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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. 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]].
=== Prime harmonics ===
 
{{Harmonics in equal|72|columns=11}}
=== Prime harmonics ===
{{Harmonics in equal|72|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}
{{Harmonics in equal|72|columns=9}}
 
{{Harmonics in equal|72|columns=9|start=10|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 ===
 
Since 72 factors into primes as {{nowrap| 2<sup>3</sup> × 3<sup>2</sup> }}, 72edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36 }}. [[144edo]], which doubles it, provides a possible correction to its approximate harmonic 13, though unlike 72 it is not consistent to the [[13-odd-limit]].
=== Subsets and supersets ===
 
Since 72 factors into primes as {{nowrap| 2<sup>3</sup> × 3<sup>2</sup> }}, 72edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36 }}. [[144edo]], which doubles it, provides a possible correction to its approximate harmonic 13, though unlike 72 it is not consistent to the [[13-odd-limit]].
== Intervals ==
 
{| class="wikitable center-all right-2 left-3"
== Intervals ==
|-
{| class="wikitable center-1 right-2"
! #
|-
! Cents
! #
! Approximate ratios<ref group="note">{{sg|limit=19-limit}} For lower limits see [[Table of 72edo intervals]].</ref>
! Cents
! colspan="3" | [[Ups and downs notation]]
! 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" | [[SKULO interval names|SKULO interval names and notation]]
! [[Kite's ups and downs notation|Ups and downs notation]]
! (K, S, U)  
|-
|-
| 0
| 0
| 0.0
| 0.0
| [[1/1]]
| 1/1
| {{UDnote|step=0}}
|-
| 1
| 16.7
| [[81/80]], [[91/90]], [[99/98]], [[100/99]], [[105/104]]
| {{UDnote|step=1}}
|-
| 2
| 33.3
| [[45/44]], [[49/48]], [[50/49]], [[55/54]], [[64/63]]
| {{UDnote|step=2}}
|-
| 3
| 50.0
| [[33/32]], [[36/35]], [[40/39]]
| {{UDnote|step=3}}
|-
| 4
| 66.7
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=4}}
|-
| 5
| 83.3
| [[20/19]], [[21/20]], [[22/21]]
| {{UDnote|step=5}}
|-
| 6
| 100.0
| [[17/16]], [[18/17]], [[19/18]]
| {{UDnote|step=6}}
|-
| 7
| 116.7
| [[15/14]], [[16/15]]
| {{UDnote|step=7}}
|-
| 8
| 133.3
| [[13/12]], [[14/13]], [[27/25]]
| {{UDnote|step=8}}
|-
| 9
| 150.0
| [[12/11]]
| {{UDnote|step=9}}
|-
| 10
| 166.7
| [[11/10]], [[21/19]]
| {{UDnote|step=10}}
|-
| 11
| 183.3
| [[10/9]]
| {{UDnote|step=11}}
|-
| 12
| 200.0
| [[9/8]], [[19/17]]
| {{UDnote|step=12}}
|-
| 13
| 216.7
| [[17/15]], [[25/22]]
| {{UDnote|step=13}}
|-
| 14
| 233.3
| [[8/7]]
| {{UDnote|step=14}}
|-
| 15
| 250.0
| [[15/13]], [[22/19]]
| {{UDnote|step=15}}
|-
| 16
| 266.7
| [[7/6]]
| {{UDnote|step=16}}
|-
| 17
| 283.3
| [[13/11]], [[20/17]]
| {{UDnote|step=17}}
|-
| 18
| 300.0
| [[19/16]], [[25/21]], [[32/27]]
| {{UDnote|step=18}}
|-
| 19
| 316.7
| [[6/5]]
| {{UDnote|step=19}}
|-
| 20
| 333.3
| [[17/14]], ''[[39/32]]'', [[40/33]]
| {{UDnote|step=20}}
|-
| 21
| 350.0
| [[11/9]], [[27/22]]
| {{UDnote|step=21}}
|-
| 22
| 366.7
| [[16/13]], [[21/17]], [[26/21]]
| {{UDnote|step=22}}
|-
| 23
| 383.3
| [[5/4]]
| {{UDnote|step=23}}
|-
| 24
| 400.0
| [[24/19]]
| {{UDnote|step=24}}
|-
| 25
| 416.7
| [[14/11]], [[19/15]]
| {{UDnote|step=25}}
|-
| 26
| 433.3
| [[9/7]]
| {{UDnote|step=26}}
|-
| 27
| 450.0
| [[13/10]], [[22/17]]
| {{UDnote|step=27}}
|-
| 28
| 466.7
| [[17/13]], [[21/16]]
| {{UDnote|step=28}}
|-
| 29
| 483.3
| [[33/25]]
| {{UDnote|step=29}}
|-
| 30
| 500.0
| [[4/3]]
| {{UDnote|step=30}}
|-
| 31
| 516.7
| [[27/20]]
| {{UDnote|step=31}}
|-
| 32
| 533.3
| [[15/11]], [[19/14]], ''[[26/19]]''
| {{UDnote|step=32}}
|-
| 33
| 550.0
| [[11/8]]
| {{UDnote|step=33}}
|-
| 34
| 566.7
| [[18/13]], [[25/18]]
| {{UDnote|step=34}}
|-
| 35
| 583.3
| [[7/5]]
| {{UDnote|step=35}}
|-
| 36
| 600.0
| [[17/12]], [[24/17]]
| {{UDnote|step=36}}
|-
| 37
| 616.7
| [[10/7]]
| {{UDnote|step=37}}
|-
| 38
| 633.3
| [[13/9]], [[36/25]]
| {{UDnote|step=38}}
|-
| 39
| 650.0
| [[16/11]]
| {{UDnote|step=39}}
|-
| 40
| 666.7
| ''[[19/13]]'', [[22/15]], [[28/19]]
| {{UDnote|step=40}}
|-
| 41
| 683.3
| [[40/27]]
| {{UDnote|step=41}}
|-
| 42
| 700.0
| [[3/2]]
| {{UDnote|step=42}}
|-
| 43
| 716.7
| [[50/33]]
| {{UDnote|step=43}}
|-
| 44
| 733.3
| [[26/17]], [[32/21]]
| {{UDnote|step=44}}
|-
| 45
| 750.0
| [[17/11]], [[20/13]]
| {{UDnote|step=45}}
|-
| 46
| 766.7
| [[14/9]]
| {{UDnote|step=46}}
|-
| 47
| 783.3
| [[11/7]], [[30/19]]
| {{UDnote|step=47}}
|-
| 48
| 800.0
| [[19/12]]
| {{UDnote|step=48}}
|-
| 49
| 816.7
| [[8/5]]
| {{UDnote|step=49}}
|-
| 50
| 833.3
| [[13/8]], [[21/13]], [[34/21]]
| {{UDnote|step=50}}
|-
| 51
| 850.0
| [[18/11]], [[44/27]]
| {{UDnote|step=51}}
|-
| 52
| 866.7
| [[28/17]], [[33/20]], ''[[64/39]]''
| {{UDnote|step=52}}
|-
| 53
| 883.3
| [[5/3]]
| {{UDnote|step=53}}
|-
| 54
| 900.0
| [[27/16]], [[32/19]], [[42/25]]
| {{UDnote|step=54}}
|-
| 55
| 916.7
| [[17/10]], [[22/13]]
| {{UDnote|step=55}}
|-
| 56
| 933.3
| [[12/7]]
| {{UDnote|step=56}}
|-
| 57
| 950.0
| [[19/11]], [[26/15]]
| {{UDnote|step=57}}
|-
| 58
| 966.7
| [[7/4]]
| {{UDnote|step=58}}
|-
| 59
| 983.3
| [[30/17]], [[44/25]]
| {{UDnote|step=59}}
|-
| 60
| 1000.0
| [[16/9]], [[34/19]]
| {{UDnote|step=60}}
|-
| 61
| 1016.7
| [[9/5]]
| {{UDnote|step=61}}
|-
| 62
| 1033.3
| [[20/11]], [[38/21]]
| {{UDnote|step=62}}
|-
| 63
| 1050.0
| [[11/6]]
| {{UDnote|step=63}}
|-
| 64
| 1066.7
| [[13/7]], [[24/13]], [[50/27]]
| {{UDnote|step=64}}
|-
| 65
| 1083.3
| [[15/8]], [[28/15]]
| {{UDnote|step=65}}
|-
| 66
| 1100.0
| [[17/9]], [[32/17]], [[36/19]]
| {{UDnote|step=66}}
|-
| 67
| 1116.7
| [[19/10]], [[21/11]], [[40/21]]
| {{UDnote|step=67}}
|-
| 68
| 1133.3
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| {{UDnote|step=68}}
|-
| 69
| 1150.0
| [[35/18]], [[39/20]], [[64/33]]
| {{UDnote|step=69}}
|-
| 70
| 1166.7
| [[49/25]], [[55/28]], [[63/32]], [[88/45]], [[96/49]]
| {{UDnote|step=70}}
|-
| 71
| 1183.3
| [[99/50]], [[160/81]], [[180/91]], [[196/99]], [[208/105]]
| {{UDnote|step=71}}
|-
| 72
| 1200.0
| [[2/1]]
| {{UDnote|step=72}}
|}
<references group="note" />
 
