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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]]), [[Georg Friedrich Haas]], [[Ezra Sims]], [[Rick Tagawa]], [[James Tenney]], 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 [[zeta integral edo]]. The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.
72edo approximates [[11-limit]] [[just intonation]] exceptionally well. It is the second edo (after [[58edo|58]]) to be [[consistent]] in the [[17-odd-limit]], and the second edo (also after 58) to be [[consistency|distinctly consistent]] in the [[11-odd-limit]], but it is the first edo to be [[consistency|consistent to distance 2]] in the 11-odd-limit, meaning every interval in the 11-odd-limit is approximated with less than 25% [[relative interval error|relative error]] (about 4 cents). It also has pretty good accuracy for the [[19-limit]], being almost consistent to the entire [[21-odd-limit]] with the only inconsistency occurring at [[19/13]] and its [[octave complement]]. It is the ninth [[zeta integral edo]].


72edo is the only regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s, ''and'' tunes the octave just. It is also an excellent tuning for [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].
The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.


The 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 [[octave reduction|octave reduced]] [[13/1|13th harmonic]] is mapped on 50\72, an interval inherited from [[36edo]] (25\36) that is a very close approximation to [[acoustic phi]], and the [[17/1|17th]] and [[19/1|19th harmonics]] come from 12edo.  


72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]].
72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|72|columns=9}}
{{Harmonics in equal|72|columns=11}}
{{Harmonics in equal|72|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}
{{Harmonics in equal|72|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}
 
=== As a tuning of other temperaments ===
72et is the only 11-limit regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, and splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s (144edo is enfactored in the 11-limit with 72edo, so it is already covered here). It is also an excellent tuning for [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].


=== Subsets and supersets ===
=== 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, though unlike 72 it is not consistent to the [[13-odd-limit]].


