31 equal temperament

File:Syntonic tuning continuum.svg continuum at {{nobr|p5 {{=}} 696.77 cents}}

{{cite journal

|last1 = Milne |first1 = A.

|last2 = Sethares |first2 = W.A. |author2-link = William Sethares

|last3 = Plamondon |first3 = J.

|date = Winter 2007

|title = Isomorphic controllers and dynamic tuning: Invariant fingerings across a tuning continuum

|journal = Computer Music Journal

|volume = 31 |issue = 4 |pages = 15–32

|url = http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15

|via = mitpressjournals.org

}}

]]

In music, 31 equal temperament, {{nobr| 31 {{sc|ET}},}} which can also be abbreviated {{nobr| 31 {{sc|TET}}}} (31 tone {{sc|ET}}) or {{nobr|31 equal division of the octave}} (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equally-proportioned steps (equal frequency ratios). {{audio|31-tet scale on C.mid|Play}} Each step represents a frequency ratio of {{radic|2 |31}}, or 38.71 cents ({{Audio|1 step in 31-et on C.mid|Play}}).

{{nobr|31 {{sc|EDO}}}} is a very good approximation of quarter-comma meantone temperament. More generally, it is a regular diatonic tuning in which the tempered perfect fifth is equal to 696.77 cents, as shown in Figure 1. On an isomorphic keyboard, the fingering of music composed in {{nobr|31 {{sc|EDO}}}} is precisely the same as it is in any other syntonic tuning (such as {{nobr|12 equal temperament),}} so long as the notes are spelled properly—that is, with no assumption of enharmonicity.

History and use

Division of the octave into 31 steps arose naturally out of Renaissance music theory; the lesser diesis{{px2}} – the ratio of an octave to three major thirds, 128:125 or 41.06 cents – was approximately one-fifth of a tone or two-fifths of a semitone. In 1555, Nicola Vicentino proposed an extended-meantone tuning of 31 tones. In 1666, Lemme Rossi first proposed an equal temperament of this order. In 1691, having discovered it independently, scientist Christiaan Huygens wrote about it also.

{{cite web

|last = Monzo |first = Joe

|year = 2005

|title = Equal temperament

|website = Tonalsoft Encyclopedia of Microtonal Music Theory

|publisher = Joe Monzo / Tonalsoft

|url = http://tonalsoft.com/enc/e/equal-temperament.aspx#edo-table

|access-date = 28 February 2019

}}

Since the standard system of tuning at that time was quarter-comma meantone, in which the fifth is tuned to {{radic|5 |4}}, the appeal of this method was immediate, as the fifth of {{nobr|31 {{sc|EDO}}}}, at 696.77 cents, is only 0.19 cent wider than the fifth of quarter-comma meantone. Huygens not only realized this, he went farther and noted that {{nobr|31 {{sc|EDO}}}} provides an excellent approximation of septimal, or 7 limit harmony.

In the twentieth century, physicist, music theorist, and composer Adriaan Fokker, after reading Huygens's work, led a revival of interest in this system of tuning which led to a number of compositions, particularly by Dutch composers. Fokker designed the Fokker organ, a 31 tone equal-tempered organ, which was installed in Teyler's Museum in Haarlem in 1951 and moved to Muziekgebouw aan 't IJ in 2010 where it has been frequently used in concerts since it moved.

Interval size

Image:31ed2.svg

Here are the sizes of some common intervals:

class="wikitable sortable"

style="text-align:center;vertical-align:bottom;background:#FFFFB4;"

! interval name

! size
{{small|(steps)}}

! size
{{small|(cents)}}

!class="unsortable"| {{sc|midi}}
audio

! just
ratio

! just
{{small|(cents)}}

!class="unsortable"| {{sc|midi}}
audio

! error
{{small|(cents)}}

style="text-align:center;"

| octave

| 31

| 1200

|

| 2:1

| 1200

|

| 0

style="text-align:center;background:#D4D4D4;"

| minor seventh

| 26

| 1006.45

|

| 9:5

| 1017.60

|

| −11.15

style="text-align:center;background:#D4D4D4;"

| grave just minor seventh

| 26

| 1006.45

|

| 16:9{{0}}

| 996.09

|

| +10.36

style="text-align:center;"

| harmonic seventh, subminor seventh, augmented sixth

| 25

| 967.74

| {{Audio|25 steps in 31-et on C.mid|Play}}

| 7:4

| {{0}}968.83

| {{Audio|Harmonic seventh on C.mid|Play}}

| −{{0}}1.09

style="text-align:center;"

| minor sixth

| 21

| {{0}}812.90

| {{Audio|21 steps in 31-et on C.mid|Play}}

| 8:5

| {{0}}813.69

| {{Audio|Just minor sixth on C.mid|Play}}

| −{{0}}0.78

style="text-align:center;"

