Xenharmonic Wiki

72edo: Difference between revisions

← Older edit
VisualWikitext
Gud2228 (talk | contribs)
autopatrolled, emailconfirmed
227 edits
Tags: Mobile edit Mobile web edit
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
m Text replacement - "Ups and Downs Notation" to "Ups and downs notation"
 
(97 intermediate revisions by 16 users not shown)
Line 1: Line 1:
{{interwiki
{{interwiki
| de = 72edo
| de = 72-EDO
| en = 72edo
| en = 72edo
| es =  
| es =  
Line 7: Line 7:
{{Infobox ET}}
{{Infobox ET}}
{{Wikipedia|72 equal temperament}}
{{Wikipedia|72 equal temperament}}
The '''72 equal divisions of the octave''' ('''72edo'''), or '''72-tone equal temperament''' ('''72tet''', '''72et''') when viewed from a [[regular temperament]] perspective, 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 [[12edo]]. 72edo is also a superset of [[24edo]], a common and standard tuning of [[Arabic, Turkish, Persian|Arabic]] music, and has itself been used to tune Turkish music.
{{ED intro}}


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


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|72|columns=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 ===
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]]
! 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
| 1/1
| P1
| P1
| perfect unison
| perfect unison
| D
| P1
| 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
| comma-wide unison, large unison
| 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
| super unison, on unison
| SD, OD
| SD
|-
|-
| 3
| 3
| 50.000
| 50.0
| 33/32
| 33/32, 36/35, 40/39
| ^<sup>3</sup>1, <br>v<sup>3</sup>m2
| ^<sup>3</sup>1, v<sup>3</sup>m2
| trup unison,<br>trudminor 2nd
| trup unison, trudminor 2nd
| ^<sup>3</sup>D, <br>v<sup>3</sup>Eb
| ^<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
| 66.667
| 66.7
| 25/24
| 25/24, 26/25, 27/26, 28/27
| vvm2
| vvm2
| dudminor 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
| 20/19, 21/20, 22/21
| 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
| 17/16, 18/17, 19/18
| m2
| minor 2nd
| 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
| classic minor 2nd
| 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
| on minor 2nd
| 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
| neutral 2nd
| 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
| off major 2nd
| oE
| sE
|-
|-
| 11
| 11
| 183.333
| 183.3
| 10/9
| 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
| 9/8
| M2
| M2
| major 2nd
| major 2nd
| E
| M2
| 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
| large major 2nd
| LE
| KE
|-
|-
| 14
| 14
| 233.333
| 233.3
| 8/7
| 8/7
| ^^M2
| ^^M2
| dupmajor 2nd
| dupmajor 2nd
| ^^E
| ^^E
| SM2
| supermajor 2nd
| 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, hm3
| hypermajor 2nd, hypominor 3rd
| HE, hF
| UE, uF
|-
|-
| 16
| 16
| 266.667
| 266.7
| 7/6
| 7/6
| vvm3
| vvm3
| dudminor 3rd
| dudminor 3rd
| vvF
| vvF
| sm3
| subminor 3rd
| 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
| little minor 3rd
| lF
| kF
|-
|-
| 18
| 18
| 300.000
| 300.0
| 25/21
| 19/16, 25/21, 32/27
| m3
| minor 3rd
| 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
| classic minor 3rd
| 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
| on minor third
| 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
| neutral 3rd
| 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
| off major 3rd
| oF#
| sF#
|-
|-
| 23
| 23
| 383.333
| 383.3
| 5/4
| 5/4
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
| kM3
| classic major 3rd
| kF#
| kF#
|-
|-
| 24
| 24
| 400.000
| 400.0
| 44/35
| 24/19
| M3
| M3
| major 3rd
| major 3rd
| F#
| M3
| major 3rd
| F#
| F#
| F#
|-
|-
| 25
| 25
| 416.667
| 416.7
| 14/11
| 14/11
| ^M3
| ^M3
| upmajor 3rd
| upmajor 3rd
| ^F#
| ^F#
| LM3
| large major 3rd
| LF#
| KF#
|-
|-
| 26
| 26
| 433.333
| 433.3
| 9/7
| 9/7
| ^^M3
| ^^M3
| dupmajor 3rd
| dupmajor 3rd
| ^^F#
| ^^F#
| SM3
| supermajor 3rd
| SF#
| SF#
|-
|-
| 27
| 27
| 450.000
| 450.0
| 35/27, 13/10
| 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
| trupmajor 3rd, trud 4th
| ^<sup>3</sup>F#, v<sup>3</sup>G
| ^<sup>3</sup>F#, 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
| 17/13, 21/16
| vv4
| vv4
| dud 4th
| dud 4th
| vvG
| vvG
| s4
| sub 4th
| sG
| sG
|-
|-
| 29
| 29
| 483.333
| 483.3
| 33/25
| 33/25
| v4
| v4
| down 4th
| down 4th
| vG
| vG
| l4
| little 4th
| lG
| kG
|-
|-
| 30
| 30
| 500.000
| 500.0
| 4/3
| 4/3
| P4
| P4
| perfect 4th
| perfect 4th
| G
| P4
| perfect 4th
| G
| G
| G
|-
|-
| 31
| 31
| 516.667
| 516.7
| 27/20
| 27/20
| ^4
| ^4
| up 4th
| up 4th
| ^G
