Editing Just intonation (section)

==Diatonic scale==
{{Main|Five-limit tuning}}

[[File:Primary triads in C.png|thumb|Primary triads in C[[File:Primary triads in C just.mid]]]]
[[File:Just intonation diatonic scale derivation.png|thumb|right|Just tuned diatonic scale derivation.<ref name="C&G"/>]]

The prominent notes of a given scale may be tuned so that their frequencies form (relatively) small whole number ratios.

The 5&nbsp;limit diatonic major scale is tuned in such a way that major triads on the [[Tonic (music)|tonic]], [[subdominant]], and [[Dominant (music)|dominant]] are tuned in the proportion 4:5:6, and minor triads on the [[mediant]] and [[submediant]] are tuned in the proportion 10:12:15. Because of the two sizes of [[major second|wholetone]] – 9:8 (greater wholetone) and 10:9 (lesser wholetone) – the [[supertonic]] must be microtonally lowered by a [[syntonic comma]] to form a pure minor triad.

The 5&nbsp;limit diatonic major scale ([[Ptolemy's intense diatonic scale]]) on C is shown in the table below:<ref name="C&G">{{cite book |first1=Murray |last1=Campbell |first2=Clive |last2=Greated |year=2001 |orig-year=1987 |title=The Musician's Guide to Acoustics |place=London, UK & New York, NY |publisher=Oxford University Press |pages=172–173 |edition=reprint of 1st |isbn=978-0-19-816505-7}}</ref><ref>{{cite book |last=Wright |first=David |year=2009 |title=Mathematics and Music |series=Mathematical World |volume=28 |place=Providence, RI |publisher=[[American Mathematical Society]] |pages=140–141 |isbn=978-0-8218-4873-9 }}</ref><ref name=Johnson2006>{{cite book |last=Johnston |first=Ben |author-link=Ben Johnston (composer) |year=2006 |orig-year=2003 |article=A notation system for extended just intonation |title=''Maximum Clarity'' and Other Writings on Music |editor-last=Gilmore |editor-first=Bob |editor-link=Bob Gilmore |place=Urbana and Chicago, IL |publisher=University of Illinois Press |pages=77–88 |isbn=978-0-252-03098-7 }}</ref>{{rp|page=78}}<ref>{{cite book |last=Partch |first=Harry |author-link=Harry Partch |year=1979 |title=Genesis of a Music |title-link=Genesis of a Music |pages=165, 73 |isbn=978-0-306-80106-8 }}</ref>
:{| class="wikitable" style="text-align:center;"
|+ [[Ptolemy's intense diatonic scale|Diatonic, justly tuned, major scale]]
! rowspan="4" | Note
! style="width:6em;" | Name
! colspan="2" style="width: 3em" | C
! colspan="2" style="width: 3em" | D
! colspan="2" style="width: 3em" | E
! colspan="2" style="width: 3em" | F
! colspan="2" style="width: 3em" | G
! colspan="2" style="width: 3em" | A
! colspan="2" style="width: 3em" | B
! colspan="2" style="width: 3em" | C
|-
! Ratio from&nbsp;C
| colspan="2" style="background: #fbb" | 1:1
| colspan="2" style="background: #8f6" | 9:8
| colspan="2" style="background: #aeb" | 5:4
| colspan="2" style="background: #fe8" | 4:3
| colspan="2" style="background: #fc8" | 3:2
| colspan="2" style="background: #fe8" | 5:3
| colspan="2" style="background: #8f6" | 15:8
| colspan="2" style="background: #fe8" | 2:1
|-
! {{small|Harmonic of nominal fundamental&nbsp;F}}
| colspan="2" | 24
| colspan="2" | 27
| colspan="2" | 30
| colspan="2" | 32
| colspan="2" | 36
| colspan="2" | 40
| colspan="2" | 45
| colspan="2" | 48
|-
! [[cents (music)|Cents]]
| colspan="2" | 0
| colspan="2" | 204
| colspan="2" | 386
| colspan="2" | 498
| colspan="2" | 702
| colspan="2" | 884
| colspan="2" | 1088
| colspan="2" | 1200
|- <!--- this row inserted to avoid bug in Chrome causing misalignment of columns -- >
| || || || || || || || || || || || || || || || || ||  -->
| colspan="18" | 
|-
! rowspan="3" | Step
! Symbol
| rowspan="3" style="background-color: white; width:1.5em" | 
| colspan="2" style="width:3em" | {{math|T}} 
| colspan="2" style="width:3em" | {{math|t}} 
| colspan="2" style="width:3em" | {{math|s}} 
| colspan="2" style="width:3em" | {{math|T}} 
| colspan="2" style="width:3em" | {{math|t}} 
| colspan="2" style="width:3em" | {{math|T}} 
| colspan="2" style="width:3em" | {{math|s}} 
| rowspan="3" style="background-color: white; width: 1.5em" | 
|-
! Ratio
| colspan="2" style="background: #8f6" | 9:8
| colspan="2" style="background: #5f3" | 10:9
| colspan="2" style="background: #2b2" | 16:15
| colspan="2" style="background: #8f6" | 9:8
| colspan="2" style="background: #5f3" | 10:9
| colspan="2" style="background: #8f6" | 9:8
| colspan="2" style="background: #2a2" | 16:15
|-
! [[cents (music)|Cents]]
| colspan="2" | 204
| colspan="2" | 182
| colspan="2" | 112
| colspan="2" | 204
| colspan="2" | 182
| colspan="2" | 204
| colspan="2" | 112
|}
[[File:C_major_just_intonation_vs_equal_temperament.svg|thumb|Intervals of the C&nbsp;major scale in just intonation, compared with [[12 equal temperament|equal temperament]] (blue)]]
In this example the interval from D up to A would be a [[Wolf interval|wolf fifth]] with the ratio {{sfrac| 40 | 27 }}, about 680&nbsp;[[cents (music)|cents]]; noticeably smaller than the 702&nbsp;cents of the pure {{sfrac| 3 | 2 }} ratio. This is mentioned by [[Heinrich Schenker|Schenker]] in reference to the teaching of Bruckner.<ref>{{cite book |first=Heinrich |last=Schenker |author-link=Heinrich Schenker |year=2001 |orig-year=1935 |title=Free Composition |title-link=Free Composition |publisher=Pendragon Press |isbn=9781576470749 |at=p.&nbsp;8, footnote&nbsp;14 |quote=Bruckner used to teach that even the sixth tone of the diatonic scale was dissonant and therefore had to be resolved downward. }}</ref>

