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__FORCETOC__
{{interwiki
{{interwiki
| de = 72edo
| de = 72-EDO
| en = 72edo
| en = 72edo
| es =  
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}}
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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]].
 
The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.
 
The [[octave 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]].


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.
=== 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)}}


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.
=== 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]].  


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]].
=== 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}}
|-
| 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
|-
|-
! Degrees
! #
! 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
| minor 3rd
| F
| m3
| m3
| minor 3rd
| 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
| perfect 4th
| G
| P4
| P4
| perfect 4th
| 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
| m6
| minor 6th
| minor 6th
| Bb
| m6
| 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
| minor 7th
| C
| m7
| m7
| minor 7th
| 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]].  
 
=== 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.
 
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.
 
== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}
 
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt-szg}}
 
=== 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.
{{Ups and downs sharpness}}
 
Half-sharps and half-flats can be used to avoid triple arrows:
{{Ups and downs sharpness|72|true}}


== Just approximation ==
=== 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]].


=== Selected just intervals ===
==== Evo flavor ====
{| class="wikitable center-all"
{{Sagittal chart|Evo}}
! colspan="2" |
 
! prime 2
==== Evo-SZ flavor ====
! prime 3
{{Sagittal chart|Evo-SZ}}
! prime 5
 
! prime 7
==== Revo flavor ====
! prime 11
{{Sagittal chart}}
! prime 13
 
! prime 17
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:
! prime 19
 
! prime 23
<div class="noresize">
! prime 29
[[File:72edo Sagittal.png]]
! prime 31
</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 [[the Riemann zeta function and tuning#The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.
 
[[File:plot72.png|alt=plot72.png|plot72.png]]
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
! rowspan="2" |Error
! rowspan="2" | [[Subgroup]]
! absolute (¢)
! rowspan="2" | [[Comma list]]
| 0.000
! rowspan="2" | [[Mapping]]
| -1.955
! rowspan="2" | Optimal<br>8ve stretch (¢)
| -2.980
! colspan="2" | Tuning error
| -2.159
| -1.318
| -7.194
| -4.955
| +2.487
| +5.059
| +3.756
| +4.964
|-
|-
! [[Relative error|relative]] (%)
! [[TE error|Absolute]] (¢)
| 0.0
! [[TE simple badness|Relative]] (%)
| -11.7
| -17.9
| -13.0
| -7.9
| -43.2
| -29.7
| +14.9
| +30.4
| +22.5
| +29.8
|}
 
=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 72et.
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 11-limit
! 13-limit
! 17-limit
! 19-limit
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3.5
| +0.617
| 15625/15552, 531441/524288
| {{Mapping| 72 114 167 }}
| +0.839
| +0.839
| 0.594
| 3.56
|-
| 2.3.5.7
| 225/224, 1029/1024, 4375/4374
| {{Mapping| 72 114 167 202 }}
| +0.822
| +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
| +0.734
| 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
| +0.936
| 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
| +0.975
| 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
| +0.780
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 0.594
| 0.515
| 0.493
| 0.638
| 0.599
| 0.762
| 0.762
|-
! [[TE simple badness|relative]] (%)
| 3.70
| 3.56
| 3.09
| 2.96
| 3.82
| 3.59
| 4.57
| 4.57
|}
|}
* 72et has a lower relative error than any previous ETs in the 7-, 11-, 13-, 17-, and 19-limit. The next ET that does better in these subgroups is 99, 270, 224, 494, and 217, respectively.
* 72et has lower relative errors than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next equal temperaments doing better in these subgroups are [[99edo|99]], [[270edo|270]], [[224edo|224]], [[494edo|494]], and [[217edo|217]], respectively.
 
=== 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]]
 
== Commas ==


=== Commas ===
Commas tempered out by 72edo include…
Commas tempered out by 72edo include…


{| class="wikitable center-1 center-2 right-4"
{| class="commatable wikitable center-1 center-2 right-4"
! [[Harmonic limit|Prime<br>Limit]]
|-
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
Line 737: Line 1,400:
| 3
| 3
| [[531441/524288|(12 digits)]]
| [[531441/524288|(12 digits)]]
| {{Monzo|-19 12 }}
| {{Monzo| -19 12 }}
| 23.46
| 23.46
| Pythagorean comma
| Pythagorean comma
Line 751: Line 1,414:
| {{Monzo| -25 7 6 }}
| {{Monzo| -25 7 6 }}
| 31.57
| 31.57
| [[Ampersand]]
| [[Ampersand comma]]
|-
|-
| 5
| 5
Line 769: Line 1,432:
| {{Monzo| -5 2 2 -1 }}
| {{Monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Septimal kleisma, Marvel comma
| Marvel comma
|-
|-
| 7
| 7
| [[1029/1024]]
| [[1029/1024]]
|
| {{Monzo| -10 1 0 3 }}
|
| 8.43
|
| Gamelisma
|-
|-
| 7
| 7
Line 791: Line 1,454:
| 7
| 7
| [[16875/16807]]
| [[16875/16807]]
|
| {{Monzo| 0 3 4 -5 }}
|
| 6.99
|
| Mirkwai comma
|-
|-
| 7
| 7
| [[19683/19600]]
| [[19683/19600]]
|
| {{Monzo| -4 9 -2 -2 }}
|
| 7.32
|
| Cataharry comma
|-
|-
| 7
| 7
Line 839: Line 1,502:
| 11
| 11
| [[1375/1372]]
| [[1375/1372]]
|
| {{Monzo| -2 0 3 -3 1 }}
|
| 3.78
|
| Moctdel comma
|-
|-
| 11
| 11
Line 851: Line 1,514:
| 11
| 11
| [[4000/3993]]
| [[4000/3993]]
|
| {{Monzo| 5 -1 3 0 -3 }}
|
| 3.03
|
| Wizardharry comma
|-
|-
| 11
| 11
| [[6250/6237]]
| [[6250/6237]]
|
| {{Monzo| 1 -4 5 -1 -1 }}
|
| 3.60
|
| Liganellus comma
|-
|-
| 11
| 11
Line 865: Line 1,528:
| {{Monzo| -3 4 -2 -2 2 }}
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Kalisma, Gauss' comma
| Kalisma
|-
| 11
| <abbr title="1771561/1769472">(14 digits)</abbr>
| {{Monzo| 16 -3 0 0 6 }}
| 2.04
| [[Nexus comma]]
|-
|-
| 13
| 13
| [[169/168]]
| [[169/168]]
|
| {{Monzo| -3 -1 0 -1 0 2 }}
|
| 10.27
|
| Buzurgisma
|-
|-
| 13
| 13
Line 889: Line 1,558:
| {{Monzo| 2 -1 0 1 -2 1 }}
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.76
| 4.76
| Gentle comma
| Minor minthma
|-
|-
| 13
| 13
| [[625/624]]
| [[625/624]]
|
| {{Monzo| -4 -1 4 0 0 -1 }}
|
| 2.77
|
| Tunbarsma
|-
|-
| 13
| 13
Line 901: Line 1,570:
| {{Monzo| 2 -3 -2 0 0 2 }}
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| 2.56
| Island comma, Parizeksma
| Island comma
 
|-
|-
| 13
| 13
| [[729/728]]
| [[729/728]]
|
| {{Monzo| -3 6 0 -1 0 -1 }}
|
| 2.38
|
| Squbema
|-
|-
| 13
| 13
Line 918: Line 1,586:
| 13
| 13
| [[1575/1573]]
| [[1575/1573]]
|
| {{Monzo| 2 2 1 -2 -1 }}
|
| 2.20
|
| Nicola
|-
|-
| 13
| 13
| [[1716/1715]]
| [[1716/1715]]
|
| {{Monzo| 2 1 -1 -3 1 1 }}
|
| 1.01
|
| Lummic comma
|-
|-
| 13
| 13
| [[2080/2079]]
| [[2080/2079]]
|
| {{Monzo| 5 -3 1 -1 -1 1 }}
|
| 0.83
|
| Ibnsinma
|-
|-
| 13
| 13
| [[6656/6655]]
| [[6656/6655]]
|
| {{Monzo| 9 0 -1 0 -3 1 }}
|
| 0.26012
|
| Jacobin comma
|}
|}
<references/>
<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
| 8
| 1\72
| 34\72<br>(2\72)
| [[Octoid]]
| 566.7<br>(33.3)
| 168/121<br>(55/54)
| [[Octowerck]] / octowerckis
|-
|-
| 8
| 8
| 2\72
| 35\72<br>(1\72)
| [[Octowerck]]
| 583.3<br>(16.7)
|-
| 7/5<br>(100/99)
| 8
| [[Octoid]] / octopus
| 4\72
|
|-
|-
| 9
| 9
| 1\72
| 19\72<br>(3\72)
|  
| 316.7<br>(50.0)
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimnic / 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
| |  
|  
|}
|}
== Instruments ==
If one can get six 12edo instruments tuned a twelfth-tone apart, it is possible to use these instruments in combination to play the full gamut of 72edo (see Music).
One can also use a skip fretting system:
* [[Skip fretting system 72 2 27]]
Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo:
* [[Lumatone mapping for 72edo]]


== Music ==
== Music ==
[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]
* [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]]


[[Category:Equal divisions of the octave]]
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