| perfect fifth

| 18

| {{0}}696.77

| {{Audio|18 steps in 31-et on C.mid|Play}}

| 3:2

| {{0}}701.96

| {{Audio|Just perfect fifth on C.mid|Play}}

| −{{0}}5.19

style="text-align:center;"

| greater septimal tritone, diminished fifth

| 16

| 619.35

|

| 10:7{{0}}

| {{0}}617.49

|

| +{{0}}1.87

style="text-align:center;"

| lesser septimal tritone, augmented fourth

| 15

| {{0}}580.65

| {{Audio|15 steps in 31-et on C.mid|Play}}

| 7:5

| {{0}}582.51

| {{Audio|Lesser septimal tritone on C.mid|Play}}

| −{{0}}1.86

style="text-align:center;"

| undecimal tritone, half augmented fourth, 11th harmonic

| 14

| {{0}}541.94

| {{Audio|14 steps in 31-et on C.mid|Play}}

| 11:8{{0}}

| {{0}}551.32

| {{Audio|Eleventh harmonic on C.mid|Play}}

| −{{0}}9.38

style="text-align:center;"

| perfect fourth

| 13

| {{0}}503.23

| {{Audio|13 steps in 31-et on C.mid|Play}}

| 4:3

| {{0}}498.04

| {{Audio|Just perfect fourth on C.mid|Play}}

| +{{0}}5.19

style="text-align:center;"

| septimal narrow fourth, half diminished fourth

| 12

| {{0}}464.52

| {{Audio|12 steps in 31-et on C.mid|Play}}

| 21:16

| {{0}}470.78

| {{Audio|Twenty-first harmonic on C.mid|Play}}

| −{{0}}6.26

style="text-align:center;background:#D4D4D4;"

| tridecimal augmented third, and greater major third

| 12

| {{0}}464.52

| {{Audio|12 steps in 31-et on C.mid|Play}}

| 13:10

| {{0}}454.21

| {{Audio|Tridecimal major third on C.mid|Play}}

| +10.31

style="text-align:center;"

| septimal major third

| 11

| {{0}}425.81

| {{Audio|11 steps in 31-et on C.mid|Play}}

| 9:7

| {{0}}435.08

| {{Audio|Septimal major third on C.mid|Play}}

| −{{0}}9.27

style="text-align:center;"

| diminished fourth

| 11

| {{0}}425.81

| {{Audio|11 steps in 31-et on C.mid|Play}}

| 32:25

| {{0}}427.37

| {{Audio|Just diminished fourth on C.mid|Play}}

| −{{0}}1.56

style="text-align:center;"

| undecimal major third

| 11

| {{0}}425.81

| {{Audio|11 steps in 31-et on C.mid|Play}}

| 14:11

| {{0}}417.51

| {{Audio|Undecimal major third on C.mid|Play}}

| +{{0}}8.30

style="text-align:center;"

| major third

| 10

| {{0}}387.10

| {{Audio|10 steps in 31-et on C.mid|Play}}

| 5:4

| {{0}}386.31

| {{Audio|Just major third on C.mid|Play}}

| +{{0}}0.79

style="text-align:center;background:#D4D4D4;"

| tridecimal neutral third

| {{0}}9

| {{0}}348.39

| {{Audio|9 steps in 31-et on C.mid|Play}}

| 16:13

| {{0}}359.47

| {{Audio|Tridecimal neutral third on C.mid|Play}}

| −11.09

style="text-align:center;"

| undecimal neutral third

| {{0}}9

| {{0}}348.39

| {{Audio|9 steps in 31-et on C.mid|Play}}

| 11:9{{0}}

| {{0}}347.41

| {{Audio|Undecimal neutral third on C.mid|Play}}

| +{{0}}0.98

style="text-align:center;"

| minor third

| {{0}}8

| {{0}}309.68

| {{Audio|8 steps in 31-et on C.mid|Play}}

| 6:5

| {{0}}315.64

| {{Audio|Just minor third on C.mid|Play}}

| −{{0}}5.96

style="text-align:center;"

| septimal minor third

| {{0}}7

| {{0}}270.97

| {{Audio|7 steps in 31-et on C.mid|Play}}

| 7:6

| {{0}}266.87

| {{Audio|Septimal minor third on C.mid|Play}}

| +{{0}}4.10

style="text-align:center;"

| septimal whole tone

| {{0}}6

| {{0}}232.26

| {{Audio|6 steps in 31-et on C.mid|Play}}

| 8:7

| {{0}}231.17

| {{Audio|Septimal major second on C.mid|Play}}

| +{{0}}1.09

style="text-align:center;background:#D4D4D4;"

| whole tone, major tone

| {{0}}5

| {{0}}193.55

| {{Audio|5 steps in 31-et on C.mid|Play}}

| 9:8

| {{0}}203.91

| {{Audio|Major tone on C.mid|Play}}

| −10.36

style="text-align:center;"

