@@ Line 6: / Line 6: @@
 | ja =
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-= Theory =
+== Theory ==
-<b>72-tone equal temperament</b>, or <b>72-edo</b>, divides the octave into 72 steps or ''moria''. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in 12-tone equal temperament. 72-tone is also a superset of [[24edo|24-tone equal temperament]], a common and standard tuning of [[Arabic,_Turkish,_Persian|Arabic]] music, and has itself been used to tune Turkish music.
+'''72-tone equal temperament''', or '''72-edo''', divides the octave into 72 steps or ''moria''. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in 12-tone equal temperament. 72-tone is also a superset of [[24edo|24-tone equal temperament]], a common and standard tuning of [[Arabic,_Turkish,_Persian|Arabic]] music, and has itself been used to tune Turkish music.
-Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with [[96edo|96-edo]]), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.
+Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with [[96edo]]), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.
--tone equal temperament approximates [[11-limit|11-limit just intonation]] exceptionally well, is consistent in the [[17-limit|17-limit]], and is the ninth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other 5-limit major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.
+-tone equal temperament approximates [[11-limit]] [[just intonation]] exceptionally well, is consistent in the [[17-limit]], and is the ninth [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.
-is an excellent tuning for [[Gamelismic_clan|miracle temperament]], especially the 11-limit version, and the related rank three temperament [[Marvel_family#Prodigy|prodigy]], and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.
+is an excellent tuning for [[Gamelismic_clan #Miracle|miracle temperament]], especially the 11-limit version, and the related rank three temperament [[Marvel_family #Prodigy|prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].
-=Intervals=
+== Intervals ==
+{| class="wikitable center-all right-2 left-3"
-{| class="wikitable"
 |-
-| | degrees
+! Degrees
-| | cents value
+! Cents
-| | approximate ratios (17-limit)
+! Approximate Ratios (17-limit)
-| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]]
+! colspan="3" | [[Ups and Downs Notation]]
 |-
 | 0
-|0.000
+| 0.000
-| | 1/1
+| 1/1
-| style="text-align:center;" | P1
+| P1
-| style="text-align:center;" | perfect unison
+| perfect unison
-| style="text-align:center;" | D
+| D
 |-
-| | 1
+| 1
-| | 16.667
+| 16.667
-| | 81/80
+| 81/80
-| style="text-align:center;" | ^1
+| ^1
-| style="text-align:center;" | up unison
+| up unison
-| style="text-align:center;" | ^D
+| ^D
 |-
-| | 2
+| 2
-| | 33.333
+| 33.333
-| | 45/44
+| 45/44
-| style="text-align:center;" | ^^
+| ^^
-| style="text-align:center;" | double-up unison
+| double-up unison
-| style="text-align:center;" | ^^D
+| ^^D
 |-
-| | 3
+| 3
-| | 50
+| 50.000
-| | 33/32
+| 33/32
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>1, v<span style="font-size: 90%; vertical-align: super;">3</span>m2
+| ^<sup>3</sup>1, <br>v<sup>3</sup>m2
-| style="text-align:center;" | triple-up unison,
+| triple-up unison,<br>triple-down minor 2nd
+| ^<sup>3</sup>D, <br>v<sup>3</sup>Eb
-triple-down minor 2nd
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>D, v<span style="font-size: 90%; vertical-align: super;">3</span>Eb
 |-
-| | 4
+| 4
-| | 66.667
+| 66.667
-| | 25/24
+| 25/24
-| style="text-align:center;" | vvm2
+| vvm2
-| style="text-align:center;" | double-downminor 2nd
+| double-downminor 2nd
-| style="text-align:center;" | vvEb
+| vvEb
 |-
-| | 5
+| 5
-| | 83.333
+| 83.333
-| | 21/20
+| 21/20
-| style="text-align:center;" | vm2
+| vm2
-| style="text-align:center;" | downminor 2nd
+| downminor 2nd
-| style="text-align:center;" | vEb
+| vEb
 |-
-| | 6
+| 6
-| | 100
+| 100.000
-| | 35/33, 17/16, 18/17
+| 35/33, 17/16, 18/17
-| style="text-align:center;" | m2
+| m2
-| style="text-align:center;" | minor 2nd
+| minor 2nd
-| style="text-align:center;" | Eb
