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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.


72edo is the only regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s, ''and'' tunes the octave just. 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]].
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 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.
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.
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{{Harmonics in equal|72|columns=9}}
{{Harmonics in equal|72|columns=9}}
{{Harmonics in equal|72|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}
{{Harmonics in equal|72|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}
=== Octave stretch ===
72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 72 factors into 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.
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.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degrees
! #
! Cents
! Cents
! Approximate ratios<ref group="note">{{sg|limit=17-limit}} For lower limits see [[Table of 72edo intervals]].</ref>
! Approximate ratios<ref group="note">{{sg|limit=19-limit}} For lower limits see [[Table of 72edo intervals]].</ref>
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]
! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]
Line 54: Line 57:
| 1
| 1
| 16.7
| 16.7
| 81/80
| 81/80, 91/90, 99/98, 100/99, 105/104
| ^1
| ^1
| up unison
| up unison
Line 65: Line 68:
| 2
| 2
| 33.3
| 33.3
| 45/44, 64/63
| 45/44, 49/48, 50/49, 55/54, 64/63
| ^^
| ^^
| dup unison
| dup unison
Line 76: Line 79:
| 3
| 3
| 50.0
| 50.0
| 33/32
| 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 90:
| 4
| 4
| 66.7
| 66.7
| 25/24
| 25/24, 26/25, 27/26, 28/27
| vvm2
| vvm2
| dudminor 2nd
| dudminor 2nd
Line 98: Line 101:
| 5
| 5
| 83.3
| 83.3
| 21/20
| 20/19, 21/20, 22/21
| vm2
| vm2
| downminor 2nd
| downminor 2nd
Line 109: Line 112:
| 6
| 6
| 100.0
| 100.0
| 35/33, 17/16, 18/17
| 17/16, 18/17, 19/18
| m2
| m2
| minor 2nd
| minor 2nd
Line 131: Line 134:
| 8
| 8
| 133.3
| 133.3
| 27/25, 13/12, 14/13
| 13/12, 14/13, 27/25
| ^^m2, v~2
| ^^m2, v~2
| dupminor 2nd, downmid 2nd
| dupminor 2nd, downmid 2nd
Line 186: Line 189:
| 13
| 13
| 216.7
| 216.7
| 25/22, 17/15
| 17/15, 25/22
| ^M2
| ^M2
| upmajor 2nd
| upmajor 2nd
Line 208: Line 211:
| 15
| 15
| 250.0
| 250.0
| 81/70, 15/13
| 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 230: Line 233:
| 17
| 17
| 283.3
| 283.3
| 33/28, 13/11, 20/17
| 13/11, 20/17
| vm3
| vm3
| downminor 3rd
| downminor 3rd
Line 241: Line 244:
| 18
| 18
| 300.0
| 300.0
| 25/21
| 19/16, 25/21, 32/27
| m3
| m3
| minor 3rd
| minor 3rd
Line 263: Line 266:
| 20
| 20
| 333.3
| 333.3
| 40/33, 17/14
| 17/14, 39/32, 40/33
| ^^m3, v~3
| ^^m3, v~3
| dupminor 3rd, downmid 3rd
| dupminor 3rd, downmid 3rd
Line 274: Line 277:
| 21
| 21
| 350.0
| 350.0
| 11/9
| 11/9, 27/22
| ~3
| ~3
| mid 3rd
| mid 3rd
Line 285: Line 288:
| 22
| 22
| 366.7
| 366.7
| 99/80, 16/13, 21/17
| 16/13, 21/17, 26/21
| ^~3, vvM3
| ^~3, vvM3
| upmid 3rd, dudmajor 3rd
| upmid 3rd, dudmajor 3rd
Line 307: Line 310:
| 24
| 24
| 400.0
| 400.0
| 44/35
| 24/19
| M3
| M3
| major 3rd
| major 3rd
Line 340: Line 343:
| 27
| 27
| 450.0
| 450.0
| 35/27, 13/10
