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


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


72-tone equal temperament approximates [[11-limit|11-limit just intonation]] exceptionally well, is consistent in the [[17-limit|17-limit]], and is the ninth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.
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]].


72 is an excellent tuning for [[Gamelismic_clan|miracle temperament]], especially the 11-limit version, and the related rank three temperament [[Marvel_family#Prodigy|prodigy]], and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.
== 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]].  


=Commas=
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.


Commas tempered out by 72edo include...
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]].


{| class="wikitable"
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.
|-
! | 3-limit
|-
| |<nowiki> Pythagorean comma = 531441/524288 = |-19 12</nowiki>&gt;
|}


{| class="wikitable"
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]].
|-
! | 5-limit
|-
| |<nowiki> kleisma = 15625/15552 = |-6 -5 6</nowiki>&gt;


ampersand = 34171875/33554432 = |-25 7 6&gt;
=== Prime harmonics ===
{{Harmonics in equal|72|columns=9}}
{{Harmonics in equal|72|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}


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


ennealimma = 7629394531250/7625597484987 = |1 -27 18&gt;
=== 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.


{| class="wikitable"
== Intervals ==
{| class="wikitable center-all right-2 left-3"
|-
! #
! Cents
! 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" | [[SKULO interval names|SKULO interval names and notation]]
! (K, S, U)
|-
| 0
| 0.0
| 1/1
| P1
| perfect unison
| D
| P1
| perfect unison
| D
| D
|-
| 1
| 16.7
| 81/80, 91/90, 99/98, 100/99, 105/104
| ^1
| up unison
| ^D
| K1, L1
| comma-wide unison, large unison
| KD, LD
| KD
|-
| 2
| 33.3
| 45/44, 49/48, 50/49, 55/54, 64/63
| ^^
| dup unison
| ^^D
| S1, O1
| super unison, on unison
| SD, OD
| SD
|-
|-
! | 7-limit
| 3
! | 11-limit
| 50.0
! | 13-limit
| 33/32, 36/35, 40/39
| ^<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
225/224
| 25/24, 26/25, 27/26, 28/27
 
| vvm2
1029/1024
| dudminor 2nd
 
| vvEb
2401/2400
| kkA1, sm2
 
| classic aug unison, subminor 2nd
4375/4374
| kkD#, sEb
 
| sD#, (kkD#), sEb
16875/16807
 
19683/19600
 
420175/419904
 
250047/250000
| | .......................
 
243/242
 
385/384
 
441/440
 
540/539
 
1375/1372
 
3025/3024
 
4000/3993
 
6250/6237
 
9801/9800
| | .......................
 
169/168
 
325/324
 
351/350
 
364/363
 
625/624
 
676/675
 
729/728
 
1001/1000
 
1575/1573
 
1716/1715
 
2080/2079
 
6656/6655
|}
 
=Temperaments=
 
It 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.
 
See also [[List_of_edo-distinct_72et_rank_two_temperaments|List of edo-distinct 72et rank two temperaments]].
 
=Harmonic Scale=
Mode 8 of the harmonic series -- [[overtone_scales|overtones 8 through 16]], octave repeating -- is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).
 
{| class="wikitable"
|-
|-
| | Overtones in "Mode 8":
| 5
| | 8
| 83.3
| |
| 20/19, 21/20, 22/21
| | 9
| vm2
| |
| downminor 2nd
| | 10
| vEb
| |
| kA1, lm2
| | 11
| comma-narrow aug unison, little minor 2nd
| |
| kD#, lEb
| | 12
| kD#, kEb
| |
| | 13
| |
| | 14
| |
| | 15
| |
| | 16
|-
|-
| | ...as JI Ratio from 1/1:
| 6
| | 1/1
| 100.0
| |
| 17/16, 18/17, 19/18
| | 9/8
| m2
| |
| minor 2nd
| | 5/4
| Eb
| |
| m2
| | 11/8
| minor 2nd
| |
| Eb
| | 3/2
| Eb
| |
| | 13/8
| |
| | 7/4
| |
| | 15/8
| |
| | 2/1
|-
|-
| | ...in cents:
| 7
| | 0
| 116.7
| |
| 15/14, 16/15
| | 203.9
| ^m2
| |
| upminor 2nd
| | 386.3
| ^Eb
| |
| Km2
| | 551.3
| classic minor 2nd
| |
| KEb
| | 702.0
| KEb
| |
| | 840.5
| |
| | 968.8
| |
| | 1088.3
| |
| | 1200.0
|-
|-
| | Nearest degree of 72edo:
| 8
| | 0
| 133.3
| |
| 13/12, 14/13, 27/25
| | 12
| ^^m2, v~2
| |
| dupminor 2nd, downmid 2nd
| | 23
| ^^Eb
| |
| Om2
| | 33
| on minor 2nd
| |
| OEb
| | 42
| SEb
| |
| | 50
| |
| | 58
| |
| | 65
| |
| | 72
|-
|-
| | ...in cents:
| 9
| | 0
| 150.0
| |
| 12/11
| | 200.0
| ~2
| |
| mid 2nd
| | 383.3
| v<sup>3</sup>E
| |
| N2
| | 550.0
| neutral 2nd
| |
| UEb/uE
| | 700.0
| UEb/uE
| |
| | 833.3
| |
| | 966.7
| |
| | 1083.3
| |
| | 1200.0
|-
|-
| | Steps as Freq. Ratio:
| 10
| |
| 166.7
| | 9:8
| 11/10
| |
| ^~2, vvM2
| | 10:9
| upmid 2nd, dudmajor 2nd
| |
| vvE
| | 11:10
| oM2
| |
| off major 2nd
| | 12:11
| oE
| |
| sE
| | 13:12
| |
| | 14:13
| |
| | 15:14
| |
| | 16:15
| |
|-
|-
| | ...in cents:
| 11
| |
| 183.3
| | 203.9
| 10/9
| |
| vM2
| | 182.4
| downmajor 2nd
| |
| vE
| | 165.0
| kM2
| |
| classic/comma-narrow major 2nd
| | 150.6
| kE
| |
| kE
| | 138.6
| |
| | 128.3
| |
| | 119.4
| |
| | 111.7
| |
|-
|-
| | Nearest degree of 72edo:
| 12
| |
| 200.0
| | 12
| 9/8
| |
| M2
| | 11
| major 2nd
| |
| E
| | 10
| M2
| |
| major 2nd
| | 9
| E
| |
| E
| | 8
| |
| | 8
| |
| | 7
| |
| | 7
| |
|-
|-
| | ...in cents:
| 13
| |
| 216.7
| | 200.0
| 17/15, 25/22
| |
| ^M2
| | 183.3
| upmajor 2nd
| |
| ^E
| | 166.7
| LM2
| |
| large major 2nd
| | 150.0
| LE
| |
| KE
| | 133.3
| |
| | 133.3
| |
| | 116.7
| |
| | 116.7
| |
|}
 