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-1 right-2"
|-
|-
! 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">As a 19-limit temperament, inconsistent intervals in ''italic''. For a table of intervals by prime limit, see [[Table of 72edo intervals]].</ref>
! colspan="3" | [[Ups and downs notation]]
! [[Kite's ups and downs notation|Ups and downs notation]]
! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]
! (K, S, U)
|-
|-
| 0
| 0
| 0.0
| 0.0
| 1/1
| [[1/1]]
| P1
| {{UDnote|step=0}}
| perfect unison
| D
| P1
| perfect unison
| D
| D
|-
|-
| 1
| 1
| 16.7
| 16.7
| 81/80
| [[81/80]], [[91/90]], [[99/98]], [[100/99]], [[105/104]]
| ^1
| {{UDnote|step=1}}
| up unison
| ^D
| K1, L1
| comma-wide unison, large unison
| KD, LD
| KD
|-
|-
| 2
| 2
| 33.3
| 33.3
| 45/44, 64/63
| [[45/44]], [[49/48]], [[50/49]], [[55/54]], [[64/63]]
| ^^
| {{UDnote|step=2}}
| dup unison
| ^^D
| S1, O1
| super unison, on unison
| SD, OD
| SD
|-
|-
| 3
| 3
| 50.0
| 50.0
| 33/32
| [[33/32]], [[36/35]], [[40/39]]
| ^<sup>3</sup>1, v<sup>3</sup>m2
| {{UDnote|step=3}}
| trup unison, trudminor 2nd
|-
| ^<sup>3</sup>D, v<sup>3</sup>Eb
| 4
| U1, H1, hm2
| uber unison, hyper unison, hypominor 2nd
| UD, HD, uEb
| UD, uEb
|-
| 4
| 66.7
| 66.7
| 25/24
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| vvm2
| {{UDnote|step=4}}
| dudminor 2nd
| vvEb
| kkA1, sm2
| classic aug unison, subminor 2nd
| kkD#, sEb
| sD#, (kkD#), sEb
|-
|-
| 5
| 5
| 83.3
| 83.3
| 21/20
| [[20/19]], [[21/20]], [[22/21]]
| vm2
| {{UDnote|step=5}}
| downminor 2nd
| vEb
| kA1, lm2
| comma-narrow aug unison, little minor 2nd
| kD#, lEb
| kD#, kEb
|-
|-
| 6
| 6
| 100.0
| 100.0
| 35/33, 17/16, 18/17
| [[17/16]], [[18/17]], [[19/18]]
| m2
| {{UDnote|step=6}}
| minor 2nd
|-
| Eb
| 7
| m2
| 116.7
| minor 2nd
| [[15/14]], [[16/15]]
| Eb
| {{UDnote|step=7}}
| Eb
|-
| 7
| 116.7
| 15/14, 16/15
| ^m2
| upminor 2nd
| ^Eb
| Km2
| classic minor 2nd
| KEb
| KEb
|-
|-
| 8
| 8
| 133.3
| 133.3
| 27/25, 13/12, 14/13
| [[13/12]], [[14/13]], [[27/25]]
| ^^m2, v~2
| {{UDnote|step=8}}
| dupminor 2nd, downmid 2nd
| ^^Eb
| Om2
| on minor 2nd
| OEb
| SEb
|-
|-
| 9
| 9
| 150.0
| 150.0
| 12/11
| [[12/11]]
| ~2
| {{UDnote|step=9}}
| mid 2nd
|-
| v<sup>3</sup>E
| 10
| N2
| 166.7
| neutral 2nd
| [[11/10]], [[21/19]]
| UEb/uE
| {{UDnote|step=10}}
| UEb/uE
|-
| 10
| 166.7
| 11/10
| ^~2, vvM2
| upmid 2nd, dudmajor 2nd
| vvE
| oM2
| off major 2nd
| oE
| sE
|-
|-
| 11
| 11
| 183.3
| 183.3
| 10/9
| [[10/9]]
| vM2
| {{UDnote|step=11}}
| downmajor 2nd
| vE
| kM2
| classic/comma-narrow major 2nd
| kE
| kE
|-
|-
| 12
| 12
| 200.0
| 200.0
| 9/8
| [[9/8]], [[19/17]]
| M2
| {{UDnote|step=12}}
| major 2nd
|-
| E
| 13
| M2
| 216.7
| major 2nd
| [[17/15]], [[25/22]]
| E
| {{UDnote|step=13}}
| E
|-
| 13
| 216.7
| 25/22, 17/15
| ^M2
| upmajor 2nd
| ^E
| LM2
| large major 2nd
| LE
| KE
|-
|-
| 14
| 14
| 233.3
| 233.3
| 8/7
| [[8/7]]
| ^^M2
| {{UDnote|step=14}}
| dupmajor 2nd
| ^^E
| SM2
| supermajor 2nd
| SE
| SE
|-
|-
| 15
| 15
| 250.0
| 250.0
| 81/70,  15/13
| [[15/13]], [[22/19]]
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| {{UDnote|step=15}}
| trupmajor 2nd,<br>trudminor 3rd
|-
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
| 16
| HM2, hm3
| 266.7
| hypermajor 2nd, hypominor 3rd
| [[7/6]]
| HE, hF
| {{UDnote|step=16}}
| UE, uF
|-
| 16
| 266.7
| 7/6
| vvm3
| dudminor 3rd
| vvF
| sm3
| subminor 3rd
| sF
| sF
|-
|-
| 17
| 17
| 283.3
| 283.3
| 33/28, 13/11, 20/17
| [[13/11]], [[20/17]]
| vm3
| {{UDnote|step=17}}
| downminor 3rd
| vF
| lm3
| little minor 3rd
| lF
| kF
|-
|-
| 18
| 18
| 300.0
| 300.0
| 25/21
| [[19/16]], [[25/21]], [[32/27]]
| m3
| {{UDnote|step=18}}
| minor 3rd
| F
| m3
| minor 3rd
| F
| F
|-
|-
| 19
| 19
| 316.7
| 316.7
| 6/5
| [[6/5]]
| ^m3
| {{UDnote|step=19}}
| upminor 3rd
| ^F
| Km3
| classic minor 3rd
| KF
| KF
|-
|-
| 20
| 20
| 333.3
| 333.3
| 40/33, 17/14
| [[17/14]], ''[[39/32]]'', [[40/33]]
| ^^m3, v~3
| {{UDnote|step=20}}
| dupminor 3rd, downmid 3rd
| ^^F
| Om3
| on minor third
| OF
| SF
|-
|-
| 21
| 21
| 350.0
| 350.0
| 11/9
| [[11/9]], [[27/22]]
| ~3
| {{UDnote|step=21}}
| mid 3rd
| ^<sup>3</sup>F
| N3
| neutral 3rd
| UF/uF#
| UF/uF#
|-
|-
| 22
| 22
| 366.7
| 366.7
| 99/80, 16/13, 21/17
| [[16/13]], [[21/17]], [[26/21]]
| ^~3, vvM3
| {{UDnote|step=22}}
| upmid 3rd, dudmajor 3rd
|-
| vvF#
| 23
| oM3
| 383.3
| off major 3rd
| [[5/4]]
| oF#
| {{UDnote|step=23}}
| sF#
|-
| 23
| 383.3
| 5/4
| vM3
| downmajor 3rd
| vF#
| kM3
| classic major 3rd
| kF#
| kF#
|-
|-
| 24
| 24
| 400.0
| 400.0
| 44/35
| [[24/19]]
| M3
| {{UDnote|step=24}}
| major 3rd
| F#
| M3
| major 3rd
| F#
| F#
|-
|-
| 25
| 25
| 416.7
| 416.7
| 14/11
| [[14/11]], [[19/15]]
| ^M3
| {{UDnote|step=25}}
| upmajor 3rd
|-
| ^F#
| 26
| LM3
| 433.3
| large major 3rd
| [[9/7]]
| LF#
| {{UDnote|step=26}}
| KF#
|-
| 26
| 433.3
| 9/7
| ^^M3
| dupmajor 3rd
| ^^F#
| SM3
| supermajor 3rd
| SF#
| SF#
|-
|-
| 27
| 27
| 450.0
| 450.0
| 35/27, 13/10
| [[13/10]], [[22/17]]
| ^<sup>3</sup>M3, v<sup>3</sup>4
| {{UDnote|step=27}}
| trupmajor 3rd, trud 4th
| ^<sup>3</sup>F#, v<sup>3</sup>G
| HM3, h4
| hypermajor 3rd, hypo 4th
| HF#, hG
| UF#, uG
|-
|-
| 28
| 28
| 466.7
| 466.7
| 21/16, 17/13
| [[17/13]], [[21/16]]
| vv4
| {{UDnote|step=28}}
| dud 4th
|-
| vvG
| 29
| s4
| 483.3
| sub 4th
| [[33/25]]
| sG
| {{UDnote|step=29}}
| sG
|-
| 29
| 483.3
| 33/25
| v4
| down 4th
| vG
| l4
| little 4th
| lG
| kG
|-
|-
| 30
| 30
| 500.0
| 500.0
| 4/3
| [[4/3]]
| P4
| {{UDnote|step=30}}
| perfect 4th
| G
| P4
| perfect 4th
| G
| G
|-
|-
| 31
| 31
| 516.7
| 516.7
| 27/20
| [[27/20]]
| ^4
| {{UDnote|step=31}}
| up 4th
|-
| ^G
| 32
| K4
| 533.3
| comma-wide 4th
| [[15/11]], [[19/14]], ''[[26/19]]''
| KG
| {{UDnote|step=32}}
| KG
|-
|-
| 32
| 533.3
| 15/11
| ^^4, v~4
| dup 4th, downmid 4th
| ^^G
| O4
| on 4th
| OG
| SG
|-
| 33
| 33
| 550.0
| 550.0
| 11/8
| [[11/8]]
| ~4
| {{UDnote|step=33}}
| mid 4th
| ^<sup>3</sup>G
| U4/N4
| uber 4th / neutral 4th
| UG
| UG
|-
|-
| 34
| 34
| 566.7
| 566.7
| 25/18, 18/13
| [[18/13]], [[25/18]]
| ^~4, vvA4
| {{UDnote|step=34}}
| upmid 4th, dudaug 4th
|-
| vvG#
| 35
| kkA4, sd5
| classic aug 4th, sub dim 5th
| kkG#, sAb
| SG#, (kkG#), sAb
|-
| 35
| 583.3
| 583.3
| 7/5
| [[7/5]]
| vA4, vd5
| {{UDnote|step=35}}
| downaug 4th, <br>downdim 5th
| vG#, vAb
| kA4, ld5
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, kAb
|-
|-
| 36
| 36
| 600.0
| 600.0
| 99/70, 17/12
| [[17/12]], [[24/17]]
| A4, d5
| {{UDnote|step=36}}
| aug 4th, dim 5th
| G#, Ab
| A4, d5
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
|-
|-
| 37
| 37
| 616.7
| 616.7
| 10/7
| [[10/7]]
| ^A4, ^d5
| {{UDnote|step=37}}
| upaug 4th, updim 5th
|-
| ^G#, ^Ab
| 38
| LA4, Kd5
| 633.3
| large aug 4th, comma-wide dim 5th
| [[13/9]], [[36/25]]
| LG#, KAb
| {{UDnote|step=38}}
| KG#, KAb
|-
| 38
| 633.3
| 36/25, 13/9
| v~5, ^^d5
| downmid 5th, <br>dupdim 5th
| ^^Ab
| SA4, KKd5
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, SAb, (KKAb)
|-
|-
| 39
| 39
| 650.0
| 650.0
| 16/11
| [[16/11]]
| ~5
| {{UDnote|step=39}}
| mid 5th
| v<sup>3</sup>A
| u5/N5
| unter 5th / neutral 5th
| uA
| uA
|-
|-
| 40
| 40
| 666.7
| 666.7
| 22/15
| ''[[19/13]]'', [[22/15]], [[28/19]]
| vv5, ^~5
| {{UDnote|step=40}}
| dud 5th, upmid 5th
| vvA
| o5
| off 5th
| oA
| sA
|-
|-
| 41
| 41
| 683.3
| 683.3
| 40/27
| [[40/27]]
| v5
| {{UDnote|step=41}}
| down 5th
| vA
| k5
| comma-narrow 5th
| kA
| kA
|-
|-
| 42
| 42
| 700.0
| 700.0
| 3/2
| [[3/2]]
| P5
| {{UDnote|step=42}}
| perfect 5th
| A
| P5
| perfect 5th
| A
| A
|-
|-
| 43
| 43
| 716.7
| 716.7
| 50/33
| [[50/33]]
| ^5
| {{UDnote|step=43}}
| up 5th
| ^A
| L5
| large fifth
| LA
| KA
|-
|-
| 44
| 44
| 733.3
| 733.3
| 32/21
| [[26/17]], [[32/21]]
| ^^5
| {{UDnote|step=44}}
| dup 5th
|-
| ^^A
| 45
| S5
| 750.0
| super fifth
| [[17/11]], [[20/13]]
| SA
| {{UDnote|step=45}}
| SA
|-
| 45
| 750.0
| 54/35, 17/11
| ^<sup>3</sup>5, v<sup>3</sup>m6
| trup 5th, trudminor 6th
| ^<sup>3</sup>A, v<sup>3</sup>Bb
| H5, hm6
| hyper fifth, hypominor 6th
| HA, hBb
| UA, uBb
|-
|-
| 46
| 46
| 766.7
| 766.7
| 14/9
| [[14/9]]
| vvm6
| {{UDnote|step=46}}
| dudminor 6th
| vvBb
| sm6
| superminor 6th
| sBb
| sBb
|-
|-
| 47
| 47
| 783.3
| 783.3
| 11/7
| [[11/7]], [[30/19]]
| vm6
| {{UDnote|step=47}}
| downminor 6th
| vBb
| lm6
| little minor 6th
| lBb
| kBb
|-
|-
| 48
| 48
| 800.0
| 800.0
| 35/22
| [[19/12]]
| m6
| {{UDnote|step=48}}
| minor 6th
| Bb
| m6
| minor 6th
| Bb
| Bb
|-
|-
| 49
| 49
| 816.7
| 816.7
| 8/5
| [[8/5]]
| ^m6
| {{UDnote|step=49}}
| upminor 6th
| ^Bb
| Km6
| classic minor 6th
| kBb
| kBb
|-
|-
| 50
| 50
| 833.3
| 833.3
| 81/50, 13/8
| [[13/8]], [[21/13]], [[34/21]]
| ^^m6, v~6
| {{UDnote|step=50}}
| dupminor 6th, downmid 6th
| ^^Bb
| Om6
| on minor 6th
| oBb
| sBb
|-
|-
| 51
| 51
| 850.0
| 850.0
| 18/11
| [[18/11]], [[44/27]]
| ~6
| {{UDnote|step=51}}
| mid 6th
| v<sup>3</sup>B
| N6
| neutral 6th
| UBb, uB
| UBb, uB
|-
|-
| 52
| 52
| 866.7
| 866.7
| 33/20, 28/17
| [[28/17]], [[33/20]], ''[[64/39]]''
| ^~6, vvM6
| {{UDnote|step=52}}
| upmid 6th, dudmajor 6th
| vvB
| oM6
| off major 6th
| oB
| sB
|-
|-
| 53
| 53
| 883.3
| 883.3
| 5/3
| [[5/3]]
| vM6
| {{UDnote|step=53}}
| downmajor 6th
| vB
| kM6
| classic major 6th
| kB
| kB
|-
|-
| 54
| 54
| 900.0
| 900.0
| 27/16
| [[27/16]], [[32/19]], [[42/25]]
| M6
| {{UDnote|step=54}}
| major 6th
| B
| M6
| major 6th
| B
| B
|-
|-
| 55
| 55
| 916.7
| 916.7
| 56/33, 17/10
| [[17/10]], [[22/13]]
| ^M6
| {{UDnote|step=55}}
| upmajor 6th
| ^B
| LM6
| large major 6th
| LB
| KB
|-
|-
| 56
| 56
| 933.3
| 933.3
| 12/7
| [[12/7]]
| ^^M6
| {{UDnote|step=56}}
| dupmajor 6th
| ^^B
| SM6
| supermajor 6th
| SB
| SB
|-
|-
| 57
| 57
| 950.0
| 950.0
| 121/70
| [[19/11]], [[26/15]]
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| {{UDnote|step=57}}
| trupmajor 6th,<br>trudminor 7th
| ^<sup>3</sup>B, <br>v<sup>3</sup>C
| HM6, hm7
| hypermajor 6th, hypominor 7th
| HB, hC
| UB, uC
|-
|-
| 58
| 58
| 966.7
| 966.7
| 7/4
| [[7/4]]
| vvm7
| {{UDnote|step=58}}
| dudminor 7th
| vvC
| sm7
| subminor 7th
| sC
| sC
|-
|-
| 59
| 59
| 983.3
| 983.3
| 44/25
| [[30/17]], [[44/25]]
| vm7
| {{UDnote|step=59}}
| downminor 7th
|-
| vC
| 60
| lm7
| 1000.0
| little minor 7th
| [[16/9]], [[34/19]]
| lC
| {{UDnote|step=60}}
| kC
|-
| 60
| 1000.0
| 16/9
| m7
| minor 7th
| C
| m7
| minor 7th
| C
| C
|-
|-
| 61
| 61
| 1016.7
| 1016.7
| 9/5
| [[9/5]]
| ^m7
| {{UDnote|step=61}}
| upminor 7th
| ^C
| Km7
| classic/comma-wide minor 7th
| KC
| KC
|-
|-
| 62
| 62
| 1033.3
| 1033.3
| 20/11
| [[20/11]], [[38/21]]
| ^^m7, v~7
| {{UDnote|step=62}}
| dupminor 7th, downmid 7th
| ^^C
| Om7
| on minor 7th
| OC
| SC
|-
|-
| 63
| 63
| 1050.0
| 1050.0
| 11/6
| [[11/6]]
| ~7
| {{UDnote|step=63}}
| mid 7th
| ^<sup>3</sup>C
| N7, hd8
| neutral 7th, hypo dim 8ve
| UC/uC#, hDb
| UC/uC#, uDb
|-
|-
| 64
| 64
| 1066.7
| 1066.7
| 50/27
| [[13/7]], [[24/13]], [[50/27]]
| ^~7, vvM7
| {{UDnote|step=64}}
| upmid 7th, dudmajor 7th
| vvC#
| oM7, sd8
| off major 7th, sub dim 8ve
| oC#, sDb
| sC#, sDb
|-
|-
| 65
| 65
| 1083.3
| 1083.3
| 15/8
| [[15/8]], [[28/15]]
| vM7
| {{UDnote|step=65}}
| downmajor 7th
| vC#
| kM7, ld8
| classic major 7th, little dim 8ve
| kC#, lDb
| kC#, kDb
|-
|-
| 66
| 66
| 1100.0
| 1100.0
| 66/35, 17/9
| [[17/9]], [[32/17]], [[36/19]]
| M7
| {{UDnote|step=66}}
| major 7th
|-
| C#
| 67
| M7, d8
| 1116.7
| major 7th, dim 8ve
| [[19/10]], [[21/11]], [[40/21]]
| C#, Db
| {{UDnote|step=67}}
| C#, Db
|-
| 67
| 1116.7
| 21/11
| ^M7
| upmajor 7th
| ^C#
| LM7, Kd8
| large major 7th, comma-wide dim 8ve
| LC#, KDb
| KC#, KDb
|-
|-
| 68
| 68
| 1133.3
| 1133.3
| 27/14, 48/25
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| ^^M7
| {{UDnote|step=68}}
| dupmajor 7th
| ^^C#
| SM7, KKd8
| supermajor 7th, classic dim 8ve
| SC#, KKDb
| SC#, SDb, (KKDb)
|-
|-
| 69
| 69
| 1150.0
| 1150.0
| 35/18
| [[35/18]], [[39/20]], [[64/33]]
| ^<sup>3</sup>M7, v<sup>3</sup>8
| {{UDnote|step=69}}
| trupmajor 7th, trud octave
| ^<sup>3</sup>C#, v<sup>3</sup>D
| HM7, u8, h8
| hypermajor 7th, unter 8ve, hypo 8ve
| HC#, uD, hD
| UC#, uDb, uD
|-
|-
| 70
| 70
| 1166.7
| 1166.7
| 49/25
| [[49/25]], [[55/28]], [[63/32]], [[88/45]], [[96/49]]
| vv8
| {{UDnote|step=70}}
| dud octave
| vvD
| s8, o8
| sub 8ve, off 8ve
| sD, oD
| sD
|-
|-
| 71
| 71
| 1183.3
| 1183.3
| 99/50
| [[99/50]], [[160/81]], [[180/91]], [[196/99]], [[208/105]]
| v8
| {{UDnote|step=71}}
| down octave
| vD
| k8, l8
| comma-narrow 8ve, little 8ve
| kD, lD
| kD
|-
|-
| 72
| 72
| 1200.0
| 1200.0
| 2/1
| [[2/1]]
| P8
| {{UDnote|step=72}}
| perfect octave
| D
| P8
| perfect octave
| D
| D
|}
|}
<references group="note" />


