72edo: Difference between revisions

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

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

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]].  

The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.

The 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 center-all right-2 left-3"

! #

! Approximate ratios<ref group="note">{{sg|limit=19-limit}} For lower limits see [[Table of 72edo intervals]].</ref>

| 0

| 1/1

| P1

| perfect unison

| D

| 1

| comma-wide unison, large unison

| 2

| 45/44, 49/48, 50/49, 55/54, 64/63

| dup unison

| S1, O1

| super unison, on unison

| SD, OD

| SD

| 3

| 50.0

| 33/32, 36/35, 40/39

| 4

| 66.7

| 25/24, 26/25, 27/26, 28/27

| vvEb

| kkA1, sm2

| classic aug unison, subminor 2nd

| kkD#, sEb

| sD#, (kkD#), sEb

| 5

| 83.3

| 20/19, 21/20, 22/21

| vm2

| downminor 2nd

| vEb

| kA1, lm2

| comma-narrow aug unison, little minor 2nd

| kD#, lEb

| kD#, kEb

| 6

| 100.0

| 17/16, 18/17, 19/18

| m2

| minor 2nd

| Eb

| m2

| minor 2nd

| Eb

| Eb

| 7

| 116.7

| 15/14, 16/15

| ^m2

| upminor 2nd

| ^Eb

| Km2

| classic minor 2nd

| KEb

| KEb

| 8

| 133.3

| 13/12, 14/13, 27/25

| ^^m2, v~2

| dupminor 2nd, downmid 2nd

| ^^Eb

| Om2

| on minor 2nd

| OEb

| SEb

| 9

| 150.0

| 12/11

| ~2

| mid 2nd

| v³E

| N2

| neutral 2nd

| UEb/uE

| UEb/uE

| 10

| 166.7

| 11/10

| ^~2, vvM2

| upmid 2nd, dudmajor 2nd

| vvE

| oM2

| off major 2nd

| oE

| sE

| 11

| 183.3

| 10/9

| vM2

| downmajor 2nd

| vE

| kM2

| classic/comma-narrow major 2nd

| kE

| kE

| 12

| 200.0

| 9/8

| M2

| major 2nd

| E

| M2

| major 2nd

| E

| E

| 13

| 216.7

| 17/15, 25/22

| ^M2

| upmajor 2nd

| ^E

| LM2

| large major 2nd

| LE

| KE

| 14

| 233.3

| 8/7

| ^^M2

| dupmajor 2nd

| ^^E

| SM2

| supermajor 2nd

| SE

| SE

| 15

| 250.0

| 15/13, 22/19

| ^³M2, <br>v³m3

| trupmajor 2nd,<br>trudminor 3rd

| ^³E, <br>v³F

| HM2, hm3

| hypermajor 2nd, hypominor 3rd

| HE, hF

| UE, uF

| 16

| 266.7

| 7/6

| vvm3

| dudminor 3rd

| vvF

| sm3

| subminor 3rd

| sF

| sF

| 17

| 283.3

| 13/11, 20/17

| vm3

| downminor 3rd

| vF

| lm3

| little minor 3rd

| lF

| kF

| 18

| 300.0