+| Eb
 |-
-| | 7
+| 7
-| | 116.667
+| 116.667
-| | 15/14, 16/15
+| 15/14, 16/15
-| style="text-align:center;" | ^m2
+| ^m2
-| style="text-align:center;" | upminor 2nd
+| upminor 2nd
-| style="text-align:center;" | ^Eb
+| ^Eb
 |-
-| | 8
+| 8
-| | 133.333
+| 133.333
-| | 27/25, 13/12, 14/13
+| 27/25, 13/12, 14/13
-| style="text-align:center;" | v~2
+| v~2
-| style="text-align:center;" | downmid 2nd
+| downmid 2nd
-| style="text-align:center;" | ^^Eb
+| ^^Eb
 |-
-| | 9
+| 9
-| | 150
+| 150.000
-| | 12/11
+| 12/11
-| style="text-align:center;" | ~2
+| ~2
-| style="text-align:center;" | mid 2nd
+| mid 2nd
-| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>E
+| v<sup>3</sup>E
 |-
-| | 10
+| 10
-| | 166.667
+| 166.667
-| | 11/10
+| 11/10
-| style="text-align:center;" | ^~2
+| ^~2
-| style="text-align:center;" | upmid 2nd
+| upmid 2nd
-| style="text-align:center;" | vvE
+| vvE
 |-
-| | 11
+| 11
-| | 183.333
+| 183.333
-| | 10/9
+| 10/9
-| style="text-align:center;" | vM2
+| vM2
-| style="text-align:center;" | downmajor 2nd
+| downmajor 2nd
-| style="text-align:center;" | vE
+| vE
 |-
-| | 12
+| 12
-| | 200
+| 200.000
-| | 9/8
+| 9/8
-| style="text-align:center;" | M2
+| M2
-| style="text-align:center;" | major 2nd
+| major 2nd
-| style="text-align:center;" | E
+| E
 |-
-| | 13
+| 13
-| | 216.667
+| 216.667
-| | 25/22, 17/15
+| 25/22, 17/15
-| style="text-align:center;" | ^M2
+| ^M2
-| style="text-align:center;" | upmajor 2nd
+| upmajor 2nd
-| style="text-align:center;" | ^E
+| ^E
 |-
-| | 14
+| 14
-| | 233.333
+| 233.333
-| | 8/7
+| 8/7
-| style="text-align:center;" | ^^M2
+| ^^M2
-| style="text-align:center;" | double-upmajor 2nd
+| double-upmajor 2nd
-| style="text-align:center;" | ^^E
+| ^^E
 |-
-| | 15
+| 15
-| | 250
+| 250.000
-| | 81/70,  15/13
+| 81/70,  15/13
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M2, v<span style="font-size: 90%; vertical-align: super;">3</span>m3
+| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
-| style="text-align:center;" | triple-up major 2nd,
+| triple-up major 2nd,<br>triple-down minor 3rd
+| ^<sup>3</sup>E, <br>v<sup>3</sup>F
-triple-down minor 3rd
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>E, v<span style="font-size: 90%; vertical-align: super;">3</span>F
 |-
-| | 16
+| 16
-| | 266.667
+| 266.667
-| | 7/6
+| 7/6
-| style="text-align:center;" | vvm3
+| vvm3
-| style="text-align:center;" | double-downminor 3rd
+| double-downminor 3rd
-| style="text-align:center;" | vvF
+| vvF
 |-
-| | 17
+| 17
-| | 283.333
+| 283.333
-| | 33/28, 13/11, 20/17
+| 33/28, 13/11, 20/17
-| style="text-align:center;" | vm3
+| vm3
-| style="text-align:center;" | downminor 3rd
+| downminor 3rd
-| style="text-align:center;" | vF
+| vF
 |-
-| | 18
+| 18
-| | 300
+| 300.000
-| | 25/21
+| 25/21
-| style="text-align:center;" | m3
+| m3
-| style="text-align:center;" | minor 3rd
+| minor 3rd
-| style="text-align:center;" | F
+| F
 |-
-| | 19
+| 19
-| | 316.667
+| 316.667
-| | 6/5
+| 6/5
-| style="text-align:center;" | ^m3
+| ^m3
-| style="text-align:center;" | upminor 3rd
+| upminor 3rd
-| style="text-align:center;" | ^F
+| ^F
 |-
-| | 20
+| 20
-| | 333.333
+| 333.333
-| | 40/33, 17/14
+| 40/33, 17/14
-| style="text-align:center;" | v~3
+| v~3
-| style="text-align:center;" | downmid 3rd
+| downmid 3rd
-| style="text-align:center;" | ^^F
+| ^^F
 |-
-| | 21
+| 21
-| | 350
+| 350.000
-| | 11/9
+| 11/9
-| style="text-align:center;" | ~3
+| ~3
-| style="text-align:center;" | mid 3rd
+| mid 3rd
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>F
+| ^<sup>3</sup>F
 |-
-| | 22
+| 22
-| | 366.667
+| 366.667
-| | 99/80, 16/13, 21/17
+| 99/80, 16/13, 21/17
-| style="text-align:center;" | ^~3
+| ^~3
-| style="text-align:center;" | upmid 3rd
+| upmid 3rd
-| style="text-align:center;" | vvF#
+| vvF#
 |-
-| | 23
+| 23
-| | 383.333
+| 383.333
-| | 5/4
+| 5/4
-| style="text-align:center;" | vM3
+| vM3
-| style="text-align:center;" | downmajor 3rd
+| downmajor 3rd
-| style="text-align:center;" | vF#
+| vF#
 |-
-| | 24
+| 24
-| | 400
+| 400.000
-| | 44/35
+| 44/35
-| style="text-align:center;" | M3
+| M3
-| style="text-align:center;" | major 3rd
