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{{interwiki
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{{Infobox ET}}
{{Wikipedia|72 equal temperament}}
{{ED intro}}
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]].
== Theory ==
== Theory ==
'''72-tone equal temperament''', or '''72-edo''', divides the octave into 72 steps or ''moria''. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in 12-tone equal temperament. 72-tone is also a superset of [[24edo|24-tone equal temperament]], a common and standard tuning of [[Arabic,_Turkish,_Persian|Arabic]] music, and has itself been used to tune Turkish music.
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]].


Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with [[96edo]]), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.
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.


72-tone equal temperament approximates [[11-limit]] [[just intonation]] exceptionally well, is consistent in the [[17-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 12-tone, 72, 42 and 30 steps respectively, but the major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.
The [[octave 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.  


72 is an excellent tuning for [[Gamelismic_clan #Miracle|miracle temperament]], especially the 11-limit version, and the related rank three temperament [[Marvel_family #Prodigy|prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].
72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]].
 
=== Prime harmonics ===
{{Harmonics in equal|72|columns=11}}
{{Harmonics in equal|72|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}
 
=== As a tuning of other temperaments ===
72et is the only 11-limit regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, and splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s (144edo is enfactored in the 11-limit with 72edo, so it is already covered here). It is also an excellent tuning for [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].
 
=== Subsets and supersets ===
Since 72 factors into primes as {{nowrap| 2<sup>3</sup> × 3<sup>2</sup> }}, 72edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36 }}. [[144edo]], which doubles it, provides a possible correction to its approximate harmonic 13, though unlike 72 it is not consistent to the [[13-odd-limit]].


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! 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>
! [[Kite's ups and downs notation|Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 16.7
| [[81/80]], [[91/90]], [[99/98]], [[100/99]], [[105/104]]
| {{UDnote|step=1}}
|-
| 2
| 33.3
| [[45/44]], [[49/48]], [[50/49]], [[55/54]], [[64/63]]
| {{UDnote|step=2}}
|-
| 3
| 50.0
| [[33/32]], [[36/35]], [[40/39]]
| {{UDnote|step=3}}
|-
| 4
| 66.7
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=4}}
|-
| 5
| 83.3
| [[20/19]], [[21/20]], [[22/21]]
| {{UDnote|step=5}}
|-
| 6
| 100.0
| [[17/16]], [[18/17]], [[19/18]]
| {{UDnote|step=6}}
|-
| 7
| 116.7
| [[15/14]], [[16/15]]
| {{UDnote|step=7}}
|-
| 8
| 133.3
| [[13/12]], [[14/13]], [[27/25]]
| {{UDnote|step=8}}
|-
| 9
| 150.0
| [[12/11]]
| {{UDnote|step=9}}
|-
| 10
| 166.7
| [[11/10]], [[21/19]]
| {{UDnote|step=10}}
|-
| 11
| 183.3
| [[10/9]]
| {{UDnote|step=11}}
|-
| 12
| 200.0
| [[9/8]], [[19/17]]
| {{UDnote|step=12}}
|-
| 13
| 216.7
| [[17/15]], [[25/22]]
| {{UDnote|step=13}}
|-
| 14
| 233.3
| [[8/7]]
| {{UDnote|step=14}}
|-
| 15
| 250.0
| [[15/13]], [[22/19]]
| {{UDnote|step=15}}
|-
| 16
| 266.7
| [[7/6]]
| {{UDnote|step=16}}
|-
| 17
| 283.3
| [[13/11]], [[20/17]]
| {{UDnote|step=17}}
|-
| 18
| 300.0
| [[19/16]], [[25/21]], [[32/27]]
| {{UDnote|step=18}}
|-
| 19
| 316.7
| [[6/5]]
| {{UDnote|step=19}}
|-
| 20
| 333.3
| [[17/14]], ''[[39/32]]'', [[40/33]]
| {{UDnote|step=20}}
|-
| 21
| 350.0
| [[11/9]], [[27/22]]
| {{UDnote|step=21}}
|-
| 22
| 366.7
| [[16/13]], [[21/17]], [[26/21]]
| {{UDnote|step=22}}
|-
| 23
| 383.3
| [[5/4]]
| {{UDnote|step=23}}
|-
| 24
| 400.0
| [[24/19]]
| {{UDnote|step=24}}
|-
| 25
| 416.7
| [[14/11]], [[19/15]]
| {{UDnote|step=25}}
|-
| 26
| 433.3
| [[9/7]]
| {{UDnote|step=26}}
|-
| 27
| 450.0
| [[13/10]], [[22/17]]
| {{UDnote|step=27}}
|-
| 28
| 466.7
| [[17/13]], [[21/16]]
| {{UDnote|step=28}}
|-
| 29
| 483.3
| [[33/25]]
| {{UDnote|step=29}}
|-
| 30
| 500.0
| [[4/3]]
| {{UDnote|step=30}}
|-
| 31
| 516.7
| [[27/20]]
| {{UDnote|step=31}}
|-
| 32
| 533.3
| [[15/11]], [[19/14]], ''[[26/19]]''
| {{UDnote|step=32}}
|-
| 33
| 550.0
| [[11/8]]
| {{UDnote|step=33}}
|-
| 34
| 566.7
| [[18/13]], [[25/18]]
| {{UDnote|step=34}}
|-
| 35
| 583.3
| [[7/5]]
| {{UDnote|step=35}}
|-
| 36
| 600.0
| [[17/12]], [[24/17]]
| {{UDnote|step=36}}
|-
| 37
| 616.7
| [[10/7]]
| {{UDnote|step=37}}
|-
| 38
| 633.3
| [[13/9]], [[36/25]]
| {{UDnote|step=38}}
|-
| 39
| 650.0
| [[16/11]]
| {{UDnote|step=39}}
|-
| 40
| 666.7
| ''[[19/13]]'', [[22/15]], [[28/19]]
| {{UDnote|step=40}}
|-
| 41
| 683.3
| [[40/27]]
| {{UDnote|step=41}}
|-
| 42
| 700.0
| [[3/2]]
| {{UDnote|step=42}}
|-
| 43
| 716.7
| [[50/33]]
| {{UDnote|step=43}}
|-
| 44
| 733.3
| [[26/17]], [[32/21]]
| {{UDnote|step=44}}
|-
| 45
| 750.0
| [[17/11]], [[20/13]]
| {{UDnote|step=45}}
|-
| 46
| 766.7
| [[14/9]]
| {{UDnote|step=46}}
|-
| 47
| 783.3
| [[11/7]], [[30/19]]
| {{UDnote|step=47}}
|-
| 48
| 800.0
| [[19/12]]
| {{UDnote|step=48}}
|-
| 49
| 816.7
| [[8/5]]
| {{UDnote|step=49}}
|-
| 50
| 833.3
| [[13/8]], [[21/13]], [[34/21]]
| {{UDnote|step=50}}
|-
| 51
| 850.0
| [[18/11]], [[44/27]]
| {{UDnote|step=51}}
|-
| 52
| 866.7
| [[28/17]], [[33/20]], ''[[64/39]]''
| {{UDnote|step=52}}
|-
| 53
| 883.3
| [[5/3]]
| {{UDnote|step=53}}
|-
| 54
| 900.0
| [[27/16]], [[32/19]], [[42/25]]
| {{UDnote|step=54}}
|-
| 55
| 916.7
| [[17/10]], [[22/13]]
| {{UDnote|step=55}}
|-
| 56
| 933.3
| [[12/7]]
| {{UDnote|step=56}}
|-
| 57
| 950.0
| [[19/11]], [[26/15]]
| {{UDnote|step=57}}
|-
| 58
| 966.7
| [[7/4]]
| {{UDnote|step=58}}
|-
| 59
| 983.3
| [[30/17]], [[44/25]]
| {{UDnote|step=59}}
|-
| 60
| 1000.0
| [[16/9]], [[34/19]]
| {{UDnote|step=60}}
|-
| 61
| 1016.7
| [[9/5]]
| {{UDnote|step=61}}
|-
| 62
| 1033.3
| [[20/11]], [[38/21]]
| {{UDnote|step=62}}
|-
| 63
| 1050.0
| [[11/6]]
| {{UDnote|step=63}}
|-
| 64
| 1066.7
| [[13/7]], [[24/13]], [[50/27]]
| {{UDnote|step=64}}
|-
| 65
| 1083.3
| [[15/8]], [[28/15]]
| {{UDnote|step=65}}
|-
| 66
| 1100.0
| [[17/9]], [[32/17]], [[36/19]]
| {{UDnote|step=66}}
|-
| 67
| 1116.7
| [[19/10]], [[21/11]], [[40/21]]
| {{UDnote|step=67}}
|-
| 68
| 1133.3
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| {{UDnote|step=68}}
|-
| 69
| 1150.0
| [[35/18]], [[39/20]], [[64/33]]
| {{UDnote|step=69}}
|-
| 70
| 1166.7
| [[49/25]], [[55/28]], [[63/32]], [[88/45]], [[96/49]]
| {{UDnote|step=70}}
|-
| 71
| 1183.3
| [[99/50]], [[160/81]], [[180/91]], [[196/99]], [[208/105]]
| {{UDnote|step=71}}
|-
|-
! Degrees
| 72
| 1200.0
| [[2/1]]
| {{UDnote|step=72}}
|}
<references group="note" />
 