| 19/16, 25/21, 32/27

| m3

| minor 3rd

| F

| m3

| minor 3rd

| F

| F

| 19

| 316.7

| 6/5

| ^m3

| upminor 3rd

| ^F

| Km3

| classic minor 3rd

| KF

| KF

| 20

| 333.3

| 17/14, 39/32, 40/33

| ^^m3, v~3

| dupminor 3rd, downmid 3rd

| ^^F

| Om3

| on minor third

| SF

| 21

| 350.0

| 11/9, 27/22

| ~3

| mid 3rd

| ^³F

| N3

| neutral 3rd

| UF/uF#

| UF/uF#

| 22

| 366.7

| 16/13, 21/17, 26/21

| ^~3, vvM3

| upmid 3rd, dudmajor 3rd

| vvF#

| oM3

| off major 3rd

| oF#

| sF#

| 23

| 383.3

| 5/4

| vM3

| downmajor 3rd

| vF#

| kM3

| classic major 3rd

| kF#

| kF#

| 24

| 400.0

| 24/19

| M3

| major 3rd

| F#

| M3

| major 3rd

| F#

| F#

| 25

| 416.7

| 14/11

| ^M3

| upmajor 3rd

| ^F#

| LM3

| large major 3rd

| LF#

| KF#

| 26

| 433.3

| 9/7

| ^^M3

| dupmajor 3rd

| ^^F#

| SM3

| supermajor 3rd

| SF#

| SF#

| 27

| 450.0

| 13/10, 22/17

| ^³M3, v³4

| trupmajor 3rd, trud 4th

| ^³F#, v³G

| HM3, h4

| hypermajor 3rd, hypo 4th

| HF#, hG

| UF#, uG

| 28

| 466.7

| 17/13, 21/16

| vv4

| dud 4th

| vvG

| s4

| sub 4th

| sG

| sG

| 29

| 483.3

| 33/25

| v4

| down 4th

| vG

| l4

| little 4th

| lG

| kG

| 30

| 500.0

| 4/3

| P4

| perfect 4th

| G

| P4

| G

| 31

| 516.7

| 27/20

| ^4

| up 4th

| ^G

| K4

| comma-wide 4th

| KG

| KG

| 32

| 533.3

| 15/11, 19/14, ''26/19''

| ^^4, v~4

| dup 4th, downmid 4th

| ^^G

| O4

| on 4th

| OG

| SG

| 33

| 550.0

| 11/8

| ~4

| mid 4th

| ^³G

| U4/N4

| uber 4th / neutral 4th

| UG

| UG

| 34

| 566.7

| 18/13, 25/18

| ^~4, vvA4

| upmid 4th, dudaug 4th

| vvG#

| kkA4, sd5

| classic aug 4th, sub dim 5th

| kkG#, sAb

| SG#, (kkG#), sAb

| 35

| 583.3

| 7/5

| vA4, vd5

| downaug 4th, <br>downdim 5th

| vG#, vAb

| kA4, ld5

| comma-narrow aug 4th, little dim 5th

| kG#, kAb

| 36

| 600.0

| 17/12, 24/17

| A4, d5

| aug 4th, dim 5th

| G#, Ab

| A4, d5

| aug 4th, dim 5th

| G#, Ab

| G#, Ab

| 37

| 616.7

| 10/7

| ^A4, ^d5

| upaug 4th, updim 5th

| ^G#, ^Ab

| large aug 4th, comma-wide dim 5th

| LG#, KAb

| KG#, KAb

| 38

| 633.3

| 13/9, 36/25

| v~5, ^^d5

| downmid 5th, <br>dupdim 5th

| ^^Ab

| SA4, KKd5

| super aug 4th, classic dim 5th

| SG#, KKAb

| SG#, SAb, (KKAb)

