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== Theory ==
== Theory ==
72edo approximates [[11-limit]] [[just intonation]] exceptionally well. It is [[consistent]] in the [[17-odd-limit]] and is the ninth [[zeta integral edo]]. It is the second edo (after [[58edo|58]]) to be [[consistency|distinctly consistent]] in the [[11-odd-limit]], the first edo to be [[consistency|consistent to distance 2]] in the 11-odd-limit, meaning every interval in the 11-odd-limit is approximated with less than 25% relative error (about 4 cents), and the first edo to be consistent in the 12- and 13-[[odd prime sum limit|odd-prime-sum-limit]].  
72edo approximates [[11-limit]] [[just intonation]] exceptionally well. It is the second edo (after [[58edo|58]]) to be [[consistent]] in the [[17-odd-limit]], and the second edo (also after 58) to be [[consistency|distinctly consistent]] in the [[11-odd-limit]], but it is the first edo to be [[consistency|consistent to distance 2]] in the 11-odd-limit, meaning every interval in the 11-odd-limit is approximated with less than 25% [[relative interval error|relative error]] (about 4 cents). It also has pretty good accuracy for the [[19-limit]], being almost consistent to the entire [[21-odd-limit]] with the only inconsistency occurring at [[19/13]] and its [[octave complement]]. It is the ninth [[zeta integral edo]].


The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo 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.
The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo 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.


72et is the only 11-limit regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, and splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s (144edo is enfactored in the 11-limit with 72edo, so it is already covered here). It is also an excellent tuning for [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].
The [[octave reduction|octave reduced]] [[13/1|13th harmonic]] is mapped on 50\72, an interval inherited from [[36edo]] (25\36) that is a very close approximation to [[acoustic phi]], and the [[17/1|17th]] and [[19/1|19th harmonics]] come from 12edo.  
 
The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament.


72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]].
72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]].
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{{Harmonics in equal|72|columns=11}}
{{Harmonics in equal|72|columns=11}}
{{Harmonics in equal|72|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}
{{Harmonics in equal|72|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}
=== As a tuning of other temperaments ===
72et is the only 11-limit regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, and splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s (144edo is enfactored in the 11-limit with 72edo, so it is already covered here). It is also an excellent tuning for [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].