=== Interval quality and chord names in color notation ===
=== Proposed interval names and solfèges ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
{| class="wikitable center-all right-2 left-4 left-7 mw-collapsible mw-collapsed"
 
|+ style="font-size: 105%; white-space: nowrap;" | Table of proposed interval names and solfèges
{| class="wikitable center-all"
|-
|-
! Quality
! #
! [[Color notation|Color]]
! Cents
! Monzo format
! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]]
! Examples
! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]
! (K, S, U)
|-
|-
| dudminor
| 0
| zo
| 0.0
| (a b 0 1)
| P1
| 7/6, 7/4
| perfect unison
| D
| P1
| perfect unison
| D
| D
|-
|-
| minor
| 1
| fourthward wa
| 16.7
| (a b), b < -1
| ^1
| 32/27, 16/9
| up unison
|-
| ^D
| upminor
| K1, L1
| gu
| comma-wide unison, large unison
| (a b -1)
| KD, LD
| 6/5, 9/5
| KD
|-
|-
| rowspan="2" | dupminor, <br>downmid
| 2
| luyo
| 33.3
| (a b 1 0 -1)
| ^^
| 15/11
| dup unison
| ^^D
| S1, O1
| super unison, on unison
| SD, OD
| SD
|-
|-
| tho
| 3
| (a b 0 0 0 1)
| 50.0
| 13/8, 13/9
| ^<sup>3</sup>1, v<sup>3</sup>m2
|-
| trup unison, trudminor 2nd
| rowspan="2" | mid
| ^<sup>3</sup>D, v<sup>3</sup>Eb
| ilo
| U1, H1, hm2
| (a b 0 0 1)
| uber unison, hyper unison, hypominor 2nd
| 11/9, 11/6
| UD, HD, uEb
| UD, uEb
|-
|-
| lu
| 4
| (a b 0 0 -1)
| 66.7
| 12/11, 18/11
| vvm2
| dudminor 2nd
| vvEb
| kkA1, sm2
| classic aug unison, subminor 2nd
| kkD#, sEb
| sD#, (kkD#), sEb
|-
|-
| rowspan="2" | upmid, <br>dudmajor
| 5
| logu
| 83.3
| (a b -1 0 1)
| vm2
| 11/10
| downminor 2nd
| vEb
| kA1, lm2
| comma-narrow aug unison, little minor 2nd
| kD#, lEb
| kD#, kEb
|-
|-
| thu
| 6
| (a b 0 0 0 -1)
| 100.0
| 16/13, 18/13
| m2
| minor 2nd
| Eb
| m2
| minor 2nd
| Eb
| Eb
|-
|-
| downmajor
| 7
| yo
| 116.7
| (a b 1)
| ^m2
| 5/4, 5/3
| upminor 2nd
| ^Eb
| Km2
| classic minor 2nd
| KEb
| KEb
|-
|-
| major
| 8
| fifthward wa
| 133.3
| (a b), b > 1
| ^^m2, v~2
| 9/8, 27/16
| dupminor 2nd, downmid 2nd
| ^^Eb
| Om2
| on minor 2nd
| OEb
| SEb
|-
|-
| dupmajor
| 9
| ru
| 150.0
| (a b 0 -1)
| ~2
| 9/7, 12/7
| mid 2nd
| v<sup>3</sup>E
| N2
| neutral 2nd
| UEb/uE
| UEb/uE
|-
|-
| rowspan="2" | trupmajor, <br>trudminor
| 10
| thogu
| 166.7
| (a b -1 0 0 1)
| ^~2, vvM2
| 13/10
| upmid 2nd, dudmajor 2nd
|-
| vvE
| thuyo
| oM2
| (a b 1 0 0 -1)
| off major 2nd
| 15/13
| oE
|}
| sE
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:
 