=== Proposed interval names and solfèges ===
{| 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
|-
! #
! 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)  
|-
| 0
| 0.0
| P1
| P1
| perfect unison
| perfect unison
Line 54: Line 429:
| 1
| 1
| 16.7
| 16.7
| 81/80, 91/90, 99/98, 100/99, 105/104
| ^1
| ^1
| up unison
| up unison
Line 65: Line 439:
| 2
| 2
| 33.3
| 33.3
| 45/44, 49/48, 50/49, 55/54, 64/63
| ^^
| ^^
| dup unison
| dup unison
Line 76: Line 449:
| 3
| 3
| 50.0
| 50.0
| 33/32, 36/35, 40/39
| ^<sup>3</sup>1, v<sup>3</sup>m2
| ^<sup>3</sup>1, v<sup>3</sup>m2
| trup unison, trudminor 2nd
| trup unison, trudminor 2nd
Line 87: Line 459:
| 4
| 4
| 66.7
| 66.7
| 25/24, 26/25, 27/26, 28/27
| vvm2
| vvm2
| dudminor 2nd
| dudminor 2nd
Line 98: Line 469:
| 5
| 5
| 83.3
| 83.3
| 20/19, 21/20, 22/21
| vm2
| vm2
| downminor 2nd
| downminor 2nd
Line 109: Line 479:
| 6
| 6
| 100.0
| 100.0
| 17/16, 18/17, 19/18
| m2
| m2
| minor 2nd
| minor 2nd
Line 120: Line 489:
| 7
| 7
| 116.7
| 116.7
| 15/14, 16/15
| ^m2
| ^m2
| upminor 2nd
| upminor 2nd
Line 131: Line 499:
| 8
| 8
| 133.3
| 133.3
| 13/12, 14/13, 27/25
| ^^m2, v~2
| ^^m2, v~2
| dupminor 2nd, downmid 2nd
| dupminor 2nd, downmid 2nd
Line 142: Line 509:
| 9
| 9
| 150.0
| 150.0
| 12/11
| ~2
| ~2
| mid 2nd
| mid 2nd
Line 153: Line 519:
| 10
| 10
| 166.7
| 166.7
| 11/10
| ^~2, vvM2
| ^~2, vvM2
| upmid 2nd, dudmajor 2nd
| upmid 2nd, dudmajor 2nd
Line 164: Line 529:
| 11
| 11
| 183.3
| 183.3
| 10/9
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
Line 175: Line 539:
| 12
| 12
| 200.0
| 200.0
| 9/8, 19/17
| M2
| M2
| major 2nd
| major 2nd
Line 186: Line 549:
| 13
| 13
| 216.7
| 216.7
| 17/15, 25/22
| ^M2
| ^M2
| upmajor 2nd
| upmajor 2nd
Line 197: Line 559:
| 14
| 14
| 233.3
| 233.3
| 8/7
| ^^M2
| ^^M2
| dupmajor 2nd
| dupmajor 2nd
Line 208: Line 569:
| 15
| 15
| 250.0
| 250.0
| 15/13, 22/19
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| trupmajor 2nd,<br>trudminor 3rd
| trupmajor 2nd,<br>trudminor 3rd
Line 219: Line 579:
| 16
| 16
| 266.7
| 266.7
| 7/6
| vvm3
| vvm3
| dudminor 3rd
| dudminor 3rd
Line 230: Line 589:
| 17
| 17
| 283.3
| 283.3
| 13/11, 20/17
| vm3
| vm3
| downminor 3rd
| downminor 3rd
Line 241: Line 599:
| 18
| 18
| 300.0
| 300.0
| 19/16, 25/21, 32/27
| m3
| m3
| minor 3rd
| minor 3rd
Line 252: Line 609:
| 19
| 19
| 316.7
| 316.7
| 6/5
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
Line 263: Line 619:
| 20
| 20
| 333.3
| 333.3
| 17/14, 39/32, 40/33
| ^^m3, v~3
| ^^m3, v~3
| dupminor 3rd, downmid 3rd
| dupminor 3rd, downmid 3rd
Line 274: Line 629:
| 21
| 21
| 350.0
| 350.0
| 11/9, 27/22
| ~3
| ~3
| mid 3rd
| mid 3rd
Line 285: Line 639:
| 22
| 22
| 366.7
| 366.7
| 16/13, 21/17, 26/21
| ^~3, vvM3
| ^~3, vvM3
| upmid 3rd, dudmajor 3rd
| upmid 3rd, dudmajor 3rd
Line 296: Line 649:
| 23
| 23
| 383.3
| 383.3
| 5/4
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
Line 307: Line 659:
| 24
| 24
| 400.0
| 400.0
| 24/19
| M3
| M3
| major 3rd
| major 3rd
Line 318: Line 669:
| 25
| 25
| 416.7
| 416.7
| 14/11, 19/15
| ^M3
| ^M3
| upmajor 3rd
| upmajor 3rd
Line 329: Line 679:
| 26
| 26
| 433.3
| 433.3
| 9/7
| ^^M3
| ^^M3
| dupmajor 3rd
| dupmajor 3rd
Line 340: Line 689:
| 27
| 27
| 450.0
| 450.0
| 13/10, 22/17
| ^<sup>3</sup>M3, v<sup>3</sup>4
| ^<sup>3</sup>M3, v<sup>3</sup>4
| trupmajor 3rd, trud 4th
| trupmajor 3rd, trud 4th
Line 351: Line 699:
| 28
| 28
| 466.7
| 466.7
| 17/13, 21/16
| vv4
| vv4
| dud 4th
| dud 4th
Line 362: Line 709:
| 29
| 29
| 483.3
| 483.3
| 33/25
| v4
| v4
| down 4th
| down 4th
Line 373: Line 719:
| 30
| 30
| 500.0
| 500.0
| 4/3
| P4
| P4
| perfect 4th
| perfect 4th
Line 384: Line 729:
| 31
| 31
| 516.7
| 516.7
| 27/20
| ^4
| ^4
| up 4th
| up 4th
Line 395: Line 739:
| 32
| 32
| 533.3
| 533.3
| 15/11, 19/14, ''26/19''
| ^^4, v~4
| ^^4, v~4
| dup 4th, downmid 4th
| dup 4th, downmid 4th
Line 406: Line 749:
| 33
| 33
| 550.0
| 550.0
| 11/8
| ~4
| ~4
| mid 4th
| mid 4th
Line 417: Line 759:
| 34
| 34
| 566.7
| 566.7
| 18/13, 25/18
| ^~4, vvA4
| ^~4, vvA4
| upmid 4th, dudaug 4th
| upmid 4th, dudaug 4th
Line 428: Line 769:
| 35
| 35
| 583.3
| 583.3
| 7/5
| vA4, vd5
| vA4, vd5
| downaug 4th, <br>downdim 5th
| downaug 4th, <br>downdim 5th
Line 439: Line 779:
| 36
| 36
| 600.0
| 600.0
| 17/12, 24/17
| A4, d5
| A4, d5
| aug 4th, dim 5th
| aug 4th, dim 5th
Line 450: Line 789:
| 37
| 37
| 616.7
| 616.7
| 10/7
| ^A4, ^d5
| ^A4, ^d5
| upaug 4th, updim 5th
| upaug 4th, updim 5th
Line 461: Line 799:
| 38
| 38
| 633.3
| 633.3
| 13/9, 36/25
| v~5, ^^d5
| v~5, ^^d5
| downmid 5th, <br>dupdim 5th
| downmid 5th, <br>dupdim 5th
Line 472: Line 809:
| 39
| 39
| 650.0
| 650.0
| 16/11
| ~5
| ~5
| mid 5th
| mid 5th
Line 483: Line 819:
| 40
| 40
| 666.7
| 666.7
| ''19/13'', 22/15, 28/19
| vv5, ^~5
| vv5, ^~5
| dud 5th, upmid 5th
| dud 5th, upmid 5th
Line 494: Line 829:
| 41
| 41
| 683.3
| 683.3
| 40/27
| v5
| v5
| down 5th
| down 5th
Line 505: Line 839:
| 42
| 42
| 700.0
| 700.0
| 3/2
| P5
| P5
| perfect 5th
| perfect 5th
Line 516: Line 849:
| 43
| 43
| 716.7
| 716.7
| 50/33
| ^5
| ^5
| up 5th
| up 5th
Line 527: Line 859:
| 44
| 44
| 733.3
| 733.3
| 26/17, 32/21
| ^^5
| ^^5
| dup 5th
| dup 5th
Line 538: Line 869:
| 45
| 45
| 750.0
| 750.0
| 17/11, 20/13
| ^<sup>3</sup>5, v<sup>3</sup>m6
| ^<sup>3</sup>5, v<sup>3</sup>m6
| trup 5th, trudminor 6th
| trup 5th, trudminor 6th
Line 549: Line 879:
| 46
| 46
| 766.7
| 766.7
| 14/9
| vvm6
| vvm6
| dudminor 6th
| dudminor 6th
Line 560: Line 889:
| 47
| 47
| 783.3
| 783.3
| 11/7, 30/19
| vm6
| vm6
| downminor 6th
| downminor 6th
Line 571: Line 899:
| 48
| 48
| 800.0
| 800.0
| 19/12
| m6
| m6
| minor 6th
| minor 6th
Line 582: Line 909:
| 49
| 49
| 816.7
| 816.7
| 8/5
| ^m6
| ^m6
| upminor 6th
| upminor 6th
Line 593: Line 919:
| 50
| 50
| 833.3
| 833.3
| 13/8, 21/13, 34/21
| ^^m6, v~6
| ^^m6, v~6
| dupminor 6th, downmid 6th
| dupminor 6th, downmid 6th
Line 604: Line 929:
| 51
| 51
| 850.0
| 850.0
| 18/11, 44/27
| ~6
| ~6
| mid 6th
| mid 6th
Line 615: Line 939:
| 52
| 52
| 866.7
| 866.7
| 28/17, 33/20, 64/39
| ^~6, vvM6
| ^~6, vvM6
| upmid 6th, dudmajor 6th
| upmid 6th, dudmajor 6th
Line 626: Line 949:
| 53
| 53
| 883.3
| 883.3
| 5/3
| vM6
| vM6
| downmajor 6th
| downmajor 6th
Line 637: Line 959:
| 54
| 54
| 900.0
| 900.0
| 27/16, 32/19, 42/25
| M6
| M6
| major 6th
| major 6th
Line 648: Line 969:
| 55
| 55
| 916.7
| 916.7
| 17/10, 22/13
| ^M6
| ^M6
| upmajor 6th
| upmajor 6th
Line 659: Line 979:
| 56
| 56
| 933.3
| 933.3
| 12/7
| ^^M6
| ^^M6
| dupmajor 6th
| dupmajor 6th
Line 670: Line 989:
| 57
| 57
| 950.0
| 950.0
| 19/11, 26/15
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| trupmajor 6th,<br>trudminor 7th
| trupmajor 6th,<br>trudminor 7th
Line 681: Line 999:
| 58
| 58
| 966.7
| 966.7
| 7/4
| vvm7
| vvm7
| dudminor 7th
| dudminor 7th
Line 692: Line 1,009:
| 59
| 59
| 983.3
| 983.3
| 30/17, 44/25
| vm7
| vm7
| downminor 7th
| downminor 7th
Line 703: Line 1,019:
| 60
| 60
| 1000.0
| 1000.0
| 16/9, 34/19
| m7
| m7
| minor 7th
| minor 7th
Line 714: Line 1,029:
| 61
| 61
| 1016.7
| 1016.7
| 9/5
| ^m7
| ^m7
| upminor 7th
| upminor 7th
Line 725: Line 1,039:
| 62
| 62
| 1033.3
| 1033.3
| 20/11
| ^^m7, v~7
| ^^m7, v~7
| dupminor 7th, downmid 7th
| dupminor 7th, downmid 7th
Line 736: Line 1,049:
| 63
| 63
| 1050.0
| 1050.0
| 11/6
| ~7
| ~7
| mid 7th
| mid 7th
Line 747: Line 1,059:
| 64
| 64
| 1066.7
| 1066.7
| 13/7, 24/13, 50/27
| ^~7, vvM7
| ^~7, vvM7
| upmid 7th, dudmajor 7th
| upmid 7th, dudmajor 7th
Line 758: Line 1,069:
| 65
| 65
| 1083.3
| 1083.3
| 15/8, 28/15
| vM7
| vM7
| downmajor 7th
| downmajor 7th
Line 769: Line 1,079:
| 66
| 66
| 1100.0
| 1100.0
| 17/9, 32/17, 36/19
| M7
| M7
| major 7th
| major 7th
Line 780: Line 1,089:
| 67
| 67
| 1116.7
| 1116.7
| 19/10, 21/11, 40/21
| ^M7
| ^M7
| upmajor 7th
| upmajor 7th
Line 791: Line 1,099:
| 68
| 68
| 1133.3
| 1133.3
| 25/13, 27/14, 48/25, 52/27
| ^^M7
| ^^M7
| dupmajor 7th
| dupmajor 7th
Line 802: Line 1,109:
| 69
| 69
| 1150.0
| 1150.0
| 35/18, 39/20, 64/33
| ^<sup>3</sup>M7, v<sup>3</sup>8
| ^<sup>3</sup>M7, v<sup>3</sup>8
| trupmajor 7th, trud octave
| trupmajor 7th, trud octave
Line 813: Line 1,119:
| 70
| 70
| 1166.7
| 1166.7
| 49/25, 55/28, 63/32, 88/45, 96/49
| vv8
| vv8
| dud octave
| dud octave
Line 824: Line 1,129:
| 71
| 71
| 1183.3
| 1183.3
| 99/50, 160/81, 180/91, 196/99, 208/105
| v8
| v8
| down octave
| down octave
Line 835: Line 1,139:
| 72
| 72
| 1200.0
| 1200.0
| 2/1
| P8
| P8
| perfect octave
| perfect octave
Line 844: Line 1,147:
| D
| D
|}
|}
<references group="note" />