=== Proposed interval names and solfèges ===
{| class="wikitable center-all right-2 left-4 left-7 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Table of proposed interval names and solfèges
|-
! #
! Cents
! Cents
! Approximate Ratios (17-limit)
! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]]
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]
! (K, S, U)
|-
|-
| 0
| 0
| 0.000
| 0.0
| 1/1
| P1
| perfect unison
| D
| P1
| P1
| perfect unison
| perfect unison
| D
| D
| D
|-
|-
| 1
| 1
| 16.667
| 16.7
| 81/80
| ^1
| ^1
| up unison
| up unison
| ^D
| ^D
| K1, L1
| comma-wide unison, large unison
| KD, LD
| KD
|-
|-
| 2
| 2
| 33.333
| 33.3
| 45/44
| ^^
| ^^
| double-up unison
| dup unison
| ^^D
| ^^D
| S1, O1
| super unison, on unison
| SD, OD
| SD
|-
|-
| 3
| 3
| 50.000
| 50.0
| 33/32
| ^<sup>3</sup>1, v<sup>3</sup>m2
| ^<sup>3</sup>1, <br>v<sup>3</sup>m2
| trup unison, trudminor 2nd
| triple-up unison,<br>triple-down minor 2nd
| ^<sup>3</sup>D, v<sup>3</sup>Eb
| ^<sup>3</sup>D, <br>v<sup>3</sup>Eb
| U1, H1, hm2
| uber unison, hyper unison, hypominor 2nd
| UD, HD, uEb
| UD, uEb
|-
|-
| 4
| 4
| 66.667
| 66.7
| 25/24
| vvm2
| vvm2
| double-downminor 2nd
| dudminor 2nd
| vvEb
| vvEb
| kkA1, sm2
| classic aug unison, subminor 2nd
| kkD#, sEb
| sD#, (kkD#), sEb
|-
|-
| 5
| 5
| 83.333
| 83.3
| 21/20
| vm2
| vm2
| downminor 2nd
| downminor 2nd
| vEb
| vEb
| kA1, lm2
| comma-narrow aug unison, little minor 2nd
| kD#, lEb
| kD#, kEb
|-
|-
| 6
| 6
| 100.000
| 100.0
| 35/33, 17/16, 18/17
| m2
| minor 2nd
| Eb
| m2
| m2
| minor 2nd
| minor 2nd
| Eb
| Eb
| Eb
|-
|-
| 7
| 7
| 116.667
| 116.7
| 15/14, 16/15
| ^m2
| ^m2
| upminor 2nd
| upminor 2nd
| ^Eb
| ^Eb
| Km2
| classic minor 2nd
| KEb
| KEb
|-
|-
| 8
| 8
| 133.333
| 133.3
| 27/25, 13/12, 14/13
| ^^m2, v~2
| v~2
| dupminor 2nd, downmid 2nd
| downmid 2nd
| ^^Eb
| ^^Eb
| Om2
| on minor 2nd
| OEb
| SEb
|-
|-
| 9
| 9
| 150.000
| 150.0
| 12/11
| ~2
| ~2
| mid 2nd
| mid 2nd
| v<sup>3</sup>E
| v<sup>3</sup>E
| N2
| neutral 2nd
| UEb/uE
| UEb/uE
|-
|-
| 10
| 10
| 166.667
| 166.7
| 11/10
| ^~2, vvM2
| ^~2
| upmid 2nd, dudmajor 2nd
| upmid 2nd
| vvE
| vvE
| oM2
| off major 2nd
| oE
| sE
|-
|-
| 11
| 11
| 183.333
| 183.3
| 10/9
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
| vE
| vE
| kM2
| classic/comma-narrow major 2nd
| kE
| kE
|-
|-
| 12
| 12
| 200.000
| 200.0
| 9/8
| M2
| major 2nd
| E
| M2
| M2
| major 2nd
| major 2nd
| E
| E
| E
|-
|-
| 13
| 13
| 216.667
| 216.7
| 25/22, 17/15
| ^M2
| ^M2
| upmajor 2nd
| upmajor 2nd
| ^E
| ^E
| LM2
| large major 2nd
| LE
| KE
|-
|-
| 14
| 14
| 233.333
| 233.3
| 8/7
| ^^M2
| ^^M2
| double-upmajor 2nd
| dupmajor 2nd
| ^^E
| ^^E
| SM2
| supermajor 2nd
| SE
| SE
|-
|-
| 15
| 15
| 250.000
| 250.0
| 81/70,  15/13
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| triple-up major 2nd,<br>triple-down minor 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
| hypermajor 2nd, hypominor 3rd
| HE, hF
| UE, uF
|-
|-
| 16
| 16
| 266.667
| 266.7
| 7/6
| vvm3
| vvm3
| double-downminor 3rd
| dudminor 3rd
| vvF
| vvF
| sm3
| subminor 3rd
| sF
| sF
|-
|-
| 17
| 17
| 283.333
| 283.3
| 33/28, 13/11, 20/17
| vm3
| vm3
| downminor 3rd
| downminor 3rd
| vF
| vF
| lm3
| little minor 3rd
| lF
| kF
|-
|-
| 18
| 18
| 300.000
| 300.0
| 25/21
| m3
| m3
| minor 3rd
| minor 3rd
| F
| m3
| minor 3rd
| F
| F
| F
|-
|-
| 19
| 19
| 316.667
| 316.7
| 6/5
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
| ^F
| ^F
| Km3
| classic minor 3rd
| KF
| KF
|-
|-
| 20
| 20
| 333.333
| 333.3
| 40/33, 17/14
| ^^m3, v~3
| v~3
| dupminor 3rd, downmid 3rd
| downmid 3rd
| ^^F
| ^^F
| Om3
| on minor third
| OF
| SF
|-
|-
| 21
| 21
| 350.000
| 350.0
| 11/9
| ~3
| ~3
| mid 3rd
| mid 3rd
| ^<sup>3</sup>F
| ^<sup>3</sup>F
| N3
| neutral 3rd
| UF/uF#
| UF/uF#
|-
|-
| 22
| 22
| 366.667
| 366.7
| 99/80, 16/13, 21/17
| ^~3, vvM3
| ^~3
| upmid 3rd, dudmajor 3rd
| upmid 3rd
| vvF#
| vvF#
| oM3
