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{{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<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|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 50: Line 56:
|-
|-
| 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 61: Line 67:
|-
|-
| 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 72: Line 78:
|-
|-
| 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 83: Line 89:
|-
|-
| 4
| 4
| 66.667
| 66.7
| 25/24
| 25/24, 26/25, 27/26, 28/27
| vvm2
| vvm2
| dudminor 2nd
| dudminor 2nd
Line 94: Line 100:
|-
|-
| 5
| 5
| 83.333
| 83.3
| 21/20
| 20/19, 21/20, 22/21
| vm2
| vm2
| downminor 2nd
| downminor 2nd
Line 105: Line 111:
|-
|-
| 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 116: Line 122:
|-
|-
| 7
| 7
| 116.667
| 116.7
| 15/14, 16/15
| 15/14, 16/15
| ^m2
| ^m2
Line 127: Line 133:
|-
|-
| 8
| 8
| 133.333
| 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 138: Line 144:
|-
|-
| 9
| 9
| 150.000
| 150.0
| 12/11
| 12/11
| ~2
| ~2
Line 149: Line 155:
|-
|-
| 10
| 10
| 166.667
| 166.7
| 11/10
| 11/10
| ^~2, vvM2
| ^~2, vvM2
Line 160: Line 166:
|-
|-
| 11
| 11
| 183.333
| 183.3
| 10/9
| 10/9
| vM2
| vM2
Line 171: Line 177:
|-
|-
| 12
| 12
| 200.000
| 200.0
| 9/8
| 9/8
| M2
| M2
Line 182: Line 188:
|-
|-
| 13
| 13
| 216.667
| 216.7
| 25/22, 17/15
| 17/15, 25/22
| ^M2
| ^M2
| upmajor 2nd
| upmajor 2nd
Line 193: Line 199:
|-
|-
| 14
| 14
| 233.333
| 233.3
| 8/7
| 8/7
| ^^M2
| ^^M2
Line 204: Line 210:
|-
|-
| 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
| ^<sup>3</sup>E, <br />v<sup>3</sup>F
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
| HM2, hm3
| HM2, hm3
| hypermajor 2nd, hypominor 3rd
| hypermajor 2nd, hypominor 3rd
Line 215: Line 221:
|-
|-
| 16
| 16
| 266.667
| 266.7
| 7/6
| 7/6
| vvm3
| vvm3
Line 226: Line 232:
|-
|-
| 17
| 17
| 283.333
| 283.3
| 33/28, 13/11, 20/17
| 13/11, 20/17
| vm3
| vm3
| downminor 3rd
| downminor 3rd
Line 237: Line 243:
|-
|-
| 18
| 18
| 300.000
| 300.0
| 25/21
| 19/16, 25/21, 32/27
| m3
| m3
| minor 3rd
| minor 3rd
Line 248: Line 254:
|-
|-
| 19
| 19
| 316.667
| 316.7
| 6/5
| 6/5
| ^m3
| ^m3
Line 259: Line 265:
|-
|-
| 20
| 20
| 333.333
| 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 270: Line 276:
|-
|-
| 21
| 21
| 350.000
| 350.0
| 11/9
| 11/9, 27/22
| ~3
| ~3
| mid 3rd
| mid 3rd
Line 281: Line 287:
|-
|-
| 22
| 22
| 366.667
| 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 292: Line 298:
|-
|-
| 23
| 23
| 383.333
| 383.3
| 5/4
| 5/4
| vM3
| vM3
Line 303: Line 309:
|-
|-
| 24
| 24
| 400.000
| 400.0
| 44/35
| 24/19
| M3
| M3
| major 3rd
| major 3rd
Line 314: Line 320:
|-
|-
| 25
| 25
| 416.667
| 416.7
| 14/11
| 14/11
| ^M3
| ^M3
Line 325: Line 331:
|-
|-
| 26
| 26
| 433.333
| 433.3
| 9/7
| 9/7
| ^^M3
| ^^M3
Line 336: Line 342:
|-
|-
| 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 347: Line 353:
|-
|-
| 28
| 28
| 466.667
| 466.7
| 21/16, 17/13
| 17/13, 21/16
| vv4
| vv4
| dud 4th
| dud 4th
Line 358: Line 364:
|-
|-
| 29
| 29
| 483.333
| 483.3
| 33/25
| 33/25
| v4
| v4
Line 369: Line 375:
|-
|-
| 30
| 30
| 500.000
| 500.0
| 4/3
| 4/3
| P4
| P4
Line 380: Line 386:
|-
|-
| 31
| 31
| 516.667
| 516.7
| 27/20
| 27/20
| ^4
| ^4
Line 391: Line 397:
|-
|-
| 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 402: Line 408:
|-
|-
| 33
| 33
| 550.000
| 550.0
| 11/8
| 11/8
| ~4
| ~4
Line 413: Line 419:
|-
|-
| 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 424: Line 430:
|-
|-
| 35
| 35
| 583.333
| 583.3
| 7/5
| 7/5
| vA4, vd5
| vA4, vd5
| downaug 4th,  
| downaug 4th, <br>downdim 5th
downdim 5th
| vG#, vAb
| vG#, vAb
| kA4, ld5
| kA4, ld5
Line 436: Line 441:
|-
|-
| 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 447: Line 452:
|-
|-
| 37
| 37
| 616.667
| 616.7
| 10/7
| 10/7
| ^A4, ^d5
| ^A4, ^d5
Line 458: Line 463:
|-
|-
| 38
| 38
| 633.333
| 633.3
| 36/25, 13/9
| 13/9, 36/25
| v~5, ^^d5
| v~5, ^^d5
| downmid 5th,  
| downmid 5th, <br>dupdim 5th
dupdim 5th
| ^^Ab
| ^^Ab
| SA4, KKd5
| SA4, KKd5
Line 470: Line 474:
|-
|-
| 39
| 39
| 650.000
| 650.0
| 16/11
| 16/11
| ~5
| ~5
Line 481: Line 485:
|-
|-
| 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 492: Line 496:
|-
|-
| 41
| 41
| 683.333
| 683.3
| 40/27
| 40/27
| v5
| v5
Line 503: Line 507:
|-
|-
| 42
| 42
| 700.000
| 700.0
| 3/2
| 3/2
| P5
| P5
Line 514: Line 518:
|-
|-
| 43
| 43
| 716.667
| 716.7
| 50/33
| 50/33
| ^5
| ^5
Line 525: Line 529:
|-
|-
| 44
| 44
| 733.333
| 733.3
| 32/21
| 26/17, 32/21
| ^^5
| ^^5
| dup 5th
| dup 5th
Line 536: Line 540:
|-
|-
| 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 547: Line 551:
|-
|-
| 46
| 46
| 766.667
| 766.7
| 14/9
| 14/9
| vvm6
| vvm6
Line 558: Line 562:
|-
|-
| 47
| 47
| 783.333
| 783.3
| 11/7
| 11/7
| vm6
| vm6
Line 569: Line 573:
|-
|-
| 48
| 48
| 800.000
| 800.0
| 35/22
| 19/12
| m6
| m6
| minor 6th
| minor 6th
Line 580: Line 584:
|-
|-
| 49
| 49
| 816.667
| 816.7
| 8/5
| 8/5
| ^m6
| ^m6
Line 591: Line 595:
|-
|-
| 50
| 50
| 833.333
| 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 602: Line 606:
|-
|-
| 51
| 51
| 850.000
| 850.0
| 18/11
| 18/11, 44/27
| ~6
| ~6
| mid 6th
| mid 6th
Line 613: Line 617:
|-
|-
| 52
| 52
| 866.667
| 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 624: Line 628:
|-
|-
| 53
| 53
| 883.333
| 883.3
| 5/3
| 5/3
| vM6
| vM6
Line 635: Line 639:
|-
|-
| 54
| 54
| 900.000
| 900.0
| 27/16
| 27/16, 32/19, 42/25
| M6
| M6
| major 6th
| major 6th
Line 646: Line 650:
|-
|-
| 55
| 55
| 916.667
| 916.7
| 56/33, 17/10
| 17/10, 22/13
| ^M6
| ^M6
| upmajor 6th
| upmajor 6th
Line 657: Line 661:
|-
|-
| 56
| 56
| 933.333
| 933.3
| 12/7
| 12/7
| ^^M6
| ^^M6
Line 668: Line 672:
|-
|-
| 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
| ^<sup>3</sup>B, <br />v<sup>3</sup>C
| ^<sup>3</sup>B, <br>v<sup>3</sup>C
| HM6, hm7
| HM6, hm7
| hypermajor 6th, hypominor 7th
| hypermajor 6th, hypominor 7th
Line 679: Line 683:
|-
|-
| 58
| 58
| 966.667
| 966.7
| 7/4
| 7/4
| vvm7
| vvm7
Line 690: Line 694:
|-
|-
| 59
| 59
| 983.333
| 983.3
| 44/25
| 30/17, 44/25
| vm7
| vm7
| downminor 7th
| downminor 7th
Line 701: Line 705:
|-
|-
| 60
| 60
| 1000.000
| 1000.0
| 16/9
| 16/9
| m7
| m7
Line 712: Line 716:
|-
|-
| 61
| 61
| 1016.667
| 1016.7
| 9/5
| 9/5
| ^m7
| ^m7
Line 723: Line 727:
|-
|-
| 62
| 62
| 1033.333
| 1033.3
| 20/11
| 20/11
| ^^m7, v~7
| ^^m7, v~7
Line 734: Line 738:
|-
|-
| 63
| 63
| 1050.000
| 1050.0
| 11/6
| 11/6
| ~7
| ~7
Line 745: Line 749:
|-
|-
| 64
| 64
| 1066.667
| 1066.7
| 50/27
| 13/7, 24/13, 50/27
| ^~7, vvM7
| ^~7, vvM7
| upmid 7th, dudmajor 7th
| upmid 7th, dudmajor 7th
Line 756: Line 760:
|-
|-
| 65
| 65
| 1083.333
| 1083.3
| 15/8
| 15/8, 28/15
| vM7
| vM7
| downmajor 7th
| downmajor 7th
Line 767: Line 771:
|-
|-
| 66
| 66
| 1100.000
| 1100.0
| 66/35, 17/9
| 17/9, 32/17, 36/19
| M7
| M7
| major 7th
| major 7th
Line 778: Line 782:
|-
|-
| 67
| 67
| 1116.667
| 1116.7
| 21/11
| 19/10, 21/11, 40/21
| ^M7
| ^M7