| 39

| 650.0

| 16/11

| ~5

| mid 5th

| v³A

| u5/N5

| unter 5th / neutral 5th

| uA

| uA

| 40

| 666.7

| ''19/13'', 22/15, 28/19

| vv5, ^~5

| dud 5th, upmid 5th

| vvA

| o5

| off 5th

| oA

| sA

| 41

| 683.3

| 40/27

| v5

| down 5th

| vA

| k5

| comma-narrow 5th

| kA

| kA

| 42

| 700.0

| 3/2

| P5

| perfect 5th

| A

| P5

| A

| 43

| 716.7

| 50/33

| ^5

| up 5th

| ^A

| L5

| large fifth

| LA

| KA

| 44

| 733.3

| 26/17, 32/21

| ^^5

| dup 5th

| ^^A

| S5

| super fifth

| SA

| SA

| 45

| 750.0

| 17/11, 20/13

| ^³5, v³m6

| trup 5th, trudminor 6th

| ^³A, v³Bb

| H5, hm6

| hyper fifth, hypominor 6th

| HA, hBb

| UA, uBb

| 46

| 766.7

| 14/9

| vvm6

| dudminor 6th

| vvBb

| sm6

| superminor 6th

| sBb

| sBb

| 47

| 783.3

| 11/7

| vm6

| downminor 6th

| vBb

| lm6

| little minor 6th

| lBb

| kBb

| 48

| 800.0

| 19/12

| m6

| minor 6th

| Bb

| m6

| minor 6th

| Bb

| Bb

| 49

| 816.7

| 8/5

| ^m6

| upminor 6th

| ^Bb

| Km6

| classic minor 6th

| kBb

| kBb

| 50

| 833.3

| 13/8, 21/13, 34/21

| ^^m6, v~6

| dupminor 6th, downmid 6th

| Om6

| on minor 6th

| oBb

| sBb

| 51

| 850.0

| 18/11, 44/27

| ~6

| mid 6th

| v³B

| N6

| neutral 6th

| UBb, uB

| UBb, uB

| 52

| 866.7

| 28/17, 33/20, 64/39

| ^~6, vvM6

| upmid 6th, dudmajor 6th

| oM6

| off major 6th

| oB

| sB

| 53

| 883.3

| 5/3

| vM6

| downmajor 6th

| vB

| kM6

| classic major 6th

| kB

| kB

| 54

| 900.0

| 27/16, 32/19, 42/25

| major 6th

| B

| M6

| major 6th

| B

| B

| 55

| 916.7

| 17/10, 22/13

| ^M6

| upmajor 6th

| ^B

| large major 6th

| LB

| KB

| 56

| 933.3

| 12/7

| ^^M6

| dupmajor 6th

| ^^B

| SM6

| supermajor 6th

| SB

| SB

| 57

| 950.0

| 19/11, 26/15

| ^³M6, <br>v³m7

| trupmajor 6th,<br>trudminor 7th

| ^³B, <br>v³C

| HM6, hm7

| hypermajor 6th, hypominor 7th

| HB, hC

| UB, uC

| 58

| 966.7

| 7/4

| vvm7

| dudminor 7th

| vvC

| sm7

| subminor 7th

| sC

| sC

| 59

| 983.3

| 30/17, 44/25

| vm7

| downminor 7th

| vC

| lm7

| little minor 7th

| lC

| kC

| 60

| 1000.0

| 16/9

| m7

| minor 7th

| C

| m7

| minor 7th

| C

| C

| 61

| 1016.7

| 9/5

| ^m7

| upminor 7th

| ^C

| Km7

| classic/comma-wide minor 7th

| KC

| KC

| 62

| 1033.3

| 20/11

| ^^m7, v~7

| dupminor 7th, downmid 7th

| ^^C

| Om7

| on minor 7th

| OC

| SC

| 63

| 1050.0

| 11/6

| ~7

| mid 7th

| ^³C

| N7, hd8

| neutral 7th, hypo dim 8ve

| UC/uC#, hDb

| UC/uC#, uDb

| 64

| 1066.7

| 13/7, 24/13, 50/27

| ^~7, vvM7

| upmid 7th, dudmajor 7th

| vvC#

| oM7, sd8

| off major 7th, sub dim 8ve

| oC#, sDb

| sC#, sDb

| 65

| 1083.3

| 15/8, 28/15

| vM7

| downmajor 7th

| vC#

| kM7, ld8

| classic major 7th, little dim 8ve

| kC#, lDb

| kC#, kDb

| 66

| 1100.0

| 17/9, 32/17, 36/19

| M7

| major 7th

| C#

| M7, d8

| major 7th, dim 8ve

| C#, Db

| C#, Db

| 67

| 1116.7

| 19/10, 21/11, 40/21

| ^M7

| upmajor 7th

| ^C#

| large major 7th, comma-wide dim 8ve

| LC#, KDb

| KC#, KDb

| 68

| 1133.3

| 25/13, 27/14, 48/25, 52/27

| ^^M7

| dupmajor 7th

| ^^C#

| SM7, KKd8

| supermajor 7th, classic dim 8ve

| SC#, KKDb

| SC#, SDb, (KKDb)

| 69

| 1150.0

| 35/18, 39/20, 64/33

| ^³M7, v³8

| trupmajor 7th, trud octave

| ^³C#, v³D

| HM7, u8, h8

| hypermajor 7th, unter 8ve, hypo 8ve

| HC#, uD, hD

| UC#, uDb, uD

| 70

| 1166.7

| 49/25, 55/28, 63/32, 88/45, 96/49

| vv8

| dud octave

| vvD

| s8, o8

| sub 8ve, off 8ve

| sD

| 71

| 1183.3

| 99/50, 160/81, 180/91, 196/99, 208/105

| v8

| down octave

| vD

| k8, l8

| comma-narrow 8ve, little 8ve

| kD, lD

| kD

| 72

| 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:

{| class="wikitable center-all"

|-