| off major 3rd
| oF#
| sF#
|-
|-
| 23
| 23
| 383.333
| 383.3
| 5/4
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
| kM3
| classic major 3rd
| kF#
| kF#
|-
|-
| 24
| 24
| 400.000
| 400.0
| 44/35
| M3
| major 3rd
| F#
| M3
| M3
| major 3rd
| major 3rd
| F#
| F#
| F#
|-
|-
| 25
| 25
| 416.667
| 416.7
| 14/11
| ^M3
| ^M3
| upmajor 3rd
| upmajor 3rd
| ^F#
| ^F#
| LM3
| large major 3rd
| LF#
| KF#
|-
|-
| 26
| 26
| 433.333
| 433.3
| 9/7
| ^^M3
| ^^M3
| double-upmajor 3rd
| dupmajor 3rd
| ^^F#
| ^^F#
| SM3
| supermajor 3rd
| SF#
| SF#
|-
|-
| 27
| 27
| 450.000
| 450.0
| 35/27, 13/10
| ^<sup>3</sup>M3, v<sup>3</sup>4
| ^<sup>3</sup>M3, <br>v<sup>3</sup>4
| trupmajor 3rd, trud 4th
| triple-up major 3rd,<br>triple-down 4th
| ^<sup>3</sup>F#, v<sup>3</sup>G
| ^<sup>3</sup>F#, <br>v<sup>3</sup>G
| HM3, h4
| hypermajor 3rd, hypo 4th
| HF#, hG
| UF#, uG
|-
|-
| 28
| 28
| 466.667
| 466.7
| 21/16, 17/13
| vv4
| vv4
| double-down 4th
| dud 4th
| vvG
| vvG
| s4
| sub 4th
| sG
| sG
|-
|-
| 29
| 29
| 483.333
| 483.3
| 33/25
| v4
| v4
| down 4th
| down 4th
| vG
| vG
| l4
| little 4th
| lG
| kG
|-
|-
| 30
| 30
| 500.000
| 500.0
| 4/3
| P4
| P4
| perfect 4th
| perfect 4th
| G
| P4
| perfect 4th
| G
| G
| G
|-
|-
| 31
| 31
| 516.667
| 516.7
| 27/20
| ^4
| ^4
| up 4th
| up 4th
| ^G
| ^G
| K4
| comma-wide 4th
| KG
| KG
|-
|-
| 32
| 32
| 533.333
| 533.3
| 15/11
| ^^4, v~4
| v~4
| dup 4th, downmid 4th
| downmid 4th
| ^^G
| ^^G
| O4
| on 4th
| OG
| SG
|-
|-
| 33
| 33
| 550.000
| 550.0
| 11/8
| ~4
| ~4
| mid 4th
| mid 4th
| ^<sup>3</sup>G
| ^<sup>3</sup>G
| U4/N4
| uber 4th / neutral 4th
| UG
| UG
|-
|-
| 34
| 34
| 566.667
| 566.7
| 25/18, 18/13
| ^~4, vvA4
| ^~4
| upmid 4th, dudaug 4th
| upmid 4th
| vvG#
| vvG#
| kkA4, sd5
| classic aug 4th, sub dim 5th
| kkG#, sAb
| SG#, (kkG#), sAb
|-
|-
| 35
| 35
| 583.333
| 583.3
| 7/5
| vA4, vd5
| vA4, vd5
| downaug 4th, updim 5th
| downaug 4th, <br>downdim 5th
| vG#, vAb
| vG#, vAb
| kA4, ld5
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, kAb
|-
|-
| 36
| 36
| 600.000
| 600.0
| 99/70, 17/12
| A4, d5
| aug 4th, dim 5th
| G#, Ab
| A4, d5
| A4, d5
| aug 4th, dim 5th
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
| G#, Ab
|-
|-
| 37
| 37
| 616.667
| 616.7
| 10/7
| ^A4, ^d5
| ^A4, ^d5
| upaug 4th, downdim 5th
| upaug 4th, updim 5th
| ^G#, ^Ab
| ^G#, ^Ab
| LA4, Kd5
| large aug 4th, comma-wide dim 5th
| LG#, KAb
| KG#, KAb
|-
|-
| 38
| 38
| 633.333
| 633.3
| 36/25, 13/9
| v~5, ^^d5
| v~5
| downmid 5th, <br>dupdim 5th
| downmid 5th
| ^^Ab
| ^^Ab
| SA4, KKd5
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, SAb, (KKAb)
|-
|-
| 39
| 39
| 650.000
| 650.0
| 16/11
| ~5
| ~5
| mid 5th
| mid 5th
| v<sup>3</sup>A
| v<sup>3</sup>A
| u5/N5
| unter 5th / neutral 5th
| uA
| uA
|-
|-
| 40
| 40
| 666.667
| 666.7
| 22/15
| vv5, ^~5
| ^~5
| dud 5th, upmid 5th
| upmid 5th
| vvA
| vvA
| o5
| off 5th
| oA
| sA
|-
|-
| 41
| 41
| 683.333
| 683.3
| 40/27
| v5
| v5
| down 5th
| down 5th
| vA
| vA
| k5
| comma-narrow 5th
| kA
| kA
|-
|-
| 42
| 42
| 700.000
| 700.0
| 3/2
| P5
| perfect 5th
| A
| P5
| P5
| perfect 5th
| perfect 5th
| A
| A
| A
|-
|-
| 43
| 43
| 716.667
| 716.7
| 50/33
| ^5
| ^5
| up 5th
| up 5th
| ^A
| ^A
| L5
| large fifth
| LA
| KA
|-
|-
| 44
| 44
| 733.333
| 733.3
| 32/21
| ^^5
| ^^5
| double-up 5th
| dup 5th
| ^^A
| ^^A
| S5
| super fifth
| SA
| SA
|-
|-
| 45
| 45
| 750.000
| 750.0
| 54/35, 17/11
| ^<sup>3</sup>5, v<sup>3</sup>m6
| ^<sup>3</sup>5, <br>v<sup>3</sup>m6
| trup 5th, trudminor 6th
| triple-up 5th,<br>triple-down minor 6th
| ^<sup>3</sup>A, v<sup>3</sup>Bb
| ^<sup>3</sup>A, <br>v<sup>3</sup>Bb
| H5, hm6
| hyper fifth, hypominor 6th
| HA, hBb
| UA, uBb
|-
|-
| 46
| 46
| 766.667
| 766.7
| 14/9
| vvm6
| vvm6
| double-downminor 6th
| dudminor 6th
| vvBb
| vvBb
| sm6
| superminor 6th
| sBb
| sBb
|-
|-
| 47
| 47
| 783.333
| 783.3
| 11/7
| vm6
| vm6
| downminor 6th
| downminor 6th
| vBb
| vBb
| lm6
| little minor 6th
