72edo: Difference between revisions

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, meaning every interval in the 11-odd-limit is approximated with less than 25% relative error (about 4 cents), and the first edo to be consistent in the 12- and 13-[[odd prime sum limit|odd-prime-sum-limit]].  

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 (144edo is enfactored in the 11-limit with 72edo, so it is already covered here). It is also an excellent tuning for [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].

 

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

{| 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>

! colspan="3" | [[Ups and downs notation]]

! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]

| 1/1

| P1

| D

| 81/80, 91/90, 99/98, 100/99, 105/104

| ^1

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

| ^^

| SD

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

| ^³1, v³m2

| ^³D, v³Eb

|-

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

| vvm2

| sD#, (kkD#), sEb

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

| vm2

| comma-narrow aug unison, little minor 2nd

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

| m2

| Eb

| 15/14, 16/15

| ^m2

| KEb

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

| ^^m2, v~2

| SEb

| 12/11

| ~2

| UEb/uE

| 11/10

| ^~2, vvM2

| upmid 2nd, dudmajor 2nd

| vvE

| 10/9

| vM2

| kE

| 9/8, 19/17

| M2

| E

| 17/15, 25/22

| ^M2

| KE

| 8/7

| ^^M2

| dupmajor 2nd

|-

| 15

| 250.0

| 15/13, 22/19

| ^³M2, <br>v³m3

| trupmajor 2nd,<br>trudminor 3rd

| 7/6

| vvm3

| sF

| 13/11, 20/17

| vm3

| downminor 3rd

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

| m3

| F

| 6/5

| ^m3

| KF

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

| ^^m3, v~3

| SF

| 11/9, 27/22

| ~3

| mid 3rd

|-

| 22

| 366.7

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

| ^~3, vvM3

| upmid 3rd, dudmajor 3rd

| 5/4

| vM3

| kF#

| 24/19

| M3

| major 3rd

|-

| 25

| 416.7

| 14/11, 19/15

| ^M3

| upmajor 3rd

| 9/7

| ^^M3

| SF#

| 13/10, 22/17

| ^³M3, v³4

| hypermajor 3rd, hypo 4th

| 17/13, 21/16

| vv4

| sG

| 33/25

| v4

| kG

| 4/3

| P4

| G

| 27/20

| ^4

| ^G

|-

| 32

| 533.3

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

| ^^4, v~4

| ^^G

|-

| 11/8

| ~4

| UG

| 18/13, 25/18

| ^~4, vvA4

| classic aug 4th, sub dim 5th

|-

| 35

| 7/5

| vA4, vd5

| kG#, kAb

| 17/12, 24/17

| A4, d5

| aug 4th, dim 5th

| 10/7

| ^A4, ^d5

| ^G#, ^Ab

|-

| 38

| 633.3

| 13/9, 36/25

| v~5, ^^d5

| downmid 5th, <br>dupdim 5th

| 16/11

| ~5

| uA

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

| vv5, ^~5

| sA

| 40/27

| v5

| kA

| 3/2

| P5

| A

| 50/33

| ^5

| KA

| 26/17, 32/21

| ^^5

| dup 5th

|-

| 45

| 750.0

| 17/11, 20/13

| ^³5, v³m6

| trup 5th, trudminor 6th

| 14/9

| vvm6

| sBb

| 11/7, 30/19

| vm6

| kBb

| 19/12

| m6

| Bb

| 8/5

| ^m6

| kBb

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

| ^^m6, v~6

| sBb

| 18/11, 44/27

| ~6

| UBb, uB

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

| ^~6, vvM6

| sB

| 5/3

| vM6

| kB

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

| M6

| B

| 17/10, 22/13

| ^M6

| KB

| 12/7

| ^^M6

| SB

| 19/11, 26/15

| ^³M6, <br>v³m7

| hypermajor 6th, hypominor 7th

| 7/4

| vvm7

| sC

| 30/17, 44/25

| vm7

| kC

|-

| 60

| 1000.0

| 16/9, 34/19

| m7

| minor 7th

| 9/5

| ^m7

| KC

| 20/11

| ^^m7, v~7

| dupminor 7th, downmid 7th

| SC

| 11/6

| ~7

| UC/uC#, hDb

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

| ^~7, vvM7

| off major 7th, sub dim 8ve

| 15/8, 28/15

| vM7

| classic major 7th, little dim 8ve

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

| M7

| major 7th, dim 8ve

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

| ^M7

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

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

| ^^M7

| SC#, SDb, (KKDb)

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

| ^³M7, v³8

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

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

| vv8

| sD

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

| v8

| kD

| 2/1

| P8

| D

=== Interval quality and chord names in color notation ===

 

{| class="wikitable center-all"

! 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

| upminor

| gu

| (a b -1)

| 6/5, 9/5

| rowspan="2" | dupminor, <br>downmid

| luyo

| (a b 1 0 -1)

| 15/11

| tho

| (a b 0 0 0 1)

| 13/8, 13/9

|-

| rowspan="2" | mid

| ilo

| (a b 0 0 1)

| 11/9, 11/6

| lu

| (a b 0 0 -1)

| 12/11, 18/11

| rowspan="2" | upmid, <br>dudmajor

| logu

| (a b -1 0 1)

| 11/10

|-

| thu

| (a b 0 0 0 -1)

| 16/13, 18/13

| downmajor

| yo

| (a b 1)

| 5/4, 5/3

| major

| fifthward wa

| (a b), b > 1

| 9/8, 27/16

| dupmajor

| ru

| (a b 0 -1)

| 9/7, 12/7

| rowspan="2" | trupmajor, <br>trudminor

| thogu

| (a b -1 0 0 1)

| 13/10

| thuyo

| (a b 1 0 0 -1)

| 15/13

|}

All 72edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:

 

{| class="wikitable center-all"

! [[Color notation|Color of the 3rd]]

| 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

| ilo

| 18:22:27

| 0-21-42

| C v<span style="font-size: 90%; vertical-align: super;">3</span>E G

| C~

| C mid

| yo

| 4:5:6

| 0-23-42

| C vE G

| Cv

| C downmajor or C down

| ru

| 14:18:27

| 0-26-42

| C ^^E G

| C^^

| C dupmajor or C dup

|}

 

In 72tet, there are 6 [[ring number|rings]]. 12edo is the plain ring; thus every 6 degrees is the 3-limit.

 

* 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.

 

 

{{Ups and downs sharpness|72|true}}

 

[[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}}

 

 

This notation uses the same sagittal sequence as edos [[65edo #Sagittal notation|65-]] and [[79edo #Sagittal notation|79edo]], 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]].

 

{{Sagittal chart|Evo}}

 

==== Evo-SZ flavor ====

{{Sagittal chart|Evo-SZ}}