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Line 859: Line 1,161:
| zo
| zo
| (a b 0 1)
| (a b 0 1)
| 7/6, 7/4
| [[7/6]], [[7/4]]
|-
|-
| minor
| minor
| fourthward wa
| fourthward wa
| (a b), b < -1
| (a b), b < -1
| 32/27, 16/9
| [[32/27]], [[16/9]]
|-
|-
| upminor
| upminor
| gu
| gu
| (a b -1)
| (a b -1)
| 6/5, 9/5
| [[6/5]], [[9/5]]
|-
|-
| rowspan="2" | dupminor, <br>downmid
| rowspan="2" | dupminor, <br>downmid
| luyo
| luyo
| (a b 1 0 -1)
| (a b 1 0 -1)
| 15/11
| [[15/11]]
|-
|-
| tho
| tho
| (a b 0 0 0 1)
| (a b 0 0 0 1)
| 13/8, 13/9
| [[13/8]], [[13/9]]
|-
|-
| rowspan="2" | mid
| rowspan="2" | mid
| ilo
| ilo
| (a b 0 0 1)
| (a b 0 0 1)
| 11/9, 11/6
| [[11/9]], [[11/6]]
|-
|-
| lu
| lu
| (a b 0 0 -1)
| (a b 0 0 -1)
| 12/11, 18/11
| [[12/11]], [[18/11]]
|-
|-
| rowspan="2" | upmid, <br>dudmajor
| rowspan="2" | upmid, <br>dudmajor
| logu
| logu
| (a b -1 0 1)
| (a b -1 0 1)
| 11/10
| [[11/10]]
|-
|-
| thu
| thu
| (a b 0 0 0 -1)
| (a b 0 0 0 -1)
| 16/13, 18/13
| [[16/13]], [[18/13]]
|-
|-
| downmajor
| downmajor
| yo
| yo
| (a b 1)
| (a b 1)
| 5/4, 5/3
| [[5/4]], [[5/3]]
|-
|-
| major
| major
| fifthward wa
| fifthward wa
| (a b), b > 1
| (a b), b > 1
| 9/8, 27/16
| [[9/8]], [[27/16]]
|-
|-
| dupmajor
| dupmajor
| ru
| ru
| (a b 0 -1)
| (a b 0 -1)
| 9/7, 12/7
| [[9/7]], [[12/7]]
|-
|-
| rowspan="2" | trupmajor, <br>trudminor
| rowspan="2" | trupmajor, <br>trudminor
| thogu
| thogu
| (a b -1 0 0 1)
| (a b -1 0 0 1)
| 13/10
| [[13/10]]
|-
|-
| thuyo
| thuyo
| (a b 1 0 0 -1)
| (a b 1 0 0 -1)
| 15/13
| [[15/13]]
|}
|}
All 72edo 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 72edo 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:
Line 974: Line 1,276:


Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
* −1 degree (the down ring) corrects 81/64 to 5/4 via 80/81
* −1 degree (the down ring) corrects [[81/64]] to [[5/4]] via descending [[81/80]]
* −2 degrees (the dud ring) corrects 16/9 to 7/4 via 63/64
* −2 degrees (the dud ring) corrects [[16/9]] to [[7/4]] via descending [[64/63]]
* +3 degrees  (the trup ring) corrects 4/3 to 11/8 via 33/32
* +3 degrees  (the trup ring) corrects [[4/3]] to [[11/8]] via [[33/32]]
* +2 degrees (the dup ring) corrects 128/81 to 13/8 via 1053/1024
* +2 degrees (the dup ring) corrects [[128/81]] to [[13/8]] via [[1053/1024]]
* 0 degrees (the plain ring) corrects 256/243 to 17/16 via 4131/4096
* 0 degrees (the plain ring) corrects [[256/243]] to [[17/16]] via [[4131/4096]]
* 0 degrees (the plain ring) corrects 32/27 to 19/16 via 513/512
* 0 degrees (the plain ring) corrects [[32/27]] to [[19/16]] via [[513/512]]
Thus the product of a ratio's monzo with {{map| 0 0 -1 -2 3 2 0 0 }}, modulo 6, specifies which ring the ratio lies on.
Thus the product of a ratio's monzo with {{map| 0 0 -1 -2 3 2 0 0 }}, modulo 6, specifies which ring the ratio lies on.