| lBb
| kBb
|-
|-
| 48
| 48
| 800.000
| 800.0
| 35/22
| m6
| minor 6th
| Bb
| m6
| m6
| minor 6th
| minor 6th
| Bb
| Bb
| Bb
|-
|-
| 49
| 49
| 816.667
| 816.7
| 8/5
| ^m6
| ^m6
| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
| Km6
| classic minor 6th
| kBb
| kBb
|-
|-
| 50
| 50
| 833.333
| 833.3
| 81/50, 13/8
| ^^m6, v~6
| v~6
| dupminor 6th, downmid 6th
| downmid 6th
| ^^Bb
| ^^Bb
| Om6
| on minor 6th
| oBb
| sBb
|-
|-
| 51
| 51
| 850.000
| 850.0
| 18/11
| ~6
| ~6
| mid 6th
| mid 6th
| v<sup>3</sup>B
| v<sup>3</sup>B
| N6
| neutral 6th
| UBb, uB
| UBb, uB
|-
|-
| 52
| 52
| 866.667
| 866.7
| 33/20, 28/17
| ^~6, vvM6
| ^~6
| upmid 6th, dudmajor 6th
| upmid 6th
| vvB
| vvB
| oM6
| off major 6th
| oB
| sB
|-
|-
| 53
| 53
| 883.333
| 883.3
| 5/3
| vM6
| vM6
| downmajor 6th
| downmajor 6th
| vB
| vB
| kM6
| classic major 6th
| kB
| kB
|-
|-
| 54
| 54
| 900.000
| 900.0
| 27/16
| M6
| M6
| major 6th
| major 6th
| B
| M6
| major 6th
| B
| B
| B
|-
|-
| 55
| 55
| 916.667
| 916.7
| 56/33, 17/10
| ^M6
| ^M6
| upmajor 6th
| upmajor 6th
| ^B
| ^B
| LM6
| large major 6th
| LB
| KB
|-
|-
| 56
| 56
| 933.333
| 933.3
| 12/7
| ^^M6
| ^^M6
| double-upmajor 6th
| dupmajor 6th
| ^^B
| ^^B
| SM6
| supermajor 6th
| SB
| SB
|-
|-
| 57
| 57
| 950.000
| 950.0
| 121/70
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| triple-up major 6th,<br>triple-down minor 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
| hypermajor 6th, hypominor 7th
| HB, hC
| UB, uC
|-
|-
| 58
| 58
| 966.667
| 966.7
| 7/4
| vvm7
| vvm7
| double-downminor 7th
| dudminor 7th
| vvC
| vvC
| sm7
| subminor 7th
| sC
| sC
|-
|-
| 59
| 59
| 983.333
| 983.3
| 44/25
| vm7
| vm7
| downminor 7th
| downminor 7th
| vC
| vC
| lm7
| little minor 7th
| lC
| kC
|-
|-
| 60
| 60
| 1000.000
| 1000.0
| 16/9
| m7
| m7
| minor 7th
| minor 7th
| C
| m7
| minor 7th
| C
| C
| C
|-
|-
| 61
| 61
| 1016.667
| 1016.7
| 9/5
| ^m7
| ^m7
| upminor 7th
| upminor 7th
| ^C
| ^C
| Km7
| classic/comma-wide minor 7th
| KC
| KC
|-
|-
| 62
| 62
| 1033.333
| 1033.3
| 20/11
| ^^m7, v~7
| v~7
| dupminor 7th, downmid 7th
| downmid 7th
| ^^C
| ^^C
| Om7
| on minor 7th
| OC
| SC
|-
|-
| 63
| 63
| 1050.000
| 1050.0
| 11/6
| ~7
| ~7
| mid 7th
| mid 7th
| ^<sup>3</sup>C
| ^<sup>3</sup>C
| N7, hd8
| neutral 7th, hypo dim 8ve
| UC/uC#, hDb
| UC/uC#, uDb
|-
|-
| 64
| 64
| 1066.667
| 1066.7
| 50/27
| ^~7, vvM7
| ^~7
| upmid 7th, dudmajor 7th
| upmid 7th
| vvC#
| vvC#
| oM7, sd8
| off major 7th, sub dim 8ve
| oC#, sDb
| sC#, sDb
|-
|-
| 65
| 65
| 1083.333
| 1083.3
| 15/8
| vM7
| vM7
| downmajor 7th
| downmajor 7th
| vC#
| vC#
| kM7, ld8
| classic major 7th, little dim 8ve
| kC#, lDb
| kC#, kDb
|-
|-
| 66
| 66
| 1100.000
| 1100.0
| 66/35, 17/9
| M7
| M7
| major 7th
| major 7th
| C#
| C#
| M7, d8
| major 7th, dim 8ve
| C#, Db
| C#, Db
|-
|-
| 67
| 67
| 1116.667
| 1116.7
| 21/11
| ^M7
| ^M7
| upmajor 7th
| upmajor 7th
| ^C#
| ^C#
| LM7, Kd8
| large major 7th, comma-wide dim 8ve
| LC#, KDb
| KC#, KDb
|-
|-
| 68
| 68
| 1133.333
| 1133.3
| 27/14
| ^^M7
| ^^M7
| double-upmajor 7th
| dupmajor 7th
| ^^C#
| ^^C#
| SM7, KKd8
| supermajor 7th, classic dim 8ve
| SC#, KKDb
| SC#, SDb, (KKDb)
|-
|-
| 69
| 69
| 1150.000
| 1150.0
| 35/18
| ^<sup>3</sup>M7, v<sup>3</sup>8
| ^<sup>3</sup>M7, <br>v<sup>3</sup>8
| trupmajor 7th, trud octave
| triple-up major 7th,<br>triple-down octave
| ^<sup>3</sup>C#, v<sup>3</sup>D
| ^<sup>3</sup>C#, <br>v<sup>3</sup>D
| HM7, u8, h8
| hypermajor 7th, unter 8ve, hypo 8ve
| HC#, uD, hD
| UC#, uDb, uD
|-
|-
| 70
| 70
| 1166.667
| 1166.7
| 49/25
| vv8
| vv8
| double-down octave
| dud octave
| vvD
| vvD
| s8, o8
| sub 8ve, off 8ve
| sD, oD
| sD
|-
|-
| 71
| 71
| 1183.333
| 1183.3
| 99/50
| v8
| v8
| down octave
| down octave
| vD
| vD
| k8, l8
| comma-narrow 8ve, little 8ve
| kD, lD
| kD
|-
|-
| 72
| 72
| 1200.000
| 1200.0
| 2/1
| P8
| perfect octave
| D
| P8
| P8
| perfect octave
| perfect octave
| D
| D
| D
|}
|}


Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
=== 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"
{| class="wikitable center-all"
|-
|-
! quality
! Quality
! [[Kite's color notation|color]]
! [[Color notation|Color]]
! monzo format
! Monzo format
! examples
! Examples
|-
|-
| double-down minor
| dudminor
| zo
| zo
| {a, b, 0, 1}
| (a b 0 1)
| 7/6, 7/4
| [[7/6]], [[7/4]]
|-
|-
| minor
| minor
| fourthward wa
| fourthward wa
| {a, b}, b &lt; -1
| (a b), b < -1
| 32/27, 16/9
| [[32/27]], [[16/9]]
|-
|-
| upminor
| upminor
| gu
| gu
| {a, b, -1}
| (a b -1)
| 6/5, 9/5
| [[6/5]], [[9/5]]
|-
| rowspan="2" | dupminor, <br>downmid
| luyo
| (a b 1 0 -1)
| [[15/11]]
|-
|-
| mid
| tho
| (a b 0 0 0 1)
| [[13/8]], [[13/9]]
|-
| rowspan="2" | mid
| ilo
| ilo
| {a, b, 0, 0, 1}
| (a b 0 0 1)
| 11/9, 11/6
| [[11/9]], [[11/6]]
|-
|-
| "
| lu
| lu
| {a, b, 0, 0, -1}
| (a b 0 0 -1)
| 12/11, 18/11
| [[12/11]], [[18/11]]
|-
| rowspan="2" | upmid, <br>dudmajor
| logu
| (a b -1 0 1)
| [[11/10]]
|-
| thu
| (a b 0 0 0 -1)
| [[16/13]], [[18/13]]
|-
|-
| downmajor
| downmajor
| yo
| yo
| {a, b, 1}
| (a b 1)
| 5/4, 5/3
| [[5/4]], [[5/3]]
|-
|-
| major
| major
| fifthward wa
| fifthward wa
| {a, b}, b &gt; 1
| (a b), b > 1
| 9/8, 27/16
| [[9/8]], [[27/16]]
|-
|-
| double-up major
| dupmajor
| ru
| ru
| {a, b, 0, -1}
| (a b 0 -1)
| 9/7, 12/7
| [[9/7]], [[12/7]]
|-
| rowspan="2" | trupmajor, <br>trudminor
| thogu
| (a b -1 0 0 1)
| [[13/10]]
|-
| thuyo
| (a b 1 0 0 -1)
| [[15/13]]
|}
|}
All 72-edo 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:


{| 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 600: Line 1,240:
| C vvEb G
| C vvEb G
| Cvvm
| Cvvm
| C double-down minor
| C dudminor
|-
|-
| gu
| gu
Line 628: Line 1,268:
| C ^^E G
| C ^^E G
| C^^
| C^^
| C double-upmajor or C double-up
| 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]].  


== Just approximation ==
=== Relationship between primes and rings ===
{| class="wikitable center-all"
In 72tet, there are 6 [[ring number|rings]]. 12edo is the plain ring; thus every 6 degrees is the 3-limit.
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
! prime 23
! prime 29
! prime 31
|-
! rowspan="2" |Error
!absolute (¢)
| 0.000
| -1.955
| -2.980
| -2.159
| -1.318
| -7.194
| -4.955
| +2.487
| +5.059
| +3.756
| +4.964
|-
![[Relative error|relative]] (%)
|0.0
| -11.7
| -17.9
| -13.0
| -7.9
| -43.2
| -29.7
| +14.9
| +30.4
| +22.5
| +29.8
|}


== Commas ==
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 descending [[81/80]]
* −2 degrees (the dud ring) corrects [[16/9]] to [[7/4]] via descending [[64/63]]
* +3 degrees  (the trup ring) corrects [[4/3]] to [[11/8]] via [[33/32]]
* +2 degrees (the dup ring) corrects [[128/81]] to [[13/8]] via [[1053/1024]]
* 0 degrees (the plain ring) corrects [[256/243]] to [[17/16]] via [[4131/4096]]
* 0 degrees (the plain ring) corrects [[32/27]] to [[19/16]] via [[513/512]]
Thus the product of a ratio's monzo with {{map| 0 0 -1 -2 3 2 0 0 }}, modulo 6, specifies which ring the ratio lies on.


Commas tempered out by 72edo include…
== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}


{| class="wikitable"
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
|-
{{Sharpness-sharp6-qt-szg}}
! | 3-limit
|-
| |<nowiki> Pythagorean comma = 531441/524288 = |-19 12</nowiki>&gt;
|}


{| class="wikitable"
=== Kite's ups and downs notation ===
|-
72edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
! | 5-limit
{{Ups and downs sharpness}}
|-
| |<nowiki> kleisma = 15625/15552 = |-6 -5 6</nowiki>&gt;


ampersand = 34171875/33554432 = |-25 7 6&gt;
Half-sharps and half-flats can be used to avoid triple arrows:
{{Ups and downs sharpness|72|true}}


graviton = 129140163/128000000 = |-13 17 -6&gt;
=== 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]].


ennealimma = 7629394531250/7625597484987 = |1 -27 18&gt;
==== Evo flavor ====
|}
{{Sagittal chart|Evo}}


{| class="wikitable"
==== Evo-SZ flavor ====
|-
{{Sagittal chart|Evo-SZ}}
! | 7-limit
! | 11-limit
! | 13-limit
|-
| | ...............................


225/224
==== Revo flavor ====
{{Sagittal chart}}


1029/1024
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:


2401/2400
<div class="noresize">
[[File:72edo Sagittal.png]]
</div>


4375/4374
=== Ivan Wyschnegradsky's notation ===
{{Sharpness-sharp6-iw|72}}


16875/16807
== Approximation to JI ==
[[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]]


19683/19600
=== Interval mappings ===
{{Q-odd-limit intervals|72}}


420175/419904
=== 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.


250047/250000
[[File:plot72.png|alt=plot72.png|plot72.png]]
| | .......................


243/242
== 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]] (%)
|-
| 2.3.5
| 15625/15552, 531441/524288
| {{Mapping| 72 114 167 }}
| +0.839
| 0.594
| 3.56
|-
| 2.3.5.7
| 225/224, 1029/1024, 4375/4374
| {{Mapping| 72 114 167 202 }}
| +0.822
| 0.515
| 3.09
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 4000/3993
| {{Mapping| 72 114 167 202 249 }}
| +0.734
| 0.493
| 2.96
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 243/242, 325/324, 385/384
| {{Mapping| 72 114 167 202 249 266 }}
| +0.936
| 0.638
| 3.82
|-
| 2.3.5.7.11.13.17
| 169/168, 221/220, 225/224, 243/242, 273/272, 325/324
| {{Mapping| 72 114 167 202 249 266 294 }}
| +0.975
| 0.599
| 3.59
|-
| 2.3.5.7.11.13.17.19
| 153/152, 169/168, 210/209, 221/220, 225/224, 243/242, 273/272
| {{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.


385/384
=== Commas ===
Commas tempered out by 72edo include…


441/440
{| class="commatable wikitable center-1 center-2 right-4"
 
|-
540/539
! [[Harmonic limit|Prime<br>limit]]
 
! [[Ratio]]<ref group="note">{{rd}}</ref>
1375/1372
! [[Monzo]]
 
! [[Cents]]
3025/3024
! Name(s)
 
|-
4000/3993
| 3
 
| [[531441/524288|(12 digits)]]
6250/6237
| {{Monzo| -19 12 }}
 
| 23.46
9801/9800
| Pythagorean comma
| | .......................
|-
 
| 5
169/168
| [[15625/15552]]
 
| {{Monzo| -6 -5 6 }}
325/324
| 8.11
 
| Kleisma
351/350
|-
 
| 5
364/363
| [[34171875/33554432|(16 digits)]]
 
| {{Monzo| -25 7 6 }}
625/624
| 31.57
 
| [[Ampersand comma]]
676/675
|-
 
| 5
729/728
| [[129140163/128000000|(18 digits)]]
 
| {{Monzo| -13 17 -6 }}
1001/1000
| 15.35
 
| [[Graviton]]
1575/1573
|-
 
| 5
1716/1715
| <abbr title="7629394531250/7625597484987">(26 digits)</abbr>
 
| {{Monzo| 1 -27 18 }}
2080/2079
| 0.86
 
| [[Ennealimma]]
6656/6655
|-
| 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" />


== Temperaments ==
=== Rank-2 temperaments ===
 
* [[List of edo-distinct 72et rank two 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.
72edo provides the [[optimal patent val]] for [[miracle]] and [[wizard]] in the 7-limit, miracle, [[catakleismic]], [[bikleismic]], [[compton]], [[ennealimnic]], [[ennealiminal]], [[enneaportent]], [[marvolo]] and [[catalytic]] in the 11-limit, and catakleismic, bikleismic, compton, [[comptone]], [[enneaportent]], [[ennealim]], catalytic, marvolo, [[manna]], [[hendec]], [[lizard]], [[neominor]], [[hours]], and [[semimiracle]] in the 13-limit.