+| major 3rd
-| style="text-align:center;" | F#
+| F#
 |-
-| | 25
+| 25
-| | 416.667
+| 416.667
-| | 14/11
+| 14/11
-| style="text-align:center;" | ^M3
+| ^M3
-| style="text-align:center;" | upmajor 3rd
+| upmajor 3rd
-| style="text-align:center;" | ^F#
+| ^F#
 |-
-| | 26
+| 26
-| | 433.333
+| 433.333
-| | 9/7
+| 9/7
-| style="text-align:center;" | ^^M3
+| ^^M3
-| style="text-align:center;" | double-upmajor 3rd
+| double-upmajor 3rd
-| style="text-align:center;" | ^^F#
+| ^^F#
 |-
-| | 27
+| 27
-| | 450
+| 450.000
-| | 35/27, 13/10
+| 35/27, 13/10
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M3, v<span style="font-size: 90%; vertical-align: super;">3</span>4
+| ^<sup>3</sup>M3, <br>v<sup>3</sup>4
-| style="text-align:center;" | triple-up major 3rd,
+| triple-up major 3rd,<br>triple-down 4th
+| ^<sup>3</sup>F#, <br>v<sup>3</sup>G
-triple-down 4th
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>F#, v<span style="font-size: 90%; vertical-align: super;">3</span>G
 |-
-| | 28
+| 28
-| | 466.667
+| 466.667
-| | 21/16, 17/13
+| 21/16, 17/13
-| style="text-align:center;" | vv4
+| vv4
-| style="text-align:center;" | double-down 4th
+| double-down 4th
-| style="text-align:center;" | vvG
+| vvG
 |-
-| | 29
+| 29
-| | 483.333
+| 483.333
-| | 33/25
+| 33/25
-| style="text-align:center;" | v4
+| v4
-| style="text-align:center;" | down 4th
+| down 4th
-| style="text-align:center;" | vG
+| vG
 |-
-| | 30
+| 30
-| | 500
+| 500.000
-| | 4/3
+| 4/3
-| style="text-align:center;" | P4
+| P4
-| style="text-align:center;" | perfect 4th
+| perfect 4th
-| style="text-align:center;" | G
+| G
 |-
-| | 31
+| 31
-| | 516.667
+| 516.667
-| | 27/20
+| 27/20
-| style="text-align:center;" | ^4
+| ^4
-| style="text-align:center;" | up 4th
+| up 4th
-| style="text-align:center;" | ^G
+| ^G
 |-
-| | 32
+| 32
-| | 533.333
+| 533.333
-| | 15/11
+| 15/11
-| style="text-align:center;" | v~4
+| v~4
-| style="text-align:center;" | downmid 4th
+| downmid 4th
-| style="text-align:center;" | ^^G
+| ^^G
 |-
-| | 33
+| 33
-| | 550
+| 550.000
-| | 11/8
+| 11/8
-| style="text-align:center;" | ~4
+| ~4
-| style="text-align:center;" | mid 4th
+| mid 4th
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>G
+| ^<sup>3</sup>G
 |-
-| | 34
+| 34
-| | 566.667
+| 566.667
-| | 25/18, 18/13
+| 25/18, 18/13
-| style="text-align:center;" | ^~4
+| ^~4
-| style="text-align:center;" | upmid 4th
+| upmid 4th
-| style="text-align:center;" | vvG#
+| vvG#
 |-
-| | 35
+| 35
-| | 583.333
+| 583.333
-| | 7/5
+| 7/5
-| style="text-align:center;" | vA4, vd5
+| vA4, vd5
-| style="text-align:center;" | downaug 4th, updim 5th
+| downaug 4th, updim 5th
-| style="text-align:center;" | vG#, vAb
+| vG#, vAb
 |-
-| | 36
+| 36
-| | 600
+| 600.000
-| | 99/70, 17/12
+| 99/70, 17/12
-| style="text-align:center;" | A4, d5
+| A4, d5
-| style="text-align:center;" | aug 4th, dim 5th
+| aug 4th, dim 5th
-| style="text-align:center;" | G#, Ab
+| G#, Ab
 |-
-| | 37
+| 37
-| | 616.667
+| 616.667
-| | 10/7
+| 10/7
-| style="text-align:center;" | ^A4, ^d5
+| ^A4, ^d5
-| style="text-align:center;" | upaug 4th, downdim 5th
+| upaug 4th, downdim 5th
-| style="text-align:center;" | ^G#, ^Ab
+| ^G#, ^Ab
 |-
-| | 38
+| 38
-| | 633.333
+| 633.333
-| | 36/25, 13/9
+| 36/25, 13/9
-| style="text-align:center;" | v~5
+| v~5
-| style="text-align:center;" | downmid 5th
+| downmid 5th
-| style="text-align:center;" | ^^Ab
+| ^^Ab
 |-
-| | 39
+| 39
-| | 650
+| 650.000
-| | 16/11
+| 16/11
-| style="text-align:center;" | ~5
+| ~5
-| style="text-align:center;" | mid 5th
+| mid 5th
-| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>A
+| v<sup>3</sup>A
 |-
-| | 40
+| 40
-| | 666.667
+| 666.667
-| | 22/15
+| 22/15
-| style="text-align:center;" | ^~5
+| ^~5
-| style="text-align:center;" | upmid 5th
+| upmid 5th
-| style="text-align:center;" | vvA
+| vvA
 |-
-| | 41
+| 41
-| | 683.333
+| 683.333
-| | 40/27
+| 40/27
-| style="text-align:center;" | v5
+| v5
-| style="text-align:center;" | down 5th
+| down 5th
-| style="text-align:center;" | vA
+| vA
 |-
-| | 42
+| 42
-| | 700
+| 700.000
-| | 3/2
+| 3/2