| ^G
| K4
| comma-wide 4th
| 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
| on 4th
| 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
| uber 4th / neutral 4th
| 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
| classic aug 4th, sub dim 5th
| 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
| comma-narrow aug 4th, little dim 5th
| 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
| A4, d5
| 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
| large aug 4th, comma-wide dim 5th
| 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
| super aug 4th, classic dim 5th
| 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
| unter 5th / neutral 5th
| 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
| off 5th
| oA
| sA
|-
|-
| 41
| 41
| 683.333
| 683.3
| 40/27
| 40/27
| v5
| v5
| down 5th
| down 5th
| vA
| vA
| k5
| comma-narrow 5th
| kA
| kA
|-
|-
| 42
| 42
| 700.000
| 700.0
| 3/2
| 3/2
| P5
| P5
| perfect 5th
| perfect 5th
| A
| P5
| perfect 5th
| A
| A
| A
|-
|-
| 43
| 43
| 716.667
| 716.7
| 50/33
| 50/33
| ^5
| ^5
| up 5th
| up 5th
| ^A
| ^A
| L5
| large fifth
| LA
| KA
|-
|-
| 44
| 44
| 733.333
| 733.3
| 32/21
| 26/17, 32/21
| ^^5
| ^^5
| dup 5th
| dup 5th
| ^^A
| ^^A
| S5
| super fifth
| SA
| SA
|-
|-
| 45
| 45
| 750.000
| 750.0
| 54/35, 17/11
| 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
| ^<sup>3</sup>A, v<sup>3</sup>Bb
| ^<sup>3</sup>A, v<sup>3</sup>Bb
| H5, hm6
| hyper fifth, hypominor 6th
| HA, hBb
| UA, uBb
|-
|-
| 46
| 46
| 766.667
| 766.7
| 14/9
| 14/9
| vvm6
| vvm6
| dudminor 6th
| dudminor 6th
| vvBb
| vvBb
| sm6
| superminor 6th
| sBb
| sBb
|-
|-
| 47
| 47
| 783.333
| 783.3
| 11/7
| 11/7
| vm6
| vm6
| downminor 6th
| downminor 6th
| vBb
| vBb
| lm6
| little minor 6th
| lBb
| kBb
|-
|-
| 48
| 48
| 800.000
| 800.0
| 35/22
| 19/12
| m6
| m6
| minor 6th
| minor 6th
| Bb
| m6
| minor 6th
| Bb
| Bb
| Bb
|-
|-
| 49
| 49
| 816.667
| 816.7
| 8/5
| 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
| 13/8, 21/13, 34/21
| v~6
| ^^m6, v~6
| downmid 6th
| dupminor 6th, downmid 6th
| ^^Bb
| ^^Bb
| Om6
| on minor 6th
| 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
| neutral 6th
| 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
| off major 6th
| oB
| sB
|-
|-
| 53
| 53
| 883.333
| 883.3
| 5/3
| 5/3
| vM6
| vM6
| downmajor 6th
| downmajor 6th
| vB
| vB
| kM6
| classic major 6th
| kB
| kB
|-
|-
| 54
| 54
| 900.000
| 900.0
| 27/16
| 27/16, 32/19, 42/25
| M6
| major 6th
| 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
| large major 6th
| LB
| KB
|-
|-
| 56
| 56
| 933.333
| 933.3
| 12/7
| 12/7
| ^^M6
| ^^M6
| dupmajor 6th
| dupmajor 6th
| ^^B
| ^^B
| SM6
| supermajor 6th
| 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, hm7
| hypermajor 6th, hypominor 7th
| HB, hC
| UB, uC
|-
|-
| 58
| 58
| 966.667
| 966.7
| 7/4
| 7/4
| vvm7
| vvm7
| dudminor 7th
| dudminor 7th
| vvC
| vvC
| sm7
| subminor 7th
| sC
| sC
|-
|-
| 59
| 59
| 983.333
| 983.3
| 44/25
| 30/17, 44/25
| vm7
| vm7
| downminor 7th
| downminor 7th
| vC
| vC
| lm7
| little minor 7th
| lC
| kC
|-
|-
| 60
| 60
| 1000.000
| 1000.0
| 16/9
| 16/9
| m7
| m7
| minor 7th
| minor 7th
| C
| m7
| minor 7th
| C
| C
| C
|-
|-
| 61
| 61
| 1016.667
| 1016.7
| 9/5
| 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
| 20/11
| v~7
| ^^m7, v~7
| downmid 7th
| dupminor 7th, downmid 7th
| ^^C
| ^^C
| Om7
| on minor 7th
| 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
| neutral 7th, hypo dim 8ve
| 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
| off major 7th, sub dim 8ve
| 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
| classic major 7th, little dim 8ve
| 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
| major 7th, dim 8ve
| 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
| large major 7th, comma-wide dim 8ve
| LC#, KDb
| KC#, KDb
|-
|-
| 68
| 68
| 1133.333
| 1133.3
| 27/14
| 25/13, 27/14, 48/25, 52/27
| ^^M7
| ^^M7
| dupmajor 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
| 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
| ^<sup>3</sup>C#, v<sup>3</sup>D
| ^<sup>3</sup>C#, v<sup>3</sup>D
| HM7, u8, h8
| hypermajor 7th, unter 8ve, hypo 8ve
| HC#, uD, hD
| UC#, uDb, uD
|-
|-
| 70
| 70
| 1166.667
| 1166.7
| 49/25
| 49/25, 55/28, 63/32, 88/45, 96/49
| vv8
| vv8
| dud octave
| dud octave
| vvD
| vvD
| s8, o8
| sub 8ve, off 8ve
| 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
| comma-narrow 8ve, little 8ve
| kD, lD
| kD
|-
|-
| 72
| 72
| 1200.000
| 1200.0
| 2/1
| 2/1
| P8
| P8
| perfect octave
| perfect octave
| D
| P8
| perfect octave
| D
| D
| D
|}
|}
<references group="note" />