For a justly tuned diatonic minor scale, the mediant is tuned 6:5 and the submediant is tuned 8:5. It would include a tuning of 9:5 for the [[subtonic]]. For example, on A:
:{| class="wikitable" style="text-align: center"
|+ [[Ptolemy's intense diatonic scale|Diatonic, justly tuned, minor scale]]
! rowspan="4" | Note
! style="width:6em;" | Name
! colspan="2" style="width: 3em" | A
! colspan="2" style="width: 3em" | B
! colspan="2" style="width: 3em" | C
! colspan="2" style="width: 3em" | D
! colspan="2" style="width: 3em" | E
! colspan="2" style="width: 3em" | F
! colspan="2" style="width: 3em" | G
! colspan="2" style="width: 3em" | A
|-
! Ratio from A
| colspan="2" style="background: #fbb" | 1:1
| colspan="2" style="background: #8f6" | 9:8
| colspan="2" style="background: #fe4" | 6:5
| colspan="2" style="background: #fe8" | 4:3
| colspan="2" style="background: #fc8" | 3:2
| colspan="2" style="background: #fe4" | 8:5
| colspan="2" style="background: #fe4" | 9:5
| colspan="2" style="background: #fbb" | 2:1
|-
! {{small|Harmonic of nominal fundamental&nbsp;B{{sup|{{music|b}}}} }}
| colspan="2" | 120
| colspan="2" | 135
| colspan="2" | 144
| colspan="2" | 160
| colspan="2" | 180
| colspan="2" | 192
| colspan="2" | 216
| colspan="2" | 240
|-
! [[cents (music)|Cents]]
| colspan="2" | 0
| colspan="2" | 204
| colspan="2" | 316
| colspan="2" | 498
| colspan="2" | 702
| colspan="2" | 814
| colspan="2" | 1018
| colspan="2" | 1200
|- <!--- this row inserted to avoid bug in Chrome causing misalignment of columns -- >
| || || || || || || || || || || || || || || || || ||  -->
| colspan="18" | 
|-
! rowspan="3" | Step
! Name
| rowspan="3" style="background-color: white; width:1.5em" | 
| colspan="2" style="width:3em" | T
| colspan="2" style="width:3em" | s
| colspan="2" style="width:3em" | t
| colspan="2" style="width:3em" | T
| colspan="2" style="width:3em" | s
| colspan="2" style="width:3em" | T
| colspan="2" style="width:3em" | t
| rowspan="3" style="background-color: white; width: 1.5em" | 
|-
! Ratio
| colspan="2" style="background: #8f6" | 9:8
| colspan="2" style="background: #2a2" | 16:15
| colspan="2" style="background: #5f3" | 10:9
| colspan="2" style="background: #8f6" | 9:8
| colspan="2" style="background: #2a2" | 16:15
| colspan="2" style="background: #8f6" | 9:8
| colspan="2" style="background: #5f3" | 10:9
|-
! [[cents (music)|Cents]]
| colspan="2" | 204
| colspan="2" | 112
| colspan="2" | 182
| colspan="2" | 204
| colspan="2" | 112
| colspan="2" | 204
| colspan="2" | 182
|}