=Intervals=
 
{| class="wikitable"
|-
|-
| | degrees
| 14
| | cents value
| 233.3
| | approximate ratios (11-limit)
| 8/7
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]]
| ^^M2
| dupmajor 2nd
| ^^E
| SM2
| supermajor 2nd
| SE
| SE
|-
|-
| | 0
| 15
| | 0
| 250.0
| | 1/1
| 15/13, 22/19
| style="text-align:center;" | P1
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| style="text-align:center;" | perfect unison
| trupmajor 2nd,<br>trudminor 3rd
| style="text-align:center;" | D
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
| HM2, hm3
| hypermajor 2nd, hypominor 3rd
| HE, hF
| UE, uF
|-
|-
| | 1
| 16
| | 16.667
| 266.7
| | 81/80
| 7/6
| style="text-align:center;" | ^1
| vvm3
| style="text-align:center;" | up unison
| dudminor 3rd
| style="text-align:center;" | D^
| vvF
| sm3
| subminor 3rd
| sF
| sF
|-
|-
| | 2
| 17
| | 33.333
| 283.3
| | 45/44
| 13/11, 20/17
| style="text-align:center;" | ^^
| vm3
| style="text-align:center;" | double-up unison
| downminor 3rd
| style="text-align:center;" | D^^
| vF
| lm3
| little minor 3rd
| lF
| kF
|-
|-
| | 3
| 18
| | 50
| 300.0
| | 33/32
| 19/16, 25/21, 32/27
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>1, v<span style="font-size: 90%; vertical-align: super;">3</span>m2
| m3
| style="text-align:center;" | triple-up unison,
| minor 3rd
 
| F
triple-down minor 2nd
| m3
| style="text-align:center;" | D^<span style="font-size: 90%; vertical-align: super;">3</span>, Ebv<span style="font-size: 90%; vertical-align: super;">3</span>
| minor 3rd
| F
| F
|-
|-
| | 4
| 19
| | 66.667
| 316.7
| | 25/24
| 6/5
| style="text-align:center;" | vvm2
| ^m3
| style="text-align:center;" | double-downminor 2nd
| upminor 3rd
| style="text-align:center;" | Ebvv
| ^F
| Km3
| classic minor 3rd
| KF
| KF
|-
|-
| | 5
| 20
| | 83.333
| 333.3
| | 21/20
| 17/14, 39/32, 40/33
| style="text-align:center;" | vm2
| ^^m3, v~3
| style="text-align:center;" | downminor 2nd
| dupminor 3rd, downmid 3rd
| style="text-align:center;" | Ebv
| ^^F
| Om3
| on minor third
| OF
| SF
|-
|-
| | 6
| 21
| | 100
| 350.0
| | 35/33
| 11/9, 27/22
| style="text-align:center;" | m2
| ~3
| style="text-align:center;" | minor 2nd
| mid 3rd
| style="text-align:center;" | Eb
| ^<sup>3</sup>F
| N3
| neutral 3rd
| UF/uF#
| UF/uF#
|-
|-
| | 7
| 22
| | 116.667
| 366.7
| | 15/14
| 16/13, 21/17, 26/21
| style="text-align:center;" | ^m2
| ^~3, vvM3
| style="text-align:center;" | upminor 2nd
| upmid 3rd, dudmajor 3rd
| style="text-align:center;" | Eb^
| vvF#
| oM3
| off major 3rd
| oF#
| sF#
|-
|-
| | 8
| 23
| | 133.333
| 383.3
| | 27/25
| 5/4
| style="text-align:center;" | v~2
| vM3
| style="text-align:center;" | downmid 2nd
| downmajor 3rd
| style="text-align:center;" | Eb^^
| vF#
| kM3
| classic major 3rd
| kF#
| kF#
|-
|-
| | 9
| 24
| | 150
| 400.0
| | 12/11
| 24/19
| style="text-align:center;" | ~2
| M3
| style="text-align:center;" | mid 2nd
| major 3rd
| style="text-align:center;" | Ev<span style="font-size: 90%; vertical-align: super;">3</span>
| F#
| M3
| major 3rd
| F#
| F#
|-
|-
| | 10
| 25
| | 166.667
| 416.7
| | 11/10
| 14/11
| style="text-align:center;" | ^~2
| ^M3
| style="text-align:center;" | upmid 2nd
| upmajor 3rd
| style="text-align:center;" | Evv
| ^F#
| LM3
| large major 3rd
| LF#
| KF#
|-
|-
| | 11
| 26
| | 183.333
| 433.3
| | 10/9
| 9/7
| style="text-align:center;" | vM2
| ^^M3
| style="text-align:center;" | downmajor 2nd
| dupmajor 3rd
| style="text-align:center;" | Ev
| ^^F#
| SM3
| supermajor 3rd
| SF#
| SF#
|-
|-
| | 12
| 27
| | 200
| 450.0
| | 9/8
| 13/10, 22/17
| style="text-align:center;" | M2
| ^<sup>3</sup>M3, v<sup>3</sup>4
| style="text-align:center;" | major 2nd
| trupmajor 3rd, trud 4th
| style="text-align:center;" | E
| ^<sup>3</sup>F#, v<sup>3</sup>G
| HM3, h4
| hypermajor 3rd, hypo 4th
| HF#, hG
| UF#, uG
|-
|-
| | 13
| 28
| | 216.667
| 466.7
| | 25/22
| 17/13, 21/16
| style="text-align:center;" | ^M2
| vv4
| style="text-align:center;" | upmajor 2nd
| dud 4th
| style="text-align:center;" | E^
| vvG
| s4
| sub 4th
| sG
| sG
|-
|-
| | 14
| 29
| | 233.333
| 483.3
| | 8/7
| 33/25
| style="text-align:center;" | ^^M2
| v4
| style="text-align:center;" | double-upmajor 2nd
| down 4th
| style="text-align:center;" | E^^
| vG
| l4
| little 4th
| lG
| kG
|-
|-
| | 15
| 30
| | 250
| 500.0
| | 81/70
| 4/3
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M2, v<span style="font-size: 90%; vertical-align: super;">3</span>m3
| P4
| style="text-align:center;" | triple-up major 2nd,
| perfect 4th
 
