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Line 7: Line 7:
{{Infobox ET}}
{{Infobox ET}}
{{Wikipedia|72 equal temperament}}
{{Wikipedia|72 equal temperament}}
{{EDO intro}}
{{ED intro}}


Each step of 72edo is called a ''[[morion]]'' (plural ''moria)''. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in [[12edo]]. 72edo is also a superset of [[24edo]], a common and standard tuning of [[Arabic, Turkish, Persian music|Arabic music]], and has itself been used to tune Turkish music.
Each step of 72edo is called a ''[[morion]]'' (plural ''moria)''. This produces a twelfth-tone tuning, with the whole tone measuring 200{{c}}, the same as in [[12edo]]. 72edo is also a superset of [[24edo]], a common and standard tuning of [[Arabic, Turkish, Persian music|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]]), [[Ezra Sims]], [[James Tenney]], [[Georg Friedrich Haas]] and the jazz musician [[Joe Maneri]].
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 ==
== Theory ==
72edo approximates [[11-limit]] [[just intonation]] exceptionally well, is [[consistent]] in the [[17-odd-limit]], is the first [[Trivial temperament|non-trivial]] EDO to be consistent in the 12- and 13-[[Odd prime sum limit|odd-prime-sum-limit]], and is the ninth [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 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 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, and the first edo to be consistent in the 12- and 13-[[odd prime sum limit|odd-prime-sum-limit]].  


72edo is an excellent tuning for the [[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, 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 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.
Line 23: Line 25:


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|72|columns=11}}
{{Harmonics in equal|72|columns=9}}
{{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 *
! Approximate ratios<ref group="note">{{sg|limit=19-limit}} For lower limits see [[Table of 72edo intervals]].</ref>
! colspan="3" | [[Ups and downs notation|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]]
!(K, S, U)  
! (K, S, U)  
|-
|-
| 0
| 0
| 0.000
| 0.0
| 1/1
| 1/1
| P1
| P1
Line 47: Line 53:
| perfect unison
| perfect unison
| D
| D
|D
| D
|-
|-
| 1
| 1
| 16.667
| 16.7
| 81/80
| 81/80, 91/90, 99/98, 100/99, 105/104
| ^1
| ^1
| up unison
| up unison
Line 58: Line 64:
| comma-wide unison, large unison
| comma-wide unison, large unison
| KD, LD
| KD, LD
|KD
| KD
|-
|-
| 2
| 2
| 33.333
| 33.3
| 45/44, 64/63
| 45/44, 49/48, 50/49, 55/54, 64/63
| ^^
| ^^
| dup unison
| dup unison
Line 69: Line 75:
| super unison, on unison
| super unison, on unison
| SD, OD
| SD, OD
|SD
| SD
|-
|-
| 3
| 3
| 50.000
| 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 80: Line 86:
| uber unison, hyper unison, hypominor 2nd
| uber unison, hyper unison, hypominor 2nd
| UD, HD, uEb
| UD, HD, uEb
|UD, uEb
| UD, uEb
|-
|-
| 4
| 4
| 66.667
| 66.7
| 25/24
| 25/24, 26/25, 27/26, 28/27
| vvm2
| vvm2
| dudminor 2nd
| dudminor 2nd
Line 91: Line 97:
| classic aug unison, subminor 2nd
| classic aug unison, subminor 2nd
| kkD#, sEb
| kkD#, sEb
|sD#, (kkD#), sEb
| sD#, (kkD#), sEb
|-
|-
| 5
| 5
| 83.333
| 83.3
| 21/20
| 20/19, 21/20, 22/21
| vm2
| vm2
| downminor 2nd
| downminor 2nd
Line 102: Line 108:
| comma-narrow aug unison, little minor 2nd  
| comma-narrow aug unison, little minor 2nd  
| kD#, lEb
| kD#, lEb
|kD#, kEb
| kD#, kEb
|-
|-
| 6
| 6
| 100.000
| 100.0
| 35/33, 17/16, 18/17
| 17/16, 18/17, 19/18
| m2
| m2
| minor 2nd
| minor 2nd
Line 113: Line 119:
| minor 2nd
| minor 2nd
| Eb
| Eb
|Eb
| Eb
|-
|-
| 7
| 7
| 116.667
| 116.7
| 15/14, 16/15
| 15/14, 16/15
| ^m2
| ^m2
Line 124: Line 130:
| classic minor 2nd
| classic minor 2nd
| KEb
| KEb
|KEb
| KEb
|-
|-
| 8
| 8
| 133.333
| 133.3
| 27/25, 13/12, 14/13
| 13/12, 14/13, 27/25
| v~2
| ^^m2, v~2
| downmid 2nd
| dupminor 2nd, downmid 2nd
| ^^Eb
| ^^Eb
| Om2
| Om2
| on minor 2nd
| on minor 2nd
| OEb
| OEb
|SEb
| SEb
|-
|-
| 9
| 9
| 150.000
| 150.0
| 12/11
| 12/11
| ~2
| ~2
Line 146: Line 152:
| neutral 2nd
| neutral 2nd
| UEb/uE
| UEb/uE
|UEb/uE
| UEb/uE
|-
|-
| 10
| 10
| 166.667
| 166.7
| 11/10
| 11/10
| ^~2
| ^~2, vvM2
| upmid 2nd
| upmid 2nd, dudmajor 2nd
| vvE
| vvE
| oM2
| oM2
| off major 2nd
| off major 2nd
| oE
| oE
|sE
| sE
|-
|-
| 11
| 11
| 183.333
| 183.3
| 10/9
| 10/9
| vM2
| vM2
Line 168: Line 174:
| classic/comma-narrow major 2nd
| classic/comma-narrow major 2nd
| kE
| kE
|kE
| kE
|-
|-
| 12
| 12
| 200.000
| 200.0
| 9/8
| 9/8
| M2
| M2
Line 179: Line 185:
| major 2nd
| major 2nd
| E
| E
|E
| E
|-
|-
| 13
| 13
| 216.667
| 216.7
| 25/22, 17/15
| 17/15, 25/22
| ^M2
| ^M2
| upmajor 2nd
| upmajor 2nd
Line 190: Line 196:
| large major 2nd
| large major 2nd
| LE
| LE
|KE
| KE
|-
|-
| 14
| 14
| 233.333
| 233.3
| 8/7
| 8/7
| ^^M2
| ^^M2
Line 201: Line 207:
| supermajor 2nd
| supermajor 2nd
| SE
| SE
|SE
| SE
|-
|-
| 15
| 15
| 250.000
| 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 212: Line 218:
| hypermajor 2nd, hypominor 3rd
| hypermajor 2nd, hypominor 3rd
| HE, hF
| HE, hF
|UE, uF
| UE, uF
|-
|-
| 16
| 16
| 266.667
| 266.7
| 7/6
| 7/6
| vvm3
| vvm3
Line 223: Line 229:
| subminor 3rd
| subminor 3rd
| sF
| sF
|sF
| sF
|-
|-
| 17
| 17
| 283.333
| 283.3
| 33/28, 13/11, 20/17
| 13/11, 20/17
| vm3
| vm3
| downminor 3rd
| downminor 3rd
Line 234: Line 240:
| little minor 3rd
| little minor 3rd
| lF
| lF
|kF
| kF
|-
|-
| 18
| 18
| 300.000
| 300.0
| 25/21
| 19/16, 25/21, 32/27
| m3
| m3
| minor 3rd
| minor 3rd
Line 245: Line 251:
| minor 3rd
| minor 3rd
| F
| F
|F
| F
|-
|-
| 19
| 19
| 316.667
| 316.7
| 6/5
| 6/5
| ^m3
| ^m3
Line 256: Line 262:
| classic minor 3rd
| classic minor 3rd
| KF
| KF
|KF
| KF
|-
|-
| 20
| 20
| 333.333
| 333.3
| 40/33, 17/14
| 17/14, 39/32, 40/33
| v~3
| ^^m3, v~3
| downmid 3rd
| dupminor 3rd, downmid 3rd
| ^^F
| ^^F
| Om3
| Om3
| on minor third
| on minor third
| OF
| OF
|SF
| SF
|-
|-
| 21
| 21
| 350.000
| 350.0
| 11/9
| 11/9, 27/22
| ~3
| ~3
| mid 3rd
| mid 3rd
Line 278: Line 284:
| neutral 3rd
| neutral 3rd
| UF/uF#
| UF/uF#
|UF/uF#
| UF/uF#
|-
|-
| 22
| 22
| 366.667
| 366.7
| 99/80, 16/13, 21/17
| 16/13, 21/17, 26/21
| ^~3
| ^~3, vvM3
| upmid 3rd
| upmid 3rd, dudmajor 3rd
| vvF#
| vvF#
| oM3
| oM3
| off major 3rd
| off major 3rd
| oF#
| oF#
|sF#
| sF#
|-
|-
| 23
| 23
| 383.333
| 383.3
| 5/4
| 5/4
| vM3
| vM3
Line 300: Line 306:
| classic major 3rd
| classic major 3rd
| kF#
| kF#
|kF#
| kF#
|-
|-
| 24
| 24
| 400.000
| 400.0
| 44/35
| 24/19
| M3
| M3
| major 3rd
| major 3rd
Line 311: Line 317:
| major 3rd
| major 3rd
| F#
| F#
|F#
| F#
|-
|-
| 25
| 25
| 416.667
| 416.7
| 14/11
| 14/11
| ^M3
| ^M3
Line 322: Line 328:
| large major 3rd
| large major 3rd
| LF#
| LF#
|KF#
| KF#
|-
|-
| 26
| 26
| 433.333
| 433.3
| 9/7
| 9/7
| ^^M3
| ^^M3
Line 333: Line 339:
| supermajor 3rd
| supermajor 3rd
| SF#
| SF#
|SF#
| SF#
|-
|-
| 27
| 27
| 450.000
| 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
Line 344: Line 350:
| hypermajor 3rd, hypo 4th
| hypermajor 3rd, hypo 4th
| HF#, hG
| HF#, hG
|UF#, uG
| UF#, uG
|-
|-
| 28
| 28
| 466.667
| 466.7
| 21/16, 17/13
| 17/13, 21/16
| vv4
| vv4
| dud 4th
| dud 4th
Line 355: Line 361:
| sub 4th
| sub 4th
| sG
| sG
|sG
| sG
|-
|-
| 29
| 29
| 483.333
| 483.3
| 33/25
| 33/25
| v4
| v4
Line 366: Line 372:
| little 4th
| little 4th
| lG
| lG
|kG
| kG
|-
|-
| 30
| 30
| 500.000
| 500.0
| 4/3
| 4/3
| P4
| P4
Line 377: Line 383:
| perfect 4th
| perfect 4th
| G
| G
|G
| G
|-
|-
| 31
| 31
| 516.667
| 516.7
| 27/20
| 27/20
| ^4
| ^4
Line 388: Line 394:
| comma-wide 4th
| comma-wide 4th
| KG
| KG
|KG
| KG
|-
|-
| 32
| 32
| 533.333
| 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 399: Line 405:
| on 4th
| on 4th
| OG
| OG
|SG
| SG
|-
|-
| 33
| 33
| 550.000
| 550.0
| 11/8
| 11/8
| ~4
| ~4
Line 410: Line 416:
| uber 4th / neutral 4th
| uber 4th / neutral 4th
| UG
| UG
|UG
| UG
|-
|-
| 34
| 34
| 566.667
| 566.7
| 25/18, 18/13
| 18/13, 25/18
| ^~4, vvA4
| ^~4, vvA4
| upmid 4th, dudaug 4th
| upmid 4th, dudaug 4th
Line 421: Line 427:
| classic aug 4th, sub dim 5th
| classic aug 4th, sub dim 5th
| kkG#, sAb
| kkG#, sAb
|SG#, (kkG#), sAb
| SG#, (kkG#), sAb
|-
|-
| 35
| 35
| 583.333
| 583.3
| 7/5
| 7/5
| vA4, vd5
| vA4, vd5
| downaug 4th, downdim 5th
| downaug 4th, <br>downdim 5th
| vG#, vAb
| vG#, vAb
| kA4, ld5
| kA4, ld5
| comma-narrow aug 4th, little dim 5th
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, lAb
|kG#, kAb
| kG#, kAb
|-
|-
| 36
| 36
