Editing Pentatonic scale (section)

=== Pythagorean tuning ===
{{further|Pythagorean tuning}}
[[Ben Johnston (composer)|Ben Johnston]] gives the following Pythagorean tuning for the minor pentatonic scale:<ref>[[Ben Johnston (composer)|Ben Johnston]], "Scalar Order as a Compositional Resource", ''[[Perspectives of New Music]]'' 2, no. 2 (Spring–Summer 1964): pp. 56–76. Citation on p.&nbsp;64 {{JSTOR|832482}}.</ref>
{{Listen|filename=A minor pentatonic scale Pythagorean.mid|title=A minor pentatonic scale in Pythagorean tuning}}
{| class="wikitable" style="text-align:center"
! rowspan="4" | Note
! | Solfege
! colspan="2" | A
! colspan="2" | C
! colspan="2" | D
! colspan="2" | E
! colspan="2" | G
! colspan="2" | A
|-
! Ratio
| colspan="2" | [[unison|{{frac|1|1}}]]
| colspan="2" | [[semiditone|{{frac|32|27}}]]
| colspan="2" | [[perfect fourth|{{frac|4|3}}]]
| colspan="2" | [[perfect fifth|{{frac|3|2}}]]
| colspan="2" | [[Pythagorean minor seventh|{{frac|16|9}}]]
| colspan="2" | [[octave|{{frac|2|1}}]]
|-
! Natural
| colspan="2" | 54
| colspan="2" | 64
| colspan="2" | 72
| colspan="2" | 81
| colspan="2" | 96
| colspan="2" | 108
|-
! Audio
| colspan="2" | 1[[File:Unison on C.mid|90px|center]]
| colspan="2" | 3[[File:Pythagorean minor third on C.mid|90px|none]]
| colspan="2" | 4[[File:Just perfect fourth on C.mid|90px|none]]
| colspan="2" | 5[[File:Just perfect fifth on C.mid|90px|none]]
| colspan="2" | 7[[File:Lesser just minor seventh on C.mid|90px|none]]
| colspan="2" | 8[[File:Perfect octave on C.mid|90px|none]]
|-
! rowspan="2" | Step
! Name
| rowspan="2" style="background-color: white; colspan="1" | &nbsp;
| colspan="2" | m3
| colspan="2" | T
| colspan="2" | T
| colspan="2" | m3
| colspan="2" | T
! rowspan="2" style="background-color: white; colspan="1" | &nbsp;
|-
! Ratio
| colspan="2" | {{frac|32|27}}
| colspan="2" | {{frac|9|8}}
| colspan="2" | {{frac|9|8}}
| colspan="2" | {{frac|32|27}}
| colspan="2" | {{frac|9|8}}
|}

Naturals in that table are not the alphabetic series A to G without sharps and flats: Naturals are reciprocals of terms in the [[Harmonic series (mathematics)]], which are in practice multiples of a [[fundamental frequency]]. This may be derived by proceeding with the principle that historically gives the Pythagorean diatonic and chromatic scales, stacking perfect fifths with 3:2 frequency proportions (C–G–D–A–E). Considering the anhemitonic scale as a subset of a [[just intonation|just diatonic scale]], it is tuned thus: 20:24:27:30:36 (A–C–D–E–G = {{frac|5|6}}–{{frac|1|1}}–{{frac|9|8}}–{{frac|5|4}}–{{frac|3|2}}).