| G
triple-down minor 3rd
| P4
| style="text-align:center;" | E^<span style="font-size: 90%; vertical-align: super;">3</span>, Fv<span style="font-size: 90%; vertical-align: super;">3</span>
| perfect 4th
| G
| G
|-
|-
| | 16
| 31
| | 266.667
| 516.7
| | 7/6
| 27/20
| style="text-align:center;" | vvm3
| ^4
| style="text-align:center;" | double-downminor 3rd
| up 4th
| style="text-align:center;" | Fvv
| ^G
| K4
| comma-wide 4th
| KG
| KG
|-
|-
| | 17
| 32
| | 283.333
| 533.3
| | 33/28
| 15/11, 19/14, ''26/19''
| style="text-align:center;" | vm3
| ^^4, v~4
| style="text-align:center;" | downminor 3rd
| dup 4th, downmid 4th
| style="text-align:center;" | Fv
| ^^G
| O4
| on 4th
| OG
| SG
|-
|-
| | 18
| 33
| | 300
| 550.0
| | 25/21
| 11/8
| style="text-align:center;" | m3
| ~4
| style="text-align:center;" | minor 3rd
| mid 4th
| style="text-align:center;" | F
| ^<sup>3</sup>G
| U4/N4
| uber 4th / neutral 4th
| UG
| UG
|-
|-
| | 19
| 34
| | 316.667
| 566.7
| | 6/5
| 18/13, 25/18
| style="text-align:center;" | ^m3
| ^~4, vvA4
| style="text-align:center;" | upminor 3rd
| upmid 4th, dudaug 4th
| style="text-align:center;" | F^
| vvG#
| kkA4, sd5
| classic aug 4th, sub dim 5th
| kkG#, sAb
| SG#, (kkG#), sAb
|-
|-
| | 20
| 35
| | 333.333
| 583.3
| | 40/33
| 7/5
| style="text-align:center;" | v~3
| vA4, vd5
| style="text-align:center;" | downmid 3rd
| downaug 4th, <br>downdim 5th
| style="text-align:center;" | F^^
| vG#, vAb
| kA4, ld5
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, kAb
|-
|-
| | 21
| 36
| | 350
| 600.0
| | 11/9
| 17/12, 24/17
| style="text-align:center;" | ~3
| A4, d5
| style="text-align:center;" | mid 3rd
| aug 4th, dim 5th
| style="text-align:center;" | F^<span style="font-size: 90%; vertical-align: super;">3</span>
| G#, Ab
| A4, d5
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
|-
|-
| | 22
| 37
| | 366.667
| 616.7
| | 99/80
| 10/7
| style="text-align:center;" | ^~3
| ^A4, ^d5
| style="text-align:center;" | upmid 3rd
| upaug 4th, updim 5th
| style="text-align:center;" | F#vv
| ^G#, ^Ab
| LA4, Kd5
| large aug 4th, comma-wide dim 5th
| LG#, KAb
| KG#, KAb
|-
|-
| | 23
| 38
| | 383.333
| 633.3
| | 5/4
| 13/9, 36/25
| style="text-align:center;" | vM3
| v~5, ^^d5
| style="text-align:center;" | downmajor 3rd
| downmid 5th, <br>dupdim 5th
| style="text-align:center;" | F#v
| ^^Ab
| SA4, KKd5
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, SAb, (KKAb)
|-
|-
| | 24
| 39
| | 400
| 650.0
| | 44/35
| 16/11
| style="text-align:center;" | M3
| ~5
| style="text-align:center;" | major 3rd
| mid 5th
| style="text-align:center;" | F#
| v<sup>3</sup>A
| u5/N5
| unter 5th / neutral 5th
| uA
| uA
|-
|-
| | 25
| 40
| | 416.667
| 666.7
| | 14/11
| ''19/13'', 22/15, 28/19
| style="text-align:center;" | ^M3
| vv5, ^~5
| style="text-align:center;" | upmajor 3rd
| dud 5th, upmid 5th
| style="text-align:center;" | F#^
| vvA
| o5
| off 5th
| oA
| sA
|-
|-
| | 26
| 41
| | 433.333
| 683.3
| | 9/7
| 40/27
| style="text-align:center;" | ^^M3
| v5
| style="text-align:center;" | double-upmajor 3rd
| down 5th
| style="text-align:center;" | F#^^
| vA
| k5
| comma-narrow 5th
| kA
| kA
|-
|-
| | 27
| 42
| | 450
| 700.0
| | 35/27
| 3/2
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M3, v<span style="font-size: 90%; vertical-align: super;">3</span>4
| P5
| style="text-align:center;" | triple-up major 3rd,
| perfect 5th
 
| A
triple-down 4th
| P5
| style="text-align:center;" | F#^<span style="font-size: 90%; vertical-align: super;">3</span>, Gv<span style="font-size: 90%; vertical-align: super;">3</span>
| perfect 5th
| A
| A
|-
|-
| | 28
| 43
| | 466.667
| 716.7
| | 21/16
| 50/33
| style="text-align:center;" | vv4
| ^5
| style="text-align:center;" | double-down 4th
| up 5th
| style="text-align:center;" | Gvv
| ^A
| L5
| large fifth
| LA
| KA
|-
|-
| | 29
| 44
| | 483.333
| 733.3
| | 33/25
| 26/17, 32/21
| style="text-align:center;" | v4
| ^^5
| style="text-align:center;" | down 4th
| dup 5th
| style="text-align:center;" | Gv
| ^^A
| S5
| super fifth
| SA
| SA
|-
|-
| | 30
| 45
| | 500
| 750.0
| | 4/3
| 17/11, 20/13
| style="text-align:center;" | P4
| ^<sup>3</sup>5, v<sup>3</sup>m6
| style="text-align:center;" | perfect 4th
| trup 5th, trudminor 6th
| style="text-align:center;" | G
| ^<sup>3</sup>A, v<sup>3</sup>Bb
| H5, hm6
| hyper fifth, hypominor 6th
| HA, hBb
| UA, uBb
|-
|-
| | 31
| 46
| | 516.667
| 766.7
| | 27/20
| 14/9
| style="text-align:center;" | ^4
| vvm6
| style="text-align:center;" | up 4th
| dudminor 6th
| style="text-align:center;" | G^
| vvBb
| sm6
| superminor 6th
| sBb
| sBb
|-
|-
| | 32
| 47
| | 533.333
| 783.3
| | 15/11
| 11/7
| style="text-align:center;" | ^^4
| vm6
| style="text-align:center;" | double-up 4th
| downminor 6th
| style="text-align:center;" | G^^
| vBb
| lm6
| little minor 6th
| lBb
| kBb
|-
|-
| | 33
| 48
| | 550
| 800.0
| | 11/8
| 19/12
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>4
| m6
| style="text-align:center;" | triple-up 4th
| minor 6th
| style="text-align:center;" | G^<span style="font-size: 90%; vertical-align: super;">3</span>
| Bb
| m6
| minor 6th
| Bb
| Bb
|-
|-
| | 34
| 49
| | 566.667
| 816.7
| | 25/18
| 8/5
| style="text-align:center;" | vvA4
| ^m6
| style="text-align:center;" | double-down aug 4th
| upminor 6th
| style="text-align:center;" | G#vv
| ^Bb
| Km6
| classic minor 6th
| kBb
| kBb
|-
|-
| | 35
| 50
| | 583.333
| 833.3
| | 7/5
| 13/8, 21/13, 34/21
| style="text-align:center;" | vA4, vd5
| ^^m6, v~6
| style="text-align:center;" | downaug 4th, updim 5th
| dupminor 6th, downmid 6th
| style="text-align:center;" | G#v, Abv
| ^^Bb
| Om6
| on minor 6th
| oBb
| sBb
|-
|-
| | 36
| 51
| | 600
| 850.0
| | 99/70
| 18/11, 44/27
| style="text-align:center;" | A4, d5
| ~6
| style="text-align:center;" | aug 4th, dim 5th
| mid 6th
| style="text-align:center;" | G#, Ab
| v<sup>3</sup>B
| N6
| neutral 6th
| UBb, uB
| UBb, uB
|-
|-
| | 37
| 52
| | 616.667
| 866.7
| | 10/7
| 28/17, 33/20, 64/39
| style="text-align:center;" | ^A4, ^d5
| ^~6, vvM6
| style="text-align:center;" | upaug 4th, downdim 5th
| upmid 6th, dudmajor 6th
| style="text-align:center;" | G#^, Ab^
| vvB
| oM6
| off major 6th
| oB
| sB
|-
|-
| | 38
| 53
| | 633.333
| 883.3
| | 36/25
| 5/3
| style="text-align:center;" | ^^d5
| vM6
| style="text-align:center;" | double-updim 5th
| downmajor 6th
| style="text-align:center;" | Ab^^
| vB
| kM6
| classic major 6th
| kB
| kB
|-
|-
| | 39
| 54
| | 650
| 900.0
| | 16/11
| 27/16, 32/19, 42/25
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>5
| M6
| style="text-align:center;" | triple-down 5th
| major 6th
| style="text-align:center;" | Av<span style="font-size: 90%; vertical-align: super;">3</span>
| B
| M6
| major 6th
| B
| B
|-
|-
| | 40
| 55
| | 666.667
| 916.7
| | 22/15
| 17/10, 22/13
| style="text-align:center;" | vv5
| ^M6
| style="text-align:center;" | double-down 5th
| upmajor 6th
| style="text-align:center;" | Avv
| ^B
| LM6
| large major 6th
| LB
| KB
|-
|-
| | 41
| 56
| | 683.333
| 933.3
| | 40/27
| 12/7
| style="text-align:center;" | v5
| ^^M6
| style="text-align:center;" | down 5th
| dupmajor 6th
| style="text-align:center;" | Av
| ^^B
| SM6
| supermajor 6th
| SB
| SB
|-
|-
| | 42
| 57
| | 700
| 950.0
| | 3/2
| 19/11, 26/15
| style="text-align:center;" | P5
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| style="text-align:center;" | perfect 5th
| trupmajor 6th,<br>trudminor 7th
| style="text-align:center;" | A
| ^<sup>3</sup>B, <br>v<sup>3</sup>C
| HM6, hm7
| hypermajor 6th, hypominor 7th
| HB, hC
| UB, uC
|-
|-
| | 43
| 58
| | 716.667
| 966.7
| | 50/33
| 7/4
| style="text-align:center;" | ^5
| vvm7
| style="text-align:center;" | up 5th
| dudminor 7th
| style="text-align:center;" | A^
| vvC
| sm7
| subminor 7th
| sC
| sC
|-
|-
| | 44
| 59
| | 733.333
| 983.3
| | 32/21
| 30/17, 44/25
| style="text-align:center;" | ^^5
| vm7
| style="text-align:center;" | double-up 5th
| downminor 7th
| style="text-align:center;" | A^^
| vC
| lm7
| little minor 7th
| lC
| kC
|-
|-
| | 45
| 60
| | 750
| 1000.0
| | 54/35
| 16/9
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>5, v<span style="font-size: 90%; vertical-align: super;">3</span>m6
| m7
| style="text-align:center;" | triple-up 5th,
| minor 7th
 
