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== Theory ==
== Theory ==
72edo approximates [[11-limit]] [[just intonation]] exceptionally well. It is [[consistent]] in the [[17-odd-limit]] and is the ninth [[zeta integral edo]]. It is the second edo (after [[58edo|58]]) to be [[consistency|distinctly consistent]] in the [[11-odd-limit]], the first edo to be [[consistency|consistent to distance 2]] in the 11-odd-limit, and the first edo to be consistent in the 12- and 13-[[odd prime sum limit|odd-prime-sum-limit]].  
72edo 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, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.


72et is the only 11-limit regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, and splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s. It is also an excellent tuning for [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].
The [[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.  
 
The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament.


72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]].
72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]].
72edo (with the 71.94 [[generalized patent val]]) has the highest [[path-based goodness]] out of all generalized patent vals from 1 to 140.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|72|columns=9}}
{{Harmonics in equal|72|columns=11}}
{{Harmonics in equal|72|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}
{{Harmonics in equal|72|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}


=== Octave stretch ===
=== As a tuning of other temperaments ===
72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies. [[Path-based goodness]] recommends an octave stretch of 0.93 cents, which lies between the two options above but is closer to 186ed6.  
72et is the only 11-limit regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, and splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s (144edo is enfactored in the 11-limit with 72edo, so it is already covered here). It is also an excellent tuning for [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].  


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 72 factors into 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.
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
! Cents
! Approximate ratios<ref group="note">{{sg|limit=19-limit}} For lower limits see [[Table of 72edo intervals]].</ref>
! Approximate ratios<ref group="note">As a 19-limit temperament, inconsistent intervals in ''italic''. For a table of intervals by prime limit, see [[Table of 72edo intervals]].</ref>
! colspan="3" | [[Ups and downs notation]]
! colspan="2" |[[Kite's ups and downs notation|Ups and downs notation]]
! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]
([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>6</sup>A1 and d2)
! (K, S, U)  
|-
|-
| 0
| 0
| 0.0
| 0.0
| 1/1
| [[1/1]]
| P1
|P1
| perfect unison
| {{UDnote|step=0}}
| D
| P1
| perfect unison
| D
| D
|-
|-
| 1
| 1
| 16.7
| 16.7
| 81/80, 91/90, 99/98, 100/99, 105/104
| [[81/80]], [[91/90]], [[99/98]], [[100/99]], [[105/104]]
| ^1
|^1
| up unison
| {{UDnote|step=1}}
| ^D
| K1, L1
| comma-wide unison, large unison
| KD, LD
| KD
|-
|-
| 2
| 2
| 33.3
| 33.3
| 45/44, 49/48, 50/49, 55/54, 64/63
| [[45/44]], [[49/48]], [[50/49]], [[55/54]], [[64/63]]
| ^^
|^^
| dup unison
| {{UDnote|step=2}}
| ^^D
| S1, O1
| super unison, on unison
| SD, OD
| SD
|-
|-
| 3
| 3
| 50.0
| 50.0
| 33/32, 36/35, 40/39
| [[33/32]], [[36/35]], [[40/39]]
| ^<sup>3</sup>1, v<sup>3</sup>m2
|^<sup>3</sup>1, v<sup>3</sup>m2
| trup unison, trudminor 2nd
| {{UDnote|step=3}}
| ^<sup>3</sup>D, v<sup>3</sup>Eb
|-
| U1, H1, hm2
| uber unison, hyper unison, hypominor 2nd
| UD, HD, uEb
| UD, uEb
|-
| 4
| 4
| 66.7
| 66.7
| 25/24, 26/25, 27/26, 28/27
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| vvm2
|vvm2
| dudminor 2nd
| {{UDnote|step=4}}
| vvEb
| kkA1, sm2
| classic aug unison, subminor 2nd
| kkD#, sEb
| sD#, (kkD#), sEb
|-
|-
| 5
| 5
| 83.3
| 83.3
| 20/19, 21/20, 22/21
| [[20/19]], [[21/20]], [[22/21]]
| vm2
|vm2
| downminor 2nd
| {{UDnote|step=5}}
| vEb
| kA1, lm2
| comma-narrow aug unison, little minor 2nd
| kD#, lEb
| kD#, kEb
|-
|-
| 6
| 6
| 100.0
| 100.0
| 17/16, 18/17, 19/18
| [[17/16]], [[18/17]], [[19/18]]
| m2
|m2
| minor 2nd
| {{UDnote|step=6}}
| Eb
| m2
| minor 2nd
| Eb
| Eb
|-
|-
| 7
| 7
| 116.7
| 116.7
| 15/14, 16/15
| [[15/14]], [[16/15]]
| ^m2
|^m2
| upminor 2nd
| {{UDnote|step=7}}
| ^Eb
| Km2
| classic minor 2nd
| KEb
| KEb
|-
|-
| 8
| 8
| 133.3
| 133.3
| 13/12, 14/13, 27/25
| [[13/12]], [[14/13]], [[27/25]]
| ^^m2, v~2
|^^m2, v~2
| dupminor 2nd, downmid 2nd
| {{UDnote|step=8}}
| ^^Eb
| Om2
| on minor 2nd
| OEb
| SEb
|-
|-
| 9
| 9
| 150.0
| 150.0
| 12/11
| [[12/11]]
| ~2
|~2
| mid 2nd
| {{UDnote|step=9}}
| v<sup>3</sup>E
| N2
| neutral 2nd
| UEb/uE
| UEb/uE
|-
|-
| 10
| 10
| 166.7
| 166.7
| 11/10
| [[11/10]], [[21/19]]
| ^~2, vvM2
|^~2, vvM2
| upmid 2nd, dudmajor 2nd
| {{UDnote|step=10}}
| vvE
| oM2
| off major 2nd
| oE
| sE
|-
|-
| 11
| 11
| 183.3
| 183.3
| 10/9
| [[10/9]]
| vM2
|vM2
| downmajor 2nd
| {{UDnote|step=11}}
| vE
| kM2
| classic/comma-narrow major 2nd
| kE
| kE
|-
|-
| 12
| 12
| 200.0
| 200.0
| 9/8
| [[9/8]], [[19/17]]
| M2
|M2
| major 2nd
| {{UDnote|step=12}}
| E
| M2
| major 2nd
| E
| E
|-
|-
| 13
| 13
| 216.7
| 216.7
| 17/15, 25/22
| [[17/15]], [[25/22]]
| ^M2
|^M2
| upmajor 2nd
| {{UDnote|step=13}}
| ^E
| LM2
| large major 2nd
| LE
| KE
|-
|-
| 14
| 14
| 233.3
| 233.3
| 8/7
| [[8/7]]
| ^^M2
|^^M2
| dupmajor 2nd
| {{UDnote|step=14}}
| ^^E
|-
| SM2
| 15
| supermajor 2nd
| 250.0
| SE
| [[15/13]], [[22/19]]
| SE
|^<sup>3</sup>M2, v<sup>3</sup>m3
|-
| {{UDnote|step=15}}
| 15
| 250.0
| 15/13, 22/19
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| trupmajor 2nd,<br>trudminor 3rd
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
| HM2, hm3
| hypermajor 2nd, hypominor 3rd
| HE, hF
| UE, uF
|-
|-
| 16
| 16
| 266.7
| 266.7
| 7/6
| [[7/6]]
| vvm3
|vvm3
| dudminor 3rd
| {{UDnote|step=16}}
| vvF
| sm3
| subminor 3rd
| sF
| sF
|-
|-
| 17
| 17
| 283.3
| 283.3
| 13/11, 20/17
| [[13/11]], [[20/17]]
| vm3
|vm3
| downminor 3rd
| {{UDnote|step=17}}
| vF
|-
| lm3
| 18
| little minor 3rd
| 300.0
| lF
| [[19/16]], [[25/21]], [[32/27]]
| kF
|m3
|-
| {{UDnote|step=18}}
| 18
| 300.0
| 19/16, 25/21, 32/27
| m3
| minor 3rd
| F
| m3
| minor 3rd
| F
| F
|-
|-
| 19
| 19
| 316.7
| 316.7
| 6/5
| [[6/5]]
| ^m3
|^m3
| upminor 3rd
| {{UDnote|step=19}}
| ^F
| Km3
| classic minor 3rd
| KF
| KF
|-
|-
| 20
| 20
| 333.3
| 333.3
| 17/14, 39/32, 40/33
| [[17/14]], ''[[39/32]]'', [[40/33]]
| ^^m3, v~3
|^^m3, v~3
| dupminor 3rd, downmid 3rd
| {{UDnote|step=20}}
| ^^F
|-
| Om3
| 21
| on minor third
| 350.0
| OF
| [[11/9]], [[27/22]]
| SF
|~3
|-
| {{UDnote|step=21}}
| 21
| 350.0
| 11/9, 27/22
| ~3
| mid 3rd
| ^<sup>3</sup>F
| N3
| neutral 3rd
| UF/uF#
| UF/uF#
|-
|-
| 22
| 22
| 366.7
| 366.7
| 16/13, 21/17, 26/21
| [[16/13]], [[21/17]], [[26/21]]
| ^~3, vvM3
|^~3, vvM3
| upmid 3rd, dudmajor 3rd
| {{UDnote|step=22}}
| vvF#
| oM3
| off major 3rd
| oF#
| sF#
|-
|-
| 23
| 23
| 383.3
| 383.3
| 5/4
| [[5/4]]
| vM3
|vM3
| downmajor 3rd
| {{UDnote|step=23}}
| vF#
|-
| kM3
| 24
| classic major 3rd
| 400.0
| kF#
| [[24/19]]
| kF#
|M3
|-
| {{UDnote|step=24}}
| 24
|-
| 400.0
| 24/19
| M3
| major 3rd
| F#
| M3
| major 3rd
| F#
| F#
|-
| 25
| 25
| 416.7
| 416.7
| 14/11
| [[14/11]], [[19/15]]
| ^M3
|^M3
| upmajor 3rd
| {{UDnote|step=25}}
| ^F#
| LM3
| large major 3rd
| LF#
| KF#
|-
|-
| 26
| 26
| 433.3
| 433.3
| 9/7
| [[9/7]]
| ^^M3
|^^M3
| dupmajor 3rd
| {{UDnote|step=26}}
| ^^F#
|-
| SM3
| 27
| supermajor 3rd
| SF#
| SF#
|-
| 27
| 450.0
| 450.0
| 13/10, 22/17
| [[13/10]], [[22/17]]
| ^<sup>3</sup>M3, v<sup>3</sup>4
|^<sup>3</sup>M3, v<sup>3</sup>4
| trupmajor 3rd, trud 4th
| {{UDnote|step=27}}
| ^<sup>3</sup>F#, v<sup>3</sup>G
| HM3, h4
| hypermajor 3rd, hypo 4th
| HF#, hG
| UF#, uG
|-
|-
| 28
| 28
| 466.7
| 466.7
| 17/13, 21/16
| [[17/13]], [[21/16]]
| vv4
|vv4
| dud 4th
| {{UDnote|step=28}}
| vvG
| s4
| sub 4th
| sG
| sG
|-
|-
| 29
| 29
| 483.3
| 483.3
| 33/25
| [[33/25]]
| v4
|v4
| down 4th
| {{UDnote|step=29}}
| vG
| l4
| little 4th
| lG
| kG
|-
|-
| 30
| 30
| 500.0
| 500.0
| 4/3
| [[4/3]]
| P4
|P4
| perfect 4th
| {{UDnote|step=30}}
| G
| P4
| perfect 4th
| G
| G
|-
|-
| 31
| 31
| 516.7
| 516.7
| 27/20
| [[27/20]]
| ^4
|^4
| up 4th
| {{UDnote|step=31}}
| ^G
| K4
| comma-wide 4th
| KG
| KG
|-
|-
| 32
| 32
| 533.3
| 533.3
| 15/11, 19/14, ''26/19''
| [[15/11]], [[19/14]], ''[[26/19]]''
| ^^4, v~4
|^^4, v~4
| dup 4th, downmid 4th
| {{UDnote|step=32}}
| ^^G
| O4
| on 4th
| OG
| SG
|-
|-
| 33
| 33
| 550.0
| 550.0
| 11/8
| [[11/8]]
| ~4
|~4
| mid 4th
| {{UDnote|step=33}}
| ^<sup>3</sup>G
|-
| U4/N4
| 34
| uber 4th / neutral 4th
| 566.7
| UG
| [[18/13]], [[25/18]]
| UG
|^~4, vvA4
|-
| {{UDnote|step=34}}
| 34
| 566.7
| 18/13, 25/18
| ^~4, vvA4
| upmid 4th, dudaug 4th
| vvG#
| kkA4, sd5
| classic aug 4th, sub dim 5th
| kkG#, sAb
| SG#, (kkG#), sAb
|-
|-
| 35
| 35
| 583.3
| 583.3
| 7/5
| [[7/5]]
| vA4, vd5
|vA4, vd5
| downaug 4th, <br>downdim 5th
| {{UDnote|step=35}}
| vG#, vAb
| kA4, ld5
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, kAb
|-
|-
| 36
| 36
| 600.0
| 600.0
| 17/12, 24/17
| [[17/12]], [[24/17]]
| A4, d5
|A4, d5
| aug 4th, dim 5th
| {{UDnote|step=36}}
| G#, Ab
|-
| A4, d5
| 37
| aug 4th, dim 5th
| 616.7
| G#, Ab
| [[10/7]]
| G#, Ab
|^A4, ^d5
| {{UDnote|step=37}}
|-
| 38
| 633.3
| [[13/9]], [[36/25]]
|v~5, ^^d5
| {{UDnote|step=38}}
|-
| 39
| 650.0
| [[16/11]]
|~5
| {{UDnote|step=39}}
|-
| 40
| 666.7
| ''[[19/13]]'', [[22/15]], [[28/19]]
|vv5, ^~5
| {{UDnote|step=40}}
|-
| 41
| 683.3
| [[40/27]]
|v5
| {{UDnote|step=41}}
|-
| 42
| 700.0
| [[3/2]]
|P5
| {{UDnote|step=42}}
|-
| 43
| 716.7
| [[50/33]]
|^5
| {{UDnote|step=43}}
|-
| 44
| 733.3
| [[26/17]], [[32/21]]
|^^5
| {{UDnote|step=44}}
|-
| 45
| 750.0
| [[17/11]], [[20/13]]
|^<sup>3</sup>5, v<sup>3</sup>m6
| {{UDnote|step=45}}
|-
| 46
| 766.7
| [[14/9]]
|vvm6
| {{UDnote|step=46}}
|-
| 47
| 783.3
| [[11/7]], [[30/19]]
|vm6
| {{UDnote|step=47}}
|-
| 48
| 800.0
| [[19/12]]
|m6
| {{UDnote|step=48}}
|-
| 49
| 816.7
| [[8/5]]
|^m6
| {{UDnote|step=49}}
|-
| 50
| 833.3
| [[13/8]], [[21/13]], [[34/21]]
|^^m6, v~6
| {{UDnote|step=50}}
|-
| 51
| 850.0
| [[18/11]], [[44/27]]
|~6
| {{UDnote|step=51}}
|-
| 52
| 866.7
| [[28/17]], [[33/20]], ''[[64/39]]''
|^~6, vvM6
| {{UDnote|step=52}}
|-
| 53
| 883.3
| [[5/3]]
|vM6
| {{UDnote|step=53}}
|-
| 54
| 900.0
| [[27/16]], [[32/19]], [[42/25]]
|M6
| {{UDnote|step=54}}
|-
| 55
| 916.7
| [[17/10]], [[22/13]]
|^M6
| {{UDnote|step=55}}
|-
| 56
| 933.3
| [[12/7]]
|^^M6
| {{UDnote|step=56}}
|-
| 57
| 950.0
| [[19/11]], [[26/15]]
|^<sup>3</sup>M6, v<sup>3</sup>m7
| {{UDnote|step=57}}
|-
| 58
| 966.7
| [[7/4]]
|vvm7
| {{UDnote|step=58}}
|-
| 59
| 983.3
| [[30/17]], [[44/25]]
|vm7
| {{UDnote|step=59}}
|-
| 60
| 1000.0
| [[16/9]], [[34/19]]
|m7
| {{UDnote|step=60}}
|-
| 61
| 1016.7
| [[9/5]]
|^m7
| {{UDnote|step=61}}
|-
| 62
| 1033.3
| [[20/11]], [[38/21]]
|^^m7, v~7
| {{UDnote|step=62}}
|-
| 63
| 1050.0
| [[11/6]]
|~7
| {{UDnote|step=63}}
|-
| 64
| 1066.7
| [[13/7]], [[24/13]], [[50/27]]
|^~7, vvM7
| {{UDnote|step=64}}
|-
| 65
| 1083.3
| [[15/8]], [[28/15]]
|vM7
| {{UDnote|step=65}}
|-
| 66
| 1100.0
| [[17/9]], [[32/17]], [[36/19]]
|M7
| {{UDnote|step=66}}
|-
| 67
| 1116.7
| [[19/10]], [[21/11]], [[40/21]]
|^M7
| {{UDnote|step=67}}
|-
| 68
| 1133.3
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
|^^M7
| {{UDnote|step=68}}
|-
| 69
| 1150.0
| [[35/18]], [[39/20]], [[64/33]]
|^<sup>3</sup>M7, v<sup>3</sup>8
| {{UDnote|step=69}}
|-
| 70
| 1166.7
| [[49/25]], [[55/28]], [[63/32]], [[88/45]], [[96/49]]
|vv8
| {{UDnote|step=70}}
|-
| 71
| 1183.3
| [[99/50]], [[160/81]], [[180/91]], [[196/99]], [[208/105]]
|v8
| {{UDnote|step=71}}
|-
| 72
| 1200.0
| [[2/1]]
|P8
| {{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
! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]]
! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]
! (K, S, U)
|-
| 0
| 0.0
| P1
| perfect unison
| D
| P1
| perfect unison
| D
| D
|-
| 1
| 16.7
| ^1
| up unison
| ^D
| K1, L1
| comma-wide unison, large unison
| KD, LD
| KD
|-
| 2
| 33.3
| ^^
| dup unison
| ^^D
| S1, O1
| super unison, on unison
| SD, OD
| SD
|-
| 3
| 50.0
| ^<sup>3</sup>1, v<sup>3</sup>m2
| trup unison, trudminor 2nd
| ^<sup>3</sup>D, v<sup>3</sup>Eb
| U1, H1, hm2
| uber unison, hyper unison, hypominor 2nd
| UD, HD, uEb
| UD, uEb
|-
| 4
| 66.7
| vvm2
| dudminor 2nd
| vvEb
| kkA1, sm2
| classic aug unison, subminor 2nd
| kkD#, sEb
| sD#, (kkD#), sEb
|-
| 5
| 83.3
| vm2
| downminor 2nd
| vEb
| kA1, lm2
| comma-narrow aug unison, little minor 2nd
| kD#, lEb
| kD#, kEb
|-
| 6
| 100.0
| m2
| minor 2nd
| Eb
| m2
| minor 2nd
| Eb
| Eb
|-
| 7
| 116.7
| ^m2
| upminor 2nd
| ^Eb
| Km2
| classic minor 2nd
| KEb
| KEb
|-
| 8
| 133.3
| ^^m2, v~2
| dupminor 2nd, downmid 2nd
| ^^Eb
| Om2
| on minor 2nd
| OEb
| SEb
|-
| 9
| 150.0
| ~2
| mid 2nd
| v<sup>3</sup>E
| N2
| neutral 2nd
| UEb/uE
| UEb/uE
|-
| 10
| 166.7
| ^~2, vvM2
| upmid 2nd, dudmajor 2nd
| vvE
| oM2
| off major 2nd
| oE
| sE
|-
| 11
| 183.3
| vM2
| downmajor 2nd
| vE
| kM2
| classic/comma-narrow major 2nd
| kE
| kE
|-
| 12
| 200.0
| M2
| major 2nd
| E
| M2
| major 2nd
| E
| E
|-
| 13
| 216.7
| ^M2
| upmajor 2nd
| ^E
| LM2
| large major 2nd
| LE
| KE
|-
| 14
| 233.3
| ^^M2
| dupmajor 2nd
| ^^E
| SM2
| supermajor 2nd
| SE
| SE
|-
| 15
| 250.0
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| trupmajor 2nd,<br>trudminor 3rd
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
| HM2, hm3
| hypermajor 2nd, hypominor 3rd
| HE, hF
| UE, uF
|-
| 16
| 266.7
| vvm3
| dudminor 3rd
| vvF
| sm3
| subminor 3rd
| sF
| sF
|-
| 17
| 283.3
| vm3
| downminor 3rd
| vF
| lm3
| little minor 3rd
| lF
| kF
|-
| 18
| 300.0
| m3
| minor 3rd
| F
| m3
| minor 3rd
| F
| F
|-
| 19
| 316.7
| ^m3
| upminor 3rd
| ^F
| Km3
| classic minor 3rd
| KF
| KF
|-
| 20
| 333.3
| ^^m3, v~3
| dupminor 3rd, downmid 3rd
| ^^F
| Om3
| on minor third
| OF
| SF
|-
| 21
| 350.0
| ~3
| mid 3rd
| ^<sup>3</sup>F
| N3
| neutral 3rd
| UF/uF#
| UF/uF#
|-
| 22
| 366.7
| ^~3, vvM3
| upmid 3rd, dudmajor 3rd
| vvF#
| oM3
| off major 3rd
| oF#
| sF#
|-
| 23
| 383.3
| vM3
| downmajor 3rd
| vF#
| kM3
| classic major 3rd
| kF#
| kF#
|-
| 24
| 400.0
| M3
| major 3rd
| F#
| M3
| major 3rd
| F#
| F#
|-
| 25
| 416.7
| ^M3
| upmajor 3rd
| ^F#
| LM3
| large major 3rd
| LF#
| KF#
|-
| 26
| 433.3
| ^^M3
| dupmajor 3rd
| ^^F#
| SM3
| supermajor 3rd
| SF#
| SF#
|-
| 27
| 450.0
| ^<sup>3</sup>M3, v<sup>3</sup>4
| trupmajor 3rd, trud 4th
| ^<sup>3</sup>F#, v<sup>3</sup>G
| HM3, h4
| hypermajor 3rd, hypo 4th
| HF#, hG
| UF#, uG
|-
| 28
| 466.7
| vv4
| dud 4th
| vvG
| s4
| sub 4th
| sG
| sG
|-
| 29
| 483.3
| v4
| down 4th
| vG
| l4
| little 4th
| lG
| kG
|-
| 30
| 500.0
| P4
| perfect 4th
| G
| P4
| perfect 4th
| G
| G
|-
| 31
| 516.7
| ^4
| up 4th
| ^G
| K4
| comma-wide 4th
| KG
| KG
|-
| 32
| 533.3
| ^^4, v~4
| dup 4th, downmid 4th
| ^^G
| O4
| on 4th
| OG
| SG
|-
| 33
| 550.0
| ~4
| mid 4th
| ^<sup>3</sup>G
| U4/N4
| uber 4th / neutral 4th
| UG
| UG
|-
| 34
| 566.7
| ^~4, vvA4
| upmid 4th, dudaug 4th
| vvG#
| kkA4, sd5
| classic aug 4th, sub dim 5th
| kkG#, sAb
| SG#, (kkG#), sAb
|-
| 35
| 583.3
| vA4, vd5
| downaug 4th, <br>downdim 5th
| vG#, vAb
| kA4, ld5
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, kAb
|-
| 36
| 600.0
| A4, d5
| aug 4th, dim 5th
| G#, Ab
| A4, d5
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
|-
| 37
| 616.7
| ^A4, ^d5
| upaug 4th, updim 5th
| ^G#, ^Ab
| LA4, Kd5
| large aug 4th, comma-wide dim 5th
| LG#, KAb
| KG#, KAb
|-
| 38
| 633.3
| v~5, ^^d5
| downmid 5th, <br>dupdim 5th
| ^^Ab
| SA4, KKd5
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, SAb, (KKAb)
|-
| 39
| 650.0
| ~5
| mid 5th
| v<sup>3</sup>A
| u5/N5
| unter 5th / neutral 5th
| uA
| uA
|-
|-
| 37
| 40
| 616.7
| 666.7
| 10/7
| vv5, ^~5
| ^A4, ^d5
| dud 5th, upmid 5th
| upaug 4th, updim 5th
| vvA
| ^G#, ^Ab
| o5
| LA4, Kd5
| off 5th
| large aug 4th, comma-wide dim 5th
| oA
| LG#, KAb
| sA
| KG#, KAb
|-
|-
| 38
| 41
| 633.3
| 13/9, 36/25
| v~5, ^^d5
| downmid 5th, <br>dupdim 5th
| ^^Ab
| SA4, KKd5
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, SAb, (KKAb)
|-
| 39
| 650.0
| 16/11
| ~5
| mid 5th
| v<sup>3</sup>A
| u5/N5
| unter 5th / neutral 5th
| uA
| uA
|-
| 40
| 666.7
| ''19/13'', 22/15, 28/19
| vv5, ^~5
| dud 5th, upmid 5th
| vvA
| o5
| off 5th
| oA
| sA
|-
| 41
| 683.3
| 683.3
| 40/27
| v5
| v5
| down 5th
| down 5th
Line 510: Line 913:
| 42
| 42
| 700.0
| 700.0
| 3/2
| P5
| P5
| perfect 5th
| perfect 5th
Line 521: Line 923:
| 43
| 43
| 716.7
| 716.7
| 50/33
| ^5
| ^5
| up 5th
| up 5th
Line 532: Line 933:
| 44
| 44
| 733.3
| 733.3
| 26/17, 32/21
| ^^5
| ^^5
| dup 5th
| dup 5th
Line 543: Line 943:
| 45
| 45
| 750.0
| 750.0
| 17/11, 20/13
| ^<sup>3</sup>5, v<sup>3</sup>m6
| ^<sup>3</sup>5, v<sup>3</sup>m6
| trup 5th, trudminor 6th
| trup 5th, trudminor 6th
Line 554: Line 953:
| 46
| 46
| 766.7
| 766.7
| 14/9
| vvm6
| vvm6
| dudminor 6th
| dudminor 6th
Line 565: Line 963:
| 47
| 47
| 783.3
| 783.3
| 11/7
| vm6
| vm6
| downminor 6th
| downminor 6th
Line 576: Line 973:
| 48
| 48
| 800.0
| 800.0
| 19/12
| m6
| m6
| minor 6th
| minor 6th
Line 587: Line 983:
| 49
| 49
| 816.7
| 816.7
| 8/5
| ^m6
| ^m6
| upminor 6th
| upminor 6th
Line 598: Line 993:
| 50
| 50
| 833.3
| 833.3
| 13/8, 21/13, 34/21
| ^^m6, v~6
| ^^m6, v~6
| dupminor 6th, downmid 6th
| dupminor 6th, downmid 6th
Line 609: Line 1,003:
| 51
| 51
| 850.0
| 850.0
| 18/11, 44/27
| ~6
| ~6
| mid 6th
| mid 6th
Line 620: Line 1,013:
| 52
| 52
| 866.7
| 866.7
| 28/17, 33/20, 64/39
| ^~6, vvM6
| ^~6, vvM6
| upmid 6th, dudmajor 6th
| upmid 6th, dudmajor 6th
Line 631: Line 1,023:
| 53
| 53
| 883.3
| 883.3
| 5/3
| vM6
| vM6
| downmajor 6th
| downmajor 6th
Line 642: Line 1,033:
| 54
| 54
| 900.0
| 900.0
| 27/16, 32/19, 42/25
| M6
| M6
| major 6th
| major 6th
Line 653: Line 1,043:
| 55
| 55
| 916.7
| 916.7
| 17/10, 22/13
| ^M6
| ^M6
| upmajor 6th
| upmajor 6th
Line 664: Line 1,053:
| 56
| 56
| 933.3
| 933.3
| 12/7
| ^^M6
| ^^M6
| dupmajor 6th
| dupmajor 6th
Line 675: Line 1,063:
| 57
| 57
| 950.0
| 950.0
| 19/11, 26/15
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| trupmajor 6th,<br>trudminor 7th
| trupmajor 6th,<br>trudminor 7th
Line 686: Line 1,073:
| 58
| 58
| 966.7
| 966.7
| 7/4
| vvm7
| vvm7
| dudminor 7th
| dudminor 7th
Line 697: Line 1,083:
| 59
| 59
| 983.3
| 983.3
| 30/17, 44/25
| vm7
| vm7
| downminor 7th
| downminor 7th
Line 708: Line 1,093:
| 60
| 60
| 1000.0
| 1000.0
| 16/9
| m7
| m7
| minor 7th
| minor 7th
Line 719: Line 1,103:
| 61
| 61
| 1016.7
| 1016.7
| 9/5
| ^m7
| ^m7
| upminor 7th
| upminor 7th
Line 730: Line 1,113:
| 62
| 62
| 1033.3
| 1033.3
| 20/11
| ^^m7, v~7
| ^^m7, v~7
| dupminor 7th, downmid 7th
| dupminor 7th, downmid 7th
Line 741: Line 1,123:
| 63
| 63
| 1050.0
| 1050.0
| 11/6
| ~7
| ~7
| mid 7th
| mid 7th
Line 752: Line 1,133:
| 64
| 64
| 1066.7
| 1066.7
| 13/7, 24/13, 50/27
| ^~7, vvM7
| ^~7, vvM7
| upmid 7th, dudmajor 7th
| upmid 7th, dudmajor 7th
Line 763: Line 1,143:
| 65
| 65
| 1083.3
| 1083.3
| 15/8, 28/15
| vM7
| vM7
| downmajor 7th
| downmajor 7th
Line 774: Line 1,153:
| 66
| 66
| 1100.0
| 1100.0
| 17/9, 32/17, 36/19
| M7
| M7
| major 7th
| major 7th
Line 785: Line 1,163:
| 67
| 67
| 1116.7
| 1116.7
| 19/10, 21/11, 40/21
| ^M7
| ^M7
| upmajor 7th
| upmajor 7th
Line 796: Line 1,173:
| 68
| 68
| 1133.3
| 1133.3
| 25/13, 27/14, 48/25, 52/27
| ^^M7
| ^^M7
| dupmajor 7th
| dupmajor 7th
Line 807: Line 1,183:
| 69
| 69
| 1150.0
| 1150.0
| 35/18, 39/20, 64/33
| ^<sup>3</sup>M7, v<sup>3</sup>8
| ^<sup>3</sup>M7, v<sup>3</sup>8
| trupmajor 7th, trud octave
| trupmajor 7th, trud octave
Line 818: Line 1,193:
| 70
| 70
| 1166.7
| 1166.7
| 49/25, 55/28, 63/32, 88/45, 96/49
| vv8
| vv8
| dud octave
| dud octave
Line 829: Line 1,203:
| 71
| 71
| 1183.3
| 1183.3
| 99/50, 160/81, 180/91, 196/99, 208/105
| v8
| v8
| down octave
| down octave
Line 840: Line 1,213:
| 72
| 72
| 1200.0
| 1200.0
| 2/1
| P8
| P8
| perfect octave
| perfect octave
Line 849: Line 1,221:
| D
| D
|}
|}
<references group="note" />


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Line 864: Line 1,235:
| 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 < -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
| rowspan="2" | dupminor, <br>downmid
| luyo
| luyo
| (a b 1 0 -1)
| (a b 1 0 -1)
| 15/11
| [[15/11]]
|-
|-
| tho
| tho
| (a b 0 0 0 1)
| (a b 0 0 0 1)
| 13/8, 13/9
| [[13/8]], [[13/9]]
|-
|-
| rowspan="2" | mid
| rowspan="2" | mid
| ilo
| ilo
| (a b 0 0 1)
| (a b 0 0 1)
| 11/9, 11/6
| [[11/9]], [[11/6]]
|-
|-
| lu
| lu
| (a b 0 0 -1)
| (a b 0 0 -1)
| 12/11, 18/11
| [[12/11]], [[18/11]]
|-
|-
| rowspan="2" | upmid, <br>dudmajor
| rowspan="2" | upmid, <br>dudmajor
| logu
| logu
| (a b -1 0 1)
| (a b -1 0 1)
| 11/10
| [[11/10]]
|-
|-
| thu
| thu
| (a b 0 0 0 -1)
| (a b 0 0 0 -1)
| 16/13, 18/13
| [[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 > 1
| (a b), b > 1
| 9/8, 27/16
| [[9/8]], [[27/16]]
|-
|-
| dupmajor
| dupmajor
| ru
| ru
| (a b 0 -1)
| (a b 0 -1)
| 9/7, 12/7
| [[9/7]], [[12/7]]
|-
|-
| rowspan="2" | trupmajor, <br>trudminor
| rowspan="2" | trupmajor, <br>trudminor
| thogu
| thogu
| (a b -1 0 0 1)
| (a b -1 0 0 1)
| 13/10
| [[13/10]]
|-
|-
| thuyo
| thuyo
| (a b 1 0 0 -1)
| (a b 1 0 0 -1)
| 15/13
| [[15/13]]
|}
|}
All 72edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:
All 72edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:
Line 973: Line 1,344:
| C dupmajor or C dup
| C dupmajor or C dup
|}
|}
For a more complete list, see [[Ups and Downs Notation #Chord names in other EDOs]].  
For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]].  


=== Relationship between primes and rings ===
=== Relationship between primes and rings ===
Line 979: Line 1,350:


Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
* −1 degree (the down ring) corrects 81/64 to 5/4 via 80/81
* −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 63/64
* −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
* +3 degrees  (the trup ring) corrects [[4/3]] to [[11/8]] via [[33/32]]
* +2 degrees (the dup ring) corrects 128/81 to 13/8 via 1053/1024
* +2 degrees (the dup ring) corrects [[128/81]] to [[13/8]] via [[1053/1024]]
* 0 degrees (the plain ring) corrects 256/243 to 17/16 via 4131/4096
* 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
* 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.
Thus the product of a ratio's monzo with {{map| 0 0 -1 -2 3 2 0 0 }}, modulo 6, specifies which ring the ratio lies on.


== Notations ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
72edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6a}}
{{Sharpness-sharp6-szg}}


Half-sharps and half-flats can be used to avoid triple arrows:
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6b}}
{{Sharpness-sharp6-qt-szg}}


[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have sharps and flats with arrows borrowed from extended [[Helmholtz–Ellis notation]]:
=== Kite's ups and downs notation ===
{{Sharpness-sharp6}}
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}}


If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6-qt}}
{{Ups and downs sharpness|72|true}}


=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[65edo#Sagittal notation|65-EDO]] and [[79edo#Sagittal notation|79]], and is a superset of the notations for EDOs [[36edo#Sagittal notation|36]], [[24edo#Sagittal notation|24]], [[18edo#Sagittal notation|18]], [[12edo#Sagittal notation|12]], [[8edo#Sagittal notation|8]], and [[6edo#Sagittal notation|6]].
This notation uses the same sagittal sequence as edos [[65edo #Sagittal notation|65-]] and [[79edo #Sagittal notation|79edo]], and is a superset of the notations for edos [[36edo #Sagittal notation|36]], [[24edo #Sagittal notation|24]], [[18edo #Sagittal notation|18]], [[12edo #Sagittal notation|12]], [[8edo #Sagittal notation|8]], and [[6edo #Sagittal notation|6]].


==== Evo flavor ====
==== Evo flavor ====
<imagemap>
{{Sagittal chart|Evo}}
File:72-EDO_Evo_Sagittal.svg
 
desc none
==== Evo-SZ flavor ====
rect 80 0 300 50 [[Sagittal_notation]]
{{Sagittal chart|Evo-SZ}}
rect 300 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Evo_Sagittal.svg]]
</imagemap>


==== Revo flavor ====
==== Revo flavor ====
<imagemap>
{{Sagittal chart}}
File:72-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Revo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:72-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 711 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>


From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 72edo in the Revo flavor of Sagittal:
From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 72edo in the Revo flavor of Sagittal:  


[[File:72edo Sagittal.png|800px]]
<div class="noresize">
[[File:72edo Sagittal.png]]
</div>


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


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


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=== Zeta properties ===
=== 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.
72edo is the ninth [[zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.


[[File:plot72.png|alt=plot72.png|plot72.png]]
[[File:plot72.png|alt=plot72.png|plot72.png]]
==== Zeta peak index ====
{{ZPI
| zpi = 380
| steps = 71.9506065993786
| step size = 16.6781081733140
| tempered height = 9.157547
| pure height = 7.501282
| integral = 1.625363
| gap = 19.964746
| octave = 1200.82378847861
| consistent = 18
| distinct = 13
}}


== Regular temperament properties ==
== Regular temperament properties ==
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| 516.7
| 516.7
| 27/20
| 27/20
| [[Marvo]] / [[zarvo]]
| [[Gravity]] / [[marvo]] / [[zarvo]]
|-
|-
| 1
| 1
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| 316.7<br>(50.0)
| 316.7<br>(50.0)
| 6/5<br>(36/35)
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimnic
| [[Ennealimmal]] / ennealimnic / ennealiminal
|-
|-
| 9
| 9
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| [[Gamelstearn]]
| [[Gamelstearn]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
<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 ==
; [[Miracle]]-tempered scales
* [[Blackjack]], [[miracle_8]], [[miracle_10]], [[miracle_12]], [[miracle_12a]], [[miracle_24hi]], [[miracle_24lo]]
; [[Maeve Gutierrez]]'s scales
* [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]] (''octave reduced: 10 6 25 17 14'')
* [[Maeve Gutierrez|Gutierrez Moonglade]]: 1 4 6 1 5 2 4 7 1 4 6 1 1 4 5 1 5 1 2 3 1 1 5 1
; [[Budjarn Lambeth]]'s scales
* [[Magnetosphere scale|Magnetosphere]], [[blackened skies]], [[lost spirit]], [[moon dust]], [[5- to 10-tone scales in 72edo]]
; [[Gene Ward Smith]]'s scales
* [[Smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]]
* [[Smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]]
* [[Blackjack]], [[miracle_8]], [[miracle_10]], [[miracle_12]], [[miracle_12a]], [[miracle_24hi]], [[miracle_24lo]]
 
* [[Keenanmarvel]], [[xenakis_chrome]], [[xenakis_diat]], [[xenakis_schrome]]
; [[Iannis Xenakis]]' scales
* [[xenakis_chrome]], [[xenakis_diat]], [[xenakis_schrome]]
 
; Others
* Freivald [[Lazysunday]] scale
* [[Genus24255et72|Euler(24255) genus in 72 equal]]
* [[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]]
* [[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
* [[Keenanmarvel]]
* [[Magnetosphere scale|Magnetosphere]], [[Blackened skies]], [[Lost spirit]]
* [[Prodigy]][19]: 5 2 5 4 5 2 5 2 5 2 5 4 5 2 5 2 5 5 2
* [[5- to 10-tone scales in 72edo]]


=== Harmonic scale ===
=== Harmonic scale ===
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== Music ==
== Music ==
; [[Tolgahan Çoğulu]]
* [https://www.youtube.com/watch?v=BItT4f8MOiI ''Ne ağlarsın benim zülfü siyahım''] (from the UnTwelving 2026 convention) – transcription by [[Stephen Weigel]]
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/VwVp3RVao_k ''microtonal improvisation in 72edo''] (2025)
* [https://www.youtube.com/shorts/VwVp3RVao_k ''microtonal improvisation in 72edo''] (2025)
* [https://www.youtube.com/shorts/a07Q_zQHe9M ''72edo groove''] (2026)


; [[Ambient Esoterica]]
; [[Ambient Esoterica]]
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; [[Jake Freivald]]
; [[Jake Freivald]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday'']{{dead link}} in the [[lazysunday]] scale
* [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)}}
{{Wikipedia|In vain (Haas)}}
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* [https://www.youtube.com/watch?v=ix4yA-c-Pi8 ''Blumenstück''] (2000)
* [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])
* [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]]
; [[Claudi Meneghin]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 ''Twinkle canon – 72 edo'']{{dead link}}
* [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=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=w6Bckog1eOM ''Sicilienne in Miracle'']
* [https://www.youtube.com/watch?v=QKeZLtFHfNU ''Arietta with 5 Variations'', for Organ] (2024)
* [https://www.youtube.com/watch?v=QKeZLtFHfNU ''Arietta with 5 Variations'', for Organ] (2024)
; [https://www.youtube.com/@Mintsoda_15/videos Mintsoda_15]
* [https://www.youtube.com/watch?v=eA8S746K9_A ''Quarter-tone Impromptu-72 TET''] (2022)


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


; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* [https://www.archive.org/details/Kotekant ''Kotekant''] [https://www.archive.org/download/Kotekant/kotekant.mp3 play] (2010)
* [https://www.archive.org/details/Kotekant ''Kotekant''] [https://www.archive.org/download/Kotekant/kotekant.mp3 play] (2010)


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


; [[James Tenney]]
; [[James Tenney]]
Retrieved from "https://en.xen.wiki/w/72edo"