| 600.000
| 600.0
| 99/70, 17/12
| 17/12, 24/17
| A4, d5
| A4, d5
| aug 4th, dim 5th
| aug 4th, dim 5th
Line 443: Line 449:
| aug 4th, dim 5th
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
|G#, Ab
| G#, Ab
|-
|-
| 37
| 37
| 616.667
| 616.7
| 10/7
| 10/7
| ^A4, ^d5
| ^A4, ^d5
Line 454: Line 460:
| large aug 4th, comma-wide dim 5th
| large aug 4th, comma-wide dim 5th
| LG#, KAb
| LG#, KAb
|KG#, KAb
| KG#, KAb
|-
|-
| 38
| 38
| 633.333
| 633.3
| 36/25, 13/9
| 13/9, 36/25
| v~5, ^^d5
| v~5, ^^d5
| downmid 5th, dupdim 5th
| downmid 5th, <br>dupdim 5th
| ^^Ab
| ^^Ab
| SA4, KKd5
| SA4, KKd5
| super aug 4th, classic dim 5th
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, KKAb
|SG#, SAb, (KKAb)
| SG#, SAb, (KKAb)
|-
|-
| 39
| 39
| 650.000
| 650.0
| 16/11
| 16/11
| ~5
| ~5
Line 476: Line 482:
| unter 5th / neutral 5th
| unter 5th / neutral 5th
| uA
| uA
|uA
| uA
|-
|-
| 40
| 40
| 666.667
| 666.7
| 22/15
| ''19/13'', 22/15, 28/19
| vv5, ^~5
| vv5, ^~5
| dud 5th, upmid 5th
| dud 5th, upmid 5th
Line 487: Line 493:
| off 5th
| off 5th
| oA
| oA
|sA
| sA
|-
|-
| 41
| 41
| 683.333
| 683.3
| 40/27
| 40/27
| v5
| v5
Line 498: Line 504:
| comma-narrow 5th
| comma-narrow 5th
| kA
| kA
|kA
| kA
|-
|-
| 42
| 42
| 700.000
| 700.0
| 3/2
| 3/2
| P5
| P5
Line 509: Line 515:
| perfect 5th
| perfect 5th
| A
| A
|A
| A
|-
|-
| 43
| 43
| 716.667
| 716.7
| 50/33
| 50/33
| ^5
| ^5
Line 520: Line 526:
| large fifth
| large fifth
| LA
| LA
|KA
| KA
|-
|-
| 44
| 44
| 733.333
| 733.3
| 32/21
| 26/17, 32/21
| ^^5
| ^^5
| dup 5th
| dup 5th
Line 531: Line 537:
| super fifth
| super fifth
| SA
| SA
|SA
| SA
|-
|-
| 45
| 45
| 750.000
| 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 542: Line 548:
| hyper fifth, hypominor 6th
| hyper fifth, hypominor 6th
| HA, hBb
| HA, hBb
|UA, uBb
| UA, uBb
|-
|-
| 46
| 46
| 766.667
| 766.7
| 14/9
| 14/9
| vvm6
| vvm6
Line 553: Line 559:
| superminor 6th
| superminor 6th
| sBb
| sBb
|sBb
| sBb
|-
|-
| 47
| 47
| 783.333
| 783.3
| 11/7
| 11/7
| vm6
| vm6
Line 564: Line 570:
| little minor 6th
| little minor 6th
| lBb
| lBb
|kBb
| kBb
|-
|-
| 48
| 48
| 800.000
| 800.0
| 35/22
| 19/12
| m6
| m6
| minor 6th
| minor 6th
Line 575: Line 581:
| minor 6th
| minor 6th
| Bb
| Bb
|Bb
| Bb
|-
|-
| 49
| 49
| 816.667
| 816.7
| 8/5
| 8/5
| ^m6
| ^m6
Line 586: Line 592:
| classic minor 6th
| classic minor 6th
| kBb
| kBb
|kBb
| kBb
|-
|-
| 50
| 50
| 833.333
| 833.3
| 81/50, 13/8
| 13/8, 21/13, 34/21
| v~6
| ^^m6, v~6
| downmid 6th
| dupminor 6th, downmid 6th
| ^^Bb
| ^^Bb
| Om6
| Om6
| on minor 6th
| on minor 6th
| oBb
| oBb
|sBb
| sBb
|-
|-
| 51
| 51
| 850.000
| 850.0
| 18/11
| 18/11, 44/27
| ~6
| ~6
| mid 6th
| mid 6th
Line 608: Line 614:
| neutral 6th
| neutral 6th
| UBb, uB
| UBb, uB
|UBb, uB
| UBb, uB
|-
|-
| 52
| 52
| 866.667
| 866.7
| 33/20, 28/17
| 28/17, 33/20, 64/39
| ^~6
| ^~6, vvM6
| upmid 6th
| upmid 6th, dudmajor 6th
| vvB
| vvB
| oM6
| oM6
| off major 6th
| off major 6th
| oB
| oB
|sB
| sB
|-
|-
| 53
| 53
| 883.333
| 883.3
| 5/3
| 5/3
| vM6
| vM6
Line 630: Line 636:
| classic major 6th
| classic major 6th
| kB
| kB
|kB
| kB
|-
|-
| 54
| 54
| 900.000
| 900.0
| 27/16
| 27/16, 32/19, 42/25
| M6
| M6
| major 6th
| major 6th
| B
| B
|M6
| M6
|major 6th
| major 6th
|B
| B
|B
| B
|-
|-
| 55
| 55
| 916.667
| 916.7
| 56/33, 17/10
| 17/10, 22/13
| ^M6
| ^M6
| upmajor 6th
| upmajor 6th
Line 652: Line 658:
| large major 6th
| large major 6th
| LB
| LB
|KB
| KB
|-
|-
| 56
| 56
| 933.333
| 933.3
| 12/7
| 12/7
| ^^M6
| ^^M6
Line 663: Line 669:
| supermajor 6th
| supermajor 6th
| SB
| SB
|SB
| SB
|-
|-
| 57
| 57
| 950.000
| 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 674: Line 680:
| hypermajor 6th, hypominor 7th
| hypermajor 6th, hypominor 7th
| HB, hC
| HB, hC
|UB, uC
| UB, uC
|-
|-
| 58
| 58
| 966.667
| 966.7
| 7/4
| 7/4
| vvm7
| vvm7
Line 685: Line 691:
| subminor 7th
| subminor 7th
| sC
| sC
|sC
| sC
|-
|-
| 59
| 59
| 983.333
| 983.3
| 44/25
| 30/17, 44/25
| vm7
| vm7
| downminor 7th
| downminor 7th
Line 696: Line 702:
| little minor 7th
| little minor 7th
| lC
| lC
|kC
| kC
|-
|-
| 60
| 60
| 1000.000
| 1000.0
| 16/9
| 16/9
| m7
| m7
Line 707: Line 713:
| minor 7th
| minor 7th
| C
| C
|C
| C
|-
|-
| 61
| 61
| 1016.667
| 1016.7
| 9/5
| 9/5
| ^m7
| ^m7
Line 718: Line 724:
| classic/comma-wide minor 7th
| classic/comma-wide minor 7th
| KC
| KC
|KC
| KC
|-
|-
| 62
| 62
| 1033.333
| 1033.3
| 20/11
| 20/11
| v~7
| ^^m7, v~7
| downmid 7th
| dupminor 7th, downmid 7th
| ^^C
| ^^C
| Om7
| Om7
| on minor 7th
| on minor 7th
| OC
| OC
|SC
| SC
|-
|-
| 63
| 63
| 1050.000
| 1050.0
| 11/6
| 11/6
| ~7
| ~7
Line 740: Line 746:
| neutral 7th, hypo dim 8ve
| neutral 7th, hypo dim 8ve
| UC/uC#, hDb
| UC/uC#, hDb
|UC/uC#, uDb
| UC/uC#, uDb
|-
|-
| 64
| 64
| 1066.667
| 1066.7
| 50/27
| 13/7, 24/13, 50/27
| ^~7
| ^~7, vvM7
| upmid 7th
| upmid 7th, dudmajor 7th
| vvC#
| vvC#
| oM7, sd8
| oM7, sd8
| off major 7th, sub dim 8ve
| off major 7th, sub dim 8ve
| oC#, sDb
| oC#, sDb
|sC#, sDb
| sC#, sDb
|-
|-
| 65
| 65
| 1083.333
| 1083.3
| 15/8
| 15/8, 28/15
| vM7
| vM7
| downmajor 7th
| downmajor 7th
Line 762: Line 768:
| classic major 7th, little dim 8ve
| classic major 7th, little dim 8ve
| kC#, lDb
| kC#, lDb
|kC#, kDb
| kC#, kDb
|-
|-
| 66
| 66
| 1100.000
| 1100.0
| 66/35, 17/9
| 17/9, 32/17, 36/19
| M7
| M7
| major 7th
| major 7th
Line 773: Line 779:
| major 7th, dim 8ve
| major 7th, dim 8ve
| C#, Db
| C#, Db
|C#, Db
| C#, Db
|-
|-
| 67
| 67
| 1116.667
| 1116.7
| 21/11
| 19/10, 21/11, 40/21
| ^M7
| ^M7
| upmajor 7th
| upmajor 7th
Line 784: Line 790:
| large major 7th, comma-wide dim 8ve
| large major 7th, comma-wide dim 8ve
| LC#, KDb
| LC#, KDb
|KC#, KDb
| KC#, KDb
|-
|-
| 68
| 68
| 1133.333
| 1133.3
| 27/14, 48/25
| 25/13, 27/14, 48/25, 52/27
| ^^M7
| ^^M7
| dupmajor 7th
| dupmajor 7th
Line 795: Line 801:
| supermajor 7th, classic dim 8ve
| supermajor 7th, classic dim 8ve
| SC#, KKDb
| SC#, KKDb
|SC#, SDb, (KKDb)
| SC#, SDb, (KKDb)
|-
|-
| 69
| 69
| 1150.000
| 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 806: Line 812:
| hypermajor 7th, unter 8ve, hypo 8ve
| hypermajor 7th, unter 8ve, hypo 8ve
| HC#, uD, hD
| HC#, uD, hD
|UC#, uDb, uD
| UC#, uDb, uD
|-
|-
| 70
| 70
| 1166.667
| 1166.7
| 49/25
| 49/25, 55/28, 63/32, 88/45, 96/49
| vv8
| vv8
| dud octave
| dud octave
Line 817: Line 823:
| sub 8ve, off 8ve
| sub 8ve, off 8ve
| sD, oD
| sD, oD
|sD
| sD
|-
|-
| 71
| 71
| 1183.333
| 1183.3
| 99/50
| 99/50, 160/81, 180/91, 196/99, 208/105
| v8
| v8
| down octave
| down octave
Line 828: Line 834:
| comma-narrow 8ve, little 8ve
| comma-narrow 8ve, little 8ve
| kD, lD
| kD, lD
|kD
| kD
|-
|-
| 72
| 72
| 1200.000
| 1200.0
| 2/1
| 2/1
| P8
| P8
Line 839: Line 845:
| perfect octave
| perfect octave
| D
| D
|D
| D
|}
|}
 
