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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.
== Theory ==
 
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 harmonies 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 [[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.
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== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3 left-4"
! #
! rowspan="2" | #
! Cents
! rowspan="2" | Cents
! Approximated Ratios
! colspan="2" | Approximated Ratios
! [[Ups and Downs Notation]]
! colspan="2" rowspan="2" |[[Ups and Downs Notation]]
|-
!13-Limit
!31-Limit No-7s Extension
|-
|-
|0
|0
|0.000
|0.000
|1/1
|[[1/1]]
|
|P1
|D
|D
|-
|-
|1
|1
|13.793
|13.793
|126/125, 100/99, 91/90
|[[126/125]], [[100/99]], [[91/90]]
|
|^1
|^D
|^D
|-
|-
|2
|2
|27.586
|27.586
|81/80, 64/63, 49/48, 55/54, 65/64
|[[81/80]], [[64/63]], [[49/48]], [[55/54]], [[65/64]]
|
|^^1
|^^D
|^^D
|-
|-
|3
|3
|41.379
|41.379
|50/49, 45/44
|[[50/49]], [[45/44]], [[40/39]]
|[[39/38]]
|^<sup>3</sup>1
|^<sup>3</sup>D/v<sup>3</sup>Eb
|^<sup>3</sup>D/v<sup>3</sup>Eb
|-
|-
|4
|4
|55.172
|55.172
|28/27, 36/35, 33/32
|[[28/27]], [[36/35]], [[33/32]]
|[[34/33]], [[30/29]], [[32/31]], [[31/30]]
|vvm2
|vvEb
|vvEb
|-
|-
|5
|5
|68.966
|68.966
|25/24, 27/26, 26/25
|[[25/24]], [[27/26]], [[26/25]]
|[[24/23]]
|vm2
|vEb
|vEb
|-
|-
|6
|6
|82.759
|82.759
|21/20, 22/21
|[[21/20]], [[22/21]]
|[[20/19]], [[23/22]]
|m2
|Eb
|Eb
|-
|-
|7
|7
|96.552
|96.552
|35/33
|[[35/33]]
|[[18/17]], [[19/18]]
|^m2
|^Eb
|^Eb
|-
|-
|8
|8
|110.345
|110.345
|16/15
|[[16/15]]
|[[17/16]], [[33/31]], [[31/29]]
|^^m2
|^^Eb
|^^Eb
|-
|-
|9
|9
|124.138
|124.138
|15/14, 14/13
|[[15/14]], [[14/13]]
|[[29/27]]
|vv~2
|^<sup>3</sup>Eb
|^<sup>3</sup>Eb
|-
|-
|10
|10
|137.931
|137.931
|13/12
|[[13/12]], [[27/25]]
|[[25/23]]
|v~2
|^<sup>4</sup>Eb
|^<sup>4</sup>Eb
|-
|-
|11
|11
|151.724
|151.724
|12/11
|[[12/11]], [[35/32]]
|
|^~2
|v<sup>4</sup>E
|v<sup>4</sup>E
|-
|-
|12
|12
|165.517
|165.517
|11/10
|[[11/10]]
|[[32/29]], [[34/31]]
|^^~2
|v<sup>3</sup>E
|v<sup>3</sup>E
|-
|-
|13
|13
|179.310
|179.310
|10/9
|[[10/9]]
|
|vvM2
|vvE
|vvE
|-
|-
|14
|14
|193.103
|193.103
|28/25
|[[28/25]]
|[[19/17]], [[29/26]]
|vM2
|vE
|vE
|-
|-
|15
|15
|206.897
|206.897
|9/8
|[[9/8]]
|[[26/23]]
|M2
|E
|E
|-
|-
|16
|16
|220.690
|220.690
|25/22
|[[25/22]]
|[[17/15]], [[33/29]]
|^M2
|^E
|^E
|-
|-
|17
|17
|234.483
|234.483
|8/7
|[[8/7]]
|[[31/27]]
|^^M2
|^^E
|^^E
|-
|-
|18
|18
|248.276
|248.276
|15/13
|[[15/13]]
|[[22/19]], [[38/33]], [[23/20]]
|^<sup>3</sup>M2/v<sup>3</sup>m3
|^<sup>3</sup>E/v<sup>3</sup>F
|^<sup>3</sup>E/v<sup>3</sup>F
|-
|-
|19
|19
|262.089
|262.089
|7/6
|[[7/6]]
|[[29/25]], [[36/31]]
|vvm3
|vvF
|vvF
|-
|-