{| class="wikitable center-all"
|-
|-
! [[Color notation|Color of the 3rd]]
| 11
! JI chord
| 183.3
! Notes as edosteps
| vM2
! Notes of C chord
| downmajor 2nd
! Written name
| vE
! Spoken name
| kM2
| classic/comma-narrow major 2nd
| kE
| kE
|-
|-
| zo
| 12
| 6:7:9
| 200.0
| 0-16-42
| M2
| C vvEb G
| major 2nd
| Cvvm
| E
| C dudminor
| M2
| major 2nd
| E
| E
|-
|-
| gu
| 13
| 10:12:15
| 216.7
| 0-19-42
| ^M2
| C ^Eb G
| upmajor 2nd
| C^m
| ^E
| C upminor
| LM2
| large major 2nd
| LE
| KE
|-
|-
| ilo
| 14
| 18:22:27
| 233.3
| 0-21-42
| ^^M2
| C v<span style="font-size: 90%; vertical-align: super;">3</span>E G
| dupmajor 2nd
| C~
| ^^E
| C mid
| SM2
| supermajor 2nd
| SE
| SE
|-
|-
| yo
| 15
| 4:5:6
| 250.0
| 0-23-42
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| C vE G
| trupmajor 2nd,<br>trudminor 3rd
| Cv
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
| C downmajor or C down
| HM2, hm3
|-
| hypermajor 2nd, hypominor 3rd
| ru
| HE, hF
| 14:18:27
| UE, uF
| 0-26-42
|-
| C ^^E G
| 16
| C^^
| 266.7
| C dupmajor or C dup
| vvm3
|}
| dudminor 3rd
For a more complete list, see [[Ups and Downs Notation #Chord names in other EDOs]].
| vvF
 
| sm3
=== Relationship between primes and rings ===
| subminor 3rd
In 72tet, there are 6 [[ring number|rings]]. 12edo is the plain ring; thus every 6 degrees is the 3-limit.
| sF
 
| sF
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
| 17
* −2 degrees (the dud ring) corrects 16/9 to 7/4 via 63/64
| 283.3
* +3 degrees  (the trup ring) corrects 4/3 to 11/8 via 33/32
| vm3
* +2 degrees (the dup ring) corrects 128/81 to 13/8 via 1053/1024
| downminor 3rd
* 0 degrees (the plain ring) corrects 256/243 to 17/16 via 4131/4096
| vF
* 0 degrees (the plain ring) corrects 32/27 to 19/16 via 513/512
| lm3
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.
| little minor 3rd
 
| lF
== Notations ==
| kF
===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]].
| 18
====Evo flavor====
| 300.0
 
| m3
<imagemap>
| minor 3rd
File:72-EDO_Evo_Sagittal.svg
| F
desc none
| m3
rect 80 0 300 50 [[Sagittal_notation]]
| minor 3rd
rect 300 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| F
rect 20 80 120 106 [[81/80]]
| F
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:
 
[[File:72edo Sagittal.png|800px]]
 
=== Ups and downs notation ===
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 ===
{{sharpness-sharp6-iw}}
 
== JI approximation ==
[[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 ===
{{Q-odd-limit intervals|72}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
! rowspan="2" | [[Subgroup]]
| 19
! rowspan="2" | [[Comma list]]
| 316.7
! rowspan="2" | [[Mapping]]
| ^m3
! rowspan="2" | Optimal<br>8ve stretch (¢)
| upminor 3rd
! colspan="2" | Tuning error
| ^F
| Km3
| classic minor 3rd
| KF
| KF
|-
|-
! [[TE error|Absolute]] (¢)
| 20
! [[TE simple badness|Relative]] (%)
| 333.3
| ^^m3, v~3
| dupminor 3rd, downmid 3rd
| ^^F
| Om3
| on minor third
| OF
| SF
|-
|-
| 2.3.5
| 21
| 15625/15552, 531441/524288
| 350.0
| {{mapping| 72 114 167 }}
| ~3
| +0.839
| mid 3rd
| 0.594
| ^<sup>3</sup>F
| 3.56
| N3
| neutral 3rd
| UF/uF#
| UF/uF#
|-
|-
| 2.3.5.7
| 22
| 225/224, 1029/1024, 4375/4374
| 366.7
| {{mapping| 72 114 167 202 }}
| ^~3, vvM3
| +0.822
| upmid 3rd, dudmajor 3rd
| 0.515
| vvF#
| 3.09
| oM3
| off major 3rd
| oF#
| sF#
|-
|-
| 2.3.5.7.11
| 23
| 225/224, 243/242, 385/384, 4000/3993
| 383.3
| {{mapping| 72 114 167 202 249 }}
| vM3
| +0.734
| downmajor 3rd
| 0.493
| vF#
| 2.96
| kM3
| classic major 3rd
| kF#
| kF#
|-
|-
| 2.3.5.7.11.13
| 24
| 169/168, 225/224, 243/242, 325/324, 385/384
| 400.0
| {{mapping| 72 114 167 202 249 266 }}
| M3
| +0.936
| major 3rd
| 0.638
| F#
| 3.82
| M3
| major 3rd
| F#
| F#
|-
| 25
| 416.7
| ^M3
| upmajor 3rd
| ^F#
| LM3
| large major 3rd
| LF#
| KF#
|-
|-
| 2.3.5.7.11.13.17
| 26
| 169/168, 221/220, 225/224, 243/242, 273/272, 325/324
| 433.3
| {{mapping| 72 114 167 202 249 266 294 }}
| ^^M3
| +0.975
| dupmajor 3rd
| 0.599
| ^^F#
| 3.59
| SM3
|}
| supermajor 3rd
* 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.
| SF#
 
