87edo: Difference between revisions

The '''87 equal temperament''', often abbreviated '''87-tET''', '''87-EDO''', or '''87-ET''', is the scale derived by dividing the octave into 87 equally-sized steps, where each step represents a frequency ratio of 13.79 [[cent|cents]]. It is solid as both a [[13-limit]] (or [[15-odd-limit]]) and as a [[5-limit]] system, and of course does well enough in any limit in between. It represents the [[13-limit]] [[tonality diamond]] both uniquely and [[consistent|consistently]] (see [[87edo/13-limit detempering]]), and is the smallest equal temperament to do so.  

87et also shows some potential in limits beyond 13. The next four prime harmonics 17, 19, 23 and 29 are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they don't combine with 7, which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.  

87et [[tempering out|tempers out]] 196/195, 325/324, 352/351, 364/363, 385/384, 441/440, 625/624, 676/675, and 1001/1000 as well as the 29-comma, <46 -29|, the misty comma, <26 -12 -3|, the kleisma, 15625/15552, 245/243, 1029/1024, 3136/3125, and 5120/5103.

87et is a particularly good tuning for [[Gamelismic clan #Rodan|rodan temperament]]. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit [[POTE tuning|POTE]] generator and is close to the [[11-limit]] POTE generator also. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.

! colspan="2" | Approximated Ratios

! colspan="2" rowspan="2" |[[Ups and Downs Notation]]

!13-Limit

!31-Limit No-7s Extension

|0

|0.000

|[[1/1]]

|P1

|D

|1

|13.793

|[[126/125]], [[100/99]], [[91/90]]

|^1

|^D

|2

|27.586

|[[81/80]], [[64/63]], [[49/48]], [[55/54]], [[65/64]]

|^^1

|^^D

|3

|41.379

|[[50/49]], [[45/44]], [[40/39]]

|[[39/38]]

|^³1

|^³D/v³Eb

|4

|55.172

|[[28/27]], [[36/35]], [[33/32]]

|[[34/33]], [[30/29]], [[32/31]], [[31/30]]

|vvm2

|vvEb

|5

|68.966

|[[25/24]], [[27/26]], [[26/25]]

|[[24/23]]

|vm2

|vEb

|6

|82.759

|[[21/20]], [[22/21]]

|[[20/19]], [[23/22]]

|m2

|Eb

|7

|96.552

|[[35/33]]

|[[18/17]], [[19/18]]

|^m2

|^Eb

|8

|110.345

|[[16/15]]

|[[17/16]], [[33/31]], [[31/29]]

|^^m2

|^^Eb

|9

|124.138

|[[15/14]], [[14/13]]

|[[29/27]]

|vv~2

|^³Eb

|10

|137.931

|[[13/12]], [[27/25]]

|[[25/23]]

|v~2

|^⁴Eb

|11

|151.724

|[[12/11]], [[35/32]]

|^~2

|v⁴E

|12

|165.517

|[[11/10]]

|[[32/29]], [[34/31]]

|^^~2

|v³E

|13

|179.310

|[[10/9]]

|vvM2

|vvE

|14

|193.103

|[[28/25]]

|[[19/17]], [[29/26]]

|vM2

|vE

|15

|206.897

|[[9/8]]

|[[26/23]]

|M2

|E

|16

|220.690

|[[25/22]]

|[[17/15]], [[33/29]]

|^M2

|^E

|17

|234.483

|[[8/7]]

|[[31/27]]

|^^M2

|^^E

|18

|248.276

|[[15/13]]

|[[22/19]], [[38/33]], [[23/20]]

|^³M2/v³m3

|^³E/v³F

|19

|262.089

|[[7/6]]

|[[29/25]], [[36/31]]

|vvm3

|vvF

|20

|275.862

|[[75/64]]

|[[27/23]], [[34/29]]

|vm3

|vF

|21

|289.655

|[[32/27]], [[33/28]], [[13/11]]

|m3

|F

|22

|303.448

|[[25/21]]

|[[19/16]], [[31/26]]

|^m3

|^F

|23

|317.241

|[[6/5]]

|^^m3

|^^F

|24

|331.034

|[[40/33]]

|[[23/19]], [[29/24]]

|vv~3

|^³F

|25

|344.828

|[[11/9]], [[39/32]]

|v~3

|^⁴F

|26

|358.621

|[[27/22]], [[16/13]]

|[[38/31]]

|^~3

|v⁴F#

|27

|372.414

|[[26/21]]

|[[31/25]], [[36/29]]

|^^3

|v³F#

|28

|386.207

|[[5/4]]

|vvM3

|vvF#

|29

|400.000

|[[44/35]]

|[[34/27]], [[24/19]], [[29/23]]