|20
|20
|275.862
|275.862
|75/64
|[[75/64]]
|[[27/23]], [[34/29]]
|vm3
|vF
|vF
|-
|-
|21
|21
|289.655
|289.655
|33/28, 13/11
|[[32/27]], [[33/28]], [[13/11]]
|
|m3
|F
|F
|-
|-
|22
|22
|303.448
|303.448
|25/21
|[[25/21]]
|[[19/16]], [[31/26]]
|^m3
|^F
|^F
|-
|-
|23
|23
|317.241
|317.241
|6/5
|[[6/5]]
|
|^^m3
|^^F
|^^F
|-
|-
|24
|24
|331.034
|331.034
|63/52
|[[40/33]]
|[[23/19]], [[29/24]]
|vv~3
|^<sup>3</sup>F
|^<sup>3</sup>F
|-
|-
|25
|25
|344.828
|344.828
|11/9, 39/32
|[[11/9]], [[39/32]]
|
|v~3
|^<sup>4</sup>F
|^<sup>4</sup>F
|-
|-
|26
|26
|358.621
|358.621
|27/22, 16/13
|[[27/22]], [[16/13]]
|[[38/31]]
|^~3
|v<sup>4</sup>F#
|v<sup>4</sup>F#
|-
|-
|27
|27
|372.414
|372.414
|26/21
|[[26/21]]
|[[31/25]], [[36/29]]
|^^3
|v<sup>3</sup>F#
|v<sup>3</sup>F#
|-
|-
|28
|28
|386.207
|386.207
|5/4
|[[5/4]]
|
|vvM3
|vvF#
|vvF#
|-
|-
|29
|29
|400.000
|400.000
|63/50, 44/35
|[[44/35]]
|[[34/27]], [[24/19]], [[29/23]]
|vM3
|vF#
|vF#
|-
|-
|30
|30
|413.793
|413.793
|14/11, 33/26
|[[81/64]], [[14/11]], [[33/26]]
|[[19/15]]
|M3
|F#
|F#
|-
|-
|31
|31
|427.586
|427.586
|32/25
|[[32/25]]
|[[23/18]]
|^M3
|^F#
|^F#
|-
|-
|32
|32
|441.379
|441.379
|9/7
|[[9/7]], [[35/27]]
|[[22/17]], [[31/24]], [[40/31]]
|^^M3
|^^F#
|^^F#
|-
|-
|33
|33
|455.172
|455.172
|13/10
|[[13/10]]
|[[30/23]]
|^<sup>3</sup>M3/v<sup>3</sup>4
|^<sup>3</sup>F#/v<sup>3</sup>G
|^<sup>3</sup>F#/v<sup>3</sup>G
|-
|-
|34
|34
|468.966
|468.966
|21/16
|[[21/16]]
|[[17/13]], [[25/19]], [[38/29]]
|vv4
|vvG
|vvG
|-
|-
|35
|35
|482.759
|482.759
|33/25
|[[33/25]]
|
|v4
|vG
|vG
|-
|-
|36
|36
|496.552
|496.552
|4/3
|[[4/3]]
|
|P4
|G
|G
|-
|-
|37
|37
|510.345
|510.345
|75/56
|[[35/26]]
|[[31/23]]
|^4
|^G
|^G
|-
|-
|38
|38
|524.138
|524.138
|27/20
|[[27/20]]
|[[23/17]]
|^^4
|^^G
|^^G
|-
|-
|39
|39
|537.931
|537.931
|15/11
|[[15/11]]
|[[26/19]], [[34/25]]
|^<sup>3</sup>4
|^<sup>3</sup>G
|^<sup>3</sup>G
|-
|-
|40
|40
|551.724
|551.724
|11/8
|[[11/8]], [[48/35]]
|
|^<sup>4</sup>4
|^<sup>4</sup>G
|^<sup>4</sup>G
|-
|-
|41
|41
|565.517
|565.517
|18/13
|[[18/13]]
|[[32/23]]
|v<sup>4</sup>A4, vd5
|v<sup>4</sup>G#, vAb
|v<sup>4</sup>G#, vAb
|-
|-
|42
|42
|579.310
|579.310
|7/5, 39/28
|[[7/5]]
|[[46/33]]
|v<sup>3</sup>A4, d5
|v<sup>3</sup>G#, Ab
|v<sup>3</sup>G#, Ab
|-
|-
|43
|43
|593.103
|593.103
|45/32
|[[45/32]]
|[[24/17]], [[38/27]], [[31/22]]
|vvA4, ^d5
|vvG#, ^Ab
|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
!prime 17
!prime 19
!prime 23
!prime 29
!prime 31
|-
! rowspan="2" |Error
!absolute (¢)
|0.00
| +1.49
| -0.11
| -3.31
| +0.41
| +0.85
| +5.39
| +5.94
| +6.21
| +4.91
| -0.21
|-
!relative (%)
|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 ==
''See also: [[Detempering]]''
''Main article: [[87edo/13-limit detempering]]''