| C
triple-down minor 6th
| m7
| style="text-align:center;" | A^<span style="font-size: 90%; vertical-align: super;">3</span>, Bbv<span style="font-size: 90%; vertical-align: super;">3</span>
| minor 7th
| C
| C
|-
|-
| | 46
| 61
| | 766.667
| 1016.7
| | 14/9
| 9/5
| style="text-align:center;" | vvm6
| ^m7
| style="text-align:center;" | double-downminor 6th
| upminor 7th
| style="text-align:center;" | Bbvv
| ^C
| Km7
| classic/comma-wide minor 7th
| KC
| KC
|-
|-
| | 47
| 62
| | 783.333
| 1033.3
| | 11/7
| 20/11
| style="text-align:center;" | vm6
| ^^m7, v~7
| style="text-align:center;" | downminor 6th
| dupminor 7th, downmid 7th
| style="text-align:center;" | Bbv
| ^^C
| Om7
| on minor 7th
| OC
| SC
|-
|-
| | 48
| 63
| | 800
| 1050.0
| | 35/22
| 11/6
| style="text-align:center;" | m6
| ~7
| style="text-align:center;" | minor 6th
| mid 7th
| style="text-align:center;" | Bb
| ^<sup>3</sup>C
| N7, hd8
| neutral 7th, hypo dim 8ve
| UC/uC#, hDb
| UC/uC#, uDb
|-
|-
| | 49
| 64
| | 816.667
| 1066.7
| | 8/5
| 13/7, 24/13, 50/27
| style="text-align:center;" | ^m6
| ^~7, vvM7
| style="text-align:center;" | upminor 6th
| upmid 7th, dudmajor 7th
| style="text-align:center;" | Bb^
| vvC#
| oM7, sd8
| off major 7th, sub dim 8ve
| oC#, sDb
| sC#, sDb
|-
|-
| | 50
| 65
| | 833.333
| 1083.3
| | 81/50
| 15/8, 28/15
| style="text-align:center;" | v~6
| vM7
| style="text-align:center;" | downmid 6th
| downmajor 7th
| style="text-align:center;" | Bb^^
| vC#
| kM7, ld8
| classic major 7th, little dim 8ve
| kC#, lDb
| kC#, kDb
|-
|-
| | 51
| 66
| | 850
| 1100.0
| | 18/11
| 17/9, 32/17, 36/19
| style="text-align:center;" | ~6
| M7
| style="text-align:center;" | mid 6th
| major 7th
| style="text-align:center;" | Bv<span style="font-size: 90%; vertical-align: super;">3</span>
| C#
| M7, d8
| major 7th, dim 8ve
| C#, Db
| C#, Db
|-
|-
| | 52
| 67
| | 866.667
| 1116.7
| | 33/20
| 19/10, 21/11, 40/21
| style="text-align:center;" | ^~6
| ^M7
| style="text-align:center;" | upmid 6th
| upmajor 7th
| style="text-align:center;" | Bvv
| ^C#
| LM7, Kd8
| large major 7th, comma-wide dim 8ve
| LC#, KDb
| KC#, KDb
|-
|-
| | 53
| 68
| | 883.333
| 1133.3
| | 5/3
| 25/13, 27/14, 48/25, 52/27
| style="text-align:center;" | vM6
| ^^M7
| style="text-align:center;" | downmajor 6th
| dupmajor 7th
| style="text-align:center;" | Bv
| ^^C#
| SM7, KKd8
| supermajor 7th, classic dim 8ve
| SC#, KKDb
| SC#, SDb, (KKDb)
|-
|-
| | 54
| 69
| | 900
| 1150.0
| | 27/16
| 35/18, 39/20, 64/33
| style="text-align:center;" | M6
| ^<sup>3</sup>M7, v<sup>3</sup>8
| style="text-align:center;" | major 6th
| trupmajor 7th, trud octave
| style="text-align:center;" | B
| ^<sup>3</sup>C#, v<sup>3</sup>D
| HM7, u8, h8
| hypermajor 7th, unter 8ve, hypo 8ve
| HC#, uD, hD
| UC#, uDb, uD
|-
|-
| | 55
| 70
| | 916.667
| 1166.7
| | 56/33
| 49/25, 55/28, 63/32, 88/45, 96/49
| style="text-align:center;" | ^M6
| vv8
| style="text-align:center;" | upmajor 6th
| dud octave
| style="text-align:center;" | B^
| vvD
| s8, o8
| sub 8ve, off 8ve
| sD, oD
| sD
|-
|-
| | 56
| 71
| | 933.333
| 1183.3
| | 12/7
| 99/50, 160/81, 180/91, 196/99, 208/105
| style="text-align:center;" | ^^M6
| v8
| style="text-align:center;" | double-upmajor 6th
| down octave
| style="text-align:center;" | B^^
| vD
| k8, l8
| comma-narrow 8ve, little 8ve
| kD, lD
| kD
|-
|-
| | 57
| 72
| | 950
| 1200.0
| | 121/70
| 2/1
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M6, v<span style="font-size: 90%; vertical-align: super;">3</span>m7
| P8
| style="text-align:center;" | triple-up major 6th,
| perfect octave
| D
| P8
| perfect octave
| D
| D
|}
<references group="note" />
 