== Notations ==
== 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-sharp6a}}
{{Sharpness-sharp6-szg}}


Half-sharps and half-flats can be used to avoid triple arrows:
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6b}}
{{Sharpness-sharp6-qt-szg}}


[[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]]:
=== Kite's ups and downs notation ===
{{Sharpness-sharp6}}
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}}


If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6-qt}}
{{Ups and downs sharpness|72|true}}


=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[65edo#Sagittal notation|65-EDO]] and [[79edo#Sagittal notation|79]], and is a superset of the notations for EDOs [[36edo#Sagittal notation|36]], [[24edo#Sagittal notation|24]], [[18edo#Sagittal notation|18]], [[12edo#Sagittal notation|12]], [[8edo#Sagittal notation|8]], and [[6edo#Sagittal notation|6]].
This notation uses the same sagittal sequence as edos [[65edo #Sagittal notation|65-]] and [[79edo #Sagittal notation|79edo]], and is a superset of the notations for edos [[36edo #Sagittal notation|36]], [[24edo #Sagittal notation|24]], [[18edo #Sagittal notation|18]], [[12edo #Sagittal notation|12]], [[8edo #Sagittal notation|8]], and [[6edo #Sagittal notation|6]].


==== Evo flavor ====
==== Evo flavor ====
[[File:72-EDO_Evo_Sagittal.svg|alt=72-EDO_Evo_Sagittal.svg|72-EDO_Evo_Sagittal.svg]]
{{Sagittal chart|Evo}}
 
==== Evo-SZ flavor ====
{{Sagittal chart|Evo-SZ}}


==== Revo flavor ====
==== Revo flavor ====
[[File:72-EDO_Revo_Sagittal.svg|alt=72-EDO_Revo_Sagittal.svg|72-EDO_Revo_Sagittal.svg]]
{{Sagittal chart}}
 
==== Evo-SZ flavor ====
[[File:72-EDO_Evo-SZ_Sagittal.svg|alt=72-EDO_Evo-SZ_Sagittal.svg|72-EDO_Evo-SZ_Sagittal.svg]]


From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 72edo in the Revo flavor of Sagittal:
From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 72edo in the Revo flavor of Sagittal:  


<div class="noresize">
[[File:72edo Sagittal.png]]
[[File:72edo Sagittal.png]]
</div>


=== Ivan Wyschnegradsky's notation ===
=== Ivan Wyschnegradsky's notation ===
Line 1,365: Line 1,670:
| 516.7
| 516.7
| 27/20
| 27/20
| [[Marvo]] / [[zarvo]]
| [[Gravity]] / [[marvo]] / [[zarvo]]
|-
|-
| 1
| 1
Line 1,437: 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 1,469: 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 ==
Line 1,497: Line 1,802:
* [[JuneGloom]]
* [[JuneGloom]]
* [[Keenanmarvel]]
* [[Keenanmarvel]]
* [[Prodigy]][19]: 5 2 5 4 5 2 5 2 5 2 5 4 5 2 5 2 5 5 2


=== Harmonic scale ===
=== Harmonic scale ===
Line 1,693: Line 1,999:


; [[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)}}
Line 1,704: Line 2,010:


; [[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'']
Line 1,710: Line 2,016:


; [[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]]

Latest revision as of 07:43, 26 May 2026

← 71edo 72edo 73edo →
Prime factorization 23 × 32
Step size 16.6667 ¢ 
Fifth 42\72 (700 ¢) (→ 7\12)
Semitones (A1:m2) 6:6 (100 ¢ : 100 ¢)
Consistency limit 17
Distinct consistency limit 11
English Wikipedia has an article on:

72 equal divisions of the octave (abbreviated 72edo or 72ed2), also called 72-tone equal temperament (72tet) or 72 equal temperament (72et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 72 equal parts of about 16.7 ¢ each. Each step represents a frequency ratio of 21/72, or the 72nd root of 2.

Each step of 72edo is called a morion (plural moria). This produces a twelfth-tone tuning, with the whole tone measuring 200 ¢, the same as in 12edo. 72edo is also a superset of 24edo, a common and standard tuning of Arabic music, and has itself been used to tune Turkish music.

Composers that used 72edo include Ivan Wyschnegradsky, Julián Carrillo (who is better associated with 96edo), Georg Friedrich Haas, Ezra Sims, Rick Tagawa, James Tenney, and the jazz musician Joe Maneri.