<references group="note" />
<nowiki>*</nowiki> based on treating 72edo as a 17-limit temperament. For lower limits see [[Table of 72edo intervals]].


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Line 851: Line 856:
! Quality
! Quality
! [[Color notation|Color]]
! [[Color notation|Color]]
! Monzo Format
! Monzo format
! Examples
! Examples
|-
|-
Line 857: Line 862:
| zo
| zo
| (a b 0 1)
| (a b 0 1)
| 7/6, 7/40
| 7/6, 7/4
|-
|-
| minor
| minor
| fourthward wa
| fourthward wa
| (a b), b &lt; -1
| (a b), b < -1
| 32/27, 16/9
| 32/27, 16/9
|-
|-
Line 869: Line 874:
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| rowspan="2" |downmid
| rowspan="2" | dupminor, <br>downmid
| luyo
| luyo
| (a b 1 0 -1)
| (a b 1 0 -1)
Line 880: Line 885:
| 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
| rowspan="2" | upmid, <br>dudmajor
| logu
| logu
| (a b -1 0 1)
| (a b -1 0 1)
Line 903: Line 908:
| major
| major
| fifthward wa
| fifthward wa
| (a b), b &gt; 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
| 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:
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 915: Line 929:
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! [[Kite's color notation|Color of the 3rd]]
! [[Color notation|Color of the 3rd]]
! JI Chord
! JI chord
! Notes as Edosteps
! Notes as edosteps
! Notes of C Chord
! Notes of C chord
! Written Name
! Written name
! Spoken Name
! Spoken name
|-
|-
| zo
| zo
Line 957: 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]].  