=== Interval quality and chord names in color notation ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:


triple-down minor 7th
{| class="wikitable center-all"
| style="text-align:center;" | B^<span style="font-size: 90%; vertical-align: super;">3</span>, Cv<span style="font-size: 90%; vertical-align: super;">3</span>
|-
! Quality
! [[Color notation|Color]]
! Monzo format
! Examples
|-
| dudminor
| zo
| (a b 0 1)
| 7/6, 7/4
|-
| minor
| fourthward wa
| (a b), b < -1
| 32/27, 16/9
|-
|-
| | 58
| upminor
| | 966.667
| gu
| | 7/4
| (a b -1)
| style="text-align:center;" | vvm7
| 6/5, 9/5
| style="text-align:center;" | double-downminor 7th
| style="text-align:center;" | Cvv
|-
|-
| | 59
| rowspan="2" | dupminor, <br>downmid
| | 983.333
| luyo
| | 44/25
| (a b 1 0 -1)
| style="text-align:center;" | vm7
| 15/11
| style="text-align:center;" | downminor 7th
| style="text-align:center;" | Cv
|-
|-
| | 60
| tho
| | 1000
| (a b 0 0 0 1)
| | 16/9
| 13/8, 13/9
| style="text-align:center;" | m7
| style="text-align:center;" | minor 7th
| style="text-align:center;" | C
|-
|-
| | 61
| rowspan="2" | mid
| | 1016.667
| ilo
| | 9/5
| (a b 0 0 1)
| style="text-align:center;" | ^m7
| 11/9, 11/6
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | C^
|-
|-
| | 62
| lu
| | 1033.333
| (a b 0 0 -1)
| | 20/11
| 12/11, 18/11
| style="text-align:center;" | v~7
| style="text-align:center;" | downmid 7th
| style="text-align:center;" | C^^
|-
|-
| | 63
| rowspan="2" | upmid, <br>dudmajor
| | 1050
| logu
| | 11/6
| (a b -1 0 1)
| style="text-align:center;" | ~7
| 11/10
| style="text-align:center;" | mid 7th
| style="text-align:center;" | C^<span style="font-size: 90%; vertical-align: super;">3</span>
|-
|-
| | 64
| thu
| | 1066.667
| (a b 0 0 0 -1)
| | 50/27
| 16/13, 18/13
| style="text-align:center;" | ^~7
| style="text-align:center;" | upmin 7th
| style="text-align:center;" | C#vv
|-
|-
| | 65
| downmajor
| | 1083.333
| yo
| | 15/8
| (a b 1)
| style="text-align:center;" | vM7
| 5/4, 5/3
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | C#v
|-
|-
| | 66
| major
| | 1100
| fifthward wa
| | 66/35
| (a b), b > 1
| style="text-align:center;" | M7
| 9/8, 27/16
| style="text-align:center;" | major 7th
| style="text-align:center;" | C#
|-
|-
| | 67
| dupmajor
| | 1116.667
| ru
| | 21/11
| (a b 0 -1)
| style="text-align:center;" | ^M7
| 9/7, 12/7
| style="text-align:center;" | upmajor 7th
| style="text-align:center;" | C#^
|-
|-
| | 68
| rowspan="2" | trupmajor, <br>trudminor
| | 1133.333
| thogu
| | 27/14
| (a b -1 0 0 1)
| style="text-align:center;" | ^^M7
| 13/10
| style="text-align:center;" | double-upmajor 7th
| style="text-align:center;" | C#^^
|-
|-
| | 69
| thuyo
| | 1150
| (a b 1 0 0 -1)
| | 35/18
| 15/13
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M7, v<span style="font-size: 90%; vertical-align: super;">3</span>8
|}
| style="text-align:center;" | triple-up major 7th,
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:


triple-down octave
{| class="wikitable center-all"
| style="text-align:center;" | C#^<span style="font-size: 90%; vertical-align: super;">3</span>, Dv<span style="font-size: 90%; vertical-align: super;">3</span>
|-
! [[Color notation|Color of the 3rd]]
! JI chord
! Notes as edosteps
! Notes of C chord
! Written name
! Spoken name
|-
| zo
| 6:7:9
| 0-16-42
| C vvEb G
| Cvvm
| C dudminor
|-
|-
| | 70
| gu
| | 1166.667
| 10:12:15
| | 49/25
| 0-19-42
| style="text-align:center;" | vv8
| C ^Eb G
| style="text-align:center;" | double-down octave
| C^m
| style="text-align:center;" | Dvv
| C upminor
|-
|-
| | 71
| ilo
| | 1183.333
| 18:22:27
| | 99/50
| 0-21-42
| style="text-align:center;" | v8
| C v<span style="font-size: 90%; vertical-align: super;">3</span>E G
| style="text-align:center;" | down octave
| C~
| style="text-align:center;" | Dv
| C mid
|-
|-
| | 72
| yo
| | 1200
| 4:5:6
| | 2/1
| 0-23-42
| style="text-align:center;" | P8
| C vE G
| style="text-align:center;" | perfect octave
| Cv
| style="text-align:center;" | D
| C downmajor or C down
|-
| ru
| 14:18:27
| 0-26-42
| C ^^E G
| C^^
| C dupmajor or C dup
|}
|}
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]].


{| class="wikitable"
=== 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.
! | quality
 
! | [[Kite's color notation|color]]
Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
! | monzo format
* −1 degree (the down ring) corrects 81/64 to 5/4 via 80/81
! | examples
* −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 ==
=== 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:
 
[[File:72edo Sagittal.png|800px]]
 
=== Ivan Wyschnegradsky's notation ===
{{Sharpness-sharp6-iw|72}}
 
== Approximation to JI ==
[[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]]
 
=== Interval mappings ===
{{Q-odd-limit intervals|72}}
 
=== Zeta properties ===
72edo is the ninth [[zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[the Riemann zeta function and tuning#The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.
 