Theory

72edo approximates 11-limit just intonation exceptionally well. It is the second edo (after 58) to be consistent in the 17-odd-limit, and the second edo (also after 58) to be distinctly consistent in the 11-odd-limit, but it is the first edo to be 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). 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 reduced 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 17th and 19th harmonics come from 12edo.

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.

Prime harmonics

Approximation of prime harmonics in 72edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.96 -2.98 -2.16 -1.32 -7.19 -4.96 +2.49 +5.06 +3.76 +4.96
Relative (%) +0.0 -11.7 -17.9 -13.0 -7.9 -43.2 -29.7 +14.9 +30.4 +22.5 +29.8
Steps
(reduced) 		72
(0) 		114
(42) 		167
(23) 		202
(58) 		249
(33) 		266
(50) 		294
(6) 		306
(18) 		326
(38) 		350
(62) 		357
(69)
Approximation of prime harmonics in 72edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -1.34 +4.27 +5.15 +1.16 -6.84 +7.50 -0.22 +4.03 +3.64 +5.54 +2.13
Relative (%) -8.1 +25.6 +30.9 +7.0 -41.0 +45.0 -1.3 +24.2 +21.8 +33.3 +12.8
Steps
(reduced) 		375
(15) 		386
(26) 		391
(31) 		400
(40) 		412
(52) 		424
(64) 		427
(67) 		437
(5) 		443
(11) 		446
(14) 		454
(22)

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

Since 72 factors into primes as 23 × 32, 72edo has subset edos 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36. 144edo, which doubles it, provides a possible correction to its approximate harmonic 13, though unlike 72 it is not consistent to the 13-odd-limit.

Intervals

# Cents Approximate ratios[note 1] Ups and downs notation
0 0.0 1/1 D
1 16.7 81/80, 91/90, 99/98, 100/99, 105/104 ^D, ^E♭♭
2 33.3 45/44, 49/48, 50/49, 55/54, 64/63 ^^D, ^^E♭♭
3 50.0 33/32, 36/35, 40/39 ^3D, v3E♭
4 66.7 25/24, 26/25, 27/26, 28/27 vvD♯, vvE♭
5 83.3 20/19, 21/20, 22/21 vD♯, vE♭
6 100.0 17/16, 18/17, 19/18 D♯, E♭
7 116.7 15/14, 16/15 ^D♯, ^E♭
8 133.3 13/12, 14/13, 27/25 ^^D♯, ^^E♭
9 150.0 12/11 ^3D♯, v3E
10 166.7 11/10, 21/19 vvD𝄪, vvE
11 183.3 10/9 vD𝄪, vE
12 200.0 9/8, 19/17 E
13 216.7 17/15, 25/22 ^E, ^F♭
14 233.3 8/7 ^^E, ^^F♭
15 250.0 15/13, 22/19 ^3E, v3F
16 266.7 7/6 vvE♯, vvF
17 283.3 13/11, 20/17 vE♯, vF
18 300.0 19/16, 25/21, 32/27 F
19 316.7 6/5 ^F, ^G♭♭
20 333.3 17/14, 39/32, 40/33 ^^F, ^^G♭♭
21 350.0 11/9, 27/22 ^3F, v3G♭
22 366.7 16/13, 21/17, 26/21 vvF♯, vvG♭
23 383.3 5/4 vF♯, vG♭
24 400.0 24/19 F♯, G♭
25 416.7 14/11, 19/15 ^F♯, ^G♭
26 433.3 9/7 ^^F♯, ^^G♭
27 450.0 13/10, 22/17 ^3F♯, v3G
28 466.7 17/13, 21/16 vvF𝄪, vvG
29 483.3 33/25 vF𝄪, vG
30 500.0 4/3 G
31 516.7 27/20 ^G, ^A♭♭
32 533.3 15/11, 19/14, 26/19 ^^G, ^^A♭♭
33 550.0 11/8 ^3G, v3A♭
34 566.7 18/13, 25/18 vvG♯, vvA♭
35 583.3 7/5 vG♯, vA♭
36 600.0 17/12, 24/17 G♯, A♭
37 616.7 10/7 ^G♯, ^A♭
38 633.3 13/9, 36/25 ^^G♯, ^^A♭
39 650.0 16/11 ^3G♯, v3A
40 666.7 19/13, 22/15, 28/19 vvG𝄪, vvA
41 683.3 40/27 vG𝄪, vA
42 700.0 3/2 A
43 716.7 50/33 ^A, ^B♭♭
44 733.3 26/17, 32/21 ^^A, ^^B♭♭
45 750.0 17/11, 20/13 ^3A, v3B♭
46 766.7 14/9 vvA♯, vvB♭
47 783.3 11/7, 30/19 vA♯, vB♭
48 800.0 19/12 A♯, B♭
49 816.7 8/5 ^A♯, ^B♭
50 833.3 13/8, 21/13, 34/21 ^^A♯, ^^B♭
51 850.0 18/11, 44/27 ^3A♯, v3B
52 866.7 28/17, 33/20, 64/39 vvA𝄪, vvB
53 883.3 5/3 vA𝄪, vB
54 900.0 27/16, 32/19, 42/25 B
55 916.7 17/10, 22/13 ^B, ^C♭
56 933.3 12/7 ^^B, ^^C♭
57 950.0 19/11, 26/15 ^3B, v3C
58 966.7 7/4 vvB♯, vvC
59 983.3 30/17, 44/25 vB♯, vC
60 1000.0 16/9, 34/19 C
61 1016.7 9/5 ^C, ^D♭♭
62 1033.3 20/11, 38/21 ^^C, ^^D♭♭
63 1050.0 11/6 ^3C, v3D♭
64 1066.7 13/7, 24/13, 50/27 vvC♯, vvD♭
65 1083.3 15/8, 28/15 vC♯, vD♭
66 1100.0 17/9, 32/17, 36/19 C♯, D♭
67 1116.7 19/10, 21/11, 40/21 ^C♯, ^D♭
68 1133.3 25/13, 27/14, 48/25, 52/27 ^^C♯, ^^D♭
69 1150.0 35/18, 39/20, 64/33 ^3C♯, v3D
70 1166.7 49/25, 55/28, 63/32, 88/45, 96/49 vvC𝄪, vvD
71 1183.3 99/50, 160/81, 180/91, 196/99, 208/105 vC𝄪, vD
72 1200.0 2/1 D
  1. ↑ As a 19-limit temperament, inconsistent intervals in italic. For a table of intervals by prime limit, see Table of 72edo intervals.

