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'''88-cent equal tuning''' uses equal steps of 88 cents each. It is equivalent to 13.6364edo, and is a subset of [[150edo]] (every eleventh step).
'''88-cent equal temperament''' ('''88cET''', also known as '''1ed88¢''' or '''APS88¢''') uses equal steps of 88 [[cent]]s each. It is equivalent to 13.6364edo, and is a subset of [[150edo]] (every eleventh step).


==Theory==
== Theory ==
88-cent [[Equal-step tuning|equal tuning]] uses 88 cents, or 11\150 of an octave, to generate a [[nonoctave]] rank one scale. Since 88 cents is an excellent generator for [[Tetracot_family|octacot temperament]], it can be viewed as the generator chain of octacot, stripped of octaves. However viewed, octacot and 88-cent equal tuning are very closely related, and the chords of 88-cent tuning are listed on the page [[Chords_of_octacot|chords of octacot]]. From this it may be seen that octacot, and hence 88 cents tuning, share an abundance of [[Dyadic_chord|essentially tempered chords]].
88-cent [[Equal-step tuning|equal temperament]] uses 88 cents, or 11\150 of an octave, to generate a [[nonoctave]] rank-1 scale. Since the 88-cent step is an excellent generator for the [[octacot]] temperament, it can be viewed as the generator chain of octacot, stripped of octaves. However viewed, octacot and 88-cent equal temperament are very closely related, and the chords of 88-cent equal temperament are listed on the page [[Chords of octacot]]. From this it may be seen that octacot, and hence 88 cent equal temperament , share an abundance of [[essentially tempered chord]]s.


Eight steps of 88 cents gives 704 cents, two cents sharp of 3/2, and eighteen gives 1584 cents, two cents flat of 5/2. Taken together this tells us that (5/2)^4/(3/2)^9 = 20000/19683, the minimal diesis or tetracot comma, must be being tempered out. Eleven steps of 88 cents gives 968 cents, less than a cent flat of 7/4, and this tells us that (7/4)^8/(3/2)^11 = 5764801/5668704 must be tempered out also. Taking this, multiplying it by the tetracot comma and taking the fourth root yields 245/243, which therefore must be tempered out also. The tetracot comma and 245/243 taken together define 7-limit octacot.
Eight steps of 88 cents gives 704 cents, two cents sharp of 3/2, and eighteen gives 1584 cents, two cents flat of 5/2. Taken together this tells us that (5/2)<sup>4</sup>/(3/2)<sup>9</sup> = [[20000/19683]], the minimal diesis or tetracot comma, must be being tempered out. Eleven steps of 88 cents gives 968 cents, less than a cent