| whole tone, major second

| {{0}}5

| {{0}}193.55

| {{Audio|5 steps in 31-et on C.mid|Play}}

| 28:25

| {{0}}196.20

|

| −{{0}}2.65

style="text-align:center;"

| mean tone, major second

| {{0}}5

| {{0}}193.55

|

| {{small|{{sfrac| 1 | 2 }}}}{{sqrt|5 }}

| {{0}}193.16

|

| +{{0}}0.39

style="text-align:center;background:#D4D4D4;"

| whole tone, minor tone

| {{0}}5

| 193.55

| {{Audio|5 steps in 31-et on C.mid|Play}}

| 10:9{{0}}

| {{0}}182.40

| {{Audio|Minor tone on C.mid|Play}}

| +11.15

style="text-align:center;background:#D4D4D4;"

| greater undecimal neutral second

| {{0}}4

| {{0}}154.84

| {{Audio|4 steps in 31-et on C.mid|Play}}

| 11:10

| {{0}}165.00

|

| −10.16

style="text-align:center;"

| lesser undecimal neutral second

| {{0}}4

| {{0}}154.84

| {{Audio|4 steps in 31-et on C.mid|Play}}

| 12:11

| {{0}}150.64

| {{Audio|Lesser undecimal neutral second on C.mid|Play}}

| +{{0}}4.20

style="text-align:center;"

| septimal diatonic semitone

| {{0}}3

| {{0}}116.13

| {{Audio|3 steps in 31-et on C.mid|Play}}

| 15:14

| {{0}}119.44

| {{Audio|Septimal diatonic semitone on C.mid|Play}}

| −{{0}}3.31

style="text-align:center;"

| diatonic semitone, minor second

| {{0}}3

| {{0}}116.13

| {{Audio|3 steps in 31-et on C.mid|Play}}

| 16:15

| {{0}}111.73

| {{Audio|Just diatonic semitone on C.mid|Play}}

| +{{0}}4.40

style="text-align:center;"

| septimal chromatic semitone

| {{0}}2

| {{0}}{{0}}77.42

| {{Audio|2 steps in 31-et on C.mid|Play}}

| 21:20

| {{0}}{{0}}84.47

| {{audio|Septimal chromatic semitone on C.mid|Play}}

| −{{0}}7.05

style="text-align:center;"

| chromatic semitone, augmented unison

| {{0}}2

| {{0}}{{0}}77.42

| {{Audio|2 steps in 31-et on C.mid|Play}}

| 25:24

| {{0}}{{0}}70.67

| {{Audio|Just chromatic semitone on C.mid|Play}}

| +{{0}}6.75

style="text-align:center;"

| lesser diesis

| {{0}}1

| {{0}}{{0}}38.71

| {{Audio|1 step in 31-et on C.mid|Play}}

| 128:125

| {{0}}{{0}}41.06

| {{Audio|Septimal diesis on C.mid|Play}}

| −{{0}}2.35

style="text-align:center;"

| undecimal diesis

| {{0}}1

| {{0}}{{0}}38.71

| {{Audio|1 step in 31-et on C.mid|Play}}

| 45:44

| {{0}}{{0}}38.91

| {{Audio|Undecimal diesis on C.mid|Play}}

| −{{0}}0.20

style="text-align:center;"

| septimal diesis

| {{0}}1

| {{0}}{{0}}38.71

| {{Audio|1 step in 31-et on C.mid|Play}}

| 49:48

| {{0}}{{0}}35.70

| {{Audio|Septimal diesis on C.mid|Play}}

| +{{0}}3.01

The 31 equal temperament has a very close fit to the 7:6, 8:7, and 7:5 ratios, which have no approximate fits in 12 equal temperament and only poor fits in 19 equal temperament. The composer Joel Mandelbaum (born 1932) used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series.{{refn|name=Keislar-1991|

{{cite journal

|last=Keislar |first=Douglas

|date=Winter 1991

|title=Six American composers on nonstandard tunnings: Easley Blackwood; John Eaton; Lou Harrison; Ben Johnston; Joel Mandelbaum; William Schottstaedt

|journal=Perspectives of New Music

|volume=29 |issue=1 |pages=176–211 |JSTOR=833076

}}{{efn|

The following composers mentioned in the title of {{harvp|Keislar|1991}}'s journal article{{refn|name=Keislar-1991}} have Wikipedia articles:

: {{nobr|Easley Blackwood}}

: {{nobr|John Eaton}}

: {{nobr|Lou Harrison}}

: {{nobr|Ben Johnston}}

: {{nobr|Joel Mandelbaum}}

}}

The tuning has poor matches to both the 9:8 and 10:9 intervals (major and minor tone in just intonation); however, it has a good match for the average of the two. Practically it is very close to quarter-comma meantone.