=== Subsets and supersets ===
=== Subsets and supersets ===
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== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-1 right-2"
|-
|-
! #
! #
! Cents
! Cents
! Approximate ratios<ref group="note">{{sg|limit=19-limit}} For lower limits see [[Table of 72edo intervals]].</ref>
! Approximate ratios<ref group="note">As a 19-limit temperament, inconsistent intervals in ''italic''. For a table of intervals by prime limit, see [[Table of 72edo intervals]].</ref>
! colspan="3" | [[Ups and downs notation]]
! [[Kite's ups and downs notation|Ups and downs notation]]
! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]
! (K, S, U)
|-
|-
| 0
| 0
| 0.0
| 0.0
| [[1/1]]
| [[1/1]]
| P1
| {{UDnote|step=0}}
| perfect unison
| D
| P1
| perfect unison
| D
| D
|-
|-
| 1
| 1
| 16.7
| 16.7
| [[81/80]], [[91/90]], [[99/98]], [[100/99]], [[105/104]]
| [[81/80]], [[91/90]], [[99/98]], [[100/99]], [[105/104]]
| ^1
| {{UDnote|step=1}}
| up unison
| ^D
| K1, L1
| comma-wide unison, large unison
| KD, LD
| KD
|-
|-
| 2
| 2
| 33.3
| 33.3
| [[45/44]], [[49/48]], [[50/49]], [[55/54]], [[64/63]]
| [[45/44]], [[49/48]], [[50/49]], [[55/54]], [[64/63]]
| ^^
| {{UDnote|step=2}}
| dup unison
| ^^D
| S1, O1
| super unison, on unison
| SD, OD
| SD
|-
|-
| 3
| 3
| 50.0
| 50.0
| [[33/32]], [[36/35]], [[40/39]]
| [[33/32]], [[36/35]], [[40/39]]
| ^<sup>3</sup>1, v<sup>3</sup>m2
| {{UDnote|step=3}}
| trup unison, trudminor 2nd
|-
| ^<sup>3</sup>D, v<sup>3</sup>Eb
| 4
| U1, H1, hm2
| uber unison, hyper unison, hypominor 2nd
| UD, HD, uEb
| UD, uEb
|-
| 4
| 66.7
| 66.7
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| vvm2
| {{UDnote|step=4}}
| dudminor 2nd
| vvEb
| kkA1, sm2
| classic aug unison, subminor 2nd
| kkD#, sEb
| sD#, (kkD#), sEb
|-
|-
| 5
| 5
| 83.3
| 83.3
| [[20/19]], [[21/20]], [[22/21]]
| [[20/19]], [[21/20]], [[22/21]]
| vm2
| {{UDnote|step=5}}
| downminor 2nd
| vEb
| kA1, lm2
| comma-narrow aug unison, little minor 2nd
| kD#, lEb
| kD#, kEb
|-
|-
| 6
| 6
| 100.0
| 100.0
| [[17/16]], [[18/17]], [[19/18]]
| [[17/16]], [[18/17]], [[19/18]]
| m2
| {{UDnote|step=6}}
| minor 2nd
| Eb
| m2
| minor 2nd
| Eb
| Eb
|-
|-
| 7
| 7
| 116.7
| 116.7
| [[15/14]], [[16/15]]
| [[15/14]], [[16/15]]
| ^m2
| {{UDnote|step=7}}
| upminor 2nd
|-
| ^Eb
| Km2
| classic minor 2nd
| KEb
| KEb
|-
| 8
| 8
| 133.3
| 133.3
| [[13/12]], [[14/13]], [[27/25]]
| [[13/12]], [[14/13]], [[27/25]]
| ^^m2, v~2
| {{UDnote|step=8}}
| dupminor 2nd, downmid 2nd
| ^^Eb
| Om2
| on minor 2nd
| OEb
| SEb
|-
|-
| 9
| 9
| 150.0
| 150.0
| [[12/11]]
| [[12/11]]
| ~2
| {{UDnote|step=9}}
| mid 2nd
| v<sup>3</sup>E
| N2
| neutral 2nd
| UEb/uE
| UEb/uE
|-
|-
| 10
| 10
| 166.7
| 166.7
| [[11/10]]
| [[11/10]], [[21/19]]
| ^~2, vvM2
| {{UDnote|step=10}}
| upmid 2nd, dudmajor 2nd
|-
| vvE
| 11
| oM2
| 183.3
| off major 2nd
| [[10/9]]
| oE
| {{UDnote|step=11}}
| sE
|-
| 11
| 183.3
| [[10/9]]
| vM2
| downmajor 2nd
| vE
| kM2
| classic/comma-narrow major 2nd
| kE
| kE
|-
|-
| 12
| 12
| 200.0
| 200.0
| [[9/8]], [[19/17]]
| [[9/8]], [[19/17]]
| M2
| {{UDnote|step=12}}
| major 2nd
| E
| M2
| major 2nd
| E
| E
|-
|-
| 13
| 13
| 216.7
| 216.7
| [[17/15]], [[25/22]]
| [[17/15]], [[25/22]]
| ^M2
| {{UDnote|step=13}}
| upmajor 2nd
|-
| ^E
| 14
| LM2
| 233.3
| large major 2nd
| LE
| KE
|-
| 14
| 233.3
| [[8/7]]
| [[8/7]]
| ^^M2
| {{UDnote|step=14}}
| dupmajor 2nd
| ^^E
| SM2
| supermajor 2nd
| SE
| SE
|-