=== Remembering the pitch structure ===
=== Relationship between primes and rings ===
The pitch structure is very easy to remember. In 72tet, 12edo is the Pythagorian ring; ''id est'', every 6 degrees is the 3-limit.
In 72tet, there are 6 [[ring number|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:
Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
* -1 degree corrects 5/4 (80/81)
* −1 degree (the down ring) corrects 81/64 to 5/4 via 80/81
* -2 degrees corrects 7/4 (63/64)
* −2 degrees (the dud ring) corrects 16/9 to 7/4 via 63/64
* +3 degrees corrects 11/8 (33/32)
* +3 degrees (the trup ring) corrects 4/3 to 11/8 via 33/32
* +2 degrees corrects 13/8 (1053/1024)
* +2 degrees (the dup ring) corrects 128/81 to 13/8 via 1053/1024
* 0 degree corrects 17/16 and 19/16 (4131/4096 and 513/512)
* 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
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 ==
== Notations ==
=== Sagittal ===
=== 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 ===
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]].
 
==== Evo flavor ====
<imagemap>
File:72-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:72-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Revo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:72-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 711 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
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|800px]]
[[File:72edo Sagittal.png|800px]]


=== Ups and downs ===
=== Ivan Wyschnegradsky's notation ===
Using [[Helmholtz-Ellis notation|Helmholtz&ndash;Ellis]] accidentals, 72edo can also be notated using [[ups and downs notation]]:
{{Sharpness-sharp6-iw|72}}
{{Sharpness-sharp6|72}}


In some cases, certain notes may be best notated using semi- and sesquisharps and flats with arrows:
== Approximation to JI ==
{{Sharpness-sharp6-qt|72}}
[[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]]


== JI approximation ==
=== Interval mappings ===
[[File:24ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 24edo]]
{{Q-odd-limit intervals|72}}
=== Z function ===
 
72edo is the ninth [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[The Riemann Zeta Function and Tuning #The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.
=== 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.


[[File:plot72.png|alt=plot72.png|plot72.png]]
[[File:plot72.png|alt=plot72.png|plot72.png]]
Line 991: Line 1,058:
== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 1,002: Line 1,070:
| 2.3.5
| 2.3.5
| 15625/15552, 531441/524288
| 15625/15552, 531441/524288
| [{{val| 72 114 167 }}]
| {{Mapping| 72 114 167 }}
| +0.839
| +0.839
| 0.594
| 0.594
Line 1,009: Line 1,077:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1029/1024, 4375/4374
| 225/224, 1029/1024, 4375/4374
| [{{val| 72 114 167 202 }}]
| {{Mapping| 72 114 167 202 }}
| +0.822
| +0.822
| 0.515
| 0.515
Line 1,016: 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
| [{{val| 72 114 167 202 249 }}]
| {{Mapping| 72 114 167 202 249 }}
| +0.734
| +0.734
| 0.493
| 0.493
Line 1,023: 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
| [{{val| 72 114 167 202 249 266 }}]
| {{Mapping| 72 114 167 202 249 266 }}
| +0.936
| +0.936
| 0.638
| 0.638
Line 1,030: 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
| [{{val| 72 114 167 202 249 266 294 }}]
| {{Mapping| 72 114 167 202 249 266 294 }}
| +0.975
| +0.975
| 0.599
| 0.599
| 3.59
| 3.59
|-
| 2.3.5.7.11.13.17.19
| 153/152, 169/168, 210/209, 221/220, 225/224, 243/242, 273/272
| {{Mapping| 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 [[99edo|99]], [[270edo|270]], [[224edo|224]], [[494edo|494]], and [[217edo|217]], respectively.  
* 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 [[99edo|99]], [[270edo|270]], [[224edo|224]], [[494edo|494]], and [[217edo|217]], respectively.  


Line 1,042: Line 1,116:


{| class="commatable wikitable center-1 center-2 right-4"
{| class="commatable wikitable center-1 center-2 right-4"
! [[Harmonic limit|Prime<br>Limit]]
|-
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
Line 1,050: Line 1,125:
| 3
| 3
| [[531441/524288|(12 digits)]]
| [[531441/524288|(12 digits)]]
| {{Monzo|-19 12 }}
| {{Monzo| -19 12 }}
| 23.46
| 23.46
| Pythagorean comma
| Pythagorean comma
Line 1,064: Line 1,139:
| {{Monzo| -25 7 6 }}
| {{Monzo| -25 7 6 }}
| 31.57
| 31.57
| [[Ampersand]]
| [[Ampersand comma]]
|-
|-
| 5
| 5
Line 1,106: Line 1,181:
| {{Monzo| 0 3 4 -5 }}
| {{Monzo| 0 3 4 -5 }}
| 6.99
| 6.99
| Mirkwai
| Mirkwai comma
|-
|-
| 7
| 7
Line 1,112: Line 1,187:
| {{Monzo| -4 9 -2 -2 }}
| {{Monzo| -4 9 -2 -2 }}
| 7.32
| 7.32
| Cataharry
| Cataharry comma
|-
|-
| 7
| 7
Line 1,154: Line 1,229:
| {{Monzo| -2 0 3 -3 1 }}
| {{Monzo| -2 0 3 -3 1 }}
| 3.78
| 3.78
| Moctdel  
| Moctdel comma
|-
|-
| 11
| 11
Line 1,166: Line 1,241:
| {{Monzo| 5 -1 3 0 -3 }}
| {{Monzo| 5 -1 3 0 -3 }}
| 3.03
| 3.03
| Wizardharry  
| Wizardharry comma
|-
|-
| 11
| 11
Line 1,208: Line 1,283:
| {{Monzo| 2 -1 0 1 -2 1 }}
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.76
| 4.76
| Gentle comma
| Minor minthma
|-
|-
| 13
| 13
Line 1,258: Line 1,333:
| Jacobin comma
| Jacobin comma
|}
|}
<references/>
<references group="note" />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 1,265: Line 1,340:
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.
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.