-| style="text-align:center;" | P5
+| P5
-| style="text-align:center;" | perfect 5th
+| perfect 5th
-| style="text-align:center;" | A
+| A
 |-
-| | 43
+| 43
-| | 716.667
+| 716.667
-| | 50/33
+| 50/33
-| style="text-align:center;" | ^5
+| ^5
-| style="text-align:center;" | up 5th
+| up 5th
-| style="text-align:center;" | ^A
+| ^A
 |-
-| | 44
+| 44
-| | 733.333
+| 733.333
-| | 32/21
+| 32/21
-| style="text-align:center;" | ^^5
+| ^^5
-| style="text-align:center;" | double-up 5th
+| double-up 5th
-| style="text-align:center;" | ^^A
+| ^^A
 |-
-| | 45
+| 45
-| | 750
+| 750.000
-| | 54/35, 17/11
+| 54/35, 17/11
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>5, v<span style="font-size: 90%; vertical-align: super;">3</span>m6
+| ^<sup>3</sup>5, <br>v<sup>3</sup>m6
-| style="text-align:center;" | triple-up 5th,
+| triple-up 5th,<br>triple-down minor 6th
+| ^<sup>3</sup>A, <br>v<sup>3</sup>Bb
-triple-down minor 6th
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>A, v<span style="font-size: 90%; vertical-align: super;">3</span>Bb
 |-
-| | 46
+| 46
-| | 766.667
+| 766.667
-| | 14/9
+| 14/9
-| style="text-align:center;" | vvm6
+| vvm6
-| style="text-align:center;" | double-downminor 6th
+| double-downminor 6th
-| style="text-align:center;" | vvBb
+| vvBb
 |-
-| | 47
+| 47
-| | 783.333
+| 783.333
-| | 11/7
+| 11/7
-| style="text-align:center;" | vm6
+| vm6
-| style="text-align:center;" | downminor 6th
+| downminor 6th
-| style="text-align:center;" | vBb
+| vBb
 |-
-| | 48
+| 48
-| | 800
+| 800.000
-| | 35/22
+| 35/22
-| style="text-align:center;" | m6
+| m6
-| style="text-align:center;" | minor 6th
+| minor 6th
-| style="text-align:center;" | Bb
+| Bb
 |-
-| | 49
+| 49
-| | 816.667
+| 816.667
-| | 8/5
+| 8/5
-| style="text-align:center;" | ^m6
+| ^m6
-| style="text-align:center;" | upminor 6th
+| upminor 6th
-| style="text-align:center;" | ^Bb
+| ^Bb
 |-
-| | 50
+| 50
-| | 833.333
+| 833.333
-| | 81/50, 13/8
+| 81/50, 13/8
-| style="text-align:center;" | v~6
+| v~6
-| style="text-align:center;" | downmid 6th
+| downmid 6th
-| style="text-align:center;" | ^^Bb
+| ^^Bb
 |-
-| | 51
+| 51
-| | 850
+| 850.000
-| | 18/11
+| 18/11
-| style="text-align:center;" | ~6
+| ~6
-| style="text-align:center;" | mid 6th
+| mid 6th
-| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>B
+| v<sup>3</sup>B
 |-
-| | 52
+| 52
-| | 866.667
+| 866.667
-| | 33/20, 28/17
+| 33/20, 28/17
-| style="text-align:center;" | ^~6
+| ^~6
-| style="text-align:center;" | upmid 6th
+| upmid 6th
-| style="text-align:center;" | vvB
+| vvB
 |-
-| | 53
+| 53
-| | 883.333
+| 883.333
-| | 5/3
+| 5/3
-| style="text-align:center;" | vM6
+| vM6
-| style="text-align:center;" | downmajor 6th
+| downmajor 6th
-| style="text-align:center;" | vB
+| vB
 |-
-| | 54
+| 54
-| | 900
+| 900.000
-| | 27/16
+| 27/16
-| style="text-align:center;" | M6
+| M6
-| style="text-align:center;" | major 6th
+| major 6th
-| style="text-align:center;" | B
+| B
 |-
-| | 55
+| 55
-| | 916.667
+| 916.667
-| | 56/33, 17/10
+| 56/33, 17/10
-| style="text-align:center;" | ^M6
+| ^M6
-| style="text-align:center;" | upmajor 6th
+| upmajor 6th
-| style="text-align:center;" | ^B
+| ^B
 |-
-| | 56
+| 56
-| | 933.333
+| 933.333
-| | 12/7
+| 12/7
-| style="text-align:center;" | ^^M6
+| ^^M6
-| style="text-align:center;" | double-upmajor 6th
+| double-upmajor 6th
-| style="text-align:center;" | ^^B
+| ^^B
 |-
-| | 57
+| 57
-| | 950
+| 950.000
-| | 121/70
+| 121/70
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M6, v<span style="font-size: 90%; vertical-align: super;">3</span>m7
+| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
-| style="text-align:center;" | triple-up major 6th,
+| triple-up major 6th,<br>triple-down minor 7th
+| ^<sup>3</sup>B, <br>v<sup>3</sup>C
-triple-down minor 7th
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>B, v<span style="font-size: 90%; vertical-align: super;">3</span>C
 |-
-| | 58
+| 58
-| | 966.667
+| 966.667
-| | 7/4
+| 7/4
-| style="text-align:center;" | vvm7
+| vvm7
-| style="text-align:center;" | double-downminor 7th
+| double-downminor 7th
-| style="text-align:center;" | vvC
+| vvC
