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__FORCETOC__
{{interwiki
| de = 72-EDO
| en = 72edo
| es =
| ja =
}}
{{Infobox ET}}
{{Wikipedia|72 equal temperament}}
{{ED intro}}


<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.
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 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.
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-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.
== 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]].  


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.
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=
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]].


Commas tempered out by 72edo include...
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.


{| 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]].
 
=== 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)}}
 
=== 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 ==
{| class="wikitable center-all right-2 left-3"
|-
|-
! | 3-limit
! #
! 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)
|-
|-
| |<nowiki> Pythagorean comma = 531441/524288 = |-19 12</nowiki>&gt;
| 0
|}
| 0.0
 
| 1/1
{| class="wikitable"
| P1
| perfect unison
| D
| P1
| perfect unison
| D
| D
|-
|-
! | 5-limit
| 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
|-
|-
| |<nowiki> kleisma = 15625/15552 = |-6 -5 6</nowiki>&gt;
| 2
 
| 33.3
ampersand = 34171875/33554432 = |-25 7 6&gt;
| 45/44, 49/48, 50/49, 55/54, 64/63
 
| ^^
graviton = 129140163/128000000 = |-13 17 -6&gt;
| dup unison
 
| ^^D
ennealimma = 7629394531250/7625597484987 = |1 -27 18&gt;
| S1, O1
|}
| super unison, on unison
 
| SD, OD
{| class="wikitable"
| 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
|pions
| 8/7
|7mus
| ^^M2
| | approximate ratios (11-limit)
| dupmajor 2nd
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]]
| ^^E
| SM2
| supermajor 2nd
| SE
| SE
|-
|-
| colspan="4"| 0
| 15
| | 1/1
| 250.0
| style="text-align:center;" | P1
| 15/13, 22/19
| style="text-align:center;" | perfect unison
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
| style="text-align:center;" | D
| trupmajor 2nd,<br>trudminor 3rd
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
| HM2, hm3
| hypermajor 2nd, hypominor 3rd
| HE, hF
| UE, uF
|-
|-
| | 1
| 16
| | 16.667
| 266.7
|17.667
| 7/6
|21.333 (15.555<sub>16</sub>)
| vvm3
| | 81/80
| dudminor 3rd
| style="text-align:center;" | ^1
| vvF
| style="text-align:center;" | up unison
| sm3
| style="text-align:center;" | D^
| subminor 3rd
| sF
| sF
|-
|-
| | 2
| 17
| | 33.333
| 283.3
|35.333
| 13/11, 20/17
|42.667 (2A.AAB<sub>16</sub>)
| vm3
| | 45/44
| downminor 3rd
| style="text-align:center;" | ^^
| vF
| style="text-align:center;" | double-up unison
| lm3
| style="text-align:center;" | D^^
| little minor 3rd
| lF
| kF
|-
|-
| | 3
| 18
| | 50
| 300.0
|53
| 19/16, 25/21, 32/27
|64 (40<sub>16</sub>)
| m3
| | 33/32
| minor 3rd
| 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
| F
| style="text-align:center;" | triple-up unison,
| m3
 