| SF#
=== Commas ===
Commas tempered out by 72edo include…
 
{| class="commatable wikitable center-1 center-2 right-4"
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cents]]
! Name(s)
|-
|-
| 3
| 27
| [[531441/524288|(12 digits)]]
| 450.0
| {{Monzo| -19 12 }}
| ^<sup>3</sup>M3, v<sup>3</sup>4
| 23.46
| trupmajor 3rd, trud 4th
| Pythagorean comma
| ^<sup>3</sup>F#, v<sup>3</sup>G
|-
| HM3, h4
| 5
| hypermajor 3rd, hypo 4th
| [[15625/15552]]
| HF#, hG
| {{Monzo| -6 -5 6 }}
| UF#, uG
| 8.11
| Kleisma
|-
|-
| 5
| 28
| [[34171875/33554432|(16 digits)]]
| 466.7
| {{Monzo| -25 7 6 }}
| vv4
| 31.57
| dud 4th
| [[Ampersand]]
| vvG
| s4
| sub 4th
| sG
| sG
|-
|-
| 5
| 29
| [[129140163/128000000|(18 digits)]]
| 483.3
| {{Monzo| -13 17 -6 }}
| v4
| 15.35
| down 4th
| [[Graviton]]
| vG
| l4
| little 4th
| lG
| kG
|-
|-
| 5
| 30
| <abbr title="7629394531250/7625597484987">(26 digits)</abbr>
| 500.0
| {{Monzo| 1 -27 18 }}
| P4
| 0.86
| perfect 4th
| [[Ennealimma]]
| G
| P4
| perfect 4th
| G
| G
|-
|-
| 7
| 31
| [[225/224]]
| 516.7
| {{Monzo| -5 2 2 -1 }}
| ^4
| 7.71
| up 4th
| Marvel comma
| ^G
| K4
| comma-wide 4th
| KG
| KG
|-
|-
| 7
| 32
| [[1029/1024]]
| 533.3
| {{Monzo| -10 1 0 3 }}
| ^^4, v~4
| 8.43
| dup 4th, downmid 4th
| Gamelisma
| ^^G
| O4
| on 4th
| OG
| SG
|-
|-
| 7
| 33
| [[2401/2400]]
| 550.0
| {{Monzo| -5 -1 -2 4 }}
| ~4
| 0.72
| mid 4th
| Breedsma
| ^<sup>3</sup>G
| U4/N4
| uber 4th / neutral 4th
| UG
| UG
|-
|-
| 7
| 34
| [[4375/4374]]
| 566.7
| {{Monzo| -1 -7 4 1 }}
| ^~4, vvA4
| 0.40
| upmid 4th, dudaug 4th
| Ragisma
| vvG#
| kkA4, sd5
| classic aug 4th, sub dim 5th
| kkG#, sAb
| SG#, (kkG#), sAb
|-
|-
| 7
| 35
| [[16875/16807]]
| 583.3
| {{Monzo| 0 3 4 -5 }}
| vA4, vd5
| 6.99
| downaug 4th, <br>downdim 5th
| Mirkwai comma
| vG#, vAb
| kA4, ld5
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, kAb
|-
|-
| 7
| 36
| [[19683/19600]]
| 600.0
| {{Monzo| -4 9 -2 -2 }}
| A4, d5
| 7.32
| aug 4th, dim 5th
| Cataharry comma
| G#, Ab
| A4, d5
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
|-
|-
| 7
| 37
| <abbr title="420175/419904">(12 digits)</abbr>
| 616.7
| {{Monzo | -6 -8 2 5 }}
| ^A4, ^d5
| 1.12
| upaug 4th, updim 5th
| [[Wizma]]
| ^G#, ^Ab
| LA4, Kd5
| large aug 4th, comma-wide dim 5th
| LG#, KAb
| KG#, KAb
|-
|-
| 7
| 38
| <abbr title="250047/250000">(12 digits)</abbr>
| 633.3
| {{Monzo| -4 6 -6 3 }}
| v~5, ^^d5
| 0.33
| downmid 5th, <br>dupdim 5th
| [[Landscape comma]]
| ^^Ab
| SA4, KKd5
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, SAb, (KKAb)
|-
|-
| 11
| 39
| [[243/242]]
| 650.0
| {{Monzo| -1 5 0 0 -2}}
| ~5
| 7.14
| mid 5th
| Rastma
| v<sup>3</sup>A
| u5/N5
| unter 5th / neutral 5th
| uA
| uA
|-
|-
| 11
| 40
| [[385/384]]
| 666.7
| {{Monzo| -7 -1 1 1 1 }}
| vv5, ^~5
| 4.50
| dud 5th, upmid 5th
| Keenanisma
| vvA
| o5
| off 5th
| oA
| sA
|-
|-
| 11
| 41
| [[441/440]]
| 683.3
| {{Monzo| -3 2 -1 2 -1 }}
| v5
| 3.93
| down 5th
| Werckisma
| vA
| k5
| comma-narrow 5th
| kA
| kA
|-
|-
| 11
| 42
| [[540/539]]
| 700.0
| {{Monzo| 2 3 1 -2 -1 }}
| P5
| 3.21
| perfect 5th
| Swetisma
| A
| P5
| perfect 5th
| A
| A
|-
|-
| 11
| 43
| [[1375/1372]]
| 716.7
| {{Monzo| -2 0 3 -3 1 }}
| ^5
| 3.78
| up 5th
| Moctdel comma
| ^A
| L5
| large fifth
| LA
| KA
|-
|-
| 11
| 44
| [[3025/3024]]
| 733.3
| {{Monzo| -4 -3 2 -1 2 }}
| ^^5
| 0.57
| dup 5th
| Lehmerisma
| ^^A
| S5
| super fifth
| SA
| SA
|-
|-
| 11
| 45
| [[4000/3993]]
| 750.0
| {{Monzo| 5 -1 3 0 -3 }}
| ^<sup>3</sup>5, v<sup>3</sup>m6
| 3.03
| trup 5th, trudminor 6th
| Wizardharry comma
| ^<sup>3</sup>A, v<sup>3</sup>Bb
| H5, hm6
| hyper fifth, hypominor 6th
| HA, hBb
| UA, uBb
|-
|-
| 11
| 46
| [[6250/6237]]
| 766.7
| {{Monzo| 1 -4 5 -1 -1 }}
| vvm6
| 3.60
| dudminor 6th
| Liganellus comma
| vvBb
| sm6
| superminor 6th
| sBb
| sBb
|-
|-
| 11
| 47
| [[9801/9800]]
| 783.3
| {{Monzo| -3 4 -2 -2 2 }}
| vm6
| 0.18
| downminor 6th
| Kalisma
| vBb
| lm6
| little minor 6th
| lBb
| kBb
|-
|-
| 11
| 48
| <abbr title="1771561/1769472">(14 digits)</abbr>
| 800.0
| {{Monzo| 16 -3 0 0 6 }}
| m6
| 2.04
| minor 6th
| [[Nexus comma]]
| Bb
| m6
| minor 6th
| Bb
| Bb
|-
|-
| 13
| 49
| [[169/168]]
| 816.7
| {{Monzo| -3 -1 0 -1 0 2 }}
| ^m6
| 10.27
| upminor 6th
| Buzurgisma
| ^Bb
| Km6
| classic minor 6th
| kBb
| kBb
|-
|-
| 13
| 50
| [[325/324]]
| 833.3
| {{Monzo| -2 -4 2 0 0 1 }}
| ^^m6, v~6
| 5.34
| dupminor 6th, downmid 6th
| Marveltwin comma
| ^^Bb
| Om6
| on minor 6th
| oBb
| sBb
|-
|-
| 13
| 51
| [[351/350]]
| 850.0
| {{Monzo| -1 3 -2 -1 0 1 }}
| ~6
| 4.94
| mid 6th
| Ratwolfsma
| v<sup>3</sup>B
| N6
| neutral 6th
| UBb, uB
| UBb, uB
|-
|-
| 13
| 52
| [[364/363]]
| 866.7
| {{Monzo| 2 -1 0 1 -2 1 }}
| ^~6, vvM6
| 4.76
| upmid 6th, dudmajor 6th
| Minor minthma
| vvB
| oM6
| off major 6th
| oB
| sB
|-
|-
| 13
| 53
| [[625/624]]
| 883.3
| {{Monzo| -4 -1 4 0 0 -1 }}
| vM6
| 2.77
| downmajor 6th
| Tunbarsma
| vB
| kM6
| classic major 6th
| kB
| kB
|-
|-
| 13
| 54
| [[676/675]]
| 900.0
| {{Monzo| 2 -3 -2 0 0 2 }}
| M6
| 2.56
| major 6th
| Island comma
| B
| M6
| major 6th
| B
| B
|-
|-
| 13
| 55
| [[729/728]]
| 916.7
| {{Monzo| -3 6 0 -1 0 -1 }}
| ^M6
| 2.38
| upmajor 6th
| Squbema
| ^B
| LM6
| large major 6th
| LB
| KB
|-
|-
| 13
| 56
| [[1001/1000]]
| 933.3
| {{Monzo| -3 0 -3 1 1 1 }}
| ^^M6
| 1.73
| dupmajor 6th
| Sinbadma
| ^^B
| SM6
| supermajor 6th
| SB
| SB
|-
|-
| 13
| 57
| [[1575/1573]]
| 950.0
| {{Monzo| 2 2 1 -2 -1 }}
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| 2.20
| trupmajor 6th,<br>trudminor 7th
| Nicola
| ^<sup>3</sup>B, <br>v<sup>3</sup>C
| HM6, hm7
| hypermajor 6th, hypominor 7th
| HB, hC
| UB, uC
|-
|-
| 13
| 58
| [[1716/1715]]
| 966.7
| {{Monzo| 2 1 -1 -3 1 1 }}
| vvm7
| 1.01
| dudminor 7th
| Lummic comma
| vvC
| sm7
| subminor 7th
| sC
| sC
|-
|-
| 13
| 59
| [[2080/2079]]
| 983.3
| {{Monzo| 5 -3 1 -1 -1 1 }}
| vm7
| 0.83
| downminor 7th
| Ibnsinma
| vC
| lm7
| little minor 7th
| lC
| kC
|-
| 60
| 1000.0
| m7
| minor 7th
| C
| m7
| minor 7th
| C
| C
|-
|-
| 13
| 61
| [[6656/6655]]
| 1016.7
| {{Monzo| 9 0 -1 0 -3 1 }}
| ^m7
| 0.26012
| upminor 7th
| Jacobin comma
| ^C
|}
| Km7
 
| classic/comma-wide minor 7th
=== Rank-2 temperaments ===
| KC
* [[List of edo-distinct 72et rank two temperaments]]
| KC
 