[[File:plot72.png|alt=plot72.png|plot72.png]]
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
| style="text-align:center;" | double-down minor
! rowspan="2" | [[Subgroup]]
| style="text-align:center;" | zo
! rowspan="2" | [[Comma list]]
| style="text-align:center;" | {a, b, 0, 1}
! rowspan="2" | [[Mapping]]
| style="text-align:center;" | 7/6, 7/4
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| style="text-align:center;" | minor
! [[TE error|Absolute]] (¢)
| style="text-align:center;" | fourthward wa
! [[TE simple badness|Relative]] (%)
| style="text-align:center;" | {a, b}, b &lt; -1
| style="text-align:center;" | 32/27, 16/9
|-
|-
| style="text-align:center;" | upminor
| 2.3.5
| style="text-align:center;" | gu
| 15625/15552, 531441/524288
| style="text-align:center;" | {a, b, -1}
| {{Mapping| 72 114 167 }}
| style="text-align:center;" | 6/5, 9/5
| +0.839
| 0.594
| 3.56
|-
|-
| style="text-align:center;" | mid
| 2.3.5.7
| style="text-align:center;" | ilo
| 225/224, 1029/1024, 4375/4374
| style="text-align:center;" | {a, b, 0, 0, 1}
| {{Mapping| 72 114 167 202 }}
| style="text-align:center;" | 11/9, 11/6
| +0.822
| 0.515
| 3.09
|-
|-
| style="text-align:center;" | "
| 2.3.5.7.11
| style="text-align:center;" | lu
| 225/224, 243/242, 385/384, 4000/3993
| style="text-align:center;" | {a, b, 0, 0, -1}
| {{Mapping| 72 114 167 202 249 }}
| style="text-align:center;" | 12/11, 18/11
| +0.734
| 0.493
| 2.96
|-
|-
| style="text-align:center;" | downmajor
| 2.3.5.7.11.13
| style="text-align:center;" | yo
| 169/168, 225/224, 243/242, 325/324, 385/384
| style="text-align:center;" | {a, b, 1}
| {{Mapping| 72 114 167 202 249 266 }}
| style="text-align:center;" | 5/4, 5/3
| +0.936
| 0.638
| 3.82
|-
|-
| style="text-align:center;" | major
| 2.3.5.7.11.13.17
| style="text-align:center;" | fifthward wa
| 169/168, 221/220, 225/224, 243/242, 273/272, 325/324
| style="text-align:center;" | {a, b}, b &gt; 1
| {{Mapping| 72 114 167 202 249 266 294 }}
| style="text-align:center;" | 9/8, 27/16
| +0.975
| 0.599
| 3.59
|-
|-
| style="text-align:center;" | double-up major
| 2.3.5.7.11.13.17.19
| style="text-align:center;" | ru
| 153/152, 169/168, 210/209, 221/220, 225/224, 243/242, 273/272
| style="text-align:center;" | {a, b, 0, -1}
| {{Mapping| 72 114 167 202 249 266 294 306 }}
| style="text-align:center;" | 9/7, 12/7
| +0.780
| 0.762
| 4.57
|}
|}
All 72edo chords can be named using ups and downs. Here are the zo, gu, ilo, yo and ru triads:
* 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.
 
=== Commas ===
Commas tempered out by 72edo include…


{| class="wikitable"
{| class="commatable wikitable center-1 center-2 right-4"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cents]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{Monzo| -19 12 }}
| 23.46
| Pythagorean comma
|-
| 5
| [[15625/15552]]
| {{Monzo| -6 -5 6 }}
| 8.11
| Kleisma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| [[Ampersand comma]]
|-
| 5
| [[129140163/128000000|(18 digits)]]
| {{Monzo| -13 17 -6 }}
| 15.35
| [[Graviton]]
|-
| 5
| <abbr title="7629394531250/7625597484987">(26 digits)</abbr>
| {{Monzo| 1 -27 18 }}
| 0.86
| [[Ennealimma]]
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Marvel comma
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Gamelisma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Breedsma
|-
| 7
| [[4375/4374]]
| {{Monzo| -1 -7 4 1 }}
| 0.40
| Ragisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Mirkwai comma
|-
| 7
| [[19683/19600]]
| {{Monzo| -4 9 -2 -2 }}
| 7.32
| Cataharry comma
|-
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| {{Monzo | -6 -8 2 5 }}
| 1.12
| [[Wizma]]
|-
| 7
| <abbr title="250047/250000">(12 digits)</abbr>
| {{Monzo| -4 6 -6 3 }}
| 0.33
| [[Landscape comma]]
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2}}
| 7.14
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Swetisma
|-
| 11
| [[1375/1372]]
| {{Monzo| -2 0 3 -3 1 }}
| 3.78
| Moctdel comma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Lehmerisma
|-
| 11
| [[4000/3993]]
| {{Monzo| 5 -1 3 0 -3 }}
| 3.03
| Wizardharry comma
|-
| 11
| [[6250/6237]]
| {{Monzo| 1 -4 5 -1 -1 }}
| 3.60
| Liganellus comma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Kalisma
|-
| 11
| <abbr title="1771561/1769472">(14 digits)</abbr>
| {{Monzo| 16 -3 0 0 6 }}
| 2.04
| [[Nexus comma]]
|-
| 13
| [[169/168]]
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.27
| Buzurgisma
|-
| 13
| [[325/324]]
| {{Monzo| -2 -4 2 0 0 1 }}
| 5.34
| Marveltwin comma
|-
| 13
| [[351/350]]
| {{Monzo| -1 3 -2 -1 0 1 }}
| 4.94
| Ratwolfsma
|-
| 13
| [[364/363]]
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.76
| Minor minthma
|-
| 13
| [[625/624]]
| {{Monzo| -4 -1 4 0 0 -1 }}
| 2.77
| Tunbarsma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Island comma
|-
|-
! | [[Kite's color notation|color of the 3rd]]
| 13
! | JI chord
| [[729/728]]
! | notes as edosteps
| {{Monzo| -3 6 0 -1 0 -1 }}
! | notes of C chord
| 2.38
! | written name
| Squbema
! | spoken name
|-
|-
| style="text-align:center;" | zo
| 13
| style="text-align:center;" | 6:7:9
| [[1001/1000]]
| style="text-align:center;" | 0-16-42
| {{Monzo| -3 0 -3 1 1 1 }}
| style="text-align:center;" | C Ebvv G
| 1.73
| style="text-align:center;" | C.vvm
| Sinbadma
| style="text-align:center;" | C double-down minor
|-
|-
| style="text-align:center;" | gu
| 13
| style="text-align:center;" | 10:12:15
| [[1575/1573]]
| style="text-align:center;" | 0-19-42
| {{Monzo| 2 2 1 -2 -1 }}
| style="text-align:center;" | C Eb^ G
| 2.20
| style="text-align:center;" | C.^m
| Nicola
| style="text-align:center;" | C upminor
|-
|-
| style="text-align:center;" | ilo
| 13
| style="text-align:center;" | 18:22:27
| [[1716/1715]]
| style="text-align:center;" | 0-21-42
| {{Monzo| 2 1 -1 -3 1 1 }}
| style="text-align:center;" | C Ev<span style="font-size: 90%; vertical-align: super;">3</span> G
| 1.01
| style="text-align:center;" | C~
| Lummic comma
| style="text-align:center;" | C mid
|-
|-
| style="text-align:center;" | yo
| 13
| style="text-align:center;" | 4:5:6
| [[2080/2079]]
| style="text-align:center;" | 0-23-42
| {{Monzo| 5 -3 1 -1 -1 1 }}
| style="text-align:center;" | C Ev G
| 0.83
| style="text-align:center;" | C.v
| Ibnsinma
| style="text-align:center;" | C downmajor or C dot down
|-
|-
| style="text-align:center;" | ru
| 13
| style="text-align:center;" | 14:18:27
| [[6656/6655]]
| style="text-align:center;" | 0-26-42
| {{Monzo| 9 0 -1 0 -3 1 }}
| style="text-align:center;" | C E^^ G
| 0.26012
| style="text-align:center;" | C.^^
| Jacobin comma
| style="text-align:center;" | C double-upmajor or C dot double-up
|}
|}
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
<references group="note" />
 