Proposed interval names and solfèges

Table of proposed interval names and solfèges
# Cents Ups and downs notation SKULO interval names and notation (K, S, U)
0 0.0 P1 perfect unison D P1 perfect unison D D
1 16.7 ^1 up unison ^D K1, L1 comma-wide unison, large unison KD, LD KD
2 33.3 ^^ dup unison ^^D S1, O1 super unison, on unison SD, OD SD
3 50.0 ^31, v3m2 trup unison, trudminor 2nd ^3D, v3Eb U1, H1, hm2 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 v3E 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 ^3M2,
v3m3 		trupmajor 2nd,
trudminor 3rd 		^3E,
v3F 		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 ^3F 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 ^3M3, v34 trupmajor 3rd, trud 4th ^3F#, v3G 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 ^3G 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,
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,
dupdim 5th 		^^Ab SA4, KKd5 super aug 4th, classic dim 5th SG#, KKAb SG#, SAb, (KKAb)
39 650.0 ~5 mid 5th v3A 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 ^35, v3m6 trup 5th, trudminor 6th ^3A, v3Bb 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 v3B 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 ^3M6,
v3m7 		trupmajor 6th,
trudminor 7th 		^3B,
v3C 		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 ^3C 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 ^3M7, v38 trupmajor 7th, trud octave ^3C#, v3D 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:

Quality 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
dupminor,
downmid 		luyo (a b 1 0 -1) 15/11
tho (a b 0 0 0 1) 13/8, 13/9
mid ilo (a b 0 0 1) 11/9, 11/6
lu (a b 0 0 -1) 12/11, 18/11
upmid,
dudmajor 		logu (a b -1 0 1) 11/10
thu (a b 0 0 0 -1) 16/13, 18/13
downmajor yo (a b 1) 5/4, 5/3
major fifthward wa (a b), b > 1 9/8, 27/16
dupmajor ru (a b 0 -1) 9/7, 12/7
trupmajor,
trudminor 		thogu (a b -1 0 0 1) 13/10
thuyo (a b 1 0 0 -1) 15/13

All 72edo 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 dudminor
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 dupmajor or C dup

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

Relationship between primes and rings

In 72tet, there are 6 rings. 12edo is the plain ring; thus every 6 degrees is the 3-limit.

Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:

Thus the product of a ratio's monzo with ⟨0 0 -1 -2 3 2 0 0], modulo 6, specifies which ring the ratio lies on.

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats combined with quartertone accidentals and arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol
Flat symbol

If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
Flat symbol

Kite's ups and downs notation

72edo can also be notated with Kite's ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Semitones 0 1⁄6 1⁄3 1⁄2 2⁄3 5⁄6 1 1 1⁄6 1 1⁄3 1 1⁄2 1 2⁄3 1 5⁄6 2 2 1⁄6
Sharp symbol   
 
  
   
    
   
  
   
    
     
    
   
  
   
Flat symbol
 
  
   
    
   
  
   
    
     
    
   
  
   

Half-sharps and half-flats can be used to avoid triple arrows:

Semitones 0 1⁄6 1⁄3 1⁄2 2⁄3 5⁄6 1 1 1⁄6 1 1⁄3 1 1⁄2 1 2⁄3 1 5⁄6 2 2 1⁄6
Sharp symbol   
 
   
  
   
   
  
   
   
  
   
   
  
   
Flat symbol
 
   
  
   
   
  
   
   
  
   
   
  
   

Sagittal notation

This notation uses the same sagittal sequence as edos 65- and 79edo, and is a superset of the notations for edos 36, 24, 18, 12, 8, and 6.

Evo flavor

72-EDO_Evo_Sagittal.svg

Evo-SZ flavor

72-EDO_Evo-SZ_Sagittal.svg

Revo flavor

72-EDO_Revo_Sagittal.svg

From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 72edo in the Revo flavor of Sagittal:

Ivan Wyschnegradsky's notation

Semitones 0 1⁄6 1⁄3 1⁄2 2⁄3 5⁄6 1 1 1⁄6 1 1⁄3 1 1⁄2 1 2⁄3 1 5⁄6 2 2 1⁄6
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 
  
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 

Approximation to JI

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Selected intervals approximated in 72edo

Interval mappings

The following table shows how 15-odd-limit intervals are represented in 72edo. Prime harmonics are in bold.

As 72edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 72edo
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/6, 12/7 0.204 1.2
11/6, 12/11 0.637 3.8
7/5, 10/7 0.821 4.9
11/7, 14/11 0.841 5.0
9/5, 10/9 0.930 5.6
5/3, 6/5 1.025 6.2
11/8, 16/11 1.318 7.9
11/10, 20/11 1.662 10.0
9/7, 14/9 1.751 10.5
3/2, 4/3 1.955 11.7
7/4, 8/7 2.159 13.0
15/13, 26/15 2.259 13.6
11/9, 18/11 2.592 15.6
15/14, 28/15 2.776 16.7
5/4, 8/5 2.980 17.9
13/9, 18/13 3.284 19.7
15/11, 22/15 3.617 21.7
9/8, 16/9 3.910 23.5
13/10, 20/13 4.214 25.3
15/8, 16/15 4.935 29.6
13/7, 14/13 5.035 30.2
13/12, 24/13 5.239 31.4
13/11, 22/13 5.876 35.3
13/8, 16/13 7.194 43.2

Zeta properties

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

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 15625/15552, 531441/524288 [⟨72 114 167]] +0.839 0.594 3.56
2.3.5.7 225/224, 1029/1024, 4375/4374 [⟨72 114 167 202]] +0.822 0.515 3.09
2.3.5.7.11 225/224, 243/242, 385/384, 4000/3993 [⟨72 114 167 202 249]] +0.734 0.493 2.96
2.3.5.7.11.13 169/168, 225/224, 243/242, 325/324, 385/384 [⟨72 114 167 202 249 266]] +0.936 0.638 3.82
2.3.5.7.11.13.17 169/168, 221/220, 225/224, 243/242, 273/272, 325/324 [⟨72 114 167 202 249 266 294]] +0.975 0.599 3.59
2.3.5.7.11.13.17.19 153/152, 169/168, 210/209, 221/220, 225/224, 243/242, 273/272 [⟨72 114 167 202 249 266 294 306]] +0.780 0.762 4.57
  • 72et has lower relative errors than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next equal temperaments doing better in these subgroups are 99, 270, 224, 494, and 217, respectively.

