Editing Pentatonic scale (section)

== Tuning ==
=== 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}}).

=== Just intonation ===
{{further|Just intonation}}

[[File:Lou Harrison - Old Granddad pentatonic tuning.png|thumb|Just pentatonic tuning of Lou Harrison's "[[American gamelan]]", Old Granddad.<ref>{{cite journal|last1=Miller|first1=Leta E.|last2=Lieberman|first2=Fredric|author2-link=Fredric Lieberman|date=Summer 1999|title=Lou Harrison and the American Gamelan|journal=[[American Music (journal)|American Music]]|volume=17|number=2|pages=146–178 (158)|doi=10.2307/3052712 |jstor=3052712 }}</ref> This gives the proportions 24:27:30:36:40.[[File:Lou Harrison - Old Granddad pentatonic tuning.mid]]]]

{| class="wikitable"
|-
! Modes
! Ratios (just)
|-
! [[#Major pentatonic scale|Major]]
| 24:27:30:36:40
|-
! Blues major
| 24:27:32:36:40
|-
! Suspended
| 24:27:32:36:42
|-
! [[#Minor pentatonic scale|Minor]]
| 30:36:40:45:54
|-
! Blues minor
| 15:18:20:24:27
|}
(A minor seventh can be 7:4, 16:9, or 9:5; a major sixth can be 27:16 or 5:3. Both were chosen to minimize ratio parts.)

=== Other ===
Assigning precise frequency proportions to the pentatonic scales of most cultures is problematic as tuning may be variable.

[[File:Slendro on C.png|thumb|300px|Slendro approximated in Western notation.<ref>"The representations of ''slendro'' and ''pelog'' tuning systems in Western notation shown above should not be regarded in any sense as absolute. Not only is it difficult to convey non-Western scales with Western notation&nbsp;..." Jennifer Lindsay, ''Javanese Gamelan'' (Oxford and New York: Oxford University Press, 1992), pp.&nbsp;39–41. {{ISBN|0-19-588582-1}}.</ref>[[File:Slendro on C.mid]]]]
For example, the [[slendro]] anhemitonic scale and its modes of Java and Bali are said to approach, very roughly, an equally-tempered five-note scale,<ref>Lindsay (1992), p.&nbsp;38–39: "Slendro is made up of five equal, or relatively equal, intervals".</ref> but their tunings vary dramatically from [[gamelan]] to gamelan.<ref>"... in general, no two gamelan sets will have exactly the same tuning, either in pitch or in interval structure. There are no Javanese standard forms of these two tuning systems." Lindsay (1992), pp.&nbsp;39–41.</ref>

Composer [[Lou Harrison]] has been one of the most recent proponents and developers of new pentatonic scales based on historical models. Harrison and [[William Colvig]] tuned the slendro scale of the [[gamelan]] Si Betty to overtones 16:19:21:24:28{{sfn|Miller|Lieberman|1999|p=159}} ({{frac|1|1}}–{{frac|19|16}}–{{frac|21|16}}–{{frac|3|2}}–{{frac|7|4}}). They tuned the Mills gamelan so that the intervals between scale steps are [[septimal whole tone|8:7]]–[[subminor third|7:6]]–[[major second|9:8]]–8:7–7:6{{sfn|Miller|Lieberman|1999|p=161}} ({{frac|1|1}}–{{frac|8|7}}–{{frac|4|3}}–{{frac|3|2}}–[[septimal major sixth|{{frac|12|7}}]]–{{frac|2|1}} = 42:48:56:63:72)