<nowiki>*</nowiki> based on treating 72edo as a 17-limit temperament
=== Interval quality and chord names in color notation ===
 
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
Combining ups and downs notation with [[Kite's_color_notation|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
|-
|-
| dudminor
| dudminor
| 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 567: 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)
|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
| 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
Line 601: 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 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 655: 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]].
 
=== 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 80/81
* −2 degrees (the dud ring) corrects 16/9 to 7/4 via 63/64
* +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.


== 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 687: 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 694: 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 701: 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 708: 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 715: 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 is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next ETs doing better in these subgroups are 99, 270, 224, 494, and 217, respectively.  


=== Commas ===
=== Commas ===
Line 727: 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 735: 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 749: Line 1,139:
| {{Monzo| -25 7 6 }}
| {{Monzo| -25 7 6 }}
| 31.57
| 31.57
| [[Ampersand]]
| [[Ampersand comma]]
|-
|-
| 5
| 5
Line 767: Line 1,157:
| {{Monzo| -5 2 2 -1 }}
| {{Monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Septimal kleisma, Marvel comma
| Marvel comma
|-
|-
| 7
| 7
Line 791: Line 1,181:
| {{Monzo| 0 3 4 -5 }}
| {{Monzo| 0 3 4 -5 }}
| 6.99
| 6.99
| Mirkwai
| Mirkwai comma
|-
|-
| 7
| 7
Line 797: Line 1,187:
| {{Monzo| -4 9 -2 -2 }}
| {{Monzo| -4 9 -2 -2 }}
| 7.32
| 7.32
| Cataharry
| Cataharry comma
|-
|-
| 7
| 7
Line 839: 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 851: 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 863: Line 1,253:
| {{Monzo| -3 4 -2 -2 2 }}
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Kalisma, Gauss' comma
| Kalisma
|-
|-
| 11
| 11
Line 875: Line 1,265:
| {{Monzo| -3 -1 0 -1 0 2 }}
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.27
| 10.27
| Buzurgisma, Dhanvantarisma
| Buzurgisma
|-
|-
| 13
| 13
Line 893: 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 905: Line 1,295:
| {{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
Line 943: Line 1,333:
| Jacobin comma
| Jacobin comma
|}
|}
<references/>
<references group="note" />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 950: 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
| [[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
| 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 ==
* [[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]]
* [[Blackjack]], [[miracle_8]], [[miracle_10]], [[miracle_12]], [[miracle_12a]], [[miracle_24hi]], [[miracle_24lo]]
* [[keenanmarvel]], [[xenakis_chrome]], [[xenakis_diat]], [[xenakis_schrome]]
* [[Keenanmarvel]], [[xenakis_chrome]], [[xenakis_diat]], [[xenakis_schrome]]
* [[genus24255et72|Euler(24255) genus in 72 equal]]
* [[Genus24255et72|Euler(24255) genus in 72 equal]]
* [[JuneGloom]]
* [[JuneGloom]]
* [[Blackened skies|BlackenedSkies]]
* [[Aurora scale|Aurora]]
* [[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
* [[Harry Partch's 43-tone scale]]: 1 2 2 2 2 1 1 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 1 1 2 2 2 2 1
* [[Magnetosphere scale|Magnetosphere]], [[Blackened skies]], [[Lost spirit]]
* [[5- to 10-tone scales in 72edo]]


