfect fifths, for instance F–C–G–D–A–E–B, the result is [[Pythagorean tuning]]:

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;text-align:center"
|-
! style="text-align:left;" | note
| style="width: 60px;" | F
| style="width: 60px;" | C
| style="width: 60px;" | G
| style="width: 60px;" | D
| style="width: 60px;" | A
| style="width: 60px;" | E
| style="width: 60px;" | B ||
|-
! style="text-align:left;" | pitch
| {{frac|2|3}} || {{frac|1|1}} || {{frac|3|2}} || {{frac|9|4}} || {{frac|10|3}} || {{frac|5|1}} || {{frac|15|2}} ||
|-
! style="text-align:left;" | bring into main octave
| {{frac|4|3}} || {{frac|1|1}} || {{frac|3|2}} || {{frac|9|8}} || {{frac|5|3}} || {{frac|5|4}} || {{frac|15|8}} ||
|-
|
|
|
|
|
|
|
|
|
|-
! style="text-align:left;" | sort into note order
|C||D||E||F||G||A||B|| style="width:6em;" |C'
|-
! style="text-align:left;" | interval above C
| {{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}}
|}

This tuning dates to Ancient Mesopotamia<ref>{{cite book|last=Dumbrill|first=Richard J.|author-link=Richard Dumbrill (musicologist)|title=The Archaeomusicology of the Ancient Near East|page=18|url=https://www.academia.edu/875113|publisher=Tadema Press|location=London|date=1998}} The book title is of 2nd edition; the 1st edition was entitled ''The Musicology and Organology of the Ancient Near East''.</ref> (see {{slink|Music of Mesopotamia|Music theory}}), and was done by alternating ascending fifths with descending fourths (equal to an ascending fifth followed by a descending octave), resulting in the notes of a pentatonic or heptatonic scale falling within an octave.

Six of the "fifth" intervals (C–G, D–A, E–B, F–C', G–D', A–E') are all {{frac|3|2}} = 1.5 (701.955 [[Cent (music)|cents]]), but B–F' is the discordant [[tritone]], here {{frac|729|512}} = 1.423828125 (611.73 cents). Tones are each {{frac|9|8}} = 1.125 (203.91 cents) and diatonic semitones are {{frac|256|243}} ≈ 1.0535 (90.225 cents).

Extending the series of fifths to eleven fifths would result into the Pythagorean [[chromatic scale]].

===Equal temperament===
{{main|Equal temperament#Twelve-tone equal temperament}}
Equal temperament is the division of the octave in twelve equal semitones. The frequency ratio of the semitone then becomes the [[twelfth root of two]] ({{radic|2|12}} ≈ 1.059463, 100 [[Cent (music)|cents]]). The tone is the sum of two semitones. Its ratio is the sixth root of two ({{radic|2|6}} ≈ 1.122462, 200 cents). Equal temperament can be produced by a succession of tempered fifths, each of them with the ratio of 2<sup>{{frac|7|12}}</sup> ≈ 1.498307, 700 cents.

===Meantone temperament===
{{main|Meantone temperament}}
The fifths could be tempered more than in equal temperament, in order to produce better thirds. See [[quarter-comma meantone]] for a meantone temperament commonly used in the sixteenth and seventeenth centuries and sometimes after, which produces perfect major thirds.

===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]].