{| class="wikitable center-1 center-2"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator
! Generator*
! Names
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
| 1
| 1
| 1\72
| 1\72
| 16.7
| 105/104
| [[Quincy]]
| [[Quincy]]
|-
|-
| 1
| 1
| 5\72
| 5\72
| 83.3
| 21/20
| [[Marvolo]]
| [[Marvolo]]
|-
|-
| 1
| 1
| 7\72
| 7\72
| 116.7
| 15/14
| [[Miracle]] / benediction / manna
| [[Miracle]] / benediction / manna
|-
| 1
| 11\72
|
|-
| 1
| 13\72
|
|-
|-
| 1
| 1
| 17\72
| 17\72
| 283.3
| 13/11
| [[Neominor]]
| [[Neominor]]
|-
|-
| 1
| 1
| 19\72
| 19\72
| 316.7
| 6/5
| [[Catakleismic]]
| [[Catakleismic]]
|-
| 1
| 23\72
|
|-
|-
| 1
| 1
| 25\72
| 25\72
| 416.7
| 14/11
| [[Sqrtphi]]
| [[Sqrtphi]]
|-
|-
| 1
| 1
| 29\72
| 29\72
|  
| 483.3
| 45/34
| [[Hemiseven]]
|-
|-
| 1
| 1
| 31\72
| 31\72
| [[Marvo]] / zarvo
| 516.7
| 27/20
| [[Marvo]] / [[zarvo]]
|-
|-
| 1
| 1
| 35\72
| 35\72
| 583.3
| 7/5
| [[Cotritone]]
| [[Cotritone]]
|-
| 2
| 1\72
|
|-
|-
| 2
| 2
| 5\72
| 5\72
| 83.3
| 21/20
| [[Harry]]
| [[Harry]]
|-
|-
| 2
| 2
| 7\72
| 7\72
| 116.7
| 15/14
| [[Semimiracle]]
| [[Semimiracle]]
|-
|-
| 2
| 2
| 11\72
| 11\72
| 183.3
| 10/9
| [[Unidec]] / hendec
| [[Unidec]] / hendec
|-
|-
| 2
| 2
| 13\72
| 21\72<br>(19\72)
| [[Wizard]] / lizard / gizzard
| 316.7<br>(283.3)
| 6/5<br>(13/11)
| [[Bikleismic]]
|-
|-
| 2
| 2
| 19\72
| 23\72<br>(13\72)
| [[Bikleismic]]
| 383.3<br>(216.7)
| 5/4<br>(17/15)
| [[Wizard]] / lizard / gizzard
|-
|-
| 3
| 3
| 1\72
| 11\72
|  
| 183.3
| 10/9
| [[Mirkat]]
|-
|-
| 3
| 3
| 5\72
| 19\72<br>(5\72)
| 316.7<br>(83.3)
| 6/5<br>(21/20)
| [[Tritikleismic]]
| [[Tritikleismic]]
|-
| 3
| 7\72
|
|-
| 3
| 11\72
| [[Mirkat]]
|-
|-
| 4
| 4
| 1\72
| 19\72<br>(1\72)
| 316.7<br>(16.7)
| 6/5<br>(105/104)
| [[Quadritikleismic]]
| [[Quadritikleismic]]
|-
| 4
| 5\72
|
|-
| 4
| 7\72
|
|-
| 6
| 1\72
|
|-
| 6
| 5\72
|
|-
|-
| 8
| 8
| 1\72
| 34\72<br>(2\72)
| [[Octoid]]
| 566.7<br>(33.3)
| 168/121<br>(55/54)
| [[Octowerck]] / octowerckis
|-
|-
| 8
| 8
| 2\72
| 35\72<br>(1\72)
| [[Octowerck]]
| 583.3<br>(16.7)
|-
| 7/5<br>(100/99)
| 8
| [[Octoid]] / octopus
| 4\72
|
|-
|-
| 9
| 9
| 1\72
| 19\72<br>(3\72)
|  
| 316.7<br>(50.0)
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimnic
|-
|-
| 9
| 9
| 3\72
| 23\72<br>(1\72)
| [[Ennealimmal]] / ennealimnic
| 383.3<br>(16.7)
| 5/4<br>(105/104)
| [[Enneaportent]]
|-
|-
| 12
| 12
| 1\72
| 23\72<br>(1\72)
| [[Compton]]
| 383.3<br>(16.7)
| 5/4<br>(100/99)
| [[Compton]] / comptone
|-
|-
| 18
| 18
| 1\72
| 19\72<br>(1\72)
| 316.7<br>(16.7)
| 6/5<br>(105/104)
| [[Hemiennealimmal]]
| [[Hemiennealimmal]]
|-
|-
| 24
| 24
| 1\72
| 23\72<br>(1\72)
| 383.3<br>(16.7)
| 5/4<br>(105/104)
| [[Hours]]
| [[Hours]]
|-
|-
| 36
| 36
| 1\72
| 23\72<br>(1\72)
|  
| 383.3<br>(16.7)
| 5/4<br>(81/80)
| [[Gamelstearn]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
Line 1,427: Line 1,512:


=== Harmonic scale ===
=== Harmonic scale ===
Mode 8 of the harmonic series – [[overtone scale|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).
Mode 8 of the harmonic series&mdash;[[overtone scale|harmonics 8 through 16]], octave repeating&mdash;is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).