|-
| 15
| 15
| 250.0
| 250.0
| [[15/13]], [[22/19]]
| [[15/13]], [[22/19]]
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| {{UDnote|step=15}}
| trupmajor 2nd,<br>trudminor 3rd
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
| HM2, hm3
| hypermajor 2nd, hypominor 3rd
| HE, hF
| UE, uF
|-
|-
| 16
| 16
| 266.7
| 266.7
| [[7/6]]
| [[7/6]]
| vvm3
| {{UDnote|step=16}}
| dudminor 3rd
| vvF
| sm3
| subminor 3rd
| sF
| sF
|-
|-
| 17
| 17
| 283.3
| 283.3
| [[13/11]], [[20/17]]
| [[13/11]], [[20/17]]
| vm3
| {{UDnote|step=17}}
| downminor 3rd
|-
| vF
| lm3
| little minor 3rd
| lF
| kF
|-
| 18
| 18
| 300.0
| 300.0
| [[19/16]], [[25/21]], [[32/27]]
| [[19/16]], [[25/21]], [[32/27]]
| m3
| {{UDnote|step=18}}
| minor 3rd
| F
| m3
| minor 3rd
| F
| F
|-
|-
| 19
| 19
| 316.7
| 316.7
| [[6/5]]
| [[6/5]]
| ^m3
| {{UDnote|step=19}}
| upminor 3rd
| ^F
| Km3
| classic minor 3rd
| KF
| KF
|-
|-
| 20
| 20
| 333.3
| 333.3
| [[17/14]], [[39/32]], [[40/33]]
| [[17/14]], ''[[39/32]]'', [[40/33]]
| ^^m3, v~3
| {{UDnote|step=20}}
| dupminor 3rd, downmid 3rd
| ^^F
| Om3
| on minor third
| OF
| SF
|-
|-
| 21
| 21
| 350.0
| 350.0
| [[11/9]], [[27/22]]
| [[11/9]], [[27/22]]
| ~3
| {{UDnote|step=21}}
| mid 3rd
| ^<sup>3</sup>F
| N3
| neutral 3rd
| UF/uF#
| UF/uF#
|-
|-
| 22
| 22
| 366.7
| 366.7
| [[16/13]], [[21/17]], [[26/21]]
| [[16/13]], [[21/17]], [[26/21]]
| ^~3, vvM3
| {{UDnote|step=22}}
| upmid 3rd, dudmajor 3rd
| vvF#
| oM3
| off major 3rd
| oF#
| sF#
|-
|-
| 23
| 23
| 383.3
| 383.3
| [[5/4]]
| [[5/4]]
| vM3
| {{UDnote|step=23}}
| downmajor 3rd
| vF#
| kM3
| classic major 3rd
| kF#
| kF#
|-
|-
| 24
| 24
| 400.0
| 400.0
| [[24/19]]
| [[24/19]]
| M3
| {{UDnote|step=24}}
| major 3rd
| F#
| M3
| major 3rd
| F#
| F#
|-
|-
| 25
| 25
| 416.7
| 416.7
| [[14/11]], [[19/15]]
| [[14/11]], [[19/15]]
| ^M3
| {{UDnote|step=25}}
| upmajor 3rd
|-
| ^F#
| LM3
| large major 3rd
| LF#
| KF#
|-
| 26
| 26
| 433.3
| 433.3
| [[9/7]]
| [[9/7]]
| ^^M3
| {{UDnote|step=26}}
| dupmajor 3rd
| ^^F#
| SM3
| supermajor 3rd
| SF#
| SF#
|-
|-
| 27
| 27
| 450.0
| 450.0
| [[13/10]], [[22/17]]
| [[13/10]], [[22/17]]
| ^<sup>3</sup>M3, v<sup>3</sup>4
| {{UDnote|step=27}}
| trupmajor 3rd, trud 4th
| ^<sup>3</sup>F#, v<sup>3</sup>G
| HM3, h4
| hypermajor 3rd, hypo 4th
| HF#, hG
| UF#, uG
|-
|-
| 28
| 28
| 466.7
| 466.7
| [[17/13]], [[21/16]]
| [[17/13]], [[21/16]]
| vv4
| {{UDnote|step=28}}
| dud 4th
| vvG
| s4
| sub 4th
| sG
| sG
|-
|-
| 29
| 29
| 483.3
| 483.3
| [[33/25]]
| [[33/25]]
| v4
| {{UDnote|step=29}}
| down 4th
| vG
| l4
| little 4th
| lG
| kG
|-
|-
| 30
| 30
| 500.0
| 500.0
| [[4/3]]
| [[4/3]]
| P4
| {{UDnote|step=30}}
| perfect 4th
| G
| P4
| perfect 4th
| G
| G
|-
|-
| 31
| 31
| 516.7
| 516.7
| [[27/20]]
| [[27/20]]
| ^4
| {{UDnote|step=31}}
| up 4th
| ^G
| K4
| comma-wide 4th
| KG
| KG
|-
|-
| 32
| 32
| 533.3
| 533.3
| [[15/11]], [[19/14]], ''[[26/19]]''
| [[15/11]], [[19/14]], ''[[26/19]]''
| ^^4, v~4
| {{UDnote|step=32}}
| dup 4th, downmid 4th
| ^^G
| O4
| on 4th
| OG
| SG
|-
|-
| 33
| 33
| 550.0
| 550.0
| [[11/8]]
| [[11/8]]
| ~4
| {{UDnote|step=33}}
| mid 4th
|-
| ^<sup>3</sup>G
| U4/N4
| uber 4th / neutral 4th
| UG
| UG
|-
| 34
| 34
| 566.7
| 566.7
| [[18/13]], [[25/18]]
| [[18/13]], [[25/18]]
| ^~4, vvA4
| {{UDnote|step=34}}
| upmid 4th, dudaug 4th
| vvG#
| kkA4, sd5
| classic aug 4th, sub dim 5th
| kkG#, sAb
| SG#, (kkG#), sAb
|-
|-
| 35
| 35
| 583.3
| 583.3
| [[7/5]]
| [[7/5]]
| vA4, vd5
