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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]].
=== 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=11}}
 
{{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 ===
 
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
| [[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
| 0.0
| 0.0
| 1/1
| 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.


== 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 1,010: Line 1,313:
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:  


[[File:72edo Sagittal.png|frame|none]]
<div class="noresize">
[[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]]
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