This tuning can be considered a meantone temperament. It has the necessary property that a chain of its four fifths is equivalent to its major third (the syntonic comma 81:80 is tempered out), which also means that it contains a "meantone" that falls between the sizes of 10:9 and 9:8 as the combination of one of each of its chromatic and diatonic semitones.

Scale diagram

File:31edo CoF semi and sesqui.png in 31 equal temperament]]

The following are the 31 notes in the scale:

class="wikitable"

| align="center" bgcolor="#ffeeee" | Interval (cents)

| bgcolor="#000000" |

| colspan="2" align="center" bgcolor="#ffeeee" | 39

| bgcolor="#000000" |

align="center" bgcolor="#fffbee" | Note
name

| colspan="2" align="center" bgcolor="#fffbee" | A

| colspan="2" align="center" bgcolor="#fffbee" | {{nobr|G{{sup|𝄪♯}}}}
B{{sup|𝄫}}

| colspan="2" align="center" bgcolor="#fffbee" | A{{sup|♯}}

| colspan="2" align="center" bgcolor="#fffbee" | B{{sup|♭}}

| colspan="2" align="center" bgcolor="#fffbee" | A{{sup|𝄪}}
C{{sup|𝄫}}

| colspan="2" align="center" bgcolor="#fffbee" | B

| colspan="2" align="center" bgcolor="#fffbee" | C{{sup|♭}}

| colspan="2" align="center" bgcolor="#fffbee" | B{{sup|♯}}

| colspan="2" align="center" bgcolor="#fffbee" | C

| colspan="2" align="center" bgcolor="#fffbee" | B{{sup|𝄪}}
D{{sup|𝄫}}

| colspan="2" align="center" bgcolor="#fffbee" | C{{sup|♯}}

| colspan="2" align="center" bgcolor="#fffbee" | D{{sup|♭}}

| colspan="2" align="center" bgcolor="#fffbee" | C{{sup|𝄪}}
{{nobr|E{{sup|♭𝄫}}}}

| colspan="2" align="center" bgcolor="#fffbee" | D

| colspan="2" align="center" bgcolor="#fffbee" | {{nobr|C{{sup|𝄪♯}}}}
E{{sup|𝄫}}

| colspan="2" align="center" bgcolor="#fffbee" | D{{sup|♯}}

| colspan="2" align="center" bgcolor="#fffbee" | E{{sup|♭}}

| colspan="2" align="center" bgcolor="#fffbee" | D{{sup|𝄪}}
F{{sup|𝄫}}

| colspan="2" align="center" bgcolor="#fffbee" | E

| colspan="2" align="center" bgcolor="#fffbee" | F{{sup|♭}}

| colspan="2" align="center" bgcolor="#fffbee" | E{{sup|♯}}

| colspan="2" align="center" bgcolor="#fffbee" | F

| colspan="2" align="center" bgcolor="#fffbee" | E{{sup|𝄪}}
G{{sup|𝄫}}

| colspan="2" align="center" bgcolor="#fffbee" | F{{sup|♯}}

| colspan="2" align="center" bgcolor="#fffbee" | G{{sup|♭}}

| colspan="2" align="center" bgcolor="#fffbee" | F{{sup|𝄪}}
A{{sup|♭𝄫}}

| colspan="2" align="center" bgcolor="#fffbee" | G

| colspan="2" align="center" bgcolor="#fffbee" | {{nobr|F{{sup|𝄪♯}}}}
A{{sup|𝄫}}

| colspan="2" align="center" bgcolor="#fffbee" | G{{sup|♯}}

| colspan="2" align="center" bgcolor="#fffbee" | A{{sup|♭}}

| colspan="2" align="center" bgcolor="#fffbee" | G{{sup|𝄪}}
B{{sup|♭𝄫}}

| colspan="2" align="center" bgcolor="#fffbee" | A

align="center" bgcolor="#eeeeff" | Note (cents)