| {{UDnote|step=35}}
| downaug 4th, <br>downdim 5th
| vG#, vAb
| kA4, ld5
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, kAb
|-
|-
| 36
| 36
| 600.0
| 600.0
| [[17/12]], [[24/17]]
| [[17/12]], [[24/17]]
| A4, d5
| {{UDnote|step=36}}
| aug 4th, dim 5th
| G#, Ab
| A4, d5
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
|-
|-
| 37
| 37
| 616.7
| 616.7
| [[10/7]]
| [[10/7]]
| ^A4, ^d5
| {{UDnote|step=37}}
| upaug 4th, updim 5th
| ^G#, ^Ab
| LA4, Kd5
| large aug 4th, comma-wide dim 5th
| LG#, KAb
| KG#, KAb
|-
|-
| 38
| 38
| 633.3
| 633.3
| [[13/9]], [[36/25]]
| [[13/9]], [[36/25]]
| v~5, ^^d5
| {{UDnote|step=38}}
| downmid 5th, <br>dupdim 5th
| ^^Ab
| SA4, KKd5
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, SAb, (KKAb)
|-
|-
| 39
| 39
| 650.0
| 650.0
| [[16/11]]
| [[16/11]]
| ~5
| {{UDnote|step=39}}
| mid 5th
| v<sup>3</sup>A
| u5/N5
| unter 5th / neutral 5th
| uA
| uA
|-
|-
| 40
| 40
| 666.7
| 666.7
| ''[[19/13]]'', [[22/15]], [[28/19]]
| ''[[19/13]]'', [[22/15]], [[28/19]]
| vv5, ^~5
| {{UDnote|step=40}}
| dud 5th, upmid 5th
| vvA
| o5
| off 5th
| oA
| sA
|-
|-
| 41
| 41
| 683.3
| 683.3
| [[40/27]]
| [[40/27]]
| v5
| {{UDnote|step=41}}
| down 5th
|-
| vA
| k5
| comma-narrow 5th
| kA
| kA
|-
| 42
| 42
| 700.0
| 700.0
| [[3/2]]
| [[3/2]]
| P5
| {{UDnote|step=42}}
| perfect 5th
| A
| P5
| perfect 5th
| A
| A
|-
|-
| 43
| 43
| 716.7
| 716.7
| [[50/33]]
| [[50/33]]
| ^5
| {{UDnote|step=43}}
| up 5th
| ^A
| L5
| large fifth
| LA
| KA
|-
|-
| 44
| 44
| 733.3
| 733.3
| [[26/17]], [[32/21]]
| [[26/17]], [[32/21]]
| ^^5
| {{UDnote|step=44}}
| dup 5th
| ^^A
| S5
| super fifth
| SA
| SA
|-
|-
| 45
| 45
| 750.0
| 750.0
| [[17/11]], [[20/13]]
| [[17/11]], [[20/13]]
| ^<sup>3</sup>5, v<sup>3</sup>m6
| {{UDnote|step=45}}
| trup 5th, trudminor 6th
| ^<sup>3</sup>A, v<sup>3</sup>Bb
| H5, hm6
| hyper fifth, hypominor 6th
| HA, hBb
| UA, uBb
|-
|-
| 46
| 46
| 766.7
| 766.7
| [[14/9]]
| [[14/9]]
| vvm6
| {{UDnote|step=46}}
| dudminor 6th
| vvBb
| sm6
| superminor 6th
| sBb
| sBb
|-
|-
| 47
| 47
| 783.3
| 783.3
| [[11/7]], [[30/19]]
| [[11/7]], [[30/19]]
| vm6
| {{UDnote|step=47}}
| downminor 6th
| vBb
| lm6
| little minor 6th
| lBb
| kBb
|-
|-
| 48
| 48
| 800.0
| 800.0
| [[19/12]]
| [[19/12]]
| m6
| {{UDnote|step=48}}
| minor 6th
| Bb
| m6
| minor 6th
| Bb
| Bb
|-
|-
| 49
| 49
| 816.7
| 816.7
| [[8/5]]
| [[8/5]]
| ^m6
| {{UDnote|step=49}}
| upminor 6th
|-
| ^Bb
| Km6
| classic minor 6th
| kBb
| kBb
|-
| 50
| 50
| 833.3
| 833.3
| [[13/8]], [[21/13]], [[34/21]]
| [[13/8]], [[21/13]], [[34/21]]
| ^^m6, v~6
| {{UDnote|step=50}}
| dupminor 6th, downmid 6th
| ^^Bb
| Om6
| on minor 6th
| oBb
| sBb
|-
|-
| 51
| 51
| 850.0
| 850.0
| [[18/11]], [[44/27]]
| [[18/11]], [[44/27]]
| ~6
| {{UDnote|step=51}}
| mid 6th
| v<sup>3</sup>B
| N6
| neutral 6th
| UBb, uB
| UBb, uB
|-
|-
| 52
| 52
| 866.7
| 866.7
| [[28/17]], [[33/20]], [[64/39]]
| [[28/17]], [[33/20]], ''[[64/39]]''
| ^~6, vvM6
| {{UDnote|step=52}}
| upmid 6th, dudmajor 6th
| vvB
| oM6
| off major 6th
| oB
| sB
|-
|-
| 53
| 53
| 883.3
| 883.3
| [[5/3]]
| [[5/3]]
| vM6
| {{UDnote|step=53}}
| downmajor 6th
| vB
| kM6
| classic major 6th
| kB
| kB
|-
|-
| 54
| 54
| 900.0
| 900.0
| [[27/16]], [[32/19]], [[42/25]]
| [[27/16]], [[32/19]], [[42/25]]
| M6
| {{UDnote|step=54}}
| major 6th
| B
| M6
| major 6th
| B
| B
|-
|-
| 55
| 55
| 916.7
| 916.7
| [[17/10]], [[22/13]]
| [[17/10]], [[22/13]]
| ^M6
| {{UDnote|step=55}}
| upmajor 6th