| 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
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| 28
| 28
| 466.7
| 466.7
| 21/16, 17/13
| 17/13, 21/16
| vv4
| vv4
| dud 4th
| dud 4th
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| 32
| 32
| 533.3
| 533.3
| 15/11
| 15/11, 19/14, ''26/19''
| ^^4, v~4
| ^^4, v~4
| dup 4th, downmid 4th
| dup 4th, downmid 4th
Line 417: Line 420:
| 34
| 34
| 566.7
| 566.7
| 25/18, 18/13
| 18/13, 25/18
| ^~4, vvA4
| ^~4, vvA4
| upmid 4th, dudaug 4th
| upmid 4th, dudaug 4th
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| 36
| 36
| 600.0
| 600.0
| 99/70, 17/12
| 17/12, 24/17
| A4, d5
| A4, d5
| aug 4th, dim 5th
| aug 4th, dim 5th
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| 38
| 38
| 633.3
| 633.3
| 36/25, 13/9
| 13/9, 36/25
| v~5, ^^d5
| v~5, ^^d5
| downmid 5th, <br>dupdim 5th
| downmid 5th, <br>dupdim 5th
Line 483: Line 486:
| 40
| 40
| 666.7
| 666.7
| 22/15
| ''19/13'', 22/15, 28/19
| vv5, ^~5
| vv5, ^~5
| dud 5th, upmid 5th
| dud 5th, upmid 5th
Line 527: Line 530:
| 44
| 44
| 733.3
| 733.3
| 32/21
| 26/17, 32/21
| ^^5
| ^^5
| dup 5th
| dup 5th
Line 538: Line 541:
| 45
| 45
| 750.0
| 750.0
| 54/35, 17/11
| 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 571: Line 574:
| 48
| 48
| 800.0
| 800.0
| 35/22
| 19/12
| m6
| m6
| minor 6th
| minor 6th
Line 593: Line 596:
| 50
| 50
| 833.3
| 833.3
| 81/50, 13/8
| 13/8, 21/13, 34/21
| ^^m6, v~6
| ^^m6, v~6
| dupminor 6th, downmid 6th
| dupminor 6th, downmid 6th
Line 604: Line 607:
| 51
| 51
| 850.0
| 850.0
| 18/11
| 18/11, 44/27
| ~6
| ~6
| mid 6th
| mid 6th
Line 615: Line 618:
| 52
| 52
| 866.7
| 866.7
| 33/20, 28/17
| 28/17, 33/20, 64/39
| ^~6, vvM6
| ^~6, vvM6
| upmid 6th, dudmajor 6th
| upmid 6th, dudmajor 6th
Line 637: Line 640:
| 54
| 54
| 900.0
| 900.0
| 27/16
| 27/16, 32/19, 42/25
| M6
| M6
| major 6th
| major 6th
Line 648: Line 651:
| 55
| 55
| 916.7
| 916.7
| 56/33, 17/10
| 17/10, 22/13
| ^M6
| ^M6
| upmajor 6th
| upmajor 6th
Line 670: Line 673:
| 57
| 57
| 950.0
| 950.0
| 121/70
| 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 692: Line 695:
| 59
| 59
| 983.3
| 983.3
| 44/25
| 30/17, 44/25
| vm7
| vm7
| downminor 7th
| downminor 7th
Line 747: Line 750:
| 64
| 64
| 1066.7
| 1066.7
| 50/27
| 13/7, 24/13, 50/27
| ^~7, vvM7
| ^~7, vvM7
| upmid 7th, dudmajor 7th
| upmid 7th, dudmajor 7th
Line 758: Line 761:
| 65
| 65
| 1083.3
| 1083.3
| 15/8
| 15/8, 28/15
| vM7
| vM7
| downmajor 7th
| downmajor 7th
Line 769: Line 772:
| 66
| 66
| 1100.0
| 1100.0
| 66/35, 17/9
| 17/9, 32/17, 36/19
| M7
| M7
| major 7th
| major 7th
Line 780: Line 783:
| 67
| 67
| 1116.7
| 1116.7
| 21/11
| 19/10, 21/11, 40/21
| ^M7
| ^M7
| upmajor 7th
| upmajor 7th
Line 791: Line 794:
| 68
| 68
| 1133.3
| 1133.3
| 27/14, 48/25
| 25/13, 27/14, 48/25, 52/27
| ^^M7
| ^^M7
| dupmajor 7th
| dupmajor 7th
Line 802: Line 805:
| 69
| 69
| 1150.0
| 1150.0
| 35/18
| 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 816:
| 70
| 70
| 1166.7
| 1166.7
| 49/25
| 49/25, 55/28, 63/32, 88/45, 96/49
| vv8
| vv8
| dud octave
| dud octave
Line 824: Line 827:
| 71
| 71
| 1183.3
| 1183.3
| 99/50
| 99/50, 160/81, 180/91, 196/99, 208/105
| v8
| v8
| down octave
| down octave
Line 844: Line 847:
| D
| D
|}
|}
<references group="note" />