 |-
-| | 59
+| 59
-| | 983.333
+| 983.333
-| | 44/25
+| 44/25
-| style="text-align:center;" | vm7
+| vm7
-| style="text-align:center;" | downminor 7th
+| downminor 7th
-| style="text-align:center;" | vC
+| vC
 |-
-| | 60
+| 60
-| | 1000
+| 1000.000
-| | 16/9
+| 16/9
-| style="text-align:center;" | m7
+| m7
-| style="text-align:center;" | minor 7th
+| minor 7th
-| style="text-align:center;" | C
+| C
 |-
-| | 61
+| 61
-| | 1016.667
+| 1016.667
-| | 9/5
+| 9/5
-| style="text-align:center;" | ^m7
+| ^m7
-| style="text-align:center;" | upminor 7th
+| upminor 7th
-| style="text-align:center;" | ^C
+| ^C
 |-
-| | 62
+| 62
-| | 1033.333
+| 1033.333
-| | 20/11
+| 20/11
-| style="text-align:center;" | v~7
+| v~7
-| style="text-align:center;" | downmid 7th
+| downmid 7th
-| style="text-align:center;" | ^^C
+| ^^C
 |-
-| | 63
+| 63
-| | 1050
+| 1050.000
-| | 11/6
+| 11/6
-| style="text-align:center;" | ~7
+| ~7
-| style="text-align:center;" | mid 7th
+| mid 7th
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>C
+| ^<sup>3</sup>C
 |-
-| | 64
+| 64
-| | 1066.667
+| 1066.667
-| | 50/27
+| 50/27
-| style="text-align:center;" | ^~7
+| ^~7
-| style="text-align:center;" | upmid 7th
+| upmid 7th
-| style="text-align:center;" | vvC#
+| vvC#
 |-
-| | 65
+| 65
-| | 1083.333
+| 1083.333
-| | 15/8
+| 15/8
-| style="text-align:center;" | vM7
+| vM7
-| style="text-align:center;" | downmajor 7th
+| downmajor 7th
-| style="text-align:center;" | vC#
+| vC#
 |-
-| | 66
+| 66
-| | 1100
+| 1100.000
-| | 66/35, 17/9
+| 66/35, 17/9
-| style="text-align:center;" | M7
+| M7
-| style="text-align:center;" | major 7th
+| major 7th
-| style="text-align:center;" | C#
+| C#
 |-
-| | 67
+| 67
-| | 1116.667
+| 1116.667
-| | 21/11
+| 21/11
-| style="text-align:center;" | ^M7
+| ^M7
-| style="text-align:center;" | upmajor 7th
+| upmajor 7th
-| style="text-align:center;" | ^C#
+| ^C#
 |-
-| | 68
+| 68
-| | 1133.333
+| 1133.333
-| | 27/14
+| 27/14
-| style="text-align:center;" | ^^M7
+| ^^M7
-| style="text-align:center;" | double-upmajor 7th
+| double-upmajor 7th
-| style="text-align:center;" | ^^C#
+| ^^C#
 |-
-| | 69
+| 69
-| | 1150
+| 1150.000
-| | 35/18
+| 35/18
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M7, v<span style="font-size: 90%; vertical-align: super;">3</span>8
+| ^<sup>3</sup>M7, <br>v<sup>3</sup>8
-| style="text-align:center;" | triple-up major 7th,
+| triple-up major 7th,<br>triple-down octave
+| ^<sup>3</sup>C#, <br>v<sup>3</sup>D
-triple-down octave
-| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>C#, v<span style="font-size: 90%; vertical-align: super;">3</span>D
 |-
-| | 70
+| 70
-| | 1166.667
+| 1166.667
-| | 49/25
+| 49/25
-| style="text-align:center;" | vv8
+| vv8
-| style="text-align:center;" | double-down octave
+| double-down octave
-| style="text-align:center;" | vvD
+| vvD
 |-
-| | 71
+| 71
-| | 1183.333
+| 1183.333
-| | 99/50
+| 99/50
-| style="text-align:center;" | v8
+| v8
-| style="text-align:center;" | down octave
+| down octave
-| style="text-align:center;" | vD
+| vD
 |-
-| | 72
+| 72
-| | 1200
+| 1200.000
-| | 2/1
+| 2/1
-| style="text-align:center;" | P8
+| P8
-| style="text-align:center;" | perfect octave
+| perfect octave
-| style="text-align:center;" | D
+| D
 |}
 Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
-{| class="wikitable"
+{| class="wikitable center-all"
 |-
-! | quality
+! quality
-! | [[Kite's color notation|color]]
+! [[Kite's color notation|color]]
-! | monzo format
+! monzo format
-! | examples
+! examples
 |-
-| style="text-align:center;" | double-down minor
+| double-down minor
-| style="text-align:center;" | zo
+| zo
-| style="text-align:center;" | {a, b, 0, 1}
+| {a, b, 0, 1}
-| style="text-align:center;" | 7/6, 7/4
+| 7/6, 7/4
 |-
-| style="text-align:center;" | minor
+| minor
-| style="text-align:center;" | fourthward wa
+| fourthward wa
-| style="text-align:center;" | {a, b}, b &lt; -1
+| {a, b}, b &lt; -1
-| style="text-align:center;" | 32/27, 16/9
+| 32/27, 16/9
 |-
-| style="text-align:center;" | upminor
+| upminor
-| style="text-align:center;" | gu
+| gu
-| style="text-align:center;" | {a, b, -1}