| ^B
| LM6
| large major 6th
| LB
| KB
|-
|-
| 56
| 56
| 933.3
| 933.3
| [[12/7]]
| [[12/7]]
| ^^M6
| {{UDnote|step=56}}
| dupmajor 6th
| ^^B
| SM6
| supermajor 6th
| SB
| SB
|-
|-
| 57
| 57
| 950.0
| 950.0
| [[19/11]], [[26/15]]
| [[19/11]], [[26/15]]
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| {{UDnote|step=57}}
| trupmajor 6th,<br>trudminor 7th
|-
| ^<sup>3</sup>B, <br>v<sup>3</sup>C
| HM6, hm7
| hypermajor 6th, hypominor 7th
| HB, hC
| UB, uC
|-
| 58
| 58
| 966.7
| 966.7
| [[7/4]]
| [[7/4]]
| vvm7
| {{UDnote|step=58}}
| dudminor 7th
| vvC
| sm7
| subminor 7th
| sC
| sC
|-
|-
| 59
| 59
| 983.3
| 983.3
| [[30/17]], [[44/25]]
| [[30/17]], [[44/25]]
| vm7
| {{UDnote|step=59}}
| downminor 7th
| vC
| lm7
| little minor 7th
| lC
| kC
|-
|-
| 60
| 60
| 1000.0
| 1000.0
| [[16/9]], [[34/19]]
| [[16/9]], [[34/19]]
| m7
| {{UDnote|step=60}}
| minor 7th
| C
| m7
| minor 7th
| C
| C
|-
|-
| 61
| 61
| 1016.7
| 1016.7
| [[9/5]]
| [[9/5]]
| ^m7
| {{UDnote|step=61}}
| upminor 7th
| ^C
| Km7
| classic/comma-wide minor 7th
| KC
| KC
|-
|-
| 62
| 62
| 1033.3
| 1033.3
| [[20/11]]
| [[20/11]], [[38/21]]
| ^^m7, v~7
| {{UDnote|step=62}}
| dupminor 7th, downmid 7th
| ^^C
| Om7
| on minor 7th
| OC
| SC
|-
|-
| 63
| 63
| 1050.0
| 1050.0
| [[11/6]]
| [[11/6]]
| ~7
| {{UDnote|step=63}}
| mid 7th
| ^<sup>3</sup>C
| N7, hd8
| neutral 7th, hypo dim 8ve
| UC/uC#, hDb
| UC/uC#, uDb
|-
|-
| 64
| 64
| 1066.7
| 1066.7
| [[13/7]], [[24/13]], [[50/27]]
| [[13/7]], [[24/13]], [[50/27]]
| ^~7, vvM7
| {{UDnote|step=64}}
| upmid 7th, dudmajor 7th
| vvC#
| oM7, sd8
| off major 7th, sub dim 8ve
| oC#, sDb
| sC#, sDb
|-
|-
| 65
| 65
| 1083.3
| 1083.3
| [[15/8]], [[28/15]]
| [[15/8]], [[28/15]]
| vM7
| {{UDnote|step=65}}
| downmajor 7th
|-
| vC#
| kM7, ld8
| classic major 7th, little dim 8ve
| kC#, lDb
| kC#, kDb
|-
| 66
| 66
| 1100.0
| 1100.0
| [[17/9]], [[32/17]], [[36/19]]
| [[17/9]], [[32/17]], [[36/19]]
| M7
| {{UDnote|step=66}}
| major 7th
| C#
| M7, d8
| major 7th, dim 8ve
| C#, Db
| C#, Db
|-
|-
| 67
| 67
| 1116.7
| 1116.7
| [[19/10]], [[21/11]], [[40/21]]
| [[19/10]], [[21/11]], [[40/21]]
| ^M7
| {{UDnote|step=67}}
| upmajor 7th
| ^C#
| LM7, Kd8
| large major 7th, comma-wide dim 8ve
| LC#, KDb
| KC#, KDb
|-
|-
| 68
| 68
| 1133.3
| 1133.3
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| ^^M7
| {{UDnote|step=68}}
| dupmajor 7th
| ^^C#
| SM7, KKd8
| supermajor 7th, classic dim 8ve
| SC#, KKDb
| SC#, SDb, (KKDb)
|-
|-
| 69
| 69
| 1150.0
| 1150.0
| [[35/18]], [[39/20]], [[64/33]]
| [[35/18]], [[39/20]], [[64/33]]
| ^<sup>3</sup>M7, v<sup>3</sup>8
| {{UDnote|step=69}}
| trupmajor 7th, trud octave
| ^<sup>3</sup>C#, v<sup>3</sup>D
| HM7, u8, h8
| hypermajor 7th, unter 8ve, hypo 8ve
| HC#, uD, hD
| UC#, uDb, uD
|-
|-
| 70
| 70
| 1166.7
| 1166.7
| [[49/25]], [[55/28]], [[63/32]], [[88/45]], [[96/49]]
| [[49/25]], [[55/28]], [[63/32]], [[88/45]], [[96/49]]
| vv8
| {{UDnote|step=70}}
| dud octave
| vvD
| s8, o8
| sub 8ve, off 8ve
| sD, oD
| sD
|-
|-
| 71
| 71
| 1183.3
| 1183.3
| [[99/50]], [[160/81]], [[180/91]], [[196/99]], [[208/105]]
| [[99/50]], [[160/81]], [[180/91]], [[196/99]], [[208/105]]
| v8
| {{UDnote|step=71}}
| down octave
| vD
| k8, l8
| comma-narrow 8ve, little 8ve
| kD, lD
| kD
|-
|-
| 72
| 72
| 1200.0
| 1200.0
| [[2/1]]
| [[2/1]]
| P8
| {{UDnote|step=72}}
| perfect octave
| D
| P8
| perfect octave
| D
| D
|}
|}
<references group="note" />
<references group="note" />