| upmajor 7th
| upmajor 7th
Line 789: Line 793:
|-
|-
| 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 800: Line 804:
|-
|-
| 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 811: Line 815:
|-
|-
| 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 822: Line 826:
|-
|-
| 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 833: Line 837:
|-
|-
| 72
| 72
| 1200.000
| 1200.0
| 2/1
| 2/1
| P8
| P8
Line 843: Line 847:
| D
| D
|}
|}
<references group="note" />


=== 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 861: Line 866:
| 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" |dupminor,
| rowspan="2" | dupminor, <br>downmid
downmid
| luyo
| luyo
| (a b 1 0 -1)
| (a b 1 0 -1)
Line 888: Line 892:
| 12/11, 18/11
| 12/11, 18/11
|-
|-
| rowspan="2" |upmid,
| rowspan="2" | upmid, <br>dudmajor
dudmajor
| logu
| logu
| (a b -1 0 1)
| (a b -1 0 1)
Line 905: 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
|-
|-
Line 913: Line 916:
| 9/7, 12/7
| 9/7, 12/7
|-
|-
| rowspan="2" |trupmajor,
| rowspan="2" | trupmajor, <br>trudminor
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 927: 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 969: 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 ===
In 72tet, there are 6 [[Ring number|rings]]. 12edo is the plain ring; thus 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:
* &minus;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 80/81
* &minus;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 63/64
* +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
Line 984: Line 986:


== 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 ===
Using [[Helmholtz-Ellis notation|Helmholtz&ndash;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}}
{{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]]
=== 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.
[[File:plot72.png|alt=plot72.png|plot72.png]]


=== Interval mappings ===
=== Interval mappings ===
{{Q-odd-limit intervals|72}}
{{Q-odd-limit intervals|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]]


== 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 1,022: 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 1,029: 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 1,036: 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 1,043: 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 1,050: 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
| 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.  


=== Commas ===
=== Commas ===
Line 1,061: 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]]
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
Line 1,069: 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 1,083: Line 1,139:
| {{Monzo| -25 7 6 }}
| {{Monzo| -25 7 6 }}
| 31.57
| 31.57
| [[Ampersand]]
| [[Ampersand comma]]
|-
|-
| 5
| 5
Line 1,125: Line 1,181:
| {{Monzo| 0 3 4 -5 }}
| {{Monzo| 0 3 4 -5 }}
| 6.99
| 6.99
| Mirkwai
| Mirkwai comma
|-
|-
| 7
| 7
Line 1,131: Line 1,187:
| {{Monzo| -4 9 -2 -2 }}
| {{Monzo| -4 9 -2 -2 }}
| 7.32
| 7.32
| Cataharry
| Cataharry comma
|-
|-
| 7
| 7
Line 1,173: 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 1,185: 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 1,227: 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 1,277: Line 1,333:
| Jacobin comma
| Jacobin comma
|}
|}
<references group="note" />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 1,283: 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-all right-3 left-5"