{| class="wikitable"
{| class="wikitable"
|-
|-
| Harmonics in "Mode 8":
! Harmonics in "Mode 8":
| 8
| 8
|  
|  
Line 1,450: Line 1,535:
| 16
| 16
|-
|-
| …as JI Ratio from 1/1:
! …as JI Ratio from 1/1:
| 1/1
| 1/1
|  
|  
Line 1,469: Line 1,554:
| 2/1
| 2/1
|-
|-
| …in cents:
! …in cents:
| 0
| 0
|  
|  
Line 1,488: Line 1,573:
| 1200.0
| 1200.0
|-
|-
| Nearest degree of 72edo:
! Nearest degree of 72edo:
| 0
| 0
|  
|  
Line 1,507: Line 1,592:
| 72
| 72
|-
|-
| …in cents:
! …in cents:
| 0
| 0
|  
|  
Line 1,526: Line 1,611:
| 1200.0
| 1200.0
|-
|-
| Steps as Freq. Ratio:
! Steps as Freq. Ratio:
|  
|  
| 9:8
| 9:8
Line 1,545: Line 1,630:
|  
|  
|-
|-
| …in cents:
! …in cents:
|  
|  
| 203.9
| 203.9
Line 1,564: Line 1,649:
|  
|  
|-
|-
| Nearest degree of 72edo:
! Nearest degree of 72edo:
|  
|  
| 12
| 12
Line 1,583: Line 1,668:
|  
|  
|-
|-
| …in cents:
! …in cents:
|  
|  
| 200.0
| 200.0
Line 1,602: Line 1,687:
|  
|  
|}
|}
== 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:
* [[Skip fretting system 72 2 27]]
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 1,627: Line 1,724:
; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* [https://www.archive.org/details/Kotekant ''Kotekant''] [https://www.archive.org/download/Kotekant/kotekant.mp3 play] (2010)
* [https://www.archive.org/details/Kotekant ''Kotekant''] [https://www.archive.org/download/Kotekant/kotekant.mp3 play] (2010)
;[[Ivan Wyschnegradsky]]
* [https://www.youtube.com/watch?v=RCcJHCkYQ6U Arc-en-ciel, for 6 pianos in twelfth tones, Op. 37] (1956)


; [[James Tenney]]
; [[James Tenney]]

Latest revision as of 01:07, 20 August 2025

← 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 consistent in the 17-odd-limit and is the ninth zeta integral edo. It is the second edo (after 58) to be distinctly consistent in the 11-odd-limit, the first edo to be consistent to distance 2 in the 11-odd-limit, and the first edo to be consistent in the 12- and 13-odd-prime-sum-limit.

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 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.

Prime harmonics

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

Octave stretch

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 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

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.