== Rank two temperaments ==
== Rank two temperaments ==


{| class="wikitable" style="text-align: right"
{| class="wikitable center-all right-3 left-5"
|-
|-
! Periods <br> per <br> octave
! Periods <br> per <br> octave
Line 244: Line 388:
|-
|-
| 1
| 1
| style="text-align: center" | 4\87
| 4\87
| 55.172
| 55.172
| style="text-align: center" | [[33/32]]
| [[33/32]]
| style="text-align: left;" | [[Sensa]]
| [[Sensa]]
|-
|-
| 1
| 1
| style="text-align: center" | 10\87
| 10\87
| 137.931
| 137.931
| style="text-align: center" | [[13/12]]
| [[13/12]]
| style="text-align: left" | [[Quartemka]]
| [[Quartemka]]
|-
|-
| 1
| 1
| style="text-align: center" | 14\87
| 14\87
| 193.103
| 193.103
| style="text-align: center" | [[28/25]]
| [[28/25]]
| style="text-align: left" | [[Luna]] / [[Hemithirds]]
| [[Luna]] / [[Hemithirds]]
|-
|-
| 1
| 1
| style="text-align: center" | 17\87
| 17\87
| 234.483
| 234.483
| style="text-align: center" | [[8/7]]
| [[8/7]]
| style="text-align: left" | [[Rodan]]
| [[Rodan]]
|-
|-
| 1
| 1
| style="text-align: center" | 23\87
| 23\87
| 317.241
| 317.241
| style="text-align: center" | [[6/5]]
| [[6/5]]
| style="text-align: left" | [[Hanson]] / [[Countercata]] / [[Metakleismic]]
| [[Hanson]] / [[Countercata]] / [[Metakleismic]]
|-
|-
| 1
| 1
| style="text-align: center" | 32\87
| 32\87
| 441.379
| 441.379
| style="text-align: center" | [[9/7]]
| [[9/7]]
| style="text-align: left" | [[Clyde]]
| [[Clyde]]
|-
|-
| 1
| 1
| style="text-align: center" | 38\87
| 38\87
| 524.138
| 524.138
| style="text-align: center" | [[65/48]]
| [[65/48]]
| style="text-align: left" | [[Widefourth]]
| [[Widefourth]]
|-
|-
| 1
| 1
| style="text-align: center" | 40\87
| 40\87
| 551.724
| 551.724
| style="text-align: center" | [[11/8]]
| [[11/8]]
| style="text-align: left" | [[Emkay]]
| [[Emkay]]
|-
|-
| 3
| 3
| style="text-align: center" | 23\87
| 23\87
| 317.241
| 317.241
| style="text-align: center" | [[6/5]]
| [[6/5]]
| style="text-align: left" | [[Tritikleismic]]
| [[Tritikleismic]]
|-
|-
| 29
| 29
| style="text-align: center" | 28\87
| 28\87
| 386.207
| 386.207
| style="text-align: center" | [[5/4]]
| [[5/4]]
| style="text-align: left" | [[Mystery]]
| [[Mystery]]
|}
|}


Line 309: Line 453:
* [[M&N temperaments|494&amp;87]] &lt;&lt;51 -1 -133 11 32 ... ||
* [[M&N temperaments|494&amp;87]] &lt;&lt;51 -1 -133 11 32 ... ||


== 13-limit detempering of 87et ==
== Scales ==
 
=== 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 ====
{| class="wikitable center-all"
|Overtones
|8
|9
|10
|11
|12
|13
|14
|15
|16
|-
|JI Ratios
|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
|1200.0
|-
|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
|}
* The scale in adjacent steps is 15, 13, 12, 11, 10, 9, 9, 8.
 
==== Mode 16 ====
{| class="wikitable center-all"
|Odd overtones
|17
|19
|21
|23
|25
|27
|29
|31
|-
|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
|}
* The scale in adjacent steps is 8, 7, 7, 6, 6, 6, 6, 5, 5, 5, 5, 4, 5, 4, 4, 4.
 
* 25 and 31 are close matches.


''See also: [[Detempering]]''
* 21 is a little bit flat, but still decent.


''Main article: [[87edo/13-limit detempering]]''
* The others (17, 19, 23, 27 and 29) are extremely sharp, but the intervals between them are close.


== Music ==
== Music ==
Retrieved from "https://en.xen.wiki/w/87edo"