=== Harmonic scale ===
=== 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 1,135: Line 1,535:
| 16
| 16
|-
|-
| …as JI Ratio from 1/1:
! …as JI Ratio from 1/1:
| 1/1
| 1/1
|  
|  
Line 1,154: Line 1,554:
| 2/1
| 2/1
|-
|-
| …in cents:
! …in cents:
| 0
| 0
|  
|  
Line 1,173: Line 1,573:
| 1200.0
| 1200.0
|-
|-
| Nearest degree of 72edo:
! Nearest degree of 72edo:
| 0
| 0
|  
|  
Line 1,192: Line 1,592:
| 72
| 72
|-
|-
| …in cents:
! …in cents:
| 0
| 0
|  
|  
Line 1,211: Line 1,611:
| 1200.0
| 1200.0
|-
|-
| Steps as Freq. Ratio:
! Steps as Freq. Ratio:
|  
|  
| 9:8
| 9:8
Line 1,230: Line 1,630:
|  
|  
|-
|-
| …in cents:
! …in cents:
|  
|  
| 203.9
| 203.9
Line 1,249: Line 1,649:
|  
|  
|-
|-
| Nearest degree of 72edo:
! Nearest degree of 72edo:
|  
|  
| 12
| 12
Line 1,268: Line 1,668:
|  
|  
|-
|-
| …in cents:
! …in cents:
|  
|  
| 200.0
| 200.0
Line 1,288: Line 1,688:
|}
|}


== Remembering the pitch structure ==
== Instruments ==
The pitch structure is very easy to remember. In 72tet, 12edo is the Pythagorian ring; ''id est'', every 6 degrees is the 3-limit.
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).
 
Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:


-1 degree corrects 5/4 (80/81)
One can also use a skip fretting system:
* [[Skip fretting system 72 2 27]]


-2 degrees corrects 7/4 (63/64)
Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo:
* [[Lumatone mapping for 72edo]]


+3 degrees corrects 11/8 (33/32)
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/VwVp3RVao_k ''microtonal improvisation in 72edo''] (2025)


+2 degrees corrects 13/8 (1053/1024)
; [[Ambient Esoterica]]
* [https://www.youtube.com/watch?v=seWcDAoQjxY ''Goetic Synchronities''] (2023)
* [https://www.youtube.com/watch?v=CrcdM1e2b6Q ''Rainy Day Generative Pillow''] (2024)


0 degree corrects 17/16 and 19/16 (4131/4096 and 513/512)
; [[Jake Freivald]]
 
== Music ==
'''[[Jake Freivald]]'''
* [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday'']{{dead link}} in the [[lazysunday]] scale
* [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday'']{{dead link}} in the [[lazysunday]] scale


'''[[Georg Friedrich Haas]]'''
{{Wikipedia|In vain (Haas)}}
; [[Georg Friedrich Haas]]
* [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])


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


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


'''[[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)


'''[[James Tenney]]'''
;[[Ivan Wyschnegradsky]]
* [https://www.youtube.com/watch?v=RCcJHCkYQ6U Arc-en-ciel, for 6 pianos in twelfth tones, Op. 37] (1956)
 
; [[James Tenney]]
* [https://www.youtube.com/watch?v=jGsxqU1PhZs&list=OLAK5uy_mKyMEMZW7noeLncJnu-JT65go8w7403DA ''Changes for Six Harps'']
* [https://www.youtube.com/watch?v=jGsxqU1PhZs&list=OLAK5uy_mKyMEMZW7noeLncJnu-JT65go8w7403DA ''Changes for Six Harps'']


== See also ==
; [[Xeno Ov Eleas]]
* [[Joe Maneri]]
* [https://www.youtube.com/watch?v=cx7I0NWem5w ''Χenomorphic Ghost Storm''] (2022)


== External links ==
== External links ==
Line 1,330: 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]]
[[Category:Compton]]
[[Category:Marvel]]
[[Category:Marvel]]
[[Category:Miracle]]
[[Category:Miracle]]
[[Category:Moria]]
[[Category:Prodigy]]
[[Category:Prodigy]]
[[Category:Wizard]]
[[Category:Wizard]]
Retrieved from "https://en.xen.wiki/w/72edo"