| colspan="2" align="center" bgcolor="#eeeeff" | 0

| colspan="2" align="center" bgcolor="#eeeeff" | 39

| colspan="2" align="center" bgcolor="#eeeeff" | 77

| colspan="2" align="center" bgcolor="#eeeeff" | 116

| colspan="2" align="center" bgcolor="#eeeeff" | 155

| colspan="2" align="center" bgcolor="#eeeeff" | 194

| colspan="2" align="center" bgcolor="#eeeeff" | 232

| colspan="2" align="center" bgcolor="#eeeeff" | 271

| colspan="2" align="center" bgcolor="#eeeeff" | 310

| colspan="2" align="center" bgcolor="#eeeeff" | 348

| colspan="2" align="center" bgcolor="#eeeeff" | 387

| colspan="2" align="center" bgcolor="#eeeeff" | 426

| colspan="2" align="center" bgcolor="#eeeeff" | 465

| colspan="2" align="center" bgcolor="#eeeeff" | 503

| colspan="2" align="center" bgcolor="#eeeeff" | 542

| colspan="2" align="center" bgcolor="#eeeeff" | 581

| colspan="2" align="center" bgcolor="#eeeeff" | 619

| colspan="2" align="center" bgcolor="#eeeeff" | 658

| colspan="2" align="center" bgcolor="#eeeeff" | 697

| colspan="2" align="center" bgcolor="#eeeeff" | 735

| colspan="2" align="center" bgcolor="#eeeeff" | 774

| colspan="2" align="center" bgcolor="#eeeeff" | 813

| colspan="2" align="center" bgcolor="#eeeeff" | 852

| colspan="2" align="center" bgcolor="#eeeeff" | 890

| colspan="2" align="center" bgcolor="#eeeeff" | 929

| colspan="2" align="center" bgcolor="#eeeeff" | 968

| colspan="2" align="center" bgcolor="#eeeeff" | 1006

| colspan="2" align="center" bgcolor="#eeeeff" | 1045

| colspan="2" align="center" bgcolor="#eeeeff" | 1084

| colspan="2" align="center" bgcolor="#eeeeff" | 1123

| colspan="2" align="center" bgcolor="#eeeeff" | 1161

| colspan="2" align="center" bgcolor="#eeeeff" | 1200

The five "double flat" notes and five "double sharp" notes may be replaced by half sharps and half flats, similar to the quarter tone system:

class="wikitable"

| align="center" bgcolor="#ffeeee" | Interval (cents)

| bgcolor="#000000" |

| colspan="2" align="center" bgcolor="#ffeeee" | 39

| bgcolor="#000000" |

align="center" bgcolor="#fffbee" | Note name

| colspan="2" align="center" bgcolor="#fffbee" | A

| colspan="2" align="center" bgcolor="#fffbee" | A{{music| t}}

| colspan="2" align="center" bgcolor="#fffbee" | A♯

| colspan="2" align="center" bgcolor="#fffbee" | B♭

| colspan="2" align="center" bgcolor="#fffbee" | B{{music| d}}

| colspan="2" align="center" bgcolor="#fffbee" | B

| colspan="2" align="center" bgcolor="#fffbee" | B{{music| t}}

| colspan="2" align="center" bgcolor="#fffbee" | C{{music| d}}

| colspan="2" align="center" bgcolor="#fffbee" | C

| colspan="2" align="center" bgcolor="#fffbee" | C{{music| t}}

| colspan="2" align="center" bgcolor="#fffbee" | C♯

| colspan="2" align="center" bgcolor="#fffbee" | D♭

| colspan="2" align="center" bgcolor="#fffbee" | D{{music| d}}

| colspan="2" align="center" bgcolor="#fffbee" | D

| colspan="2" align="center" bgcolor="#fffbee" | D{{music| t}}

| colspan="2" align="center" bgcolor="#fffbee" | D♯

| colspan="2" align="center" bgcolor="#fffbee" | E♭

| colspan="2" align="center" bgcolor="#fffbee" | E{{music| d}}

| colspan="2" align="center" bgcolor="#fffbee" | E

| colspan="2" align="center" bgcolor="#fffbee" | E{{music| t}}

| colspan="2" align="center" bgcolor="#fffbee" | F{{music| d}}

| colspan="2" align="center" bgcolor="#fffbee" | F

| colspan="2" align="center" bgcolor="#fffbee" | F{{music| t}}

| colspan="2" align="center" bgcolor="#fffbee" | F♯

| colspan="2" align="center" bgcolor="#fffbee" | G♭

| colspan="2" align="center" bgcolor="#fffbee" | G{{music| d}}

| colspan="2" align="center" bgcolor="#fffbee" | G

| colspan="2" align="center" bgcolor="#fffbee" | G{{music| t}}

| colspan="2" align="center" bgcolor="#fffbee" | G♯

| colspan="2" align="center" bgcolor="#fffbee" | A♭

| colspan="2" align="center" bgcolor="#fffbee" | A{{music| d}}

| colspan="2" align="center" bgcolor="#fffbee" | A

align="center" bgcolor="#eeeeff" | Note (cents)