=== Interval quality and chord names in color notation ===
=== Proposed interval names and solfèges ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
{| class="wikitable center-all right-2 left-4 left-7 mw-collapsible mw-collapsed"
 
|+ style="font-size: 105%; white-space: nowrap;" | Table of proposed interval names and solfèges
{| class="wikitable center-all"
|-
! #
! Cents
! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]]
! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]
! (K, S, U)
|-
|-
! Quality
| 0
! [[Color notation|Color]]
| 0.0
! Monzo format
| P1
! Examples
| perfect unison
|-
| D
| dudminor
| P1
| zo
| perfect unison
| (a b 0 1)
| D
| [[7/6]], [[7/4]]
| D
|-
|-
| minor
| 1
| fourthward wa
| 16.7
| (a b), b < -1
| ^1
| [[32/27]], [[16/9]]
| up unison
|-
| ^D
| upminor
| K1, L1
| gu
| comma-wide unison, large unison
| (a b -1)
| KD, LD
| [[6/5]], [[9/5]]
| KD
|-
|-
| rowspan="2" | dupminor, <br>downmid
| 2
| luyo
| 33.3
| (a b 1 0 -1)
| ^^
| [[15/11]]
| dup unison
|-
| ^^D
| tho
| S1, O1
| (a b 0 0 0 1)
| super unison, on unison
| [[13/8]], [[13/9]]
| SD, OD
|-
| SD
| rowspan="2" | mid
|-
| ilo
| 3
| (a b 0 0 1)
| 50.0
| [[11/9]], [[11/6]]
| ^<sup>3</sup>1, v<sup>3</sup>m2
|-
| trup unison, trudminor 2nd
| lu
| ^<sup>3</sup>D, v<sup>3</sup>Eb
| (a b 0 0 -1)
| U1, H1, hm2
| [[12/11]], [[18/11]]
| uber unison, hyper unison, hypominor 2nd
| UD, HD, uEb
| UD, uEb
|-
| 4
| 66.7
| vvm2
| dudminor 2nd
| vvEb
| kkA1, sm2
| classic aug unison, subminor 2nd
| kkD#, sEb
| sD#, (kkD#), sEb
|-
| 5
| 83.3
| vm2
| downminor 2nd
| vEb
| kA1, lm2
| comma-narrow aug unison, little minor 2nd
| kD#, lEb
| kD#, kEb
|-
| 6
| 100.0
| m2
| minor 2nd
| Eb
| m2
| minor 2nd
| Eb
| Eb
|-
| 7
| 116.7
| ^m2
| upminor 2nd
| ^Eb
| Km2
| classic minor 2nd
| KEb
| KEb
|-
| 8
| 133.3
| ^^m2, v~2
| dupminor 2nd, downmid 2nd
| ^^Eb
| Om2
| on minor 2nd
| OEb
| SEb
|-
| 9
| 150.0
| ~2
| mid 2nd
| v<sup>3</sup>E
| N2
| neutral 2nd
| UEb/uE
| UEb/uE
|-
| 10
| 166.7
| ^~2, vvM2
| upmid 2nd, dudmajor 2nd
| vvE
| oM2
| off major 2nd
| oE
| sE
|-
| 11
| 183.3
| vM2
| downmajor 2nd
| vE
| kM2
| classic/comma-narrow major 2nd
| kE
| kE
|-
| 12
| 200.0
| M2
| major 2nd
| E
| M2
| major 2nd
| E
| E
|-
| 13
| 216.7
| ^M2
| upmajor 2nd
| ^E
| LM2
| large major 2nd
| LE
| KE
|-
| 14
| 233.3
| ^^M2
| dupmajor 2nd
| ^^E
| SM2
| supermajor 2nd
| SE
| SE
|-
| 15
| 250.0
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| trupmajor 2nd,<br>trudminor 3rd
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
| HM2, hm3
| hypermajor 2nd, hypominor 3rd
| HE, hF
| UE, uF
|-
| 16
| 266.7
| vvm3
| dudminor 3rd
| vvF
| sm3
| subminor 3rd
| sF
| sF
|-
| 17
| 283.3
| vm3
| downminor 3rd
| vF
| lm3
| little minor 3rd
| lF
| kF
|-
| 18
| 300.0
| m3
| minor 3rd
| F
| m3
| minor 3rd
| F
| F
|-
| 19
| 316.7
| ^m3
| upminor 3rd
| ^F
| Km3
| classic minor 3rd
| KF
| KF
|-
| 20
| 333.3
| ^^m3, v~3
| dupminor 3rd, downmid 3rd
| ^^F
| Om3
| on minor third
| OF
| SF
|-
| 21
| 350.0
| ~3
| mid 3rd
| ^<sup>3</sup>F
| N3
| neutral 3rd
| UF/uF#
| UF/uF#
|-
| 22
| 366.7
| ^~3, vvM3
| upmid 3rd, dudmajor 3rd
| vvF#
| oM3
| off major 3rd
| oF#
| sF#
|-
| 23
| 383.3
| vM3
| downmajor 3rd
| vF#
| kM3
| classic major 3rd
| kF#
| kF#
|-
| 24
| 400.0
| M3
| major 3rd
| F#
| M3
| major 3rd
| F#
| F#
|-
| 25
| 416.7
| ^M3
| upmajor 3rd
| ^F#
| LM3
| large major 3rd
| LF#
| KF#
|-
| 26
| 433.3
| ^^M3
| dupmajor 3rd
| ^^F#
| SM3
| supermajor 3rd
| SF#
| SF#
|-
| 27
| 450.0
| ^<sup>3</sup>M3, v<sup>3</sup>4
| trupmajor 3rd, trud 4th
| ^<sup>3</sup>F#, v<sup>3</sup>G
| HM3, h4
| hypermajor 3rd, hypo 4th
| HF#, hG
| UF#, uG
|-
| 28
| 466.7
| vv4
| dud 4th
| vvG
| s4
| sub 4th
| sG
| sG
|-
| 29
| 483.3
| v4
| down 4th
| vG
| l4
| little 4th
| lG
| kG
|-
| 30
| 500.0
| P4
| perfect 4th
| G
| P4
| perfect 4th
| G
| G
|-
| 31
| 516.7
| ^4
| up 4th
| ^G
| K4
| comma-wide 4th
| KG
| KG
|-
| 32
| 533.3
| ^^4, v~4
| dup 4th, downmid 4th
| ^^G
| O4
| on 4th
| OG
| SG
