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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.


72-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.
72-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.


72 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.
72 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
|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:
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:
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"
{| 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]].
72edo 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"
{| class="wikitable"
Line 806: Line 942:
| | 16
| | 16
|-
|-
| | ...as JI Ratio from 1/1:
| | …as JI Ratio from 1/1:
| | 1/1
| | 1/1
| |  
| |  
Line 825: Line 961:
| | 2/1
| | 2/1
|-
|-
| | ...in cents:
| | …in cents:
| | 0
| | 0
| |  
| |  
Line 863: Line 999:
| | 72
| | 72
|-
|-
| | ...in cents:
| | …in cents:
| | 0
| | 0
| |  
| |  
Line 901: Line 1,037:
| |  
| |  
|-
|-
| | ...in cents:
| | …in cents:
| |  
| |  
| | 203.9
| | 203.9
Line 959: Line 1,095:
|}
|}


=Linear temperaments=
== Z function ==
 
72edo 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=
72edo 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]]
[[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]]
[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
''[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]]
[[Category:Edo]]

Revision as of 08:04, 15 August 2020

Theory

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 24-tone equal temperament, a common and standard tuning of 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), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.

72-tone equal temperament approximates 11-limit just intonation exceptionally well, is consistent in the 17-limit, and is the ninth 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.

72 is an excellent tuning for miracle temperament, especially the 11-limit version, and the related rank three temperament prodigy, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.

Intervals

Degrees Cents Approximate Ratios (17-limit) Ups and Downs Notation
0 0.000 1/1 P1 perfect unison D
1 16.667 81/80 ^1 up unison ^D
2 33.333 45/44 ^^ double-up unison ^^D
3 50.000 33/32 ^31,
v3m2 		triple-up unison,
triple-down minor 2nd 		^3D,
v3Eb
4 66.667 25/24 vvm2 double-downminor 2nd vvEb
5 83.333 21/20 vm2 downminor 2nd vEb
6 100.000 35/33, 17/16, 18/17 m2 minor 2nd Eb
7 116.667 15/14, 16/15 ^m2 upminor 2nd ^Eb
8 133.333 27/25, 13/12, 14/13 v~2 downmid 2nd ^^Eb
9 150.000 12/11 ~2 mid 2nd v3E
10 166.667 11/10 ^~2 upmid 2nd vvE
11 183.333 10/9 vM2 downmajor 2nd vE
12 200.000 9/8 M2 major 2nd E
13 216.667 25/22, 17/15 ^M2 upmajor 2nd ^E
14 233.333 8/7 ^^M2 double-upmajor 2nd ^^E
15 250.000 81/70, 15/13 ^3M2,
v3m3 		triple-up major 2nd,
triple-down minor 3rd 		^3E,
v3F
16 266.667 7/6 vvm3 double-downminor 3rd vvF
17 283.333 33/28, 13/11, 20/17 vm3 downminor 3rd vF
18 300.000 25/21 m3 minor 3rd F
19 316.667 6/5 ^m3 upminor 3rd ^F
20 333.333 40/33, 17/14 v~3 downmid 3rd ^^F
21 350.000 11/9 ~3 mid 3rd ^3F
22 366.667 99/80, 16/13, 21/17 ^~3 upmid 3rd vvF#
23 383.333 5/4 vM3 downmajor 3rd vF#
24 400.000 44/35 M3 major 3rd F#
25 416.667 14/11 ^M3 upmajor 3rd ^F#
26 433.333 9/7 ^^M3 double-upmajor 3rd ^^F#
27 450.000 35/27, 13/10 ^3M3,
v34 		triple-up major 3rd,
triple-down 4th 		^3F#,
v3G
28 466.667 21/16, 17/13 vv4 double-down 4th vvG
29 483.333 33/25 v4 down 4th vG
30 500.000 4/3 P4 perfect 4th G
31 516.667 27/20 ^4 up 4th ^G
32 533.333 15/11 v~4 downmid 4th ^^G
33 550.000 11/8 ~4 mid 4th ^3G
34 566.667 25/18, 18/13 ^~4 upmid 4th vvG#
35 583.333 7/5 vA4, vd5 downaug 4th, updim 5th vG#, vAb
36 600.000 99/70, 17/12 A4, d5 aug 4th, dim 5th G#, Ab
37 616.667 10/7 ^A4, ^d5 upaug 4th, downdim 5th ^G#, ^Ab
38 633.333 36/25, 13/9 v~5 downmid 5th ^^Ab
39 650.000 16/11 ~5 mid 5th v3A
40 666.667 22/15 ^~5 upmid 5th vvA
41 683.333 40/27 v5 down 5th vA
42 700.000 3/2 P5 perfect 5th A
43 716.667 50/33 ^5 up 5th ^A
44 733.333 32/21 ^^5 double-up 5th ^^A
45 750.000 54/35, 17/11 ^35,
v3m6 		triple-up 5th,
triple-down minor 6th 		^3A,
v3Bb
46 766.667 14/9 vvm6 double-downminor 6th vvBb
47 783.333 11/7 vm6 downminor 6th vBb
48 800.000 35/22 m6 minor 6th Bb
49 816.667 8/5 ^m6 upminor 6th ^Bb
50 833.333 81/50, 13/8 v~6 downmid 6th ^^Bb
51 850.000 18/11 ~6 mid 6th v3B
52 866.667 33/20, 28/17 ^~6 upmid 6th vvB
53 883.333 5/3 vM6 downmajor 6th vB
54 900.000 27/16 M6 major 6th B
55 916.667 56/33, 17/10 ^M6 upmajor 6th ^B
56 933.333 12/7 ^^M6 double-upmajor 6th ^^B
57 950.000 121/70 ^3M6,
v3m7 		triple-up major 6th,
triple-down minor 7th 		^3B,
v3C
58 966.667 7/4 vvm7 double-downminor 7th vvC
59 983.333 44/25 vm7 downminor 7th vC
60 1000.000 16/9 m7 minor 7th C
61 1016.667 9/5 ^m7 upminor 7th ^C
62 1033.333 20/11 v~7 downmid 7th ^^C
63 1050.000 11/6 ~7 mid 7th ^3C
64 1066.667 50/27 ^~7 upmid 7th vvC#
65 1083.333 15/8 vM7 downmajor 7th vC#
66 1100.000 66/35, 17/9 M7 major 7th C#
67 1116.667 21/11 ^M7 upmajor 7th ^C#
68 1133.333 27/14 ^^M7 double-upmajor 7th ^^C#
69 1150.000 35/18 ^3M7,
v38 		triple-up major 7th,
triple-down octave 		^3C#,
v3D
70 1166.667 49/25 vv8 double-down octave vvD
71 1183.333 99/50 v8 down octave vD
72 1200.000 2/1 P8 perfect octave D