=== Rank-2 temperaments ===
* [[List of edo-distinct 72et rank two temperaments]]


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


{| class="wikitable"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! | Periods per octave
! Periods<br>per 8ve
! | Generator
! Generator*
! | Names
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
| | 1
| 1
| | 1\72
| 1\72
| | [[Quincy|quincy]]
| 16.7
| 105/104
| [[Quincy]]
|-
|-
| | 1
| 1
| | 5\72
| 5\72
| | [[marvolo|marvolo]]
| 83.3
| 21/20
| [[Marvolo]]
|-
|-
| | 1
| 1
| | 7\72
| 7\72
| | [[Miracle|miracle]]/benediction/manna
| 116.7
| 15/14
| [[Miracle]] / benediction / manna
|-
|-
| | 1
| 1
| | 11\72
| 17\72
| |  
| 283.3
| 13/11
| [[Neominor]]
|-
|-
| | 1
| 1
| | 13\72
| 19\72
| |  
| 316.7
| 6/5
| [[Catakleismic]]
|-
|-
| | 1
| 1
| | 17\72
| 25\72
| | [[Neominor|neominor]]
| 416.7
| 14/11
| [[Sqrtphi]]
|-
|-
| | 1
| 1
| | 19\72
| 29\72
| | [[catakleismic|catakleismic]]
| 483.3
| 45/34
| [[Hemiseven]]
|-
|-
| | 1
| 1
| | 23\72
| 31\72
| |  
| 516.7
| 27/20
| [[Marvo]] / [[zarvo]]
|-
|-
| | 1
| 1
| | 25\72
| 35\72
| | [[Sqrtphi|sqrtphi]]
| 583.3
| 7/5
| [[Cotritone]]
|-
|-
| | 1
| 2
| | 29\72
| 5\72
| |  
| 83.3
| 21/20
| [[Harry]]
|-
|-
| | 1
| 2
| | 31\72
| 7\72
| | [[Marvo|marvo]]/zarvo
| 116.7
| 15/14
| [[Semimiracle]]
|-
|-
| | 1
| 2
| | 35\72
| 11\72
| | [[cotritone|cotritone]]
| 183.3
| 10/9
| [[Unidec]] / hendec
|-
|-
| | 2
| 2
| | 1\72
| 21\72<br>(19\72)
| |  
| 316.7<br>(283.3)
| 6/5<br>(13/11)
| [[Bikleismic]]
|-
|-
| | 2
| 2
| | 5\72
| 23\72<br>(13\72)
| | [[Harry|harry]]
| 383.3<br>(216.7)
| 5/4<br>(17/15)
| [[Wizard]] / lizard / gizzard
|-
|-
| | 2
| 3
| | 7\72
| 11\72
| |  
| 183.3
| 10/9
| [[Mirkat]]
|-
|-
| | 2
| 3
| | 11\72
| 19\72<br>(5\72)
| | [[Unidec|unidec]]/hendec
| 316.7<br>(83.3)
| 6/5<br>(21/20)
| [[Tritikleismic]]
|-
|-
| | 2
| 4
| | 13\72
| 19\72<br>(1\72)
| | [[wizard|wizard]]/lizard/gizzard
| 316.7<br>(16.7)
| 6/5<br>(105/104)
| [[Quadritikleismic]]
|-
|-
| | 2
| 8
| | 17\72
| 34\72<br>(2\72)
| |
| 566.7<br>(33.3)
| 168/121<br>(55/54)
| [[Octowerck]] / octowerckis
|-
|-
| | 3
| 8
| | 1\72
| 35\72<br>(1\72)
| |  
| 583.3<br>(16.7)
| 7/5<br>(100/99)
| [[Octoid]] / octopus
|-
|-
| | 3
| 9
| | 5\72
| 19\72<br>(3\72)
| | [[Tritikleismic|tritikleismic]]
| 316.7<br>(50.0)
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimnic
|-
|-
| | 3
| 9
| | 7\72
| 23\72<br>(1\72)
| |  
| 383.3<br>(16.7)
| 5/4<br>(105/104)
| [[Enneaportent]]
|-
|-
| | 3
| 12
| | 11\72
| 23\72<br>(1\72)
| | [[Mirkat|mirkat]]
| 383.3<br>(16.7)
| 5/4<br>(100/99)
| [[Compton]] / comptone
|-
|-
| | 4
| 18
| | 1\72
| 19\72<br>(1\72)
| | [[Quadritikleismic|quadritikleismic]]
| 316.7<br>(16.7)
| 6/5<br>(105/104)
| [[Hemiennealimmal]]
|-
|-
| | 4
| 24
| | 5\72
| 23\72<br>(1\72)
| |
| 383.3<br>(16.7)
| 5/4<br>(105/104)
| [[Hours]]
|-
|-
| | 4
| 36
| | 7\72
| 23\72<br>(1\72)
| |  
| 383.3<br>(16.7)
|-
| 5/4<br>(81/80)
| | 6
| [[Gamelstearn]]
| | 1\72
|}
| |
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
|-
 
| | 6
== Scales ==
| | 5\72
* [[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]]
* [[Genus24255et72|Euler(24255) genus in 72 equal]]
* [[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
* [[Magnetosphere scale|Magnetosphere]], [[Blackened skies]], [[Lost spirit]]
* [[5- to 10-tone scales in 72edo]]
 