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 left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br>per 8ve
| 62
! Generator*
| 1033.3
! Cents*
| ^^m7, v~7
! Associated<br>ratio*
| dupminor 7th, downmid 7th
! Temperament
| ^^C
| Om7
| on minor 7th
| OC
| SC
|-
|-
| 1
| 63
| 1\72
| 1050.0
| 16.7
| ~7
| 105/104
| mid 7th
| [[Quincy]]
| ^<sup>3</sup>C
| N7, hd8
| neutral 7th, hypo dim 8ve
| UC/uC#, hDb
| UC/uC#, uDb
|-
|-
| 1
| 64
| 5\72
| 1066.7
| 83.3
| ^~7, vvM7
| 21/20
| upmid 7th, dudmajor 7th
| [[Marvolo]]
| vvC#
| oM7, sd8
| off major 7th, sub dim 8ve
| oC#, sDb
| sC#, sDb
|-
|-
| 1
| 65
| 7\72
| 1083.3
| 116.7
| vM7
| 15/14
| downmajor 7th
| [[Miracle]] / benediction / manna
| vC#
| kM7, ld8
| classic major 7th, little dim 8ve
| kC#, lDb
| kC#, kDb
|-
|-
| 1
| 66
| 17\72
| 1100.0
| 283.3
| M7
| 13/11
| major 7th
| [[Neominor]]
| C#
| M7, d8
| major 7th, dim 8ve
| C#, Db
| C#, Db
|-
|-
| 1
| 67
| 19\72
| 1116.7
| 316.7
| ^M7
| 6/5
| upmajor 7th
| [[Catakleismic]]
| ^C#
| LM7, Kd8
| large major 7th, comma-wide dim 8ve
| LC#, KDb
| KC#, KDb
|-
|-
| 1
| 68
| 25\72
| 1133.3
| 416.7
| ^^M7
| 14/11
| dupmajor 7th
| [[Sqrtphi]]
| ^^C#
| SM7, KKd8
| supermajor 7th, classic dim 8ve
| SC#, KKDb
| SC#, SDb, (KKDb)
|-
|-
| 1
| 69
| 29\72
| 1150.0
| 483.3
| ^<sup>3</sup>M7, v<sup>3</sup>8
| 45/34
| trupmajor 7th, trud octave
| [[Hemiseven]]
| ^<sup>3</sup>C#, v<sup>3</sup>D
| HM7, u8, h8
| hypermajor 7th, unter 8ve, hypo 8ve
| HC#, uD, hD
| UC#, uDb, uD
|-
|-
| 1
| 70
| 31\72
| 1166.7
| 516.7
| vv8
| 27/20
| dud octave
| [[Marvo]] / [[zarvo]]
| vvD
| s8, o8
| sub 8ve, off 8ve
| sD, oD
| sD
|-
|-
| 1
| 71
| 35\72
| 1183.3
| 583.3
| v8
| 7/5
| down octave
| [[Cotritone]]
| vD
| k8, l8
| comma-narrow 8ve, little 8ve
| kD, lD
| kD
|-
|-
| 2
| 72
| 5\72
| 1200.0
| 83.3
| P8
| 21/20
| perfect octave
| [[Harry]]
| D
|-
| P8
| 2
| perfect octave
| 7\72
| D
| 116.7
| D
| 15/14
|}
| [[Semimiracle]]
 
=== Interval quality and chord names in color notation ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
 
{| class="wikitable center-all"
|-
|-
| 2
! Quality
| 11\72
! [[Color notation|Color]]
| 183.3
! Monzo format
| 10/9
! Examples
| [[Unidec]] / hendec
|-
|-
| 2
| dudminor
| 21\72<br>(19\72)
| zo
| 316.7<br>(283.3)
| (a b 0 1)
| 6/5<br>(13/11)
| [[7/6]], [[7/4]]
| [[Bikleismic]]
|-
|-
| 2
| minor
| 23\72<br>(13\72)
| fourthward wa
| 383.3<br>(216.7)
| (a b), b < -1
| 5/4<br>(17/15)
| [[32/27]], [[16/9]]
| [[Wizard]] / lizard / gizzard
|-
| upminor
| gu
| (a b -1)
| [[6/5]], [[9/5]]
|-
|-
| 3
| rowspan="2" | dupminor, <br>downmid
| 11\72
| luyo
| 183.3
| (a b 1 0 -1)
| 10/9
| [[15/11]]
| [[Mirkat]]
|-
|-
| 3
| tho
| 19\72<br>(5\72)
| (a b 0 0 0 1)
| 316.7<br>(83.3)
| [[13/8]], [[13/9]]
| 6/5<br>(21/20)
| [[Tritikleismic]]
|-
|-
| 4
| rowspan="2" | mid
| 19\72<br>(1\72)
| ilo
| 316.7<br>(16.7)
| (a b 0 0 1)
| 6/5<br>(105/104)
| [[11/9]], [[11/6]]
| [[Quadritikleismic]]
|-
|-
| 8
| lu
| 34\72<br>(2\72)
| (a b 0 0 -1)
| 566.7<br>(33.3)
| [[12/11]], [[18/11]]
| 168/121<br>(55/54)
| [[Octowerck]] / octowerckis
|-
|-
| 8
| rowspan="2" | upmid, <br>dudmajor
| 35\72<br>(1\72)
| logu
| 583.3<br>(16.7)
| (a b -1 0 1)
| 7/5<br>(100/99)
| [[11/10]]
| [[Octoid]] / octopus
|-
|-
| 9
| thu
| 19\72<br>(3\72)
| (a b 0 0 0 -1)
| 316.7<br>(50.0)
| [[16/13]], [[18/13]]
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimnic
|-
|-
| 9
| downmajor
| 23\72<br>(1\72)
| yo
| 383.3<br>(16.7)
| (a b 1)
| 5/4<br>(105/104)
| [[5/4]], [[5/3]]
| [[Enneaportent]]
|-
|-
| 12
| major
| 23\72<br>(1\72)
| fifthward wa
| 383.3<br>(16.7)
| (a b), b > 1
| 5/4<br>(100/99)
| [[9/8]], [[27/16]]
| [[Compton]] / comptone
|-
|-
| 18
| dupmajor
| 19\72<br>(1\72)
| ru
| 316.7<br>(16.7)
| (a b 0 -1)
| 6/5<br>(105/104)
| [[9/7]], [[12/7]]
| [[Hemiennealimmal]]
|-
|-
| 24
| rowspan="2" | trupmajor, <br>trudminor
| 23\72<br>(1\72)
| thogu
| 383.3<br>(16.7)
| (a b -1 0 0 1)
| 5/4<br>(105/104)
| [[13/10]]
| [[Hours]]
|-
|-
| 36
| thuyo
| 23\72<br>(1\72)
| (a b 1 0 0 -1)
| 383.3<br>(16.7)
| [[15/13]]
| 5/4<br>(81/80)
| [[Decades]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
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:


== Scales ==
{| class="wikitable center-all"
* [[Smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]]
* [[Blackjack]], [[miracle_8]], [[miracle_10]], [[miracle_12]], [[miracle_12a]], [[miracle_24hi]], [[miracle_24lo]]
* [[Keenanmarvel]], [[xenakis_chrome]], [[xenakis_diat]], [[xenakis_schrome]]
* [[Genus24255et72|Euler(24255) genus in 72 equal]]
* [[JuneGloom]]
* [[Harry Partch's 43-tone scale]]: 1 2 2 2 2 1 1 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 1 1 2 2 2 2 1
* [[Magnetosphere scale|Magnetosphere]], [[Blackened skies]], [[Lost spirit]]
* [[5- to 10-tone scales in 72edo]]
 