| minor 3rd
triple-down minor 2nd
| F
| 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>
| F
|-
|-
| | 4
| 19
| | 66.667
| 316.7
|70.667
| 6/5
|85.333 (55.555<sub>16</sub>)
| ^m3
| | 25/24
| upminor 3rd
| style="text-align:center;" | vvm2
| ^F
| style="text-align:center;" | double-downminor 2nd
| Km3
| style="text-align:center;" | Ebvv
| classic minor 3rd
| KF
| KF
|-
|-
| | 5
| 20
| | 83.333
| 333.3
|88.333
| 17/14, 39/32, 40/33
|106.667 (6A.AAB<sub>16</sub>)
| ^^m3, v~3
| | 21/20
| dupminor 3rd, downmid 3rd
| style="text-align:center;" | vm2
| ^^F
| style="text-align:center;" | downminor 2nd
| Om3
| style="text-align:center;" | Ebv
| on minor third
| OF
| SF
|-
|-
| | 6
| 21
| | 100
| 350.0
|106
| 11/9, 27/22
|128 (80<sub>16</sub>)
| ~3
| | 35/33
| mid 3rd
| style="text-align:center;" | m2
| ^<sup>3</sup>F
| style="text-align:center;" | minor 2nd
| N3
| style="text-align:center;" | Eb
| neutral 3rd
| UF/uF#
| UF/uF#
|-
|-
| | 7
| 22
| | 116.667
| 366.7
|123.667
| 16/13, 21/17, 26/21
|149.333 (95.555<sub>16</sub>)
| ^~3, vvM3
| | 15/14
| upmid 3rd, dudmajor 3rd
| style="text-align:center;" | ^m2
| vvF#
| style="text-align:center;" | upminor 2nd
| oM3
| style="text-align:center;" | Eb^
| off major 3rd
| oF#
| sF#
|-
|-
| | 8
| 23
| | 133.333
| 383.3
|141.333
| 5/4
|170.667 (AA.AAB<sub>16</sub>)
| vM3
| | 27/25
| downmajor 3rd
| style="text-align:center;" | v~2
| vF#
| style="text-align:center;" | downmid 2nd
| kM3
| style="text-align:center;" | Eb^^
| classic major 3rd
| kF#
| kF#
|-
|-
| | 9
| 24
| | 150
| 400.0
|159
| 24/19
|192 (C0<sub>16</sub>)
| M3
| | 12/11
| major 3rd
| style="text-align:center;" | ~2
| F#
| style="text-align:center;" | mid 2nd
| M3
| style="text-align:center;" | Ev<span style="font-size: 90%; vertical-align: super;">3</span>
| major 3rd
| F#
| F#
|-
|-
| | 10
| 25
| | 166.667
| 416.7
|176.667
| 14/11
|213.333 (D5.555<sub>16</sub>)
| ^M3
| | 11/10
| upmajor 3rd
| style="text-align:center;" | ^~2
| ^F#
| style="text-align:center;" | upmid 2nd
| LM3
| style="text-align:center;" | Evv
| large major 3rd
| LF#
| KF#
|-
|-
| | 11
| 26
| | 183.333
| 433.3
|194.333
| 9/7
|234.667 (EA.AAB<sub>16</sub>)
| ^^M3
| | 10/9
| dupmajor 3rd
| style="text-align:center;" | vM2
| ^^F#
| style="text-align:center;" | downmajor 2nd
| SM3
| style="text-align:center;" | Ev
| supermajor 3rd
| SF#
| SF#
|-
|-
| | 12
| 27
| | 200
| 450.0
|212
| 13/10, 22/17
|256 (100<sub>16</sub>)
| ^<sup>3</sup>M3, v<sup>3</sup>4
| | 9/8
| trupmajor 3rd, trud 4th
| style="text-align:center;" | M2
| ^<sup>3</sup>F#, v<sup>3</sup>G
| style="text-align:center;" | major 2nd
| HM3, h4
| style="text-align:center;" | E
| hypermajor 3rd, hypo 4th
| HF#, hG
| UF#, uG
|-
|-
| | 13
| 28
| | 216.667
| 466.7
|229.667
| 17/13, 21/16
|277.333 (115.555<sub>16</sub>)
| vv4
| | 25/22
| dud 4th
| style="text-align:center;" | ^M2
| vvG
| style="text-align:center;" | upmajor 2nd
| s4
| style="text-align:center;" | E^
| sub 4th
| sG
| sG
|-
|-
| | 14
| 29
| | 233.333
| 483.3
|247.333
| 33/25
|298.667 (12A.AAB<sub>16</sub>)
| v4
| | 8/7
| down 4th
| style="text-align:center;" | ^^M2
| vG
| style="text-align:center;" | double-upmajor 2nd
| l4
| style="text-align:center;" | E^^
| little 4th
| lG
| kG
|-
|-
| | 15
| 30
| | 250
| 500.0
|265
| 4/3
|320 (140<sub>16</sub>)
| P4
| | 81/70
| perfect 4th
| 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
| G
| style="text-align:center;" | triple-up major 2nd,
| P4
 