| colspan="2" align="center" bgcolor="#eeeeff" | 0

| colspan="2" align="center" bgcolor="#eeeeff" | 39

| colspan="2" align="center" bgcolor="#eeeeff" | 77

| colspan="2" align="center" bgcolor="#eeeeff" | 116

| colspan="2" align="center" bgcolor="#eeeeff" | 155

| colspan="2" align="center" bgcolor="#eeeeff" | 194

| colspan="2" align="center" bgcolor="#eeeeff" | 232

| colspan="2" align="center" bgcolor="#eeeeff" | 271

| colspan="2" align="center" bgcolor="#eeeeff" | 310

| colspan="2" align="center" bgcolor="#eeeeff" | 348

| colspan="2" align="center" bgcolor="#eeeeff" | 387

| colspan="2" align="center" bgcolor="#eeeeff" | 426

| colspan="2" align="center" bgcolor="#eeeeff" | 465

| colspan="2" align="center" bgcolor="#eeeeff" | 503

| colspan="2" align="center" bgcolor="#eeeeff" | 542

| colspan="2" align="center" bgcolor="#eeeeff" | 581

| colspan="2" align="center" bgcolor="#eeeeff" | 619

| colspan="2" align="center" bgcolor="#eeeeff" | 658

| colspan="2" align="center" bgcolor="#eeeeff" | 697

| colspan="2" align="center" bgcolor="#eeeeff" | 735

| colspan="2" align="center" bgcolor="#eeeeff" | 774

| colspan="2" align="center" bgcolor="#eeeeff" | 813

| colspan="2" align="center" bgcolor="#eeeeff" | 852

| colspan="2" align="center" bgcolor="#eeeeff" | 890

| colspan="2" align="center" bgcolor="#eeeeff" | 929

| colspan="2" align="center" bgcolor="#eeeeff" | 968

| colspan="2" align="center" bgcolor="#eeeeff" | 1006

| colspan="2" align="center" bgcolor="#eeeeff" | 1045

| colspan="2" align="center" bgcolor="#eeeeff" | 1084

| colspan="2" align="center" bgcolor="#eeeeff" | 1123

| colspan="2" align="center" bgcolor="#eeeeff" | 1161

| colspan="2" align="center" bgcolor="#eeeeff" | 1200

= [[Mode (music)|Modes]] =

== [[Ionian mode]] ([[major scale]]) ==

class="wikitable"

! Key signature

! colspan="7" |Scale

! Number of
sharps

!

! Key signature

! colspan="7" |Scale

! Number of
flats

C major

|C

|D

|E

|F

|G

|A

|B

|0 (no sharps or flats)