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

quality color monzo format examples
double-down minor zo {a, b, 0, 1} 7/6, 7/4
minor fourthward wa {a, b}, b < -1 32/27, 16/9
upminor gu {a, b, -1} 6/5, 9/5
mid ilo {a, b, 0, 0, 1} 11/9, 11/6
" lu {a, b, 0, 0, -1} 12/11, 18/11
downmajor yo {a, b, 1} 5/4, 5/3
major fifthward wa {a, b}, b > 1 9/8, 27/16
double-up major ru {a, b, 0, -1} 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:

color of the 3rd JI chord notes as edosteps notes of C chord written name spoken name
zo 6:7:9 0-16-42 C vvEb G Cvvm C double-down minor
gu 10:12:15 0-19-42 C ^Eb G C^m C upminor
ilo 18:22:27 0-21-42 C v3E G C~ C mid
yo 4:5:6 0-23-42 C vE G Cv C downmajor or C down
ru 14:18:27 0-26-42 C ^^E G C^^ C double-upmajor or C double-up

For a more complete list, see Ups and Downs Notation #Chord names in other EDOs.

Just approximation

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23 prime 29 prime 31
error (¢) 0.000 -1.955 -2.980 -2.159 -1.318 -7.194 -4.955 +2.487 +5.059 +3.755 -4.964

Commas

Commas tempered out by 72edo include…

3-limit
Pythagorean comma = 531441/524288 = |-19 12>
5-limit
kleisma = 15625/15552 = |-6 -5 6>

ampersand = 34171875/33554432 = |-25 7 6>

graviton = 129140163/128000000 = |-13 17 -6>

ennealimma = 7629394531250/7625597484987 = |1 -27 18>

7-limit 11-limit 13-limit
...............................

225/224

1029/1024

2401/2400

4375/4374

16875/16807

19683/19600

420175/419904

250047/250000

.......................

243/242

385/384

441/440

540/539

1375/1372

3025/3024

4000/3993

6250/6237

9801/9800

.......................

169/168

325/324

351/350

364/363

625/624

676/675

729/728

1001/1000

1575/1573

1716/1715

2080/2079

6656/6655

Temperaments

72edo 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.

Periods
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

Scales

Harmonic Scale

Mode 8 of the harmonic series – 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).

Overtones in "Mode 8": 8 9 10 11 12 13 14 15 16
…as JI Ratio from 1/1: 1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8 2/1
…in cents: 0 203.9 386.3 551.3 702.0 840.5 968.8 1088.3 1200.0
Nearest degree of 72edo: 0 12 23 33 42 50 58 65 72
…in cents: 0 200.0 383.3 550.0 700.0 833.3 966.7 1083.3 1200.0
Steps as Freq. Ratio: 9:8 10:9 11:10 12:11 13:12 14:13 15:14 16:15
…in cents: 203.9 182.4 165.0 150.6 138.6 128.3 119.4 111.7
Nearest degree of 72edo: 12 11 10 9 8 8 7 7
...in cents: 200.0 183.3 166.7 150.0 133.3 133.3 116.7 116.7

Z function

72edo is the ninth zeta integral edo, as well as being a peak and gap edo, and the maximum value of the 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.

plot72.png

Music

Kotekant play by Gene Ward Smith

Twinkle canon – 72 edo by Claudi Meneghin

Lazy Sunday by Jake Freivald in the lazysunday scale.

June Gloom #9 by Prent Rodgers

External links

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