Intervals

# Cents Approximate ratios[note 1] Ups and downs notation SKULO interval names and notation (K, S, U)
0 0.0 1/1 P1 perfect unison D P1 perfect unison D D
1 16.7 81/80, 91/90, 99/98, 100/99, 105/104 ^1 up unison ^D K1, L1 comma-wide unison, large unison KD, LD KD
2 33.3 45/44, 49/48, 50/49, 55/54, 64/63 ^^ dup unison ^^D S1, O1 super unison, on unison SD, OD SD
3 50.0 33/32, 36/35, 40/39 ^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 25/24, 26/25, 27/26, 28/27 vvm2 dudminor 2nd vvEb kkA1, sm2 classic aug unison, subminor 2nd kkD#, sEb sD#, (kkD#), sEb
5 83.3 20/19, 21/20, 22/21 vm2 downminor 2nd vEb kA1, lm2 comma-narrow aug unison, little minor 2nd kD#, lEb kD#, kEb
6 100.0 17/16, 18/17, 19/18 m2 minor 2nd Eb m2 minor 2nd Eb Eb
7 116.7 15/14, 16/15 ^m2 upminor 2nd ^Eb Km2 classic minor 2nd KEb KEb
8 133.3 13/12, 14/13, 27/25 ^^m2, v~2 dupminor 2nd, downmid 2nd ^^Eb Om2 on minor 2nd OEb SEb
9 150.0 12/11 ~2 mid 2nd v3E N2 neutral 2nd UEb/uE UEb/uE
10 166.7 11/10 ^~2, vvM2 upmid 2nd, dudmajor 2nd vvE oM2 off major 2nd oE sE
11 183.3 10/9 vM2 downmajor 2nd vE kM2 classic/comma-narrow major 2nd kE kE
12 200.0 9/8 M2 major 2nd E M2 major 2nd E E
13 216.7 17/15, 25/22 ^M2 upmajor 2nd ^E LM2 large major 2nd LE KE
14 233.3 8/7 ^^M2 dupmajor 2nd ^^E SM2 supermajor 2nd SE SE
15 250.0 15/13, 22/19 ^3M2,
v3m3 		trupmajor 2nd,
trudminor 3rd 		^3E,
v3F 		HM2, hm3 hypermajor 2nd, hypominor 3rd HE, hF UE, uF
16 266.7 7/6 vvm3 dudminor 3rd vvF sm3 subminor 3rd sF sF
17 283.3 13/11, 20/17 vm3 downminor 3rd vF lm3 little minor 3rd lF kF
18 300.0 19/16, 25/21, 32/27 m3 minor 3rd F m3 minor 3rd F F
19 316.7 6/5 ^m3 upminor 3rd ^F Km3 classic minor 3rd KF KF
20 333.3 17/14, 39/32, 40/33 ^^m3, v~3 dupminor 3rd, downmid 3rd ^^F Om3 on minor third OF SF
21 350.0 11/9, 27/22 ~3 mid 3rd ^3F N3 neutral 3rd UF/uF# UF/uF#
22 366.7 16/13, 21/17, 26/21 ^~3, vvM3 upmid 3rd, dudmajor 3rd vvF# oM3 off major 3rd oF# sF#
23 383.3 5/4 vM3 downmajor 3rd vF# kM3 classic major 3rd kF# kF#
24 400.0 24/19 M3 major 3rd F# M3 major 3rd F# F#
25 416.7 14/11 ^M3 upmajor 3rd ^F# LM3 large major 3rd LF# KF#
26 433.3 9/7 ^^M3 dupmajor 3rd ^^F# SM3 supermajor 3rd SF# SF#
27 450.0 13/10, 22/17 ^3M3, v34 trupmajor 3rd, trud 4th ^3F#, v3G HM3, h4 hypermajor 3rd, hypo 4th HF#, hG UF#, uG
28 466.7 17/13, 21/16 vv4 dud 4th vvG s4 sub 4th sG sG
29 483.3 33/25 v4 down 4th vG l4 little 4th lG kG
30 500.0 4/3 P4 perfect 4th G P4 perfect 4th G G
31 516.7 27/20 ^4 up 4th ^G K4 comma-wide 4th KG KG
32 533.3 15/11, 19/14, 26/19 ^^4, v~4 dup 4th, downmid 4th ^^G O4 on 4th OG SG
33 550.0 11/8 ~4 mid 4th ^3G U4/N4 uber 4th / neutral 4th UG UG
34 566.7 18/13, 25/18 ^~4, vvA4 upmid 4th, dudaug 4th vvG# kkA4, sd5 classic aug 4th, sub dim 5th kkG#, sAb SG#, (kkG#), sAb
35 583.3 7/5 vA4, vd5 downaug 4th,
downdim 5th 		vG#, vAb kA4, ld5 comma-narrow aug 4th, little dim 5th kG#, lAb kG#, kAb
36 600.0 17/12, 24/17 A4, d5 aug 4th, dim 5th G#, Ab A4, d5 aug 4th, dim 5th G#, Ab G#, Ab
37 616.7 10/7 ^A4, ^d5 upaug 4th, updim 5th ^G#, ^Ab LA4, Kd5 large aug 4th, comma-wide dim 5th LG#, KAb KG#, KAb
38 633.3 13/9, 36/25 v~5, ^^d5 downmid 5th,
dupdim 5th 		^^Ab SA4, KKd5 super aug 4th, classic dim 5th SG#, KKAb SG#, SAb, (KKAb)
39 650.0 16/11 ~5 mid 5th v3A u5/N5 unter 5th / neutral 5th uA uA
40 666.7 19/13, 22/15, 28/19 vv5, ^~5 dud 5th, upmid 5th vvA o5 off 5th oA sA
41 683.3 40/27 v5 down 5th vA k5 comma-narrow 5th kA kA
42 700.0 3/2 P5 perfect 5th A P5 perfect 5th A A
43 716.7 50/33 ^5 up 5th ^A L5 large fifth LA KA
44 733.3 26/17, 32/21 ^^5 dup 5th ^^A S5 super fifth SA SA
45 750.0 17/11, 20/13 ^35, v3m6 trup 5th, trudminor 6th ^3A, v3Bb H5, hm6 hyper fifth, hypominor 6th HA, hBb UA, uBb
46 766.7 14/9 vvm6 dudminor 6th vvBb sm6 superminor 6th sBb sBb
47 783.3 11/7 vm6 downminor 6th vBb lm6 little minor 6th lBb kBb
48 800.0 19/12 m6 minor 6th Bb m6 minor 6th Bb Bb
49 816.7 8/5 ^m6 upminor 6th ^Bb Km6 classic minor 6th kBb kBb
50 833.3 13/8, 21/13, 34/21 ^^m6, v~6 dupminor 6th, downmid 6th ^^Bb Om6 on minor 6th oBb sBb
51 850.0 18/11, 44/27 ~6 mid 6th v3B N6 neutral 6th UBb, uB UBb, uB
52 866.7 28/17, 33/20, 64/39 ^~6, vvM6 upmid 6th, dudmajor 6th vvB oM6 off major 6th oB sB
53 883.3 5/3 vM6 downmajor 6th vB kM6 classic major 6th kB kB
54 900.0 27/16, 32/19, 42/25 M6 major 6th B M6 major 6th B B
55 916.7 17/10, 22/13 ^M6 upmajor 6th ^B LM6 large major 6th LB KB
56 933.3 12/7 ^^M6 dupmajor 6th ^^B SM6 supermajor 6th SB SB
57 950.0 19/11, 26/15 ^3M6,
v3m7 		trupmajor 6th,
trudminor 7th 		^3B,
v3C 		HM6, hm7 hypermajor 6th, hypominor 7th HB, hC UB, uC
58 966.7 7/4 vvm7 dudminor 7th vvC sm7 subminor 7th sC sC
59 983.3 30/17, 44/25 vm7 downminor 7th vC lm7 little minor 7th lC kC
60 1000.0 16/9 m7 minor 7th C m7 minor 7th C C
61 1016.7 9/5 ^m7 upminor 7th ^C Km7 classic/comma-wide minor 7th KC KC
62 1033.3 20/11 ^^m7, v~7 dupminor 7th, downmid 7th ^^C Om7 on minor 7th OC SC
63 1050.0 11/6 ~7 mid 7th ^3C N7, hd8 neutral 7th, hypo dim 8ve UC/uC#, hDb UC/uC#, uDb
64 1066.7 13/7, 24/13, 50/27 ^~7, vvM7 upmid 7th, dudmajor 7th vvC# oM7, sd8 off major 7th, sub dim 8ve oC#, sDb sC#, sDb
65 1083.3 15/8, 28/15 vM7 downmajor 7th vC# kM7, ld8 classic major 7th, little dim 8ve kC#, lDb kC#, kDb
66 1100.0 17/9, 32/17, 36/19 M7 major 7th C# M7, d8 major 7th, dim 8ve C#, Db C#, Db
67 1116.7 19/10, 21/11, 40/21 ^M7 upmajor 7th ^C# LM7, Kd8 large major 7th, comma-wide dim 8ve LC#, KDb KC#, KDb
68 1133.3 25/13, 27/14, 48/25, 52/27 ^^M7 dupmajor 7th ^^C# SM7, KKd8 supermajor 7th, classic dim 8ve SC#, KKDb SC#, SDb, (KKDb)
69 1150.0 35/18, 39/20, 64/33 ^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 49/25, 55/28, 63/32, 88/45, 96/49 vv8 dud octave vvD s8, o8 sub 8ve, off 8ve sD, oD sD
71 1183.3 99/50, 160/81, 180/91, 196/99, 208/105 v8 down octave vD k8, l8 comma-narrow 8ve, little 8ve kD, lD kD
72 1200.0 2/1 P8 perfect octave D P8 perfect octave D D
  1. ↑ Based on treating 72edo as a 19-limit temperament; other approaches are also possible. For lower limits see Table of 72edo intervals.

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:

  • −1 degree (the down ring) corrects 81/64 to 5/4 via 80/81
  • −2 degrees (the dud ring) corrects 16/9 to 7/4 via 63/64
  • +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
  • 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

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.

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.

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

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

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

Alternative ups and downs have sharps and flats with arrows borrowed from extended Helmholtz–Ellis notation:

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  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

Sagittal notation

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

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

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 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
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

Scales

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
Claudi Meneghin
Prent Rodgers
Gene Ward Smith
Ivan Wyschnegradsky
James Tenney
Xeno Ov Eleas

External links

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