+| {a, b, -1}
-| style="text-align:center;" | 6/5, 9/5
+| 6/5, 9/5
 |-
-| style="text-align:center;" | mid
+| mid
-| style="text-align:center;" | ilo
+| ilo
-| style="text-align:center;" | {a, b, 0, 0, 1}
+| {a, b, 0, 0, 1}
-| style="text-align:center;" | 11/9, 11/6
+| 11/9, 11/6
 |-
-| style="text-align:center;" | "
+| "
-| style="text-align:center;" | lu
+| lu
-| style="text-align:center;" | {a, b, 0, 0, -1}
+| {a, b, 0, 0, -1}
-| style="text-align:center;" | 12/11, 18/11
+| 12/11, 18/11
 |-
-| style="text-align:center;" | downmajor
+| downmajor
-| style="text-align:center;" | yo
+| yo
-| style="text-align:center;" | {a, b, 1}
+| {a, b, 1}
-| style="text-align:center;" | 5/4, 5/3
+| 5/4, 5/3
 |-
-| style="text-align:center;" | major
+| major
-| style="text-align:center;" | fifthward wa
+| fifthward wa
-| style="text-align:center;" | {a, b}, b &gt; 1
+| {a, b}, b &gt; 1
-| style="text-align:center;" | 9/8, 27/16
+| 9/8, 27/16
 |-
-| style="text-align:center;" | double-up major
+| double-up major
-| style="text-align:center;" | ru
+| ru
-| style="text-align:center;" | {a, b, 0, -1}
+| {a, b, 0, -1}
-| style="text-align:center;" | 9/7, 12/7
+| 9/7, 12/7
 |}
 All 72-edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:
-{| class="wikitable"
+{| class="wikitable center-all"
 |-
-! | [[Kite's color notation|color of the 3rd]]
+! [[Kite's color notation|color of the 3rd]]
-! | JI chord
+! JI chord
-! | notes as edosteps
+! notes as edosteps
-! | notes of C chord
+! notes of C chord
-! | written name
+! written name
-! | spoken name
+! spoken name
 |-
-| style="text-align:center;" | zo
+| zo
-| style="text-align:center;" | 6:7:9
+| 6:7:9
-| style="text-align:center;" | 0-16-42
+| 0-16-42
-| style="text-align:center;" | C vvEb G
+| C vvEb G
-| style="text-align:center;" | Cvvm
+| Cvvm
-| style="text-align:center;" | C double-down minor
+| C double-down minor
 |-
-| style="text-align:center;" | gu
+| gu
-| style="text-align:center;" | 10:12:15
+| 10:12:15
-| style="text-align:center;" | 0-19-42
+| 0-19-42
-| style="text-align:center;" | C ^Eb G
+| C ^Eb G
-| style="text-align:center;" | C^m
+| C^m
-| style="text-align:center;" | C upminor
+| C upminor
 |-
-| style="text-align:center;" | ilo
+| ilo
-| style="text-align:center;" | 18:22:27
+| 18:22:27
-| style="text-align:center;" | 0-21-42
+| 0-21-42
-| style="text-align:center;" | C v<span style="font-size: 90%; vertical-align: super;">3</span>E G
+| C v<span style="font-size: 90%; vertical-align: super;">3</span>E G
-| style="text-align:center;" | C~
+| C~
-| style="text-align:center;" | C mid
+| C mid
 |-
-| style="text-align:center;" | yo
+| yo
-| style="text-align:center;" | 4:5:6
+| 4:5:6
-| style="text-align:center;" | 0-23-42
+| 0-23-42
-| style="text-align:center;" | C vE G
+| C vE G
-| style="text-align:center;" | Cv
+| Cv
-| style="text-align:center;" | C downmajor or C down
+| C downmajor or C down
 |-
-| style="text-align:center;" | ru
+| ru
-| style="text-align:center;" | 14:18:27
+| 14:18:27
-| style="text-align:center;" | 0-26-42
+| 0-26-42
-| style="text-align:center;" | C ^^E G
+| C ^^E G
-| style="text-align:center;" | C^^
+| C^^
-| style="text-align:center;" | C double-upmajor or C double-up
+| C double-upmajor or C double-up
 |}
-For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
+For a more complete list, see [[Ups and Downs Notation #Chord names in other EDOs]].
-===Approximations to prime harmonics===
-{| class="wikitable" style="text-align:center;"
+== Just approximation ==
+{| class="wikitable center-all"
 !
-!prime 2
+! prime 2
-!prime 3
+! prime 3
-!prime 5
+! prime 5
-!prime 7
+! prime 7
-!prime 11
+! prime 11
-!prime 13
+! prime 13
-!prime 17
+! prime 17
-!prime 19
+! prime 19
-!prime 23
+! prime 23
-!prime 29
+! prime 29
-!prime 31
+! prime 31
 |-
-!error
+! error (¢)
-|0.0¢
+| 0.000
-| -1.955¢
+| -1.955
-| -2.980¢
+| -2.980
-| -2.159¢
+| -2.159
-| -1.318¢
+| -1.318
-| -7.194¢
+| -7.194
-| -4.955¢
+| -4.955
-| +2.487¢
+| +2.487
-| +5.059¢
+| +5.059
-| +3.755¢
+| +3.755
-| -4.964¢
+| -4.964
 |}
-=Commas=
+== Commas ==
-Commas tempered out by 72edo include...