|-
| 33
| 550.0
| ~4
| mid 4th
| ^<sup>3</sup>G
| U4/N4
| uber 4th / neutral 4th
| UG
| UG
|-
| 34
| 566.7
| ^~4, vvA4
| upmid 4th, dudaug 4th
| vvG#
| kkA4, sd5
| classic aug 4th, sub dim 5th
| kkG#, sAb
| SG#, (kkG#), sAb
|-
| 35
| 583.3
| vA4, vd5
| downaug 4th, <br>downdim 5th
| vG#, vAb
| kA4, ld5
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, kAb
|-
| 36
| 600.0
| A4, d5
| aug 4th, dim 5th
| G#, Ab
| A4, d5
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
|-
| 37
| 616.7
| ^A4, ^d5
| upaug 4th, updim 5th
| ^G#, ^Ab
| LA4, Kd5
| large aug 4th, comma-wide dim 5th
| LG#, KAb
| KG#, KAb
|-
| 38
| 633.3
| v~5, ^^d5
| downmid 5th, <br>dupdim 5th
| ^^Ab
| SA4, KKd5
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, SAb, (KKAb)
|-
| 39
| 650.0
| ~5
| mid 5th
| v<sup>3</sup>A
| u5/N5
| unter 5th / neutral 5th
| uA
| uA
|-
| 40
| 666.7
| vv5, ^~5
| dud 5th, upmid 5th
| vvA
| o5
| off 5th
| oA
| sA
|-
| 41
| 683.3
| v5
| down 5th
| vA
| k5
| comma-narrow 5th
| kA
| kA
|-
| 42
| 700.0
| P5
| perfect 5th
| A
| P5
| perfect 5th
| A
| A
|-
| 43
| 716.7
| ^5
| up 5th
| ^A
| L5
| large fifth
| LA
| KA
|-
| 44
| 733.3
| ^^5
| dup 5th
| ^^A
| S5
| super fifth
| SA
| SA
|-
| 45
| 750.0
| ^<sup>3</sup>5, v<sup>3</sup>m6
| trup 5th, trudminor 6th
| ^<sup>3</sup>A, v<sup>3</sup>Bb
| H5, hm6
| hyper fifth, hypominor 6th
| HA, hBb
| UA, uBb
|-
| 46
| 766.7
| vvm6
| dudminor 6th
| vvBb
| sm6
| superminor 6th
| sBb
| sBb
|-
| 47
| 783.3
| vm6
| downminor 6th
| vBb
| lm6
| little minor 6th
| lBb
| kBb
|-
| 48
| 800.0
| m6
| minor 6th
| Bb
| m6
| minor 6th
| Bb
| Bb
|-
| 49
| 816.7
| ^m6
| upminor 6th
| ^Bb
| Km6
| classic minor 6th
| kBb
| kBb
|-
| 50
| 833.3
| ^^m6, v~6
| dupminor 6th, downmid 6th
| ^^Bb
| Om6
| on minor 6th
| oBb
| sBb
|-
| 51
| 850.0
| ~6
| mid 6th
| v<sup>3</sup>B
| N6
| neutral 6th
| UBb, uB
| UBb, uB
|-
| 52
| 866.7
| ^~6, vvM6
| upmid 6th, dudmajor 6th
| vvB
| oM6
| off major 6th
| oB
| sB
|-
| 53
| 883.3
| vM6
| downmajor 6th
| vB
| kM6
| classic major 6th
| kB
| kB
|-
| 54
| 900.0
| M6
| major 6th
| B
| M6
| major 6th
| B
| B
|-
| 55
| 916.7
| ^M6
| upmajor 6th
| ^B
| LM6
| large major 6th
| LB
| KB
|-
| 56
| 933.3
| ^^M6
| dupmajor 6th
| ^^B
| SM6
| supermajor 6th
| SB
| SB
|-
| 57
| 950.0
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| trupmajor 6th,<br>trudminor 7th
| ^<sup>3</sup>B, <br>v<sup>3</sup>C
| HM6, hm7
| hypermajor 6th, hypominor 7th
| HB, hC
| UB, uC
|-
| 58
| 966.7
| vvm7
| dudminor 7th
| vvC
| sm7
| subminor 7th
| sC
| sC
|-
| 59
| 983.3
| vm7
| downminor 7th
| vC
| lm7
| little minor 7th
| lC
| kC
|-
| 60
| 1000.0
| m7
| minor 7th
| C
| m7
| minor 7th
| C
| C
|-
| 61
| 1016.7
| ^m7
| upminor 7th
| ^C
| Km7
| classic/comma-wide minor 7th
| KC
| KC
|-
| 62
| 1033.3
| ^^m7, v~7
| dupminor 7th, downmid 7th
| ^^C
| Om7
| on minor 7th
| OC
| SC
|-
| 63
| 1050.0
| ~7
| mid 7th
| ^<sup>3</sup>C
| N7, hd8
| neutral 7th, hypo dim 8ve
| UC/uC#, hDb
| UC/uC#, uDb
|-
| 64
| 1066.7
| ^~7, vvM7
| upmid 7th, dudmajor 7th
| vvC#
| oM7, sd8
| off major 7th, sub dim 8ve
| oC#, sDb
| sC#, sDb
|-
| 65
| 1083.3
| vM7
| downmajor 7th
| vC#
| kM7, ld8
| classic major 7th, little dim 8ve
| kC#, lDb
| kC#, kDb
|-
| 66
| 1100.0
| M7
| major 7th
| C#
| M7, d8
| major 7th, dim 8ve
| C#, Db
| C#, Db
|-
| 67
| 1116.7
| ^M7
| upmajor 7th
| ^C#
| LM7, Kd8
| large major 7th, comma-wide dim 8ve
| LC#, KDb
| KC#, KDb
|-
| 68
| 1133.3
| ^^M7
| dupmajor 7th
| ^^C#
| SM7, KKd8
| supermajor 7th, classic dim 8ve
| SC#, KKDb
| SC#, SDb, (KKDb)
|-
| 69
| 1150.0
| ^<sup>3</sup>M7, v<sup>3</sup>8
| trupmajor 7th, trud octave
| ^<sup>3</sup>C#, v<sup>3</sup>D
| HM7, u8, h8
| hypermajor 7th, unter 8ve, hypo 8ve
| HC#, uD, hD
| UC#, uDb, uD
|-
| 70
| 1166.7
| vv8
| dud octave
| vvD
| s8, o8
| sub 8ve, off 8ve
| sD, oD
| sD
|-
| 71
| 1183.3
| v8
| down octave
| vD
| k8, l8
| comma-narrow 8ve, little 8ve
| kD, lD
| kD
|-
| 72
| 1200.0
| P8
| perfect octave
| D
| P8
| perfect octave
| D
| D
|}
 