|vM3

|vF#

|30

|413.793

|[[81/64]], [[14/11]], [[33/26]]

|[[19/15]]

|M3

|F#

|31

|427.586

|[[32/25]]

|[[23/18]]

|^M3

|^F#

|32

|441.379

|[[9/7]], [[35/27]]

|[[22/17]], [[31/24]], [[40/31]]

|^^M3

|^^F#

|33

|455.172

|[[13/10]]

|[[30/23]]

|^³M3/v³4

|^³F#/v³G

|34

|468.966

|[[21/16]]

|[[17/13]], [[25/19]], [[38/29]]

|vv4

|vvG

|35

|482.759

|[[33/25]]

|v4

|vG

|36

|496.552

|[[4/3]]

|P4

|G

|37

|510.345

|[[35/26]]

|[[31/23]]

|^4

|^G

|38

|524.138

|[[27/20]]

|[[23/17]]

|^^4

|^^G

|39

|537.931

|[[15/11]]

|[[26/19]], [[34/25]]

|^³4

|^³G

|40

|551.724

|[[11/8]], [[48/35]]

|^⁴4

|^⁴G

|41

|565.517

|[[18/13]]

|[[32/23]]

|v⁴A4, vd5

|v⁴G#, vAb

|42

|579.310

|[[7/5]]

|[[46/33]]

|v³A4, d5

|v³G#, Ab

|43

|593.103

|[[45/32]]

|[[24/17]], [[38/27]], [[31/22]]

|vvA4, ^d5

|vvG#, ^Ab

=== Selected just intervals by error ===

{| class="wikitable center-all"

! colspan="2" |

!prime 2

!prime 3

!prime 5

!prime 7

!prime 11

!prime 13

! rowspan="2" |Error

|0.00

| +1.49

| -0.11

| -3.31

| +0.41

| +0.85

| +5.39

| +5.94

| +6.21

| +4.91

| -0.21

|0.0

| +10.8

| -0.8

| -24.0

| +2.9

| +6.2

| +39.1

| +43.0

| +45.0

| +35.6

| -1.5

== 13-limit detempering of 87et ==

 

 

''Main article: [[87edo/13-limit detempering]]''

 

{| class="wikitable center-all right-3 left-5"

! Periods <br> per <br> octave

! Generator

! Cents

! Associated <br> ratio

| [[33/32]]

| [[Sensa]]

| [[13/12]]

| [[28/25]]

| [[Luna]] / [[Hemithirds]]

| [[8/7]]

| [[Rodan]]

| [[6/5]]

| [[Hanson]] / [[Countercata]] / [[Metakleismic]]

| [[9/7]]

| [[65/48]]

| [[11/8]]

| [[Emkay]]

| 23\87

| 317.241

| [[6/5]]

| 28\87

| 386.207

| [[5/4]]

87 can serve as a MOS in these:

* [[M&N temperaments|270&amp;87]] &lt;&lt;24 -9 -66 12 27 ... ||

* [[M&N temperaments|494&amp;87]] &lt;&lt;51 -1 -133 11 32 ... ||

=== Harmonic Scale ===

87edo accurately approximates the mode 8 of [[harmonic series]], and the only intervals not distinct are 14/13 and 15/14. It does mode 16 fairly decent, with the only anomaly at 28/27 (4 steps) and 29/28 (5 steps).  

==== Mode 8 ====

|1/1

|9/8

|5/4

|11/8

|3/2

|13/8

|7/4

|15/8

|2/1

|… in cents

|0.0

|203.9

|386.3

|551.3

|702.0

|840.5

|968.8

|1088.3

|Degrees in 87edo

|0

|15

|28

|40

|51

|61

|70

|79

|87

|… in cents

|0.0

|206.9

|386.2

|551.7

|703.5

|841.4

|965.5

|1089.7

|1200.0

==== Mode 16 ====

|JI Ratios

|17/16

|19/16

|21/16

|23/16

|25/16

|27/16

|29/16

|31/16

|… in cents

|105.0

|297.5

|470.8

|628.3

|772.6

|905.9

|1029.6

|1145.0

|Degrees in 87edo

|8

|22

|34

|46

|56

|66

|75

|83

|… in cents

|110.3

|303.4

|469.0

|634.5

|772.4

|910.3

|1034.5

|1144.8

* 25 and 31 are close matches.  

* 21 is a little bit flat, but still decent.

* The others (17, 19, 23, 27 and 29) are extremely sharp, but the intervals between them are close.

* [http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] by [[Gene Ward Smith]]

[[Category:theory]]

[[Category:edo]]

[[Category:87edo]]

[[Category:listen]]

[[Category:clyde]]

[[Category:countercata]]

[[Category:hemithirds]]

[[Category:mystery]]

[[Category:tritikleismic]]