+Commas tempered out by 72edo include…
 {| class="wikitable"
@@ Line 765: / Line 754: @@
 |}
-=Temperaments=
+== Temperaments ==
-It provides the optimal patent val for miracle and wizard in the 7-limit, miracle, catakleismic, bikleismic, compton, ennealimnic, ennealiminal, enneaportent, marvolo and catalytic in the 11-limit, and catakleismic, bikleismic, compton, comptone, enneaportent, ennealim, catalytic, marvolo, manna, hendec, lizard, neominor, hours, and semimiracle in the 13-limit.
+* [[List of edo-distinct 72et rank two temperaments]]
-See also [[List_of_edo-distinct_72et_rank_two_temperaments|List of edo-distinct 72et rank two temperaments]].
+edo provides the [[optimal patent val]] for [[miracle]] and [[wizard]] in the 7-limit, miracle, [[catakleismic]], [[bikleismic]], [[compton]], [[ennealimnic]], [[ennealiminal]], [[enneaportent]], [[marvolo]] and [[catalytic]] in the 11-limit, and catakleismic, bikleismic, compton, [[comptone]], [[enneaportent]], [[ennealim]], catalytic, marvolo, [[manna]], [[hendec]], [[lizard]], [[neominor]], [[hours]], and [[semimiracle]] in the 13-limit.
-=Scales=
+{| class="wikitable center-1 center-2"
-[[smithgw72a|smithgw72a]], [[smithgw72b|smithgw72b]], [[smithgw72c|smithgw72c]], [[smithgw72d|smithgw72d]], [[smithgw72e|smithgw72e]], [[smithgw72f|smithgw72f]], [[smithgw72g|smithgw72g]], [[smithgw72h|smithgw72h]], [[smithgw72i|smithgw72i]], [[smithgw72j|smithgw72j]]
+|-
+! Periods<br>per octave
+! Generator
+! Names
+|-
+| 1
+| 1\72
+| [[Quincy]]
+|-
+| 1
+| 5\72
+| [[Marvolo]]
+|-
+| 1
+| 7\72
+| [[Miracle]]/benediction/manna
+|-
+| 1
+| 11\72
+|
+|-
+| 1
+| 13\72
+|
+|-
+| 1
+| 17\72
+| [[Neominor]]
+|-
+| 1
+| 19\72
+| [[Catakleismic]]
+|-
+| 1
+| 23\72
+|
+|-
+| 1
+| 25\72
+| [[Sqrtphi]]
+|-
+| 1
+| 29\72
+|
+|-
+| 1
+| 31\72
+| [[Marvo]]/zarvo
+|-
+| 1
+| 35\72
+| [[Cotritone]]
+|-
+| 2
+| 1\72
+|
+|-
+| 2
+| 5\72
+| [[Harry]]
+|-
+| 2
+| 7\72
+|
+|-
+| 2
+| 11\72
+| [[Unidec]]/hendec
+|-
+| 2
+| 13\72
+| [[Wizard]]/lizard/gizzard
+|-
+| 2
+| 17\72
+|
+|-
+| 3
+| 1\72
+|
+|-
+| 3
+| 5\72
+| [[Tritikleismic]]
+|-
+| 3
+| 7\72
+|
+|-
+| 3
+| 11\72
+| [[Mirkat]]
+|-
+| 4
+| 1\72
+| [[Quadritikleismic]]
+|-
+| 4
+| 5\72
+|
+|-
+| 4
+| 7\72
+|
+|-
+| 6
+| 1\72
+|
+|-
+| 6
+| 5\72
+|
+|-
+| 8
+| 1\72
+| [[Octoid]]
+|-
+| 8
+| 2\72
+| [[Octowerck]]
+|-
+| 8
+| 4\72
+|
+|-
+| 9
+| 1\72
+|
+|-
+| 9
+| 3\72
+| [[Ennealimmal]]/ennealimmic
+|-
+| 12
+| 1\72
+| [[Compton]]
+|-
+| 18
+| 1\72
+| [[Hemiennealimmal]]
+|-
+| 24
+| 1\72
+| [[Hours]]
+|-
+| 36
+| 1\72
+|
+|}
-[[blackjack|blackjack]], [[miracle_8|miracle_8]], [[miracle_10|miracle_10]], [[miracle_12|miracle_12]], [[miracle_12a|miracle_12a]], [[miracle_24hi|miracle_24hi]], [[miracle_24lo|miracle_24lo]]
+== Scales ==
+* [[smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]]
+* [[blackjack]], [[miracle_8]], [[miracle_10]], [[miracle_12]], [[miracle_12a]], [[miracle_24hi]], [[miracle_24lo]]
+* [[keenanmarvel]], [[xenakis_chrome]], [[xenakis_diat]], [[xenakis_schrome]]
+* [[genus24255et72|Euler(24255) genus in 72 equal]]
+* [[JuneGloom]]
-[[keenanmarvel|keenanmarvel]], [[xenakis_chrome|xenakis_chrome]], [[xenakis_diat|xenakis_diat]], [[xenakis_schrome|xenakis_schrome]]
+=== Harmonic Scale ===
+Mode 8 of the harmonic series – [[overtone_scales|overtones 8 through 16]], octave repeating – is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).
-[[genus24255et72|Euler(24255) genus in 72 equal]]
-[[JuneGloom|JuneGloom]]
-==Harmonic Scale==
-Mode 8 of the harmonic series -- [[overtone_scales|overtones 8 through 16]], octave repeating -- is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).