=== Harmonic scale ===
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"
|-
|-
| Harmonics in "Mode 8":
! [[Color notation|Color of the 3rd]]
| 8
! JI chord
|
! Notes as edosteps
| 9
! Notes of C chord
|
! Written name
| 10
! Spoken name
|
| 11
|
| 12
|
| 13
|
| 14
|
| 15
|
| 16
|-
|-
| …as JI Ratio from 1/1:
| zo
| 1/1
| 6:7:9
|
| 0-16-42
| 9/8
| C vvEb G
|  
| Cvvm
| 5/4
| C dudminor
|  
|-
| 11/8
| gu
|  
| 10:12:15
| 3/2
| 0-19-42
|  
| C ^Eb G
| 13/8
| C^m
|  
| C upminor
| 7/4
|
| 15/8
|
| 2/1
|-
|-
| …in cents:
| ilo
| 0
| 18:22:27
|  
| 0-21-42
| 203.9
| C v<span style="font-size: 90%; vertical-align: super;">3</span>E G
|
| C~
| 386.3
| C mid
|  
|-
| 551.3
| yo
|
| 4:5:6
| 702.0
| 0-23-42
|  
| C vE G
| 840.5
| Cv
|  
| C downmajor or C down
| 968.8
|
| 1088.3
|  
| 1200.0
|-
|-
| Nearest degree of 72edo:
| ru
| 0
| 14:18:27
|  
| 0-26-42
| 12
| C ^^E G
|  
| C^^
| 23
| C dupmajor or C dup
|  
|}
| 33
For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]].
|  
 
| 42
=== 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.
| 50
 
|  
Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
| 58
* −1 degree (the down ring) corrects [[81/64]] to [[5/4]] via descending [[81/80]]
|
* −2 degrees (the dud ring) corrects [[16/9]] to [[7/4]] via descending [[64/63]]
| 65
* +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]]
| 72
* 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]]
| …in cents:
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.
| 0
 
|  
== Notation ==
| 200.0
=== Stein–Zimmermann–Gould notation ===
|  
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
| 383.3
{{Sharpness-sharp6-szg}}
|  
 
| 550.0
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
|  
{{Sharpness-sharp6-qt-szg}}
| 700.0
 
|  
=== Kite's ups and downs notation ===
| 833.3
72edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
|  
{{Ups and downs sharpness}}
| 966.7
 
|  
Half-sharps and half-flats can be used to avoid triple arrows:
| 1083.3
{{Ups and downs sharpness|72|true}}
|  
 
| 1200.0
=== Sagittal notation ===
This notation uses the same sagittal sequence as edos [[65edo #Sagittal notation|65-]] and [[79edo #Sagittal notation|79edo]], and is a superset of the notations for edos [[36edo #Sagittal notation|36]], [[24edo #Sagittal notation|24]], [[18edo #Sagittal notation|18]], [[12edo #Sagittal notation|12]], [[8edo #Sagittal notation|8]], and [[6edo #Sagittal notation|6]].
 
==== Evo flavor ====
{{Sagittal chart|Evo}}
 
==== Evo-SZ flavor ====
{{Sagittal chart|Evo-SZ}}
 
==== Revo flavor ====
{{Sagittal chart}}
 
From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 72edo in the Revo flavor of Sagittal:
 
<div class="noresize">
[[File:72edo Sagittal.png]]
</div>
 
=== Ivan Wyschnegradsky's notation ===
{{Sharpness-sharp6-iw|72}}
 
== Approximation to JI ==
[[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]]
 
=== Interval mappings ===
{{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 [[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 ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| Steps as Freq. Ratio:
| 2.3.5
|  
| 15625/15552, 531441/524288
| 9:8
| {{Mapping| 72 114 167 }}
|  
| +0.839
| 10:9
| 0.594
|  
| 3.56
| 11:10
|-
|  
| 2.3.5.7
| 12:11
| 225/224, 1029/1024, 4375/4374
|
| {{Mapping| 72 114 167 202 }}
| 13:12
| +0.822
|  
| 0.515
| 14:13
| 3.09
|  
| 15:14
|
| 16:15
|  
|-
|-
| …in cents:
| 2.3.5.7.11
|  
| 225/224, 243/242, 385/384, 4000/3993
| 203.9
| {{Mapping| 72 114 167 202 249 }}
|  
| +0.734
| 182.4
| 0.493
|  
| 2.96
| 165.0
|-
|  
| 2.3.5.7.11.13
| 150.6
| 169/168, 225/224, 243/242, 325/324, 385/384
|  
| {{Mapping| 72 114 167 202 249 266 }}
| 138.6
| +0.936
|  
| 0.638
| 128.3
| 3.82
|  
| 119.4
|  
| 111.7
|
|-
|-
| Nearest degree of 72edo:
| 2.3.5.7.11.13.17
|  
| 169/168, 221/220, 225/224, 243/242, 273/272, 325/324
| 12
| {{Mapping| 72 114 167 202 249 266 294 }}
|  
| +0.975
| 11
| 0.599
|  
| 3.59
| 10
|-
|  
| 2.3.5.7.11.13.17.19
| 9
| 153/152, 169/168, 210/209, 221/220, 225/224, 243/242, 273/272
|  
| {{Mapping| 72 114 167 202 249 266 294 306 }}
| 8
| +0.780
|  
| 0.762
| 8
| 4.57
|  
|}
| 7
* 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.
|
 
| 7
=== Commas ===
|  
Commas tempered out by 72edo include…
 
{| class="commatable wikitable center-1 center-2 right-4"
|-
|-
| …in cents:
! [[Harmonic limit|Prime<br>limit]]
|  
! [[Ratio]]<ref group="note">{{rd}}</ref>
| 200.0
! [[Monzo]]
|  
! [[Cents]]
| 183.3
! Name(s)
|  
|-
| 166.7
| 3
|  
| [[531441/524288|(12 digits)]]
| 150.0
| {{Monzo| -19 12 }}
|  
| 23.46
| 133.3
| Pythagorean comma
|  
|-
| 133.3
| 5
| [[15625/15552]]
| {{Monzo| -6 -5 6 }}
| 8.11
| Kleisma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| [[Ampersand comma]]
|-
| 5
| [[129140163/128000000|(18 digits)]]
| {{Monzo| -13 17 -6 }}
| 15.35
| [[Graviton]]
|-
| 5
| <abbr title="7629394531250/7625597484987">(26 digits)</abbr>
| {{Monzo| 1 -27 18 }}
| 0.86
| [[Ennealimma]]
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Marvel comma
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Gamelisma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Breedsma
|-
| 7
| [[4375/4374]]
| {{Monzo| -1 -7 4 1 }}
| 0.40
| Ragisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Mirkwai comma
|-
| 7
| [[19683/19600]]
| {{Monzo| -4 9 -2 -2 }}
| 7.32
| Cataharry comma
|-
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| {{Monzo | -6 -8 2 5 }}
| 1.12
| [[Wizma]]
|-
| 7
| <abbr title="250047/250000">(12 digits)</abbr>
| {{Monzo| -4 6 -6 3 }}
| 0.33
| [[Landscape comma]]
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2}}
| 7.14
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Swetisma
|-
| 11
| [[1375/1372]]
| {{Monzo| -2 0 3 -3 1 }}
| 3.78
| Moctdel comma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Lehmerisma
|-
| 11
| [[4000/3993]]
| {{Monzo| 5 -1 3 0 -3 }}
| 3.03
| Wizardharry comma
|-
| 11
| [[6250/6237]]
| {{Monzo| 1 -4 5 -1 -1 }}
| 3.60
| Liganellus comma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Kalisma
|-
| 11
| <abbr title="1771561/1769472">(14 digits)</abbr>
| {{Monzo| 16 -3 0 0 6 }}
| 2.04
| [[Nexus comma]]
|-
| 13
| [[169/168]]
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.27
| Buzurgisma
|-
| 13
| [[325/324]]
| {{Monzo| -2 -4 2 0 0 1 }}
| 5.34
| Marveltwin comma
|-
| 13
| [[351/350]]
| {{Monzo| -1 3 -2 -1 0 1 }}
| 4.94
| Ratwolfsma
|-
| 13
| [[364/363]]
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.76
| Minor minthma
|-
| 13
| [[625/624]]
| {{Monzo| -4 -1 4 0 0 -1 }}
| 2.77
| Tunbarsma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Island comma
|-
| 13
| [[729/728]]
| {{Monzo| -3 6 0 -1 0 -1 }}
| 2.38
| Squbema
|-
| 13
| [[1001/1000]]
| {{Monzo| -3 0 -3 1 1 1 }}
| 1.73
| Sinbadma
|-
| 13
| [[1575/1573]]
| {{Monzo| 2 2 1 -2 -1 }}
| 2.20
| Nicola
|-
| 13
| [[1716/1715]]
| {{Monzo| 2 1 -1 -3 1 1 }}
| 1.01
| Lummic comma
|-
| 13
| [[2080/2079]]
| {{Monzo| 5 -3 1 -1 -1 1 }}
| 0.83
| Ibnsinma
|-
| 13
| [[6656/6655]]
| {{Monzo| 9 0 -1 0 -3 1 }}
| 0.26012
| Jacobin comma
|}
<references group="note" />
 
=== Rank-2 temperaments ===
* [[List of edo-distinct 72et rank two 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.
 