{| class="wikitable center-1 center-2"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br>per octave
! Periods<br>per 8ve
! Generator
! Generator*
! Names
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
| 1
| 1
| 1\72
| 1\72
| 16.7
| 105/104
| [[Quincy]]
| [[Quincy]]
|-
|-
| 1
| 1
| 5\72
| 5\72
| 83.3
| 21/20
| [[Marvolo]]
| [[Marvolo]]
|-
|-
| 1
| 1
| 7\72
| 7\72
| [[Miracle]]/benediction/manna
| 116.7
|-
| 15/14
| 1
| [[Miracle]] / benediction / manna
| 11\72
|
|-
| 1
| 13\72
|
|-
|-
| 1
| 1
| 17\72
| 17\72
| 283.3
| 13/11
| [[Neominor]]
| [[Neominor]]
|-
|-
| 1
| 1
| 19\72
| 19\72
| 316.7
| 6/5
| [[Catakleismic]]
| [[Catakleismic]]
|-
| 1
| 23\72
|
|-
|-
| 1
| 1
| 25\72
| 25\72
| 416.7
| 14/11
| [[Sqrtphi]]
| [[Sqrtphi]]
|-
|-
| 1
| 1
| 29\72
| 29\72
|  
| 483.3
| 45/34
| [[Hemiseven]]
|-
|-
| 1
| 1
| 31\72
| 31\72
| [[Marvo]]/zarvo
| 516.7
| 27/20
| [[Gravity]] / [[marvo]] / [[zarvo]]
|-
|-
| 1
| 1
| 35\72
| 35\72
| 583.3
| 7/5
| [[Cotritone]]
| [[Cotritone]]
|-
| 2
| 1\72
|
|-
|-
| 2
| 2
| 5\72
| 5\72
| 83.3
| 21/20
| [[Harry]]
| [[Harry]]
|-
|-
| 2
| 2
| 7\72
| 7\72
|  
| 116.7
| 15/14
| [[Semimiracle]]
|-
|-
| 2
| 2
| 11\72
| 11\72
| [[Unidec]]/hendec
| 183.3
| 10/9
| [[Unidec]] / hendec
|-
|-
| 2
| 2
| 13\72
| 21\72<br>(19\72)
| [[Wizard]]/lizard/gizzard
| 316.7<br>(283.3)
| 6/5<br>(13/11)
| [[Bikleismic]]
|-
|-
| 2
| 2
| 17\72
| 23\72<br>(13\72)
|  
| 383.3<br>(216.7)
| 5/4<br>(17/15)
| [[Wizard]] / lizard / gizzard
|-
|-
| 3
| 3
| 1\72
| 11\72
|  
| 183.3
| 10/9
| [[Mirkat]]
|-
|-
| 3
| 3
| 5\72
| 19\72<br>(5\72)
| 316.7<br>(83.3)
| 6/5<br>(21/20)
| [[Tritikleismic]]
| [[Tritikleismic]]
|-
| 3
| 7\72
|
|-
| 3
| 11\72
| [[Mirkat]]
|-
|-
| 4
| 4
| 1\72
| 19\72<br>(1\72)
| 316.7<br>(16.7)
| 6/5<br>(105/104)
| [[Quadritikleismic]]
| [[Quadritikleismic]]
|-
| 4
| 5\72
|
|-
| 4
| 7\72
|
|-
| 6
| 1\72
|
|-
| 6
| 5\72
|
|-
| 8
| 1\72
| [[Octoid]]
|-
|-
| 8
| 8
| 2\72
| 34\72<br>(2\72)
| [[Octowerck]]
| 566.7<br>(33.3)
| 168/121<br>(55/54)
| [[Octowerck]] / octowerckis
|-
|-
| 8
| 8
| 4\72
| 35\72<br>(1\72)
|  
| 583.3<br>(16.7)
| 7/5<br>(100/99)
| [[Octoid]] / octopus
|-
|-
| 9
| 9
| 1\72
| 19\72<br>(3\72)
|  
| 316.7<br>(50.0)
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimnic / ennealiminal
|-
|-
| 9
| 9
| 3\72
| 23\72<br>(1\72)
| [[Ennealimmal]]/ennealimmic
| 383.3<br>(16.7)
| 5/4<br>(105/104)
| [[Enneaportent]]
|-
|-
| 12
| 12
| 1\72
| 23\72<br>(1\72)
| [[Compton]]
| 383.3<br>(16.7)
| 5/4<br>(100/99)
| [[Compton]] / comptone
|-
|-
| 18
| 18
| 1\72
| 19\72<br>(1\72)
| 316.7<br>(16.7)
| 6/5<br>(105/104)
| [[Hemiennealimmal]]
| [[Hemiennealimmal]]
|-
|-
| 24
| 24
| 1\72
| 23\72<br>(1\72)
| 383.3<br>(16.7)
| 5/4<br>(105/104)
| [[Hours]]
| [[Hours]]
|-
|-
| 36
| 36
| 1\72
| 23\72<br>(1\72)
|  
| 383.3<br>(16.7)
| 5/4<br>(81/80)
| [[Gamelstearn]]
|}
|}
<nowiki/>* [[Normal 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 ==
== Scales ==
* [[smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]]
; [[Miracle]]-tempered scales
* [[blackjack]], [[miracle_8]], [[miracle_10]], [[miracle_12]], [[miracle_12a]], [[miracle_24hi]], [[miracle_24lo]]
* [[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]]
; [[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]]
* [[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 ===
=== Harmonic scale ===
Mode 8 of the harmonic series – [[overtone_scales|overtones 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"
|-
|-
| | Overtones in "Mode 8":
! Harmonics in "Mode 8":
| | 8
| 8
| |  
|  
| | 9
| 9
| |  
|  
| | 10
| 10
| |  
|  
| | 11
| 11
| |  
|  
| | 12
| 12
| |  
|  
| | 13
| 13
| |  
|  
| | 14
| 14
| |  
|  
| | 15
| 15
| |  
|  
| | 16
| 16
|-
|-
| | …as JI Ratio from 1/1:
! …as JI Ratio from 1/1:
| | 1/1
| 1/1
| |  
|  
| | 9/8
| 9/8
| |  
|  
| | 5/4
| 5/4
| |  
|  
| | 11/8
| 11/8
| |  
|  
| | 3/2
| 3/2
| |  
|  
| | 13/8
| 13/8
| |  
|  
| | 7/4
| 7/4
| |  
|  
| | 15/8
| 15/8
| |  
|  
| | 2/1
| 2/1
|-
|-
| | …in cents:
! …in cents:
| | 0
| 0
| |  
|  
| | 203.9
| 203.9
| |  
|  
| | 386.3
| 386.3
| |  
|  
| | 551.3
| 551.3
| |  
|  
| | 702.0
| 702.0
| |  
|  
| | 840.5
| 840.5
| |  
|  
| | 968.8
| 968.8
| |  
|  
| | 1088.3
| 1088.3
| |  
|  
| | 1200.0
| 1200.0
|-
|-
| | Nearest degree of 72edo:
! Nearest degree of 72edo:
| | 0
| 0
| |  
|  
| | 12
| 12
| |  
|  
| | 23
| 23
| |  
|  
| | 33
| 33
| |  
|  
| | 42
| 42
| |  
|  
| | 50
| 50
| |  
|  
| | 58
| 58
| |  
|  
| | 65
| 65
| |  
|  
| | 72
| 72
|-
|-
| | …in cents:
! …in cents:
| | 0
| 0
| |  
|  
| | 200.0
| 200.0
| |  
|  
| | 383.3
| 383.3
| |  
|  
| | 550.0
| 550.0
| |  
|  
| | 700.0
| 700.0
| |  
|  
| | 833.3
| 833.3
| |  
|  
| | 966.7
| 966.7
| |  
|  
| | 1083.3
| 1083.3
| |  
|  
| | 1200.0
| 1200.0
|-
|-
| | Steps as Freq. Ratio:
! Steps as Freq. Ratio:
| |  
|  
| | 9:8
| 9:8
| |  
|  
| | 10:9
| 10:9
| |  
|  
| | 11:10
| 11:10
| |  
|  
| | 12:11
| 12:11
| |  
|  
| | 13:12
| 13:12
| |  
|  
| | 14:13
| 14:13
| |  
|  
| | 15:14
| 15:14
| |  
|  
| | 16:15
| 16:15
| |  
|  
|-
|-
| | …in cents:
! …in cents:
| |  
|  
| | 203.9
| 203.9
| |  
|  
| | 182.4
| 182.4
| |  
|  
| | 165.0
| 165.0
| |  
|  
| | 150.6
| 150.6
| |  
|  
| | 138.6
| 138.6
| |  
|  
| | 128.3
| 128.3
| |  
|  
| | 119.4
| 119.4
| |  
|  
| | 111.7
| 111.7
| |  
|  
|-
|-
| | Nearest degree of 72edo:
! Nearest degree of 72edo:
| |  
|  
| | 12
| 12
| |  
|  
| | 11
| 11
| |  
|  
| | 10
| 10
| |  
|  
| | 9
| 9
| |  
|  
| | 8
| 8
| |  
|  
| | 8
| 8
| |  
|  
| | 7
| 7
| |  
|  
| | 7
| 7
| |  
|  
|-
|-
| | ...in cents:
! …in cents:
| |  
|  
| | 200.0
| 200.0
| |  
|  
| | 183.3
| 183.3
| |  
|  
| | 166.7
| 166.7
| |  
|  
| | 150.0
| 150.0
| |  
|  
| | 133.3
| 133.3
| |  
|  
| | 133.3
| 133.3
| |  
|  
| | 116.7
| 116.7
| |  
|  
| | 116.7
| 116.7
| |  
|  
|}
|}


== Z function ==
== Instruments ==
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.
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]]


[[File:plot72.png|alt=plot72.png|plot72.png]]
Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo:
* [[Lumatone mapping for 72edo]]


== Music ==
== Music ==
[http://www.archive.org/details/Kotekant Kotekant] ''[http://www.archive.org/download/Kotekant/kotekant.mp3 play]'' by [[Gene_Ward_Smith|Gene Ward Smith]]
; [[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)
 
; [[Prent Rodgers]]
* [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]]
* [https://www.archive.org/details/Kotekant ''Kotekant''] [https://www.archive.org/download/Kotekant/kotekant.mp3 play] (2010)


''[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 Twinkle canon – 72 edo]'' by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
;[[Ivan Wyschnegradsky]]
* [https://www.youtube.com/watch?v=RCcJHCkYQ6U ''Arc-en-ciel, for 6 pianos in twelfth tones, Op. 37''] (1956)


''[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 Lazy Sunday]'' by [[Jake_Freivald|Jake Freivald]] in the [[lazysunday|lazysunday]] scale.
; [[James Tenney]]
* [https://www.youtube.com/watch?v=jGsxqU1PhZs&list=OLAK5uy_mKyMEMZW7noeLncJnu-JT65go8w7403DA ''Changes for Six Harps'']


''[http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 June Gloom #9]'' by Prent Rodgers
; [[Xeno Ov Eleas]]
* [https://www.youtube.com/watch?v=cx7I0NWem5w ''Χenomorphic Ghost Storm''] (2022)


== External links ==
== External links ==
* [[Wikipedia:72_equal_temperament|72 equal temperament - Wikipedia]]
* [http://orthodoxwiki.org/Byzantine_Chant OrthodoxWiki Article on Byzantine chant, which uses 72edo]
* [http://orthodoxwiki.org/Byzantine_Chant OrthodoxWiki Article on Byzantine chant, which uses 72edo]
* [http://en.wikipedia.org/wiki/Joe_Maneri Wikipedia article on Joe Maneri (1927-2009)]
* [http://www.ekmelic-music.org/en/ Ekmelic Music Society/Gesellschaft für Ekmelische Musik], a group of composers and researchers dedicated to 72edo music
* [http://www.ekmelic-music.org/en/ Ekmelic Music Society/Gesellschaft für Ekmelische Musik], a group of composers and researchers dedicated to 72edo music
* [http://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list
* [http://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list
* [http://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo] [[Category:Edo]] [[Category:72edo| ]] <!-- main article -->
* [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]]
 
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