Commas

Commas tempered out by 72edo include…

Prime
limit 		Ratio[note 1] Monzo Cents Name(s)
3 (12 digits) [-19 12⟩ 23.46 Pythagorean comma
5 15625/15552 [-6 -5 6⟩ 8.11 Kleisma
5 (16 digits) [-25 7 6⟩ 31.57 Ampersand comma
5 (18 digits) [-13 17 -6⟩ 15.35 Graviton
5 (26 digits) [1 -27 18⟩ 0.86 Ennealimma
7 225/224 [-5 2 2 -1⟩ 7.71 Marvel comma
7 1029/1024 [-10 1 0 3⟩ 8.43 Gamelisma
7 2401/2400 [-5 -1 -2 4⟩ 0.72 Breedsma
7 4375/4374 [-1 -7 4 1⟩ 0.40 Ragisma
7 16875/16807 [0 3 4 -5⟩ 6.99 Mirkwai comma
7 19683/19600 [-4 9 -2 -2⟩ 7.32 Cataharry comma
7 (12 digits) [-6 -8 2 5⟩ 1.12 Wizma
7 (12 digits) [-4 6 -6 3⟩ 0.33 Landscape comma
11 243/242 [-1 5 0 0 -2⟩ 7.14 Rastma
11 385/384 [-7 -1 1 1 1⟩ 4.50 Keenanisma
11 441/440 [-3 2 -1 2 -1⟩ 3.93 Werckisma
11 540/539 [2 3 1 -2 -1⟩ 3.21 Swetisma
11 1375/1372 [-2 0 3 -3 1⟩ 3.78 Moctdel comma
11 3025/3024 [-4 -3 2 -1 2⟩ 0.57 Lehmerisma
11 4000/3993 [5 -1 3 0 -3⟩ 3.03 Wizardharry comma
11 6250/6237 [1 -4 5 -1 -1⟩ 3.60 Liganellus comma
11 9801/9800 [-3 4 -2 -2 2⟩ 0.18 Kalisma
11 (14 digits) [16 -3 0 0 6⟩ 2.04 Nexus comma
13 169/168 [-3 -1 0 -1 0 2⟩ 10.27 Buzurgisma
13 325/324 [-2 -4 2 0 0 1⟩ 5.34 Marveltwin comma
13 351/350 [-1 3 -2 -1 0 1⟩ 4.94 Ratwolfsma
13 364/363 [2 -1 0 1 -2 1⟩ 4.76 Minor minthma
13 625/624 [-4 -1 4 0 0 -1⟩ 2.77 Tunbarsma
13 676/675 [2 -3 -2 0 0 2⟩ 2.56 Island comma
13 729/728 [-3 6 0 -1 0 -1⟩ 2.38 Squbema
13 1001/1000 [-3 0 -3 1 1 1⟩ 1.73 Sinbadma
13 1575/1573 [2 2 1 -2 -1⟩ 2.20 Nicola
13 1716/1715 [2 1 -1 -3 1 1⟩ 1.01 Lummic comma
13 2080/2079 [5 -3 1 -1 -1 1⟩ 0.83 Ibnsinma
13 6656/6655 [9 0 -1 0 -3 1⟩ 0.26012 Jacobin comma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

Rank-2 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.

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 1\72 16.7 105/104 Quincy
1 5\72 83.3 21/20 Marvolo
1 7\72 116.7 15/14 Miracle / benediction / manna
1 17\72 283.3 13/11 Neominor
1 19\72 316.7 6/5 Catakleismic
1 25\72 416.7 14/11 Sqrtphi
1 29\72 483.3 45/34 Hemiseven
1 31\72 516.7 27/20 Gravity / marvo / zarvo
1 35\72 583.3 7/5 Cotritone
2 5\72 83.3 21/20 Harry
2 7\72 116.7 15/14 Semimiracle
2 11\72 183.3 10/9 Unidec / hendec
2 21\72
(19\72) 		316.7
(283.3) 		6/5
(13/11) 		Bikleismic
2 23\72
(13\72) 		383.3
(216.7) 		5/4
(17/15) 		Wizard / lizard / gizzard
3 11\72 183.3 10/9 Mirkat
3 19\72
(5\72) 		316.7
(83.3) 		6/5
(21/20) 		Tritikleismic
4 19\72
(1\72) 		316.7
(16.7) 		6/5
(105/104) 		Quadritikleismic
8 34\72
(2\72) 		566.7
(33.3) 		168/121
(55/54) 		Octowerck / octowerckis
8 35\72
(1\72) 		583.3
(16.7) 		7/5
(100/99) 		Octoid / octopus
9 19\72
(3\72) 		316.7
(50.0) 		6/5
(36/35) 		Ennealimmal / ennealimnic / ennealiminal
9 23\72
(1\72) 		383.3
(16.7) 		5/4
(105/104) 		Enneaportent
12 23\72
(1\72) 		383.3
(16.7) 		5/4
(100/99) 		Compton / comptone
18 19\72
(1\72) 		316.7
(16.7) 		6/5
(105/104) 		Hemiennealimmal
24 23\72
(1\72) 		383.3
(16.7) 		5/4
(105/104) 		Hours
36 23\72
(1\72) 		383.3
(16.7) 		5/4
(81/80) 		Gamelstearn

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Octave stretch or compression

72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly stretching the octave, using tunings such as 114edt, 380zpi or 186ed6. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 380zpi and 186ed6 are milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.

Scales

Miracle-tempered scales
Maeve Gutierrez's scales
Budjarn Lambeth's scales
Gene Ward Smith's scales
Iannis Xenakis' scales
Others

Harmonic scale

Mode 8 of the harmonic series—harmonics 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).

Harmonics 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

Instruments

If one can get six 12edo instruments tuned a twelfth-tone apart, it is possible to use these instruments in combination to play the full gamut of 72edo (see Music).

One can also use a skip fretting system:

Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo:

Music

Bryan Deister
Ambient Esoterica
Jake Freivald
English Wikipedia has an article on:
Georg Friedrich Haas
Budjarn Lambeth
Claudi Meneghin
Prent Rodgers
Gene Ward Smith
Ivan Wyschnegradsky
James Tenney
Xeno Ov Eleas

External links

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