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 1\72
| 16.7
| 105/104
| [[Quincy]]
|-
| 1
| 5\72
| 83.3
| 21/20
| [[Marvolo]]
|-
| 1
| 7\72
| 116.7
| 15/14
| [[Miracle]] / benediction / manna
|-
| 1
| 17\72
| 283.3
| 13/11
| [[Neominor]]
|-
| 1
| 19\72
| 316.7
| 6/5
| [[Catakleismic]]
|-
| 1
| 25\72
| 416.7
| 14/11
| [[Sqrtphi]]
|-
| 1
| 29\72
| 483.3
| 45/34
| [[Hemiseven]]
|-
| 1
| 31\72
| 516.7
| 27/20
| [[Gravity]] / [[marvo]] / [[zarvo]]
|-
| 1
| 35\72
| 583.3
| 7/5
| [[Cotritone]]
|-
| 2
| 5\72
| 83.3
| 21/20
| [[Harry]]
|-
| 2
| 7\72
| 116.7
| 15/14
| [[Semimiracle]]
|-
| 2
| 11\72
| 183.3
| 10/9
| [[Unidec]] / hendec
|-
| 2
| 21\72<br>(19\72)
| 316.7<br>(283.3)
| 6/5<br>(13/11)
| [[Bikleismic]]
|-
| 2
| 23\72<br>(13\72)
| 383.3<br>(216.7)
| 5/4<br>(17/15)
| [[Wizard]] / lizard / gizzard
|-
| 3
| 11\72
| 183.3
| 10/9
| [[Mirkat]]
|-
| 3
| 19\72<br>(5\72)
| 316.7<br>(83.3)
| 6/5<br>(21/20)
| [[Tritikleismic]]
|-
| 4
| 19\72<br>(1\72)
| 316.7<br>(16.7)
| 6/5<br>(105/104)
| [[Quadritikleismic]]
|-
| 8
| 34\72<br>(2\72)
| 566.7<br>(33.3)
| 168/121<br>(55/54)
| [[Octowerck]] / octowerckis
|-
| 8
| 35\72<br>(1\72)
| 583.3<br>(16.7)
| 7/5<br>(100/99)
| [[Octoid]] / octopus
|-
| 9
| 19\72<br>(3\72)
| 316.7<br>(50.0)
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimnic / ennealiminal
|-
| 9
| 23\72<br>(1\72)
| 383.3<br>(16.7)
| 5/4<br>(105/104)
| [[Enneaportent]]
|-
| 12
| 23\72<br>(1\72)
| 383.3<br>(16.7)
| 5/4<br>(100/99)
| [[Compton]] / comptone
|-
| 18
| 19\72<br>(1\72)
| 316.7<br>(16.7)
| 6/5<br>(105/104)
| [[Hemiennealimmal]]
|-
| 24
| 23\72<br>(1\72)
| 383.3<br>(16.7)
| 5/4<br>(105/104)
| [[Hours]]
|-
| 36
| 23\72<br>(1\72)
| 383.3<br>(16.7)
| 5/4<br>(81/80)
| [[Gamelstearn]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
== Octave stretch or compression ==
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]], [[zpi|380zpi]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 380zpi and 186ed6 are milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.
 
== Scales ==
; [[Miracle]]-tempered scales
* [[Blackjack]], [[miracle_8]], [[miracle_10]], [[miracle_12]], [[miracle_12a]], [[miracle_24hi]], [[miracle_24lo]]
 
; [[Maeve Gutierrez]]'s scales
* [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]] (''octave reduced: 10 6 25 17 14'')
* [[Maeve Gutierrez|Gutierrez Moonglade]]: 1 4 6 1 5 2 4 7 1 4 6 1 1 4 5 1 5 1 2 3 1 1 5 1
 
; [[Budjarn Lambeth]]'s scales
* [[Magnetosphere scale|Magnetosphere]], [[blackened skies]], [[lost spirit]], [[moon dust]], [[5- to 10-tone scales in 72edo]]
 
; [[Gene Ward Smith]]'s scales
* [[Smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]]
 
; [[Iannis Xenakis]]' scales
* [[xenakis_chrome]], [[xenakis_diat]], [[xenakis_schrome]]
 
; Others
* Freivald [[Lazysunday]] scale
* [[Genus24255et72|Euler(24255) genus in 72 equal]]
* [[Harry Partch's 43-tone scale]]: 1 2 2 2 2 1 1 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 1 1 2 2 2 2 1
* [[JuneGloom]]
* [[Keenanmarvel]]
* [[Prodigy]][19]: 5 2 5 4 5 2 5 2 5 2 5 4 5 2 5 2 5 5 2
 
=== Harmonic scale ===
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"
|-
! Harmonics in "Mode 8":
| 8
|
| 9
|
| 10
|
| 11
|
| 12
|
| 13
|
| 14
|
| 15
|  
|  
| 116.7
| 16
|  
|-
| 116.7
! …as JI Ratio from 1/1:
|  
| 1/1
|}
|
 
| 9/8
== Instruments ==
|
 
| 5/4
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).
|
 
| 11/8
Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo: [[Lumatone mapping for 72edo]]
|
 
| 3/2
== Music ==
|
; [[Ambient Esoterica]]
| 13/8
* [https://www.youtube.com/watch?v=seWcDAoQjxY ''Goetic Synchronities''] (2023)
|
* [https://www.youtube.com/watch?v=CrcdM1e2b6Q ''Rainy Day Generative Pillow''] (2024)
| 7/4
 
|
; [[Jake Freivald]]
| 15/8
* [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday'']{{dead link}} in the [[lazysunday]] scale
|
 
| 2/1
{{Wikipedia|In vain (Haas)}}
|-
; [[Georg Friedrich Haas]]
! …in cents:
* [https://www.youtube.com/watch?v=ix4yA-c-Pi8 ''Blumenstück''] (2000)
| 0
* [https://youtu.be/cmX-h7_us7A ''in vain''] (2000) ([https://www.universaledition.com/georg-friedrich-haas-278/works/in-vain-7566 score])
|
| 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:
* [[Skip fretting system 72 2 27]]
 
Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo:
* [[Lumatone mapping for 72edo]]
 
== Music ==
; [[Tolgahan Çoğulu]]
* [https://www.youtube.com/watch?v=BItT4f8MOiI ''Ne ağlarsın benim zülfü siyahım''] (from the UnTwelving 2026 convention) – transcription by [[Stephen Weigel]]
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/VwVp3RVao_k ''microtonal improvisation in 72edo''] (2025)
 
; [[Ambient Esoterica]]
* [https://www.youtube.com/watch?v=seWcDAoQjxY ''Goetic Synchronities''] (2023)
* [https://www.youtube.com/watch?v=CrcdM1e2b6Q ''Rainy Day Generative Pillow''] (2024)
 
; [[Jake Freivald]]
* [https://web.archive.org/web/20201127014336/http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday''] in the [[lazysunday]] scale
 
{{Wikipedia|In vain (Haas)}}
; [[Georg Friedrich Haas]]
* [https://www.youtube.com/watch?v=ix4yA-c-Pi8 ''Blumenstück''] (2000)
* [https://youtu.be/cmX-h7_us7A ''in vain''] (2000) ([https://www.universaledition.com/georg-friedrich-haas-278/works/in-vain-7566 score])
 
; [[Budjarn Lambeth]]
* [https://youtu.be/eWMRJihZbPc ''Blackened Skies''] (2020)
 
; [[Claudi Meneghin]]
* [https://web.archive.org/web/20201127015744/http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 ''Twinkle canon – 72 edo'']
* [https://www.youtube.com/watch?v=zR0NDgh4944 ''The Miracle Canon'', 3-in-1 on a Ground]
* [https://www.youtube.com/watch?v=w6Bckog1eOM ''Sicilienne in Miracle'']
* [https://www.youtube.com/watch?v=QKeZLtFHfNU ''Arietta with 5 Variations'', for Organ] (2024)


; [[Claudi Meneghin]]
; [https://www.youtube.com/@Mintsoda_15/videos Mintsoda_15]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 ''Twinkle canon &ndash; 72 edo'']{{dead link}}
* [https://www.youtube.com/watch?v=eA8S746K9_A ''Quarter-tone Impromptu-72 TET''] (2022)
* [https://www.youtube.com/watch?v=zR0NDgh4944 ''The Miracle Canon'', 3-in-1 on a Ground]
* [https://www.youtube.com/watch?v=w6Bckog1eOM ''Sicilienne in Miracle'']
* [https://www.youtube.com/watch?v=QKeZLtFHfNU ''Arietta with 5 Variations'', for Organ] (2024)


; [[Prent Rodgers]]
; [[Prent Rodgers]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 ''June Gloom #9'']{{dead link}}
* [https://web.archive.org/web/20201127012907/http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 ''June Gloom #9'']


; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* [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]]
; [[Ivan Wyschnegradsky]]
* [https://www.youtube.com/watch?v=RCcJHCkYQ6U Arc-en-ciel, for 6 pianos in twelfth tones, Op. 37] (1956)
* [https://www.youtube.com/watch?v=RCcJHCkYQ6U ''Arc-en-ciel, for 6 pianos in twelfth tones, Op. 37''] (1956)


; [[James Tenney]]
; [[James Tenney]]
Line 1,714: Line 2,042:
* [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"