=== Harmonic scale ===
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"
|-
|-
| | 8
! Harmonics in "Mode 8":
| | 1\72
| 8
| | [[Octoid|octoid]]
|
| 9
|
| 10
|
| 11
|
| 12
|  
| 13
|  
| 14
|  
| 15
|
| 16
|-
|-
| | 8
! …as JI Ratio from 1/1:
| | 2\72
| 1/1
| | [[Octowerck|octowerck]]
|  
| 9/8
|  
| 5/4
|  
| 11/8
|
| 3/2
|  
| 13/8
|
| 7/4
|
| 15/8
|  
| 2/1
|-
|-
| | 8
! …in cents:
| | 4\72
| 0
| |  
|
| 203.9
|
| 386.3
|
| 551.3
|  
| 702.0
|
| 840.5
|
| 968.8
|  
| 1088.3
|  
| 1200.0
|-
|-
| | 9
! Nearest degree of 72edo:
| | 1\72
| 0
| |  
|
| 12
|
| 23
|
| 33
|
| 42
|
| 50
|  
| 58
|  
| 65
|  
| 72
|-
|-
| | 9
! …in cents:
| | 3\72
| 0
| | [[Ennealimmal|ennealimmal]]/ennealimmic
|  
| 200.0
|  
| 383.3
|  
| 550.0
|
| 700.0
|
| 833.3
|
| 966.7
|
| 1083.3
|
| 1200.0
|-
|-
| | 12
! Steps as Freq. Ratio:
| | 1\72
|
| | [[Compton|compton]]
| 9:8
|
| 10:9
|
| 11:10
|  
| 12:11
|  
| 13:12
|  
| 14:13
|
| 15:14
|
| 16:15
|  
|-
|-
| | 18
! …in cents:
| | 1\72
|  
| | [[Hemiennealimmal|hemiennealimmal]]
| 203.9
|  
| 182.4
|  
| 165.0
|
| 150.6
|
| 138.6
|
| 128.3
|
| 119.4
|
| 111.7
|  
|-
|-
| | 24
! Nearest degree of 72edo:
| | 1\72
|  
| | [[Hours|hours]]
| 12
|  
| 11
|  
| 10
|
| 9
|
| 8
|
| 8
|
| 7
|
| 7
|  
|-
|-
| | 36
! …in cents:
| | 1\72
|
| |  
| 200.0
|
| 183.3
|
| 166.7
|
| 150.0
|  
| 133.3
|  
| 133.3
|
| 116.7
|
| 116.7
|  
|}
|}


=Z function=
== Instruments ==
72edo is the ninth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[The_Riemann_Zeta_Function_and_Tuning#The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.
If one can get six 12edo instruments tuned a twelfth-tone apart, it is possible to use these instruments in combination to play the full gamut of 72edo (see Music).
 
One can also use a skip fretting system:
* [[Skip fretting system 72 2 27]]
 
Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo:
* [[Lumatone mapping for 72edo]]
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/VwVp3RVao_k ''microtonal improvisation in 72edo''] (2025)


[[File:plot72.png|alt=plot72.png|plot72.png]]
; [[Ambient Esoterica]]
* [https://www.youtube.com/watch?v=seWcDAoQjxY ''Goetic Synchronities''] (2023)
* [https://www.youtube.com/watch?v=CrcdM1e2b6Q ''Rainy Day Generative Pillow''] (2024)


=Music=
; [[Jake Freivald]]
[http://www.archive.org/details/Kotekant Kotekant] ''[http://www.archive.org/download/Kotekant/kotekant.mp3 play]'' by [[Gene_Ward_Smith|Gene Ward Smith]]
* [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/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 Twinkle canon – 72 edo]'' by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
{{Wikipedia|In vain (Haas)}}
; [[Georg Friedrich Haas]]
* [https://www.youtube.com/watch?v=ix4yA-c-Pi8 ''Blumenstück''] (2000)
* [https://youtu.be/cmX-h7_us7A ''in vain''] (2000) ([https://www.universaledition.com/georg-friedrich-haas-278/works/in-vain-7566 score])


''[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 Lazy Sunday]'' by [[Jake_Freivald|Jake Freivald]] in the [[lazysunday|lazysunday]] scale.
; [[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://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)


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


=Scales=
; [[Gene Ward Smith]]
[[smithgw72a|smithgw72a]], [[smithgw72b|smithgw72b]], [[smithgw72c|smithgw72c]], [[smithgw72d|smithgw72d]], [[smithgw72e|smithgw72e]], [[smithgw72f|smithgw72f]], [[smithgw72g|smithgw72g]], [[smithgw72h|smithgw72h]], [[smithgw72i|smithgw72i]], [[smithgw72j|smithgw72j]]
* [https://www.archive.org/details/Kotekant ''Kotekant''] [https://www.archive.org/download/Kotekant/kotekant.mp3 play] (2010)


[[blackjack|blackjack]], [[miracle_8|miracle_8]], [[miracle_10|miracle_10]], [[miracle_12|miracle_12]], [[miracle_12a|miracle_12a]], [[miracle_24hi|miracle_24hi]], [[miracle_24lo|miracle_24lo]]
;[[Ivan Wyschnegradsky]]
* [https://www.youtube.com/watch?v=RCcJHCkYQ6U Arc-en-ciel, for 6 pianos in twelfth tones, Op. 37] (1956)


[[keenanmarvel|keenanmarvel]], [[xenakis_chrome|xenakis_chrome]], [[xenakis_diat|xenakis_diat]], [[xenakis_schrome|xenakis_schrome]]
; [[James Tenney]]
* [https://www.youtube.com/watch?v=jGsxqU1PhZs&list=OLAK5uy_mKyMEMZW7noeLncJnu-JT65go8w7403DA ''Changes for Six Harps'']


[[genus24255et72|Euler(24255) genus in 72 equal]]
; [[Xeno Ov Eleas]]
* [https://www.youtube.com/watch?v=cx7I0NWem5w ''Χenomorphic Ghost Storm''] (2022)


[[JuneGloom|JuneGloom]]
== External links ==
* [http://orthodoxwiki.org/Byzantine_Chant OrthodoxWiki Article on Byzantine chant, which uses 72edo]
* [http://www.ekmelic-music.org/en/ Ekmelic Music Society/Gesellschaft für Ekmelische Musik], a group of composers and researchers dedicated to 72edo music
* [http://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list
* [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]]


=External links=
[[Category:Listen]]
<ul><li>[http://en.wikipedia.org/wiki/72_tone_equal_temperament Wikipedia article on 72edo]</li><li>[http://orthodoxwiki.org/Byzantine_Chant OrthodoxWiki Article on Byzantine chant, which uses 72edo]</li><li>[http://en.wikipedia.org/wiki/Joe_Maneri Wikipedia article on Joe Maneri (1927-2009)]</li><li>[http://www.ekmelic-music.org/en/ Ekmelic Music Society/Gesellschaft für Ekmelische Musik], a group of composers and researchers dedicated to 72edo music</li><li>[http://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list</li><li>[http://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo]</li></ul>     
[[Category:Compton]]
[[Category:edo]]
[[Category:Marvel]]
[[Category:listen]]
[[Category:Miracle]]
[[Category:marvel]]
[[Category:Prodigy]]
[[Category:miracle]]
[[Category:Wizard]]
[[Category:moria]]
[[Category:prodigy]]
[[Category:wizard]]
[[Category:zeta]]
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