| rowspan="10" |

| colspan="7" rowspan="10" |

| rowspan="10" |

G major

|G

|A

|B

|C

|D

|E

|F♯

|1

D major

|D

|E

|F♯

|G

|A

|B

|C♯

|2

A major

|A

|B

|C♯

|D

|E

|F♯

|G♯

|3

E major

|E

|F♯

|G♯

|A

|B

|C♯

|D♯

|4

B major

|B

|C♯

|D♯

|E

|F♯

|G♯

|A♯

|5

F♯ major

|F♯

|G♯

|A♯

|B

|C♯

|D♯

|E♯

|6

C♯ major

|C♯

|D♯

|E♯

|F♯

|G♯

|A♯

|B♯

|7

G♯ major

|G♯

|A♯

|B♯

|C♯

|D♯

|E♯

|F𝄪

|8

D♯ major

|D♯

|E♯

|F𝄪

|G♯

|A♯

|B♯

|C𝄪

|9

A♯ major

|A♯

|B♯

|F𝄪

|D♯

|E♯

|F𝄪

|G𝄪

|10

|

|C♭𝄫 major

|C♭𝄫

|D♭𝄫

|E♭𝄫

|F♭𝄫

|G♭𝄫

|A♭𝄫

|B♭𝄫

|21

E♯ major

|E♯

|F𝄪

|G𝄪

|A♯

|B♯

|F𝄪

|D𝄪

|11

|

|G♭𝄫 major

|G♭𝄫

|A♭𝄫

|B♭𝄫

|C♭𝄫

|D♭𝄫

|E♭𝄫

|F𝄫

|20

B♯ major

|B♯

|C𝄪

|D𝄪

|E♯

|F𝄪

|G𝄪

|A𝄪

|12

|

|D♭𝄫 major

|D♭𝄫

|E♭𝄫

|F𝄫

|G♭𝄫

|A♭𝄫

|B♭𝄫

|C𝄫

|19

F𝄪 major

|F𝄪

|G𝄪

|A𝄪

|B♯

|C𝄪

|D𝄪

|E𝄪

|13

|

|A♭𝄫 major

|A♭𝄫

|B♭𝄫

|C𝄫

|D♭𝄫

|E♭𝄫

|F𝄫

|G𝄫

|18

C𝄪 major

|C𝄪

|D𝄪

|E𝄪

|F𝄪

|G𝄪

|A𝄪

|B𝄪

|14

|

|E♭𝄫 major

|E♭𝄫

|F𝄫

|G𝄫

|A♭𝄫

|B♭𝄫

|C𝄫

|D𝄫

|17

G𝄪 major

|G𝄪

|A𝄪

|B𝄪

|C𝄪

|D𝄪

|E𝄪

|F♯𝄪

|15

|

|B♭𝄫 major

|B♭𝄫

|C𝄫

|D𝄫

|E♭𝄫

|F𝄫

|G𝄫

|A𝄫

|16

D𝄪 major

|D𝄪

|E𝄪

|F♯𝄪

|G𝄪

|A𝄪

|B𝄪

|C♯𝄪

|16

|

|F𝄫 major

|F𝄫

|G𝄫

|A𝄫

|B♭𝄫

|C𝄫

|D𝄫

|E𝄫

|15

A𝄪 major

|A𝄪

|B𝄪

|C♯𝄪

|D𝄪

|E𝄪

|F♯𝄪

|G♯𝄪

|17

|

|C𝄫 major

|C𝄫

|D𝄫

|E𝄫

|F𝄫

|G𝄫

|A𝄫

|B𝄫

|14

E𝄪 major

|E𝄪

|F♯𝄪

|G♯𝄪

|A𝄪

|B𝄪

|C♯𝄪

|D♯𝄪

|18

|

|G𝄫 major

|G𝄫

|A𝄫

|B𝄫

|C𝄫

|D𝄫

|E𝄫

|F♭

|13

B𝄪 major

|B𝄪

|C♯𝄪

|D♯𝄪

|E𝄪

|F♯𝄪

|G♯𝄪

|A♯𝄪

|19

|

|D𝄫 major

|D𝄫

|E𝄫

|F♭

|G𝄫

|A𝄫

|B𝄫

|C♭

|12

F♯𝄪 major

|F♯𝄪

|G♯𝄪

|A♯𝄪

|B𝄪

|C♯𝄪

|D♯𝄪

|E♯𝄪

|20

|

|A𝄫 major

|A𝄫

|B𝄫

|C♭

|D𝄫

|E𝄫

|F♭

|G♭

|11

C♯𝄪 major

|C♯𝄪

|D♯𝄪

|E♯𝄪

|F♯𝄪

|G♯𝄪

|A♯𝄪

|B♯𝄪

|21

|

|E𝄫 major

|E𝄫

|F♭

|G♭

|A𝄫

|B𝄫

|C♭

|D♭

|10

colspan="9" rowspan="10" |

| rowspan="10" |

|B𝄫 major

|B𝄫

|C♭

|D♭

|E𝄫

|F♭

|G♭

|A♭

|9

F♭ major

|F♭

|G♭

|A♭

|B𝄫

|C♭

|D♭

|E♭

|8

C♭ major

|C♭

|D♭

|E♭

|F♭

|G♭

|A♭

|B♭

|7

G♭ major

|G♭

|A♭

|B♭

|C♭

|D♭

|E♭

|F

|6

D♭ major

|D♭

|E♭

|F

|G♭

|A♭

|B♭

|C

|5

A♭ major

|A♭

|B♭

|C

|D♭

|E♭

|F

|G

|4

E♭ major

|E♭

|F

|G

|A♭

|B♭

|C

|D

|3

B♭ major

|B♭

|C

|D

|E♭

|F

|G

|A

|2

F major

|F

|G

|A

|B♭

|C

|D

|E

|1

C major

|C

|D

|E

|F

|G

|A

|B

|0 (no flats or sharps)

class="wikitable"

|+ {{left|Comparison between {{nobr|{{small|{{sfrac| 1 | 4 }}}} comma}} meantone and {{nobr|31 {{sc|EDO}}}} (values in cents, rounded to 2 decimal places) }}

|

| align="center" width="20em" | C

| align="center" width="20em" | C♯

| align="center" width="20em" | D♭

| align="center" width="20em" | D

| align="center" width="20em" | D♯

| align="center" width="20em" | E♭

| align="center" width="20em" | E

| align="center" width="20em" | E♯

| align="center" width="20em" | F

| align="center" width="20em" | F♯

| align="center" width="20em" | G♭

| align="center" width="20em" | G

| align="center" width="20em" | G♯

| align="center" width="20em" | A♭

| align="center" width="20em" | A

| align="center" width="20em" | A♯

| align="center" width="20em" | B♭

| align="center" width="20em" | B

| align="center" width="20em" | C♭

| align="center" width="20em" | C

| {{nobr|{{small|{{sfrac| 1 | 4 }}}} comma}}:

| align="center" width="20em" | 0.00

| align="center" width="20em" | 76.05

| align="center" width="20em" | 117.11

| align="center" width="20em" | 193.16

| align="center" width="20em" | 269.21

| align="center" width="20em" | 310.26

| align="center" width="20em" | 386.31

| align="center" width="20em" | 462.36

| align="center" width="20em" | 503.42

| align="center" width="20em" | 579.47

| align="center" width="20em" | 620.53

| align="center" width="20em" | 696.58

| align="center" width="20em" | 772.63

| align="center" width="20em" | 813.69

| align="center" width="20em" | 889.74

| align="center" width="20em" | 965.78

| align="center" width="20em" | 1006.84

| align="center" width="20em" | 1082.89

| align="center" width="20em" | 1123.95

| align="center" width="20em" | 1200.00

| {{nobr|31 {{sc|EDO}}}}:

| align="center" width="20em" | 0.00

| align="center" width="20em" | 77.42

| align="center" width="20em" | 116.13

| align="center" width="20em" | 193.55

| align="center" width="20em" | 270.97

| align="center" width="20em" | 309.68

| align="center" width="20em" | 387.10

| align="center" width="20em" | 464.52

| align="center" width="20em" | 503.23

| align="center" width="20em" | 580.65

| align="center" width="20em" | 619.35

| align="center" width="20em" | 696.77

| align="center" width="20em" | 774.19

| align="center" width="20em" | 812.90

| align="center" width="20em" | 890.32

| align="center" width="20em" | 967.74

| align="center" width="20em" | 1006.45

| align="center" width="20em" | 1083.87

| align="center" width="20em" | 1122.58

| align="center" width="20em" | 1200.00

Chords of 31 equal temperament

Many chords of {{nobr|31 {{sc|EDO}}}} are discussed in the article on septimal meantone temperament. Chords not discussed there include the neutral thirds triad ({{Audio|Neutral thirds triad in 31-tet.mid|Play}}), which might be written C–E{{music|d}}–G, C–D𝄪–G or C–F𝄫–G, and the Orwell tetrad, which is C–E–F𝄪–B𝄫.

File:Simple I-IV-V-I isomorphic 31-TET.png in 31 tone equal temperament.File:Simple_I-IV-V-I_isomorphic_31-TET.mid Whereas in {{nobr|12 {{sc|EDO}}}} B{{music|natural}} is 11 steps, in {{nobr|31 {{sc|EDO}}}} B{{music|natural}} is 28 steps.]]

File:Csub Cmin Cmaj Csup.ogg

Usual chords like the major chord are rendered nicely in {{nobr|31 {{sc|EDO}}}} because the third and the fifth are very well approximated. Also, it is possible to play subminor chords (where the first third is subminor) and supermajor chords (where the first third is supermajor).

File:Cmaj7 Gmin 31ET 12ET.ogg

It is also possible to render nicely the harmonic seventh chord. For example on tonic C, with {{nobr|C–E–G–A♯.}} The seventh here is different from stacking a fifth and a minor third, which instead yields B{{music|♭}} to make a dominant seventh. This difference cannot be made in 12 equal temperament.

Footnotes

References

External links

{{cite web

|title=main page

|lang=nl,en

|website=The Huygens-Fokker Foundation for Microtonal Music (huygens-fokker.org)

|url=http://www.huygens-fokker.org/

}}

{{cite web

|last = Fokker |first = Adriaan Daniël |author-link = Adriaan Fokker

|title = Equal temperament and the thirty-one-keyed organ

|website=The Huygens-Fokker Foundation for Microtonal Music (huygens-fokker.org)

|url=http://www.huygens-fokker.org/docs/fokkerorg.html

}}

{{cite web

|last = Rapoport |first = Paul

|title=About 31 tone equal temperament

|website=The Huygens-Fokker Foundation for Microtonal Music (huygens-fokker.org)

|url=http://www.huygens-fokker.org/docs/rap31.html

}}

{{cite web

|last = Terpstra |first = Siemen

|title = Toward a theory of meantone (and {{nobr|31 {{sc|ET}})}} harmony

|website=The Huygens-Fokker Foundation for Microtonal Music (huygens-fokker.org)

|url = http://www.huygens-fokker.org/docs/terp31.html

}}

{{cite book

|last=Barbieri |first=Patrizio

|year=2008

|title=Enharmonic instruments and music, 1470–1900

|place=Latina

|publisher=Il Levante Libreria Editrice

|url=http://www.patriziobarbieri.it/1.htm

|via=patriziobarbieri.it |url-status=dead

|archive-url=https://web.archive.org/web/20090215045859/http://www.patriziobarbieri.it/1.htm

|archive-date=2009-02-15 |df=dmy-all

}}

{{cite conference

|first=M. |last=Khramov

|year=2009

|title=Approximation to 7 limit just intonation in a scale of {{nobr|31 {{sc|EDO}}}}

|conference=FRSM-2009 International Symposium Frontiers of Research on Speech and Music

|pages=73–82

|publisher=ABV IIITM Gwalior

}}

{{cite web |title=31 tone equal temperament |website=31et.com |type=main page |url=http://31et.com/ }}

Category:Equal temperaments

Category:Microtonality

31 equal temperament

History and use

Interval size

Scale diagram

= [[Mode (music)|Modes]] =

== [[Ionian mode]] ([[major scale]]) ==

Chords of 31 equal temperament

Footnotes

See also

References

External links