{| 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*
! Cents*
! Cents*
! Associated<br />Ratio*
! Associated<br>ratio*
! Temperaments
! Temperament
|-
|-
| 1
| 1
Line 1,364: Line 1,422:
|-
|-
| 2
| 2
| 21\72<br />(19\72)
| 21\72<br>(19\72)
| 316.7<br />(283.3)
| 316.7<br>(283.3)
| 6/5<br />(13/11)
| 6/5<br>(13/11)
| [[Bikleismic]]
| [[Bikleismic]]
|-
|-
| 2
| 2
| 23\72<br />(13\72)
| 23\72<br>(13\72)
| 383.3<br />(216.7)
| 383.3<br>(216.7)
| 5/4<br />(17/15)
| 5/4<br>(17/15)
| [[Wizard]] / lizard / gizzard
| [[Wizard]] / lizard / gizzard
|-
|-
Line 1,382: Line 1,440:
|-
|-
| 3
| 3
| 19\72<br />(5\72)
| 19\72<br>(5\72)
| 316.7<br />(83.3)
| 316.7<br>(83.3)
| 6/5<br />(21/20)
| 6/5<br>(21/20)
| [[Tritikleismic]]
| [[Tritikleismic]]
|-
|-
| 4
| 4
| 19\72<br />(1\72)
| 19\72<br>(1\72)
| 316.7<br />(16.7)
| 316.7<br>(16.7)
| 6/5<br />(105/104)
| 6/5<br>(105/104)
| [[Quadritikleismic]]
| [[Quadritikleismic]]
|-
|-
| 8
| 8
| 34\72<br />(2\72)
| 34\72<br>(2\72)
| 566.7<br />(33.3)
| 566.7<br>(33.3)
| 168/121<br />(55/54)
| 168/121<br>(55/54)
| [[Octowerck]] / octowerckis
| [[Octowerck]] / octowerckis
|-
|-
| 8
| 8
| 35\72<br />(1\72)
| 35\72<br>(1\72)
| 583.3<br />(16.7)
| 583.3<br>(16.7)
| 7/5<br />(100/99)
| 7/5<br>(100/99)
| [[Octoid]] / octopus
| [[Octoid]] / octopus
|-
|-
| 9
| 9
| 19\72<br />(3\72)
| 19\72<br>(3\72)
| 316.7<br />(50.0)
| 316.7<br>(50.0)
| 6/5<br />(36/35)
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimnic
| [[Ennealimmal]] / ennealimnic
|-
|-
| 9
| 9
| 23\72<br />(1\72)
| 23\72<br>(1\72)
| 383.3<br />(16.7)
| 383.3<br>(16.7)
| 5/4<br />(105/104)
| 5/4<br>(105/104)
| [[Enneaportent]]
| [[Enneaportent]]
|-
|-
| 12
| 12
| 23\72<br />(1\72)
| 23\72<br>(1\72)
| 383.3<br />(16.7)
| 383.3<br>(16.7)
| 5/4<br />(100/99)
| 5/4<br>(100/99)
| [[Compton]] / comptone
| [[Compton]] / comptone
|-
|-
| 18
| 18
| 19\72<br />(1\72)
| 19\72<br>(1\72)
| 316.7<br />(16.7)
| 316.7<br>(16.7)
| 6/5<br />(105/104)
| 6/5<br>(105/104)
| [[Hemiennealimmal]]
| [[Hemiennealimmal]]
|-
|-
| 24
| 24
| 23\72<br />(1\72)
| 23\72<br>(1\72)
| 383.3<br />(16.7)
| 383.3<br>(16.7)
| 5/4<br />(105/104)
| 5/4<br>(105/104)
| [[Hours]]
| [[Hours]]
|-
|-
| 36
| 36
| 23\72<br />(1\72)
| 23\72<br>(1\72)
| 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 it is 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 1,454: 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 1,477: Line 1,535:
| 16
| 16
|-
|-
| …as JI Ratio from 1/1:
! …as JI Ratio from 1/1:
| 1/1
| 1/1
|  
|  
Line 1,496: Line 1,554:
| 2/1
| 2/1
|-
|-
| …in cents:
! …in cents:
| 0
| 0
|  
|  
Line 1,515: Line 1,573:
| 1200.0
| 1200.0
|-
|-
| Nearest degree of 72edo:
! Nearest degree of 72edo:
| 0
| 0
|  
|  
Line 1,534: Line 1,592:
| 72
| 72
|-
|-
| …in cents:
! …in cents:
| 0
| 0
|  
|  
Line 1,553: Line 1,611:
| 1200.0
| 1200.0
|-
|-
| Steps as Freq. Ratio:
! Steps as Freq. Ratio:
|  
|  
| 9:8
| 9:8
Line 1,572: Line 1,630:
|  
|  
|-
|-
| …in cents:
! …in cents:
|  
|  
| 203.9
| 203.9
Line 1,591: Line 1,649:
|  
|  
|-
|-
| Nearest degree of 72edo:
! Nearest degree of 72edo:
|  
|  
| 12
| 12
Line 1,610: Line 1,668:
|  
|  
|-
|-
| …in cents:
! …in cents:
|  
|  
| 200.0
| 200.0
Line 1,629: 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 1,654: 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]]
Line 1,667: 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]]
Retrieved from "https://en.xen.wiki/w/72edo"