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Line 967: Line 971:
| C dupmajor or C dup
| C dupmajor or C dup
|}
|}
For a more complete list, see [[Ups and Downs Notation #Chord names in other EDOs]].  
For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]].  


=== Relationship between primes and rings ===
=== Relationship between primes and rings ===
Line 982: Line 986:


== Notations ==
== Notations ==
=== Ups and downs 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.
{{Sharpness-sharp6a}}
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
[[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 ===
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-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]].
Line 1,024: Line 1,041:


[[File:72edo Sagittal.png|800px]]
[[File:72edo Sagittal.png|800px]]
=== Ups and downs notation ===
Using [[Helmholtz–Ellis]] accidentals, 72edo can also be notated using [[ups and downs notation]]:
{{Sharpness-sharp6|72}}
In some cases, certain notes may be best notated using semi- and sesquisharps and flats with arrows:
{{Sharpness-sharp6-qt|72}}


=== Ivan Wyschnegradsky's notation ===
=== Ivan Wyschnegradsky's notation ===
{{Sharpness-sharp6-iw|72}}
{{Sharpness-sharp6-iw|72}}


== JI approximation ==
== Approximation to JI ==
[[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]]
[[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]]


Line 1,042: Line 1,052:


=== Zeta properties ===
=== Zeta properties ===
72edo is the ninth [[zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[The Riemann Zeta Function and Tuning #The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.
72edo is the ninth [[zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[the Riemann zeta function and tuning#The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.


[[File:plot72.png|alt=plot72.png|plot72.png]]
[[File:plot72.png|alt=plot72.png|plot72.png]]
==== Zeta peak index ====
{{ZPI
| zpi = 380
| steps = 71.9506065993786
| step size = 16.6781081733140
| tempered height = 9.157547
| pure height = 7.501282
| integral = 1.625363
| gap = 19.964746
| octave = 1200.82378847861
| consistent = 18
| distinct = 13
}}


== Regular temperament properties ==
== Regular temperament properties ==
Line 1,074: Line 1,070:
| 2.3.5
| 2.3.5
| 15625/15552, 531441/524288
| 15625/15552, 531441/524288
| {{mapping| 72 114 167 }}
| {{Mapping| 72 114 167 }}
| +0.839
| +0.839
| 0.594
| 0.594
Line 1,081: Line 1,077:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1029/1024, 4375/4374
| 225/224, 1029/1024, 4375/4374
| {{mapping| 72 114 167 202 }}
| {{Mapping| 72 114 167 202 }}
| +0.822
| +0.822
| 0.515
| 0.515
Line 1,088: Line 1,084:
| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 4000/3993
| 225/224, 243/242, 385/384, 4000/3993
| {{mapping| 72 114 167 202 249 }}
| {{Mapping| 72 114 167 202 249 }}
| +0.734
| +0.734
| 0.493
| 0.493
Line 1,095: Line 1,091:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 169/168, 225/224, 243/242, 325/324, 385/384
| 169/168, 225/224, 243/242, 325/324, 385/384
| {{mapping| 72 114 167 202 249 266 }}
| {{Mapping| 72 114 167 202 249 266 }}
| +0.936
| +0.936
| 0.638
| 0.638
Line 1,102: Line 1,098:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 169/168, 221/220, 225/224, 243/242, 273/272, 325/324
| 169/168, 221/220, 225/224, 243/242, 273/272, 325/324
| {{mapping| 72 114 167 202 249 266 294 }}
| {{Mapping| 72 114 167 202 249 266 294 }}
| +0.975
| +0.975
| 0.599
| 0.599
Line 1,108: Line 1,104:
|-
|-
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 153/152, 210/209, 221/220, 273/272, 286/285, 351/350, 364/363
| 153/152, 169/168, 210/209, 221/220, 225/224, 243/242, 273/272
| {{mapping| 72 114 167 202 249 266 294 306 }}
| {{Mapping| 72 114 167 202 249 266 294 306 }}
| +0.780
| +0.780
| 0.599
| 0.762
| 4.57
| 4.57
|}
|}
Line 1,337: Line 1,333:
| Jacobin comma
| Jacobin comma
|}
|}
<references group="note" />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 1,500: Line 1,497:
| 383.3<br>(16.7)
| 383.3<br>(16.7)
| 5/4<br>(81/80)
| 5/4<br>(81/80)
| [[Decades]]
| [[Gamelstearn]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
Line 1,692: Line 1,689:


== Instruments ==
== 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).


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:
* [[Skip fretting system 72 2 27]]


Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo: [[Lumatone mapping for 72edo]]
Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo:
* [[Lumatone mapping for 72edo]]


== Music ==
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/VwVp3RVao_k ''microtonal improvisation in 72edo''] (2025)
; [[Ambient Esoterica]]
; [[Ambient Esoterica]]
* [https://www.youtube.com/watch?v=seWcDAoQjxY ''Goetic Synchronities''] (2023)
* [https://www.youtube.com/watch?v=seWcDAoQjxY ''Goetic Synchronities''] (2023)
Line 1,737: Line 1,740:
* [https://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo]
* [https://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo]
* [http://tonalsoft.com/enc/number/72edo.aspx 72-ed2 / 72-edo / 72-ET / 72-tone equal-temperament] on [[Tonalsoft Encyclopedia]]
* [http://tonalsoft.com/enc/number/72edo.aspx 72-ed2 / 72-edo / 72-ET / 72-tone equal-temperament] on [[Tonalsoft Encyclopedia]]
== Notes ==
<references group="note" />


[[Category:Listen]]
[[Category:Listen]]
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