 {| class="wikitable"
@@ Line 806: / Line 942: @@
 | | 16
 |-
-| | ...as JI Ratio from 1/1:
+| | …as JI Ratio from 1/1:
 | | 1/1
 | |
@@ Line 825: / Line 961: @@
 | | 2/1
 |-
-| | ...in cents:
+| | …in cents:
 | | 0
 | |
@@ Line 863: / Line 999: @@
 | | 72
 |-
-| | ...in cents:
+| | …in cents:
 | | 0
 | |
@@ Line 901: / Line 1,037: @@
 | |
 |-
-| | ...in cents:
+| | …in cents:
 | |
 | | 203.9
@@ Line 959: / Line 1,095: @@
 |}
-=Linear temperaments=
+== Z function ==
+edo is the ninth [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[The_Riemann_Zeta_Function_and_Tuning #The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.
-{| class="wikitable"
-|-
-! | Periods per octave
-! | Generator
-! | Names
-|-
-| | 1
-| | 1\72
-| | [[Quincy|quincy]]
-|-
-| | 1
-| | 5\72
-| | [[marvolo|marvolo]]
-|-
-| | 1
-| | 7\72
-| | [[Miracle|miracle]]/benediction/manna
-|-
-| | 1
-| | 11\72
-| |
-|-
-| | 1
-| | 13\72
-| |
-|-
-| | 1
-| | 17\72
-| | [[Neominor|neominor]]
-|-
-| | 1
-| | 19\72
-| | [[catakleismic|catakleismic]]
-|-
-| | 1
-| | 23\72
-| |
-|-
-| | 1
-| | 25\72
-| | [[Sqrtphi|sqrtphi]]
-|-
-| | 1
-| | 29\72
-| |
-|-
-| | 1
-| | 31\72
-| | [[Marvo|marvo]]/zarvo
-|-
-| | 1
-| | 35\72
-| | [[cotritone|cotritone]]
-|-
-| | 2
-| | 1\72
-| |
-|-
-| | 2
-| | 5\72
-| | [[Harry|harry]]
-|-
-| | 2
-| | 7\72
-| |
-|-
-| | 2
-| | 11\72
-| | [[Unidec|unidec]]/hendec
-|-
-| | 2
-| | 13\72
-| | [[wizard|wizard]]/lizard/gizzard
-|-
-| | 2
-| | 17\72
-| |
-|-
-| | 3
-| | 1\72
-| |
-|-
-| | 3
-| | 5\72
-| | [[Tritikleismic|tritikleismic]]
-|-
-| | 3
-| | 7\72
-| |
-|-
-| | 3
-| | 11\72
-| | [[Mirkat|mirkat]]
-|-
-| | 4
-| | 1\72
-| | [[Quadritikleismic|quadritikleismic]]
-|-
-| | 4
-| | 5\72
-| |
-|-
-| | 4
-| | 7\72
-| |
-|-
-| | 6
-| | 1\72
-| |
-|-
-| | 6
-| | 5\72
-| |
-|-
-| | 8
-| | 1\72
-| | [[Octoid|octoid]]
-|-
-| | 8
-| | 2\72
-| | [[Octowerck|octowerck]]
-|-
-| | 8
-| | 4\72
-| |
-|-
-| | 9
-| | 1\72
-| |
-|-
-| | 9
-| | 3\72
-| | [[Ennealimmal|ennealimmal]]/ennealimmic
-|-
-| | 12
-| | 1\72
-| | [[Compton|compton]]
-|-
-| | 18
-| | 1\72
-| | [[Hemiennealimmal|hemiennealimmal]]
-|-
-| | 24
-| | 1\72
-| | [[Hours|hours]]
-|-
-| | 36
-| | 1\72
-| |
-|}
-=Z function=
-edo is the ninth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[The_Riemann_Zeta_Function_and_Tuning#The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.
 [[File:plot72.png|alt=plot72.png|plot72.png]]
-=Music=
+== Music ==
 [http://www.archive.org/details/Kotekant Kotekant] ''[http://www.archive.org/download/Kotekant/kotekant.mp3 play]'' by [[Gene_Ward_Smith|Gene Ward Smith]]
@@ Line 1,126: / Line 1,109: @@
 ''[http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 June Gloom #9]'' by Prent Rodgers
-=External links=
+== External links ==
-<ul><li>[http://en.wikipedia.org/wiki/72_tone_equal_temperament Wikipedia article on 72edo]</li><li>[http://orthodoxwiki.org/Byzantine_Chant OrthodoxWiki Article on Byzantine chant, which uses 72edo]</li><li>[http://en.wikipedia.org/wiki/Joe_Maneri Wikipedia article on Joe Maneri (1927-2009)]</li><li>[http://www.ekmelic-music.org/en/ Ekmelic Music Society/Gesellschaft für Ekmelische Musik], a group of composers and researchers dedicated to 72edo music</li><li>[http://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list</li><li>[http://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo]</li></ul>
+* [[Wikipedia:72_equal_temperament|72 equal temperament - Wikipedia]]
+* [http://orthodoxwiki.org/Byzantine_Chant OrthodoxWiki Article on Byzantine chant, which uses 72edo]
+* [http://en.wikipedia.org/wiki/Joe_Maneri Wikipedia article on Joe Maneri (1927-2009)]
+* [http://www.ekmelic-music.org/en/ Ekmelic Music Society/Gesellschaft für Ekmelische Musik], a group of composers and researchers dedicated to 72edo music
+* [http://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list
+* [http://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo]
 [[Category:Edo]]

72edo: Difference between revisions