| perfect 4th
triple-down minor 3rd
| G
| 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>
| G
|-
|-
| | 16
| 31
| | 266.667
| 516.7
|282.333
| 27/20
|341.333 (155.555<sub>16</sub>)
| ^4
| | 7/6
| up 4th
| style="text-align:center;" | vvm3
| ^G
| style="text-align:center;" | double-downminor 3rd
| K4
| style="text-align:center;" | Fvv
| comma-wide 4th
| KG
| KG
|-
|-
| | 17
| 32
| | 283.333
| 533.3
|300.333
| 15/11, 19/14, ''26/19''
|362.667 (16A.AAB<sub>16</sub>)
| ^^4, v~4
| | 33/28
| dup 4th, downmid 4th
| style="text-align:center;" | vm3
| ^^G
| style="text-align:center;" | downminor 3rd
| O4
| style="text-align:center;" | Fv
| on 4th
| OG
| SG
|-
|-
| | 18
| 33
| | 300
| 550.0
|318
| 11/8
|384 (180<sub>16</sub>)
| ~4
| | 25/21
| mid 4th
| style="text-align:center;" | m3
| ^<sup>3</sup>G
| style="text-align:center;" | minor 3rd
| U4/N4
| style="text-align:center;" | F
| uber 4th / neutral 4th
| UG
| UG
|-
|-
| | 19
| 34
| | 316.667
| 566.7
|335.667
| 18/13, 25/18
|405.333 (195.555<sub>16</sub>)
| ^~4, vvA4
| | 6/5
| upmid 4th, dudaug 4th
| style="text-align:center;" | ^m3
| vvG#
| style="text-align:center;" | upminor 3rd
| kkA4, sd5
| style="text-align:center;" | F^
| classic aug 4th, sub dim 5th
| kkG#, sAb
| SG#, (kkG#), sAb
|-
|-
| | 20
| 35
| | 333.333
| 583.3
|353.333
| 7/5
|426.667 (1AA.AAB<sub>16</sub>)
| vA4, vd5
| | 40/33
| downaug 4th, <br>downdim 5th
| style="text-align:center;" | v~3
| vG#, vAb
| style="text-align:center;" | downmid 3rd
| kA4, ld5
| style="text-align:center;" | F^^
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, kAb
|-
|-
| | 21
| 36
| | 350
| 600.0
|371
| 17/12, 24/17
|448 (1C0<sub>16</sub>)
| A4, d5
| | 11/9
| aug 4th, dim 5th
| style="text-align:center;" | ~3
| G#, Ab
| style="text-align:center;" | mid 3rd
| A4, d5
| style="text-align:center;" | F^<span style="font-size: 90%; vertical-align: super;">3</span>
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
|-
|-
| | 22
| 37
| | 366.667
| 616.7
|388.667
| 10/7
|469.333 (1D5.555<sub>16</sub>)
| ^A4, ^d5
| | 99/80
| upaug 4th, updim 5th
| style="text-align:center;" | ^~3
| ^G#, ^Ab
| style="text-align:center;" | upmid 3rd
| LA4, Kd5
| style="text-align:center;" | F#vv
| large aug 4th, comma-wide dim 5th
| LG#, KAb
| KG#, KAb
|-
|-
| | 23
| 38
| | 383.333
| 633.3
|406.333
| 13/9, 36/25
|490.667 (1EA.AAB<sub>16</sub>)
| v~5, ^^d5
| | 5/4
| downmid 5th, <br>dupdim 5th
| style="text-align:center;" | vM3
| ^^Ab
| style="text-align:center;" | downmajor 3rd
| SA4, KKd5
| style="text-align:center;" | F#v
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, SAb, (KKAb)
|-
|-
| | 24
| 39
| | 400
| 650.0
|424
| 16/11
|512 (200<sub>16</sub>)
| ~5
| | 44/35
| mid 5th
| style="text-align:center;" | M3
| v<sup>3</sup>A
| style="text-align:center;" | major 3rd
| u5/N5
| style="text-align:center;" | F#
| unter 5th / neutral 5th
| uA
| uA
|-
|-
| | 25
| 40
| | 416.667
| 666.7
|441.667
| ''19/13'', 22/15, 28/19
|533.333 (215.555<sub>16</sub>)
| vv5, ^~5
| | 14/11
| dud 5th, upmid 5th
| style="text-align:center;" | ^M3
| vvA
| style="text-align:center;" | upmajor 3rd
| o5
| style="text-align:center;" | F#^
| off 5th
| oA
| sA
|-
|-
| | 26
| 41
| | 433.333
| 683.3
|459.333
| 40/27
|554.667 (22A.AAB<sub>16</sub>)
| v5
| | 9/7
| down 5th
| style="text-align:center;" | ^^M3
| vA
| style="text-align:center;" | double-upmajor 3rd
| k5
| style="text-align:center;" | F#^^
| comma-narrow 5th
| kA
| kA
|-
|-
| | 27
| 42
| | 450
| 700.0
|477
| 3/2
|576 (240<sub>16</sub>)
| P5
| | 35/27
| perfect 5th
| 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
| A
| style="text-align:center;" | triple-up major 3rd,
| P5
 
| perfect 5th
triple-down 4th
| A
| 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>
| A
|-
|-
| | 28
| 43
| | 466.667
| 716.7
|494.667
| 50/33
|597.333 (255.555<sub>16</sub>)
| ^5
| | 21/16
| up 5th
| style="text-align:center;" | vv4
| ^A
| style="text-align:center;" | double-down 4th
| L5
| style="text-align:center;" | Gvv
| large fifth
| LA
| KA
|-
|-
| | 29
| 44
| | 483.333
| 733.3
|512.333
| 26/17, 32/21
|618.667 (26A.AAB<sub>16</sub>)
| ^^5
| | 33/25
| dup 5th
| style="text-align:center;" | v4
| ^^A
| style="text-align:center;" | down 4th
| S5
| style="text-align:center;" | Gv
| super fifth
| SA
| SA
|-
|-
| | 30
| 45
| | 500
| 750.0
|530
| 17/11, 20/13
|640 (280<sub>16</sub>)
| ^<sup>3</sup>5, v<sup>3</sup>m6
| | 4/3
| trup 5th, trudminor 6th
| style="text-align:center;" | P4
| ^<sup>3</sup>A, v<sup>3</sup>Bb
| style="text-align:center;" | perfect 4th
| H5, hm6
| style="text-align:center;" | G
| hyper fifth, hypominor 6th
| HA, hBb
| UA, uBb
|-
|-
| | 31
| 46
| | 516.667
| 766.7
|547.667
| 14/9
|661.333 (295.555<sub>16</sub>)
| vvm6
| | 27/20
| dudminor 6th
| style="text-align:center;" | ^4
| vvBb
| style="text-align:center;" | up 4th
| sm6
| style="text-align:center;" | G^
| superminor 6th
| sBb
| sBb
|-
|-
| | 32
| 47
| | 533.333
| 783.3
|565.333
| 11/7
|682.667 (2AA.AAB<sub>16</sub>)
| vm6
| | 15/11
| downminor 6th
| style="text-align:center;" | ^^4
| vBb
| style="text-align:center;" | double-up 4th
| lm6
| style="text-align:center;" | G^^
| little minor 6th
| lBb
| kBb
|-
|-
| | 33
| 48
| | 550
| 800.0
|583
| 19/12
|704 (2C0<sub>16</sub>)
| m6
| | 11/8
| minor 6th
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>4
| Bb
| style="text-align:center;" | triple-up 4th
| m6
| style="text-align:center;" | G^<span style="font-size: 90%; vertical-align: super;">3</span>
| minor 6th
| Bb
| Bb
|-
|-
| | 34
| 49
| | 566.667
| 816.7
|600.667
| 8/5
|725.333 (2D5.555<sub>16</sub>)
| ^m6
| | 25/18
| upminor 6th
| style="text-align:center;" | vvA4
| ^Bb
| style="text-align:center;" | double-down aug 4th
| Km6
| style="text-align:center;" | G#vv
| classic minor 6th
| kBb
| kBb
|-
|-
| | 35
| 50
| | 583.333
| 833.3
|618.333
| 13/8, 21/13, 34/21
|746.667 (2EA.AAB<sub>16</sub>)
| ^^m6, v~6
| | 7/5
| dupminor 6th, downmid 6th
| style="text-align:center;" | vA4, vd5
| ^^Bb
| style="text-align:center;" | downaug 4th, updim 5th
| Om6
| style="text-align:center;" | G#v, Abv
| on minor 6th
| oBb
| sBb
|-
|-
| | 36
| 51
| | 600
| 850.0
|636
| 18/11, 44/27
|768 (300<sub>16</sub>)
| ~6
| | 99/70
| mid 6th
| style="text-align:center;" | A4, d5
| v<sup>3</sup>B
| style="text-align:center;" | aug 4th, dim 5th
| N6
| style="text-align:center;" | G#, Ab
| neutral 6th
| UBb, uB
| UBb, uB
|-
|-
| | 37
| 52
| | 616.667
| 866.7
|653.667
| 28/17, 33/20, 64/39
|789.333 (315.555<sub>16</sub>)
| ^~6, vvM6
| | 10/7
| upmid 6th, dudmajor 6th
| style="text-align:center;" | ^A4, ^d5
| vvB
| style="text-align:center;" | upaug 4th, downdim 5th
| oM6
| style="text-align:center;" | G#^, Ab^
| off major 6th
| oB
| sB
|-
|-
| | 38
| 53
| | 633.333
| 883.3
|671.333
| 5/3
|810.667 (32A.AAB<sub>16</sub>)
| vM6
| | 36/25
| downmajor 6th
| style="text-align:center;" | ^^d5
| vB
| style="text-align:center;" | double-updim 5th
| kM6
| style="text-align:center;" | Ab^^
| classic major 6th
| kB
| kB
|-
|-
| | 39
| 54
| | 650
| 900.0
|689
| 27/16, 32/19, 42/25
|832 (340<sub>16</sub>)
| M6
| | 16/11
| major 6th
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>5
| B
| style="text-align:center;" | triple-down 5th
| M6
| style="text-align:center;" | Av<span style="font-size: 90%; vertical-align: super;">3</span>
| major 6th
| B
| B
|-
|-
| | 40
| 55
| | 666.667
| 916.7
|706.667
| 17/10, 22/13
|853.333 (355.555<sub>16</sub>)
| ^M6
| | 22/15
| upmajor 6th
| style="text-align:center;" | vv5
| ^B
| style="text-align:center;" | double-down 5th
| LM6
| style="text-align:center;" | Avv
| large major 6th
| LB
| KB
|-
|-
| | 41
| 56
| | 683.333
| 933.3
|724.333
| 12/7
|874.667 (36A.AAB<sub>16</sub>)
| ^^M6
| | 40/27
| dupmajor 6th
| style="text-align:center;" | v5
| ^^B
| style="text-align:center;" | down 5th
| SM6
| style="text-align:center;" | Av
| supermajor 6th
| SB
| SB
|-
|-
| | 42
| 57
| | 700
| 950.0
|742
| 19/11, 26/15
|896 (380<sub>16</sub>)
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| | 3/2
| trupmajor 6th,<br>trudminor 7th
| style="text-align:center;" | P5
| ^<sup>3</sup>B, <br>v<sup>3</sup>C
| style="text-align:center;" | perfect 5th
| HM6, hm7
| style="text-align:center;" | A
| hypermajor 6th, hypominor 7th
| HB, hC
| UB, uC
|-
|-
| | 43
| 58
| | 716.667
| 966.7
|759.667
| 7/4
|917.333 (395.555<sub>16</sub>)
| vvm7
| | 50/33
| dudminor 7th
| style="text-align:center;" | ^5
| vvC
| style="text-align:center;" | up 5th
| sm7
| style="text-align:center;" | A^
| subminor 7th
| sC
| sC
|-
|-
| | 44
| 59
| | 733.333
| 983.3
|777.333
| 30/17, 44/25
|948.667 (3AA.AAB<sub>16</sub>)
| vm7
| | 32/21
| downminor 7th
| style="text-align:center;" | ^^5
| vC
| style="text-align:center;" | double-up 5th
| lm7
| style="text-align:center;" | A^^
| little minor 7th
| lC
| kC
|-
|-
| | 45
| 60
| | 750
| 1000.0
|795
| 16/9
|960 (3C0<sub>16</sub>)
| m7
| | 54/35
| minor 7th
| 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
| C
| style="text-align:center;" | triple-up 5th,
| m7
 
| minor 7th
triple-down minor 6th
| C
| 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>
| C
|-
|-
| | 46
| 61
| | 766.667
| 1016.7
|812.667
| 9/5
|981.333 (3D5.555<sub>16</sub>)
| ^m7
| | 14/9
| upminor 7th
| style="text-align:center;" | vvm6
| ^C
| style="text-align:center;" | double-downminor 6th
| Km7
| style="text-align:center;" | Bbvv
| classic/comma-wide minor 7th
| KC
| KC
|-
|-
| | 47
| 62
| | 783.333
| 1033.3
|830.333
| 20/11
|1002.667 (2EA.AAB<sub>16</sub>)
| ^^m7, v~7
| | 11/7
| dupminor 7th, downmid 7th
| style="text-align:center;" | vm6
| ^^C
| style="text-align:center;" | downminor 6th
| Om7
| style="text-align:center;" | Bbv
| on minor 7th
| OC
| SC
|-
|-
| | 48
| 63
| | 800
| 1050.0
|848
| 11/6
|1024 (400<sub>16</sub>)
| ~7
| | 35/22
| mid 7th
| style="text-align:center;" | m6
| ^<sup>3</sup>C
| style="text-align:center;" | minor 6th
| N7, hd8
| style="text-align:center;" | Bb
| neutral 7th, hypo dim 8ve
| UC/uC#, hDb
| UC/uC#, uDb
|-
|-
| | 49
| 64
| | 816.667
| 1066.7
|865.667
| 13/7, 24/13, 50/27
|1045.333 (415.555<sub>16</sub>)
| ^~7, vvM7
| | 8/5
| upmid 7th, dudmajor 7th
| style="text-align:center;" | ^m6
| vvC#
| style="text-align:center;" | upminor 6th
| oM7, sd8
| style="text-align:center;" | Bb^
| off major 7th, sub dim 8ve
| oC#, sDb
| sC#, sDb
|-
|-
| | 50
| 65
| | 833.333
| 1083.3
|883.333
| 15/8, 28/15
|1066.667 (42A.AAB<sub>16</sub>)
| vM7
| | 81/50
| downmajor 7th
| style="text-align:center;" | v~6
| vC#
| style="text-align:center;" | downmid 6th
| kM7, ld8
| style="text-align:center;" | Bb^^
| classic major 7th, little dim 8ve
| kC#, lDb
| kC#, kDb
|-
|-
| | 51
| 66
| | 850
| 1100.0
|901
| 17/9, 32/17, 36/19
|1088 (440<sub>16</sub>)
| M7
| | 18/11
| major 7th
| style="text-align:center;" | ~6
| C#
| style="text-align:center;" | mid 6th
| M7, d8
| style="text-align:center;" | Bv<span style="font-size: 90%; vertical-align: super;">3</span>
| major 7th, dim 8ve
| C#, Db
| C#, Db
|-
|-
| | 52
| 67
| | 866.667
| 1116.7
|918.667
| 19/10, 21/11, 40/21
|1109.333 (455.555<sub>16</sub>)
| ^M7
| | 33/20
| upmajor 7th
| style="text-align:center;" | ^~6
| ^C#
| style="text-align:center;" | upmid 6th
| LM7, Kd8
| style="text-align:center;" | Bvv
| large major 7th, comma-wide dim 8ve
| LC#, KDb
| KC#, KDb
|-
|-
| | 53
| 68
| | 883.333
| 1133.3
|936.333
| 25/13, 27/14, 48/25, 52/27
|1130.667 (46A.AAB<sub>16</sub>)
| ^^M7
| | 5/3
| dupmajor 7th
| style="text-align:center;" | vM6
| ^^C#
| style="text-align:center;" | downmajor 6th
| SM7, KKd8
| style="text-align:center;" | Bv
| supermajor 7th, classic dim 8ve
| SC#, KKDb
| SC#, SDb, (KKDb)
|-
|-
| | 54
| 69
| | 900
| 1150.0
|954
| 35/18, 39/20, 64/33
|1152 (480<sub>16</sub>)
| ^<sup>3</sup>M7, v<sup>3</sup>8
| | 27/16
| trupmajor 7th, trud octave
| style="text-align:center;" | M6
| ^<sup>3</sup>C#, v<sup>3</sup>D
| style="text-align:center;" | major 6th
| HM7, u8, h8
| style="text-align:center;" | B
| hypermajor 7th, unter 8ve, hypo 8ve
| HC#, uD, hD
| UC#, uDb, uD
|-
|-
| | 55
| 70
| | 916.667
| 1166.7
|971.667
| 49/25, 55/28, 63/32, 88/45, 96/49
|1173.333 (495.555<sub>16</sub>)
| vv8
| | 56/33
| dud octave
| style="text-align:center;" | ^M6
| vvD
| style="text-align:center;" | upmajor 6th
| s8, o8
| style="text-align:center;" | B^
| sub 8ve, off 8ve
| sD, oD
| sD
|-
|-
| | 56
| 71
| | 933.333
| 1183.3
|989.333
| 99/50, 160/81, 180/91, 196/99, 208/105
|1194.667 (4AA.AAB<sub>16</sub>)
| v8
| | 12/7
| down octave
| style="text-align:center;" | ^^M6
| vD
| style="text-align:center;" | double-upmajor 6th
| k8, l8
| style="text-align:center;" | B^^
| comma-narrow 8ve, little 8ve
| kD, lD
| kD
|-
|-
| | 57
| 72
| | 950
| 1200.0
|1007
| 2/1
|1216 (4C0<sub>16</sub>)
| P8
| | 121/70
| perfect octave
| 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
| D
| style="text-align:center;" | triple-up major 6th,
| 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
|1024.667
| (a b -1)
|1237.333 (4D5.555<sub>16</sub>)
| 6/5, 9/5
| | 7/4
| style="text-align:center;" | vvm7
| style="text-align:center;" | double-downminor 7th
| style="text-align:center;" | Cvv
|-
|-
| | 59
| rowspan="2" | dupminor, <br>downmid
| | 983.333
| luyo
|1042.333
| (a b 1 0 -1)
|1258.667 (4EA.AAB<sub>16</sub>)
| 15/11
| | 44/25
| style="text-align:center;" | vm7
| style="text-align:center;" | downminor 7th
| style="text-align:center;" | Cv
|-
|-
| | 60
| tho
| | 1000
| (a b 0 0 0 1)
|1060
| 13/8, 13/9
|1280 (500<sub>16</sub>)
| | 16/9
| style="text-align:center;" | m7
| style="text-align:center;" | minor 7th
| style="text-align:center;" | C
|-
|-
| | 61
| rowspan="2" | mid
| | 1016.667
| ilo
|1077.667
| (a b 0 0 1)
|1301.333 (515.555<sub>16</sub>)
| 11/9, 11/6
| | 9/5
| style="text-align:center;" | ^m7
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | C^
|-
|-
| | 62
| lu
| | 1033.333
| (a b 0 0 -1)
|1095.333
| 12/11, 18/11
|1322.667 (52A.AAB<sub>16</sub>)
| | 20/11
| 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
|1113
| (a b -1 0 1)
|1344 (540<sub>16</sub>)
| 11/10
| | 11/6
| style="text-align:center;" | ~7
| 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)
|1130.667
| 16/13, 18/13
|1365.333 (555.555<sub>16</sub>)
| | 50/27
| style="text-align:center;" | ^~7
| style="text-align:center;" | upmid 7th
| style="text-align:center;" | C#vv
|-
|-
| | 65
| downmajor
| | 1083.333
| yo
|1148.333
| (a b 1)
|1386.667 (56A.AAB<sub>16</sub>)
| 5/4, 5/3
| | 15/8
| style="text-align:center;" | vM7
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | C#v
|-
|-
| | 66
| major
| | 1100
| fifthward wa
|1166
| (a b), b > 1
|1408 (580<sub>16</sub>)
| 9/8, 27/16
| | 66/35
| style="text-align:center;" | M7
| style="text-align:center;" | major 7th
| style="text-align:center;" | C#
|-
|-
| | 67
| dupmajor
| | 1116.667
| ru
|1183.667
| (a b 0 -1)
|1429.333 (595.555<sub>16</sub>)
| 9/7, 12/7
| | 21/11
| style="text-align:center;" | ^M7
| style="text-align:center;" | upmajor 7th
| style="text-align:center;" | C#^
|-
|-
| | 68
| rowspan="2" | trupmajor, <br>trudminor
| | 1133.333
| thogu
|1201.333
| (a b -1 0 0 1)
|1450.667 (5AA.AAB<sub>16</sub>)
| 13/10
| | 27/14
| style="text-align:center;" | ^^M7
| style="text-align:center;" | double-upmajor 7th
| style="text-align:center;" | C#^^
|-
|-
| | 69
| thuyo
| | 1150
| (a b 1 0 0 -1)
|1219
| 15/13
|1472 (5C0<sub>16</sub>)
|}
| | 35/18
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:
| 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,


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
|-
| gu
| 10:12:15
| 0-19-42
| C ^Eb G
| C^m
| C upminor
|-
|-
| | 70
| ilo
| | 1166.667
| 18:22:27
|1236.667
| 0-21-42
|1493.333 (5D5.555<sub>16</sub>)
| C v<span style="font-size: 90%; vertical-align: super;">3</span>E G
| | 49/25
| C~
| style="text-align:center;" | vv8
| C mid
| style="text-align:center;" | double-down octave
| style="text-align:center;" | Dvv
|-
|-
| | 71
| yo
| | 1183.333
| 4:5:6
|1254.333
| 0-23-42
|1514.667 (5EA.AAB<sub>16</sub>)
| C vE G
| | 99/50
| Cv
| style="text-align:center;" | v8
| C downmajor or C down
| style="text-align:center;" | down octave
| style="text-align:center;" | Dv
|-
|-
| | 72
| ru
| | 1200
| 14:18:27
|1272
| 0-26-42
|1536 (600<sub>16</sub>)
| C ^^E G
| | 2/1
| C^^
| style="text-align:center;" | P8
| C dupmajor or C dup
| style="text-align:center;" | perfect octave
| style="text-align:center;" | D
|}
|}
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]]
Retrieved from "https://en.xen.wiki/w/72edo"