=== Interval quality and chord names in color notation ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
 
{| class="wikitable center-all"
|-
! Quality
! [[Color notation|Color]]
! Monzo format
! Examples
|-
| dudminor
| 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]]
|-
| rowspan="2" | dupminor, <br>downmid
| luyo
| (a b 1 0 -1)
| [[15/11]]
|-
| tho
| (a b 0 0 0 1)
| [[13/8]], [[13/9]]
|-
| rowspan="2" | mid
| ilo
| (a b 0 0 1)
| [[11/9]], [[11/6]]
|-
| lu
| (a b 0 0 -1)
| [[12/11]], [[18/11]]
|-
|-
| rowspan="2" | upmid, <br>dudmajor
| rowspan="2" | upmid, <br>dudmajor
Line 983: Line 1,285:


== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
72edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}
 
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt-szg}}
 
=== Kite's ups and downs notation ===
72edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
{{Ups and downs sharpness}}
{{Ups and downs sharpness}}


Half-sharps and half-flats can be used to avoid triple arrows:
Half-sharps and half-flats can be used to avoid triple arrows:
{{Ups and downs sharpness|72|true}}
{{Ups and downs sharpness|72|true}}
[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have sharps and flats with arrows borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp6}}
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt}}


=== Sagittal notation ===
=== Sagittal notation ===
Line 1,367: Line 1,670:
| 516.7
| 516.7
| 27/20
| 27/20
| [[Marvo]] / [[zarvo]]
| [[Gravity]] / [[marvo]] / [[zarvo]]
|-
|-
| 1
| 1
Line 1,439: Line 1,742:
| 316.7<br>(50.0)
| 316.7<br>(50.0)
| 6/5<br>(36/35)
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimnic
| [[Ennealimmal]] / ennealimnic / ennealiminal
|-
|-
| 9
| 9
Line 1,471: Line 1,774:
| [[Gamelstearn]]
| [[Gamelstearn]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Octave stretch or compression ==
== Octave stretch or compression ==
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; [[Jake Freivald]]
; [[Jake Freivald]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday'']{{dead link}} in the [[lazysunday]] scale
* [https://web.archive.org/web/20201127014336/http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday''] in the [[lazysunday]] scale


{{Wikipedia|In vain (Haas)}}
{{Wikipedia|In vain (Haas)}}
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; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 ''Twinkle canon – 72 edo'']{{dead link}}
* [https://web.archive.org/web/20201127015744/http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 ''Twinkle canon – 72 edo'']
* [https://www.youtube.com/watch?v=zR0NDgh4944 ''The Miracle Canon'', 3-in-1 on a Ground]
* [https://www.youtube.com/watch?v=zR0NDgh4944 ''The Miracle Canon'', 3-in-1 on a Ground]
* [https://www.youtube.com/watch?v=w6Bckog1eOM ''Sicilienne in Miracle'']
* [https://www.youtube.com/watch?v=w6Bckog1eOM ''Sicilienne in Miracle'']
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; [[Prent Rodgers]]
; [[Prent Rodgers]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 ''June Gloom #9'']{{dead link}}
* [https://web.archive.org/web/20201127012907/http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 ''June Gloom #9'']


; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
Retrieved from "https://en.xen.wiki/w/72edo"