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{{Infobox ET}}
{{Infobox ET}}
{{Wikipedia|72 equal temperament}}
{{Wikipedia|72 equal temperament}}
{{EDO intro}}
{{ED intro}}


Each step of 72edo is called a ''[[morion]]'' (plural ''moria)''. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in [[12edo]]. 72edo is also a superset of [[24edo]], a common and standard tuning of [[Arabic, Turkish, Persian music|Arabic music]], and has itself been used to tune Turkish music.
Each step of 72edo is called a ''[[morion]]'' (plural ''moria)''. This produces a twelfth-tone tuning, with the whole tone measuring 200{{c}}, the same as in [[12edo]]. 72edo is also a superset of [[24edo]], a common and standard tuning of [[Arabic, Turkish, Persian music|Arabic music]], and has itself been used to tune Turkish music.


Composers that used 72edo include [[Ivan Wyschnegradsky]], [[Julián Carrillo]] (who is better associated with [[96edo]]), [[Ezra Sims]], [[James Tenney]], [[Georg Friedrich Haas]] and the jazz musician [[Joe Maneri]].
Composers that used 72edo include [[Ivan Wyschnegradsky]], [[Julián Carrillo]] (who is better associated with [[96edo]]), [[Georg Friedrich Haas]], [[Ezra Sims]], [[Rick Tagawa]], [[James Tenney]], and the jazz musician [[Joe Maneri]].


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


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


The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament.
The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament.
Line 23: Line 25:


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|72|columns=11}}
{{Harmonics in equal|72|columns=9}}
{{Harmonics in equal|72|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}
 
=== Octave stretch ===
72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 72 factors into 2<sup>3</sup> × 3<sup>2</sup>, 72edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36 }}. [[144edo]], which doubles it, provides a possible correction to its approximate harmonic 13.  
Since 72 factors into primes as {{nowrap| 2<sup>3</sup> × 3<sup>2</sup> }}, 72edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36 }}. [[144edo]], which doubles it, provides a possible correction to its approximate harmonic 13.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degrees
! #
! Cents
! Cents
! Approximate Ratios *
! Approximate ratios<ref group="note">{{sg|limit=19-limit}} For lower limits see [[Table of 72edo intervals]].</ref>
! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]
! colspan="3" |[[SKULO interval names|SKULO interval names and notation]]
! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]
! (K, S, U)
|-
|-
| 0
| 0
| 0.000
| 0.0
| 1/1
| 1/1
| P1
| P1
| perfect unison
| perfect unison
| D
| D
|P1
| P1
|perfect unison
| perfect unison
|D
| D
| D
|-
|-
| 1
| 1
| 16.667
| 16.7
| 81/80
| 81/80, 91/90, 99/98, 100/99, 105/104
| ^1
| ^1
| up unison
| up unison
| ^D
| ^D
|K1, L1
| K1, L1
|comma-wide unison, large unison
| comma-wide unison, large unison
|KD
| KD, LD
| KD
|-
|-
| 2
| 2
| 33.333
| 33.3
| 45/44, 64/63
| 45/44, 49/48, 50/49, 55/54, 64/63
| ^^
| ^^
| dup unison
| dup unison
| ^^D
| ^^D
|S1, O1
| S1, O1
|super unison, on unison
| super unison, on unison
|SD
| SD, OD
| SD
|-
|-
| 3
| 3
| 50.000
| 50.0
| 33/32
| 33/32, 36/35, 40/39
| ^<sup>3</sup>1,  
| ^<sup>3</sup>1, v<sup>3</sup>m2
v<sup>3</sup>m2
| trup unison, trudminor 2nd
| trup unison,  
| ^<sup>3</sup>D, v<sup>3</sup>Eb
trudminor 2nd
| U1, H1, hm2
| ^<sup>3</sup>D,  
| uber unison, hyper unison, hypominor 2nd
v<sup>3</sup>Eb
| UD, HD, uEb
|U1, H1,  
| UD, uEb
hm2
|uber unison, hyper unison,  
hypominor 2nd
|UD, HD,
uEb
|-
|-
| 4
| 4
| 66.667
| 66.7
| 25/24
| 25/24, 26/25, 27/26, 28/27
| vvm2
| vvm2
| dudminor 2nd
| dudminor 2nd
| vvEb
| vvEb
|kkA1, sm2
| kkA1, sm2
|classic aug unison, subminor 2nd
| classic aug unison, subminor 2nd
|kkD#, sEb
| kkD#, sEb
| sD#, (kkD#), sEb
|-
|-
| 5
| 5
| 83.333
| 83.3
| 21/20
| 20/19, 21/20, 22/21
| vm2
| vm2
| downminor 2nd
| downminor 2nd
| vEb
| vEb
|kA1, lm2
| kA1, lm2
|comma-narrow aug unison, little minor 2nd  
| comma-narrow aug unison, little minor 2nd  
|kD#, lEb
| kD#, lEb
| kD#, kEb
|-
|-
| 6
| 6
| 100.000
| 100.0
| 35/33, 17/16, 18/17
| 17/16, 18/17, 19/18
| m2
| m2
| minor 2nd
| minor 2nd
| Eb
| Eb
|m2
| m2
|minor 2nd
| minor 2nd
|Eb
| Eb
| Eb
|-
|-
| 7
| 7
| 116.667
| 116.7
| 15/14, 16/15
| 15/14, 16/15
| ^m2
| ^m2
| upminor 2nd
| upminor 2nd
| ^Eb
| ^Eb
|Km2
| Km2
|classic minor 2nd
| classic minor 2nd
|KEb
| KEb
| KEb
|-
|-
| 8
| 8
| 133.333
| 133.3
| 27/25, 13/12, 14/13
| 13/12, 14/13, 27/25
| v~2
| ^^m2, v~2
| downmid 2nd
| dupminor 2nd, downmid 2nd
| ^^Eb
| ^^Eb
|Om2
| Om2
|on minor 2nd
| on minor 2nd
|OEb
| OEb
| SEb
|-
|-
| 9
| 9
| 150.000
| 150.0
| 12/11
| 12/11
| ~2
| ~2
| mid 2nd
| mid 2nd
| v<sup>3</sup>E
| v<sup>3</sup>E
|N2
| N2
|neutral 2nd
| neutral 2nd
|UEb/uE
| UEb/uE
| UEb/uE
|-
|-
| 10
| 10
| 166.667
| 166.7
| 11/10
| 11/10
| ^~2
| ^~2, vvM2
| upmid 2nd
| upmid 2nd, dudmajor 2nd
| vvE
| vvE
|oM2
| oM2
|off major 2nd
| off major 2nd
|oE
| oE
| sE
|-
|-
| 11
| 11
| 183.333
| 183.3
| 10/9
| 10/9
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
| vE
| vE
|kM2
| kM2
|classic/comma-narrow major 2nd
| classic/comma-narrow major 2nd
|kE
| kE
| kE
|-
|-
| 12
| 12
| 200.000
| 200.0
| 9/8
| 9/8
| M2
| M2
| major 2nd
| major 2nd
| E
| E
|M2
| M2
|major 2nd
| major 2nd
|E
| E
| E
|-
|-
| 13
| 13
| 216.667
| 216.7
| 25/22, 17/15
| 17/15, 25/22
| ^M2
| ^M2
| upmajor 2nd
| upmajor 2nd
| ^E
| ^E
|LM2
| LM2
|large major 2nd
| large major 2nd
|LE
| LE
| KE
|-
|-
| 14
| 14
| 233.333
| 233.3
| 8/7
| 8/7
| ^^M2
| ^^M2
| dupmajor 2nd
| dupmajor 2nd
| ^^E
| ^^E
|SM2
| SM2
|supermajor 2nd
| supermajor 2nd
|SE
| SE
| SE
|-
|-
| 15
| 15
| 250.000
| 250.0
| 81/70, 15/13
| 15/13, 22/19
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| trupmajor 2nd,<br>trudminor 3rd
| trupmajor 2nd,<br>trudminor 3rd
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
|HM2,
| HM2, hm3
hm3
| hypermajor 2nd, hypominor 3rd
|hypermajor 2nd,  
| HE, hF
hypominor 3rd
| UE, uF
|HE,
hF
|-
|-
| 16
| 16
| 266.667
| 266.7
| 7/6
| 7/6
| vvm3
| vvm3
| dudminor 3rd
| dudminor 3rd
| vvF
| vvF
|sm3
| sm3
|subminor 3rd
| subminor 3rd
|sF
| sF
| sF
|-
|-
| 17
| 17
| 283.333
| 283.3
| 33/28, 13/11, 20/17
| 13/11, 20/17
| vm3
| vm3
| downminor 3rd
| downminor 3rd
| vF
| vF
|lm3
| lm3
|little minor 3rd
| little minor 3rd
|lF
| lF
| kF
|-
|-
| 18
| 18
| 300.000
| 300.0
| 25/21
| 19/16, 25/21, 32/27
| m3
| m3
| minor 3rd
| minor 3rd
| F
| F
|m3
| m3
|minor 3rd
| minor 3rd
|F
| F
| F
|-
|-
| 19
| 19
| 316.667
| 316.7
| 6/5
| 6/5
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
| ^F
| ^F
|Km3
| Km3
|classic minor 3rd
| classic minor 3rd
|KF
| KF
| KF
|-
|-
| 20
| 20
| 333.333
| 333.3
| 40/33, 17/14
| 17/14, 39/32, 40/33
| v~3
| ^^m3, v~3
| downmid 3rd
| dupminor 3rd, downmid 3rd
| ^^F
| ^^F
|Om3
| Om3
|on minor third
| on minor third
|OF
| OF
| SF
|-
|-
| 21
| 21
| 350.000
| 350.0
| 11/9
| 11/9, 27/22
| ~3
| ~3
| mid 3rd
| mid 3rd
| ^<sup>3</sup>F
| ^<sup>3</sup>F
|N3
| N3
|neutral 3rd
| neutral 3rd
|UF/uF#
| UF/uF#
| UF/uF#
|-
|-
| 22
| 22
| 366.667
| 366.7
| 99/80, 16/13, 21/17
| 16/13, 21/17, 26/21
| ^~3
| ^~3, vvM3
| upmid 3rd
| upmid 3rd, dudmajor 3rd
| vvF#
| vvF#
|oM3
| oM3
|off major 3rd
| off major 3rd
|oF#
| oF#
| sF#
|-
|-
| 23
| 23
| 383.333
| 383.3
| 5/4
| 5/4
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
|kM3
| kM3
|classic major 3rd
| classic major 3rd
|kF#
| kF#
| kF#
|-
|-
| 24
| 24
| 400.000
| 400.0
| 44/35
| 24/19
| M3
| M3
| major 3rd
| major 3rd
| F#
| F#
|M3
| M3
|major 3rd
| major 3rd
|F#
| F#
| F#
|-
|-
| 25
| 25
| 416.667
| 416.7
| 14/11
| 14/11
| ^M3
| ^M3
| upmajor 3rd
| upmajor 3rd
| ^F#
| ^F#
|LM3
| LM3
|large major 3rd
| large major 3rd
|LF#
| LF#
| KF#
|-
|-
| 26
| 26
| 433.333
| 433.3
| 9/7
| 9/7
| ^^M3
| ^^M3
| dupmajor 3rd
| dupmajor 3rd
| ^^F#
| ^^F#
|SM3
| SM3
|supermajor 3rd
| supermajor 3rd
|SF#
| SF#
| SF#
|-
|-
| 27
| 27
| 450.000
| 450.0
| 35/27, 13/10
| 13/10, 22/17
| ^<sup>3</sup>M3,  
| ^<sup>3</sup>M3, v<sup>3</sup>4
v<sup>3</sup>4
| trupmajor 3rd, trud 4th
| trupmajor 3rd,  
| ^<sup>3</sup>F#, v<sup>3</sup>G
trud 4th
| HM3, h4
| ^<sup>3</sup>F#,  
| hypermajor 3rd, hypo 4th
v<sup>3</sup>G
| HF#, hG
|HM3,  
| UF#, uG
h4
|hypermajor 3rd,  
hypo 4th
|HF#,  
hG
|-
|-
| 28
| 28
| 466.667
| 466.7
| 21/16, 17/13
| 17/13, 21/16
| vv4
| vv4
| dud 4th
| dud 4th
| vvG
| vvG
|s4
| s4
|sub 4th
| sub 4th
|sG
| sG
| sG
|-
|-
| 29
| 29
| 483.333
| 483.3
| 33/25
| 33/25
| v4
| v4
| down 4th
| down 4th
| vG
| vG
|l4
| l4
|little 4th
| little 4th
|lG
| lG
| kG
|-
|-
| 30
| 30
| 500.000
| 500.0
| 4/3
| 4/3
| P4
| P4
| perfect 4th
| perfect 4th
| G
| G
|P4
| P4
|perfect 4th
| perfect 4th
|G
| G
| G
|-
|-
| 31
| 31
| 516.667
| 516.7
| 27/20
| 27/20
| ^4
| ^4
| up 4th
| up 4th
| ^G
| ^G
|K4
| K4
|comma-wide 4th
| comma-wide 4th
|KG
| KG
| KG
|-
|-
| 32
| 32
| 533.333
| 533.3
| 15/11
| 15/11, 19/14, ''26/19''
| ^^4, v~4
| ^^4, v~4
| dup 4th, downmid 4th
| dup 4th, downmid 4th
| ^^G
| ^^G
|O4
| O4
|on 4th
| on 4th
|OG
| OG
| SG
|-
|-
| 33
| 33
| 550.000
| 550.0
| 11/8
| 11/8
| ~4
| ~4
| mid 4th
| mid 4th
| ^<sup>3</sup>G
| ^<sup>3</sup>G
|U4/N4
| U4/N4
|uber 4th / neutral 4th
| uber 4th / neutral 4th
|UG
| UG
| UG
|-
|-
| 34
| 34
| 566.667
| 566.7
| 25/18, 18/13
| 18/13, 25/18
| ^~4, vvA4
| ^~4, vvA4
| upmid 4th, dudaug 4th
| upmid 4th, dudaug 4th
| vvG#
| vvG#
|kkA4, sd5
| kkA4, sd5
|classic aug 4th, sub dim 5th
| classic aug 4th, sub dim 5th
|kkG#, sAb
| kkG#, sAb
| SG#, (kkG#), sAb
|-
|-
| 35
| 35
| 583.333
| 583.3
| 7/5
| 7/5
| vA4, vd5
| vA4, vd5
| downaug 4th, downdim 5th
| downaug 4th, <br>downdim 5th
| vG#, vAb
| vG#, vAb
|kA4, ld5
| kA4, ld5
|comma-narrow aug 4th, little dim 5th
| comma-narrow aug 4th, little dim 5th
|kG#, lAb
| kG#, lAb
| kG#, kAb
|-
|-
| 36
| 36
| 600.000
| 600.0
| 99/70, 17/12
| 17/12, 24/17
| A4, d5
| A4, d5
| aug 4th, dim 5th
| aug 4th, dim 5th
| G#, Ab
| 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
| 10/7
| ^A4, ^d5
| ^A4, ^d5
| upaug 4th, updim 5th
| upaug 4th, updim 5th
| ^G#, ^Ab
| ^G#, ^Ab
|LA4, Kd5
| LA4, Kd5
|large aug 4th, comma-wide dim 5th
| large aug 4th, comma-wide dim 5th
|LG#, KAb
| LG#, KAb
| KG#, KAb
|-
|-
| 38
| 38
| 633.333
| 633.3
| 36/25, 13/9
| 13/9, 36/25
| v~5, ^^d5
| v~5, ^^d5
| downmid 5th, dupdim 5th
| downmid 5th, <br>dupdim 5th
| ^^Ab
| ^^Ab
|SA4, KKd5
| SA4, KKd5
|super aug 4th, classic dim 5th
| super aug 4th, classic dim 5th
|SG#, KKAb
| SG#, KKAb
| SG#, SAb, (KKAb)
|-
|-
| 39
| 39
| 650.000
| 650.0
| 16/11
| 16/11
| ~5
| ~5
| mid 5th
| mid 5th
| v<sup>3</sup>A
| v<sup>3</sup>A
|u5/N5
| u5/N5
|unter 5th / neutral 5th
| unter 5th / neutral 5th
|uA
| uA
| uA
|-
|-
| 40
| 40
| 666.667
| 666.7
| 22/15
| ''19/13'', 22/15, 28/19
| vv5, ^~5
| vv5, ^~5
| dud 5th, upmid 5th
| dud 5th, upmid 5th
| vvA
| vvA
|o5
| o5
|off 5th
| off 5th
|oA
| oA
| sA
|-
|-
| 41
| 41
| 683.333
| 683.3
| 40/27
| 40/27
| v5
| v5
| down 5th
| down 5th
| vA
| vA
|k5
| k5
|comma-narrow 5th
| comma-narrow 5th
|kA
| kA
| kA
|-
|-
| 42
| 42
| 700.000
| 700.0
| 3/2
| 3/2
| P5
| P5
| perfect 5th
| perfect 5th
| A
| A
|P5
| P5
|perfect 5th
| perfect 5th
|A
| A
| A
|-
|-
| 43
| 43
| 716.667
| 716.7
| 50/33
| 50/33
| ^5
| ^5
| up 5th
| up 5th
| ^A
| ^A
|L5
| L5
|large fifth
| large fifth
|LA
| LA
| KA
|-
|-
| 44
| 44
| 733.333
| 733.3
| 32/21
| 26/17, 32/21
| ^^5
| ^^5
| dup 5th
| dup 5th
| ^^A
| ^^A
|S5
| S5
|super fifth
| super fifth
|SA
| SA
| SA
|-
|-
| 45
| 45
| 750.000
| 750.0
| 54/35, 17/11
| 17/11, 20/13
| ^<sup>3</sup>5,  
| ^<sup>3</sup>5, v<sup>3</sup>m6
v<sup>3</sup>m6
| trup 5th, trudminor 6th
| trup 5th,  
| ^<sup>3</sup>A, v<sup>3</sup>Bb
trudminor 6th
| H5, hm6
| ^<sup>3</sup>A,  
| hyper fifth, hypominor 6th
v<sup>3</sup>Bb
| HA, hBb
|H5,  
| UA, uBb
hm6
|hyper fifth,  
hypominor 6th
|HA,
hBb
|-
|-
| 46
| 46
| 766.667
| 766.7
| 14/9
| 14/9
| vvm6
| vvm6
| dudminor 6th
| dudminor 6th
| vvBb
| vvBb
|sm6
| sm6
|superminor 6th
| superminor 6th
|sBb
| sBb
| sBb
|-
|-
| 47
| 47
| 783.333
| 783.3
| 11/7
| 11/7
| vm6
| vm6
| downminor 6th
| downminor 6th
| vBb
| vBb
|lm6
| lm6
|little minor 6th
| little minor 6th
|lBb
| lBb
| kBb
|-
|-
| 48
| 48
| 800.000
| 800.0
| 35/22
| 19/12
| m6
| m6
| minor 6th
| minor 6th
| Bb
| Bb
|m6
| m6
|minor 6th
| minor 6th
|Bb
| Bb
| Bb
|-
|-
| 49
| 49
| 816.667
| 816.7
| 8/5
| 8/5
| ^m6
| ^m6
| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
|Km6
| Km6
|classic minor 6th
| classic minor 6th
|kBb
| kBb
| kBb
|-
|-
| 50
| 50
| 833.333
| 833.3
| 81/50, 13/8
| 13/8, 21/13, 34/21
| v~6
| ^^m6, v~6
| downmid 6th
| dupminor 6th, downmid 6th
| ^^Bb
| ^^Bb
|Om6
| Om6
|on minor 6th
| on minor 6th
|oBb
| oBb
| sBb
|-
|-
| 51
| 51
| 850.000
| 850.0
| 18/11
| 18/11, 44/27
| ~6
| ~6
| mid 6th
| mid 6th
| v<sup>3</sup>B
| v<sup>3</sup>B
|N6
| N6
|neutral 6th
| neutral 6th
|UBb, uB
| UBb, uB
| UBb, uB
|-
|-
| 52
| 52
| 866.667
| 866.7
| 33/20, 28/17
| 28/17, 33/20, 64/39
| ^~6
| ^~6, vvM6
| upmid 6th
| upmid 6th, dudmajor 6th
| vvB
| vvB
|oM6
| oM6
|off major 6th
| off major 6th
|oB
| oB
| sB
|-
|-
| 53
| 53
| 883.333
| 883.3
| 5/3
| 5/3
| vM6
| vM6
| downmajor 6th
| downmajor 6th
| vB
| vB
|kM6
| kM6
|classic major 6th
| classic major 6th
|kB
| kB
| kB
|-
|-
| 54
| 54
| 900.000
| 900.0
| 27/16
| 27/16, 32/19, 42/25
| M6
| M6
| major 6th
| major 6th
| B
| B
|M6
| M6
|major 6th
| major 6th
|B
| B
| B
|-
|-
| 55
| 55
| 916.667
| 916.7
| 56/33, 17/10
| 17/10, 22/13
| ^M6
| ^M6
| upmajor 6th
| upmajor 6th
| ^B
| ^B
|LM6
| LM6
|large major 6th
| large major 6th
|LB
| LB
| KB
|-
|-
| 56
| 56
| 933.333
| 933.3
| 12/7
| 12/7
| ^^M6
| ^^M6
| dupmajor 6th
| dupmajor 6th
| ^^B
| ^^B
|SM6
| SM6
|supermajor 6th
| supermajor 6th
|SB
| SB
| SB
|-
|-
| 57
| 57
| 950.000
| 950.0
| 121/70
| 19/11, 26/15
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| trupmajor 6th,<br>trudminor 7th
| trupmajor 6th,<br>trudminor 7th
| ^<sup>3</sup>B, <br>v<sup>3</sup>C
| ^<sup>3</sup>B, <br>v<sup>3</sup>C
|HM6,
| HM6, hm7
hm7
| hypermajor 6th, hypominor 7th
|hypermajor 6th,
| HB, hC
hypominor 7th
| UB, uC
|HB,
hC
|-
|-
| 58
| 58
| 966.667
| 966.7
| 7/4
| 7/4
| vvm7
| vvm7
| dudminor 7th
| dudminor 7th
| vvC
| vvC
|sm7
| sm7
|subminor 7th
| subminor 7th
|sC
| sC
| sC
|-
|-
| 59
| 59
| 983.333
| 983.3
| 44/25
| 30/17, 44/25
| vm7
| vm7
| downminor 7th
| downminor 7th
| vC
| vC
|lm7
| lm7
|little minor 7th
| little minor 7th
|lC
| lC
| kC
|-
|-
| 60
| 60
| 1000.000
| 1000.0
| 16/9
| 16/9
| m7
| m7
| minor 7th
| minor 7th
| C
| C
|m7
| m7
|minor 7th
| minor 7th
|C
| C
| C
|-
|-
| 61
| 61
| 1016.667
| 1016.7
| 9/5
| 9/5
| ^m7
| ^m7
| upminor 7th
| upminor 7th
| ^C
| ^C
|Km7
| Km7
|classic/comma-wide minor 7th
| classic/comma-wide minor 7th
|KC
| KC
| KC
|-
|-
| 62
| 62
| 1033.333
| 1033.3
| 20/11
| 20/11
| v~7
| ^^m7, v~7
| downmid 7th
| dupminor 7th, downmid 7th
| ^^C
| ^^C
|Om7
| Om7
|on minor 7th
| on minor 7th
|OC
| OC
| SC
|-
|-
| 63
| 63
| 1050.000
| 1050.0
| 11/6
| 11/6
| ~7
| ~7
| mid 7th
| mid 7th
| ^<sup>3</sup>C
| ^<sup>3</sup>C
|N7, hd8
| N7, hd8
|neutral 7th, hypo dim 8ve
| neutral 7th, hypo dim 8ve
|UC/uC#, hDb
| UC/uC#, hDb
| UC/uC#, uDb
|-
|-
| 64
| 64
| 1066.667
| 1066.7
| 50/27
| 13/7, 24/13, 50/27
| ^~7
| ^~7, vvM7
| upmid 7th
| upmid 7th, dudmajor 7th
| vvC#
| vvC#
|oM7, sd8
| oM7, sd8
|off major 7th, sub dim 8ve
| off major 7th, sub dim 8ve
|oC#, sDb
| oC#, sDb
| sC#, sDb
|-
|-
| 65
| 65
| 1083.333
| 1083.3
| 15/8
| 15/8, 28/15
| vM7
| vM7
| downmajor 7th
| downmajor 7th
| vC#
| vC#
|kM7, ld8
| kM7, ld8
|classic major 7th, little dim 8ve
| classic major 7th, little dim 8ve
|kC#, lDb
| kC#, lDb
| kC#, kDb
|-
|-
| 66
| 66
| 1100.000
| 1100.0
| 66/35, 17/9
| 17/9, 32/17, 36/19
| M7
| M7
| major 7th
| major 7th
| C#
| C#
|M7, d8
| M7, d8
|major 7th, dim 8ve
| major 7th, dim 8ve
|C#, Db
| C#, Db
| C#, Db
|-
|-
| 67
| 67
| 1116.667
| 1116.7
| 21/11
| 19/10, 21/11, 40/21
| ^M7
| ^M7
| upmajor 7th
| upmajor 7th
| ^C#
| ^C#
|LM7, Kd8
| LM7, Kd8
|large major 7th, comma-wide dim 8ve
| large major 7th, comma-wide dim 8ve
|LC#, KDb
| LC#, KDb
| KC#, KDb
|-
|-
| 68
| 68
| 1133.333
| 1133.3
| 27/14, 48/25
| 25/13, 27/14, 48/25, 52/27
| ^^M7
| ^^M7
| dupmajor 7th
| dupmajor 7th
| ^^C#
| ^^C#
|SM7, KKd8
| SM7, KKd8
|supermajor 7th, classic dim 8ve
| supermajor 7th, classic dim 8ve
|SC#, KKDb
| SC#, KKDb
| SC#, SDb, (KKDb)
|-
|-
| 69
| 69
| 1150.000
| 1150.0
| 35/18
| 35/18, 39/20, 64/33
| ^<sup>3</sup>M7,  
| ^<sup>3</sup>M7, v<sup>3</sup>8
v<sup>3</sup>8
| trupmajor 7th, trud octave
| trupmajor 7th,  
| ^<sup>3</sup>C#, v<sup>3</sup>D
trud octave
| HM7, u8, h8
| ^<sup>3</sup>C#,  
| hypermajor 7th, unter 8ve, hypo 8ve
v<sup>3</sup>D
| HC#, uD, hD
|HM7,
| UC#, uDb, uD
u8, h8
|hypermajor 7th,
unter 8ve, hypo 8ve
|HC#,
uD, hD
|-
|-
| 70
| 70
| 1166.667
| 1166.7
| 49/25
| 49/25, 55/28, 63/32, 88/45, 96/49
| vv8
| vv8
| dud octave
| dud octave
| vvD
| vvD
|s8, o8
| s8, o8
|sub 8ve, off 8ve
| sub 8ve, off 8ve
|sD, oD
| sD, oD
| sD
|-
|-
| 71
| 71
| 1183.333
| 1183.3
| 99/50
| 99/50, 160/81, 180/91, 196/99, 208/105
| v8
| v8
| down octave
| down octave
| vD
| vD
|k8, l8
| k8, l8
|comma-narrow 8ve, little 8ve
| comma-narrow 8ve, little 8ve
|kD, lD
| kD, lD
| kD
|-
|-
| 72
| 72
| 1200.000
| 1200.0
| 2/1
| 2/1
| P8
| P8
| perfect octave
| perfect octave
| D
| D
|P8
| P8
|perfect octave
| perfect octave
|D
| D
| D
|}
|}
 
<references group="note" />
<nowiki>*</nowiki> based on treating 72edo as a 17-limit temperament. For lower limits see [[Table of 72edo intervals]].


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Line 807: Line 856:
! Quality
! Quality
! [[Color notation|Color]]
! [[Color notation|Color]]
! Monzo Format
! Monzo format
! Examples
! Examples
|-
|-
Line 813: Line 862:
| zo
| zo
| (a b 0 1)
| (a b 0 1)
| 7/6, 7/40
| 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
|-
|-
Line 825: Line 874:
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| rowspan="2" |downmid
| rowspan="2" | dupminor, <br>downmid
| luyo
| luyo
| (a b 1 0 -1)
| (a b 1 0 -1)
Line 836: Line 885:
| 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
| rowspan="2" | upmid, <br>dudmajor
| logu
| logu
| (a b -1 0 1)
| (a b -1 0 1)
Line 859: Line 908:
| major
| major
| fifthward wa
| fifthward wa
| (a b), b &gt; 1
| (a b), b > 1
| 9/8, 27/16
| 9/8, 27/16
|-
|-
| 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
| thogu
| (a b -1 0 0 1)
| 13/10
|-
| thuyo
| (a b 1 0 0 -1)
| 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 871: Line 929:
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! [[Kite's color notation|Color of the 3rd]]
! [[Color notation|Color of the 3rd]]
! JI Chord
! JI chord
! Notes as Edosteps
! Notes as edosteps
! Notes of C Chord
! Notes of C chord
! Written Name
! Written name
! Spoken Name
! Spoken name
|-
|-
| zo
| zo
Line 913: Line 971:
| C dupmajor or C dup
| C dupmajor or C dup
|}
|}
For a more complete list, see [[Ups and Downs Notation #Chord names in other EDOs]].  
For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]].  


=== Remembering the pitch structure ===
=== Relationship between primes and rings ===
The pitch structure is very easy to remember. In 72tet, 12edo is the Pythagorian ring; ''id est'', every 6 degrees is the 3-limit.
In 72tet, there are 6 [[ring number|rings]]. 12edo is the plain ring; thus every 6 degrees is the 3-limit.


Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
* -1 degree corrects 5/4 (80/81)
* −1 degree (the down ring) corrects 81/64 to 5/4 via 80/81
* -2 degrees corrects 7/4 (63/64)
* −2 degrees (the dud ring) corrects 16/9 to 7/4 via 63/64
* +3 degrees corrects 11/8 (33/32)
* +3 degrees (the trup ring) corrects 4/3 to 11/8 via 33/32
* +2 degrees corrects 13/8 (1053/1024)
* +2 degrees (the dup ring) corrects 128/81 to 13/8 via 1053/1024
* 0 degree corrects 17/16 and 19/16 (4131/4096 and 513/512)
* 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.


== Notations ==
== Notations ==
=== Sagittal ===
=== Ups and downs notation ===
72edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
{{Sharpness-sharp6a}}
 
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
 
[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have sharps and flats with arrows borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp6}}
 
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[65edo#Sagittal notation|65-EDO]] and [[79edo#Sagittal notation|79]], and is a superset of the notations for EDOs [[36edo#Sagittal notation|36]], [[24edo#Sagittal notation|24]], [[18edo#Sagittal notation|18]], [[12edo#Sagittal notation|12]], [[8edo#Sagittal notation|8]], and [[6edo#Sagittal notation|6]].
 
==== Evo flavor ====
<imagemap>
File:72-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:72-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Revo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:72-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 711 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 72edo in the Revo flavor of Sagittal:
From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 72edo in the Revo flavor of Sagittal:


[[File:72edo Sagittal.png|800px]]
[[File:72edo Sagittal.png|800px]]


== JI approximation ==
=== Ivan Wyschnegradsky's notation ===
=== Z function ===
{{Sharpness-sharp6-iw|72}}
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]]
== Approximation to JI ==
[[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]]


=== 15-odd-limit interval mappings ===
=== Interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 72edo. Prime harmonics are in '''bold'''. As 72edo is consistent in the 15-odd-limit, the results by direct approximation and patent val mapping are the same.
{{Q-odd-limit intervals|72}}


{{15-odd-limit|72}}
=== Zeta properties ===
72edo is the ninth [[zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[the Riemann zeta function and tuning#The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.
 
[[File:plot72.png|alt=plot72.png|plot72.png]]


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 955: Line 1,070:
| 2.3.5
| 2.3.5
| 15625/15552, 531441/524288
| 15625/15552, 531441/524288
| [{{val| 72 114 167 }}]
| {{Mapping| 72 114 167 }}
| +0.839
| +0.839
| 0.594
| 0.594
Line 962: Line 1,077:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1029/1024, 4375/4374
| 225/224, 1029/1024, 4375/4374
| [{{val| 72 114 167 202 }}]
| {{Mapping| 72 114 167 202 }}
| +0.822
| +0.822
| 0.515
| 0.515
Line 969: Line 1,084:
| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 4000/3993
| 225/224, 243/242, 385/384, 4000/3993
| [{{val| 72 114 167 202 249 }}]
| {{Mapping| 72 114 167 202 249 }}
| +0.734
| +0.734
| 0.493
| 0.493
Line 976: Line 1,091:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 169/168, 225/224, 243/242, 325/324, 385/384
| 169/168, 225/224, 243/242, 325/324, 385/384
| [{{val| 72 114 167 202 266 249 }}]
| {{Mapping| 72 114 167 202 249 266 }}
| +0.936
| +0.936
| 0.638
| 0.638
Line 983: Line 1,098:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 169/168, 221/220, 225/224, 243/242, 273/272, 325/324
| 169/168, 221/220, 225/224, 243/242, 273/272, 325/324
| [{{val| 72 114 167 202 249 266 294 }}]
| {{Mapping| 72 114 167 202 249 266 294 }}
| +0.975
| +0.975
| 0.599
| 0.599
| 3.59
| 3.59
|-
| 2.3.5.7.11.13.17.19
| 153/152, 169/168, 210/209, 221/220, 225/224, 243/242, 273/272
| {{Mapping| 72 114 167 202 249 266 294 306 }}
| +0.780
| 0.762
| 4.57
|}
|}
* 72et has lower relative errors than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next equal temperaments doing better in these subgroups are [[99edo|99]], [[270edo|270]], [[224edo|224]], [[494edo|494]], and [[217edo|217]], respectively.  
* 72et has lower relative errors than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next equal temperaments doing better in these subgroups are [[99edo|99]], [[270edo|270]], [[224edo|224]], [[494edo|494]], and [[217edo|217]], respectively.  


Line 995: Line 1,116:


{| class="commatable wikitable center-1 center-2 right-4"
{| class="commatable wikitable center-1 center-2 right-4"
! [[Harmonic limit|Prime<br>Limit]]
|-
! [[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 1,003: Line 1,125:
| 3
| 3
| [[531441/524288|(12 digits)]]
| [[531441/524288|(12 digits)]]
| {{Monzo|-19 12 }}
| {{Monzo| -19 12 }}
| 23.46
| 23.46
| Pythagorean comma
| Pythagorean comma
Line 1,017: Line 1,139:
| {{Monzo| -25 7 6 }}
| {{Monzo| -25 7 6 }}
| 31.57
| 31.57
| [[Ampersand]]
| [[Ampersand comma]]
|-
|-
| 5
| 5
Line 1,059: Line 1,181:
| {{Monzo| 0 3 4 -5 }}
| {{Monzo| 0 3 4 -5 }}
| 6.99
| 6.99
| Mirkwai
| Mirkwai comma
|-
|-
| 7
| 7
Line 1,065: Line 1,187:
| {{Monzo| -4 9 -2 -2 }}
| {{Monzo| -4 9 -2 -2 }}
| 7.32
| 7.32
| Cataharry
| Cataharry comma
|-
|-
| 7
| 7
Line 1,107: Line 1,229:
| {{Monzo| -2 0 3 -3 1 }}
| {{Monzo| -2 0 3 -3 1 }}
| 3.78
| 3.78
| Moctdel  
| Moctdel comma
|-
|-
| 11
| 11
Line 1,119: Line 1,241:
| {{Monzo| 5 -1 3 0 -3 }}
| {{Monzo| 5 -1 3 0 -3 }}
| 3.03
| 3.03
| Wizardharry  
| Wizardharry comma
|-
|-
| 11
| 11
Line 1,161: Line 1,283:
| {{Monzo| 2 -1 0 1 -2 1 }}
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.76
| 4.76
| Gentle comma
| Minor minthma
|-
|-
| 13
| 13
Line 1,211: Line 1,333:
| Jacobin comma
| Jacobin comma
|}
|}
<references/>
<references group="note" />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 1,218: Line 1,340:
72edo provides the [[optimal patent val]] for [[miracle]] and [[wizard]] in the 7-limit, miracle, [[catakleismic]], [[bikleismic]], [[compton]], [[ennealimnic]], [[ennealiminal]], [[enneaportent]], [[marvolo]] and [[catalytic]] in the 11-limit, and catakleismic, bikleismic, compton, [[comptone]], [[enneaportent]], [[ennealim]], catalytic, marvolo, [[manna]], [[hendec]], [[lizard]], [[neominor]], [[hours]], and [[semimiracle]] in the 13-limit.
72edo provides the [[optimal patent val]] for [[miracle]] and [[wizard]] in the 7-limit, miracle, [[catakleismic]], [[bikleismic]], [[compton]], [[ennealimnic]], [[ennealiminal]], [[enneaportent]], [[marvolo]] and [[catalytic]] in the 11-limit, and catakleismic, bikleismic, compton, [[comptone]], [[enneaportent]], [[ennealim]], catalytic, marvolo, [[manna]], [[hendec]], [[lizard]], [[neominor]], [[hours]], and [[semimiracle]] in the 13-limit.


{| class="wikitable center-1 center-2"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator
! Generator*
! 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
| 116.7
| 15/14
| [[Miracle]] / benediction / manna
| [[Miracle]] / benediction / manna
|-
| 1
| 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
| [[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]]
| [[Semimiracle]]
|-
|-
| 2
| 2
| 11\72
| 11\72
| 183.3
| 10/9
| [[Unidec]] / hendec
| [[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
| 19\72
| 23\72<br>(13\72)
| [[Bikleismic]]
| 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
|-
|-
| 9
| 9
| 3\72
| 23\72<br>(1\72)
| [[Ennealimmal]] / ennealimnic
| 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 lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
Line 1,380: Line 1,512:


=== Harmonic scale ===
=== Harmonic scale ===
Mode 8 of the harmonic series – [[overtone scale|harmonics 8 through 16]], octave repeating – is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).
Mode 8 of the harmonic series&mdash;[[overtone scale|harmonics 8 through 16]], octave repeating&mdash;is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).


{| class="wikitable"
{| class="wikitable"
|-
|-
| Harmonics in "Mode 8":
! Harmonics in "Mode 8":
| 8
| 8
|  
|  
Line 1,403: Line 1,535:
| 16
| 16
|-
|-
| …as JI Ratio from 1/1:
! …as JI Ratio from 1/1:
| 1/1
| 1/1
|  
|  
Line 1,422: Line 1,554:
| 2/1
| 2/1
|-
|-
| …in cents:
! …in cents:
| 0
| 0
|  
|  
Line 1,441: Line 1,573:
| 1200.0
| 1200.0
|-
|-
| Nearest degree of 72edo:
! Nearest degree of 72edo:
| 0
| 0
|  
|  
Line 1,460: Line 1,592:
| 72
| 72
|-
|-
| …in cents:
! …in cents:
| 0
| 0
|  
|  
Line 1,479: Line 1,611:
| 1200.0
| 1200.0
|-
|-
| Steps as Freq. Ratio:
! Steps as Freq. Ratio:
|  
|  
| 9:8
| 9:8
Line 1,498: Line 1,630:
|  
|  
|-
|-
| …in cents:
! …in cents:
|  
|  
| 203.9
| 203.9
Line 1,517: Line 1,649:
|  
|  
|-
|-
| Nearest degree of 72edo:
! Nearest degree of 72edo:
|  
|  
| 12
| 12
Line 1,536: Line 1,668:
|  
|  
|-
|-
| …in cents:
! …in cents:
|  
|  
| 200.0
| 200.0
Line 1,555: Line 1,687:
|  
|  
|}
|}
== Instruments ==
If one can get six 12edo instruments tuned a twelfth-tone apart, it is possible to use these instruments in combination to play the full gamut of 72edo (see Music).
One can also use a skip fretting system:
* [[Skip fretting system 72 2 27]]
Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo:
* [[Lumatone mapping for 72edo]]


== Music ==
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/VwVp3RVao_k ''microtonal improvisation in 72edo''] (2025)
; [[Ambient Esoterica]]
; [[Ambient Esoterica]]
* [https://www.youtube.com/watch?v=seWcDAoQjxY ''Goetic Synchronities''] (2023)
* [https://www.youtube.com/watch?v=seWcDAoQjxY ''Goetic Synchronities''] (2023)
Line 1,573: Line 1,717:
* [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)


; [[Prent Rodgers]]
; [[Prent Rodgers]]
Line 1,579: Line 1,724:
; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* [https://www.archive.org/details/Kotekant ''Kotekant''] [https://www.archive.org/download/Kotekant/kotekant.mp3 play] (2010)
* [https://www.archive.org/details/Kotekant ''Kotekant''] [https://www.archive.org/download/Kotekant/kotekant.mp3 play] (2010)
;[[Ivan Wyschnegradsky]]
* [https://www.youtube.com/watch?v=RCcJHCkYQ6U Arc-en-ciel, for 6 pianos in twelfth tones, Op. 37] (1956)


; [[James Tenney]]
; [[James Tenney]]
Line 1,591: Line 1,739:
* [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
* [https://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo]
* [https://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo]
* [http://tonalsoft.com/enc/number/72edo.aspx 72-ed2 / 72-edo / 72-ET / 72-tone equal-temperament] on [[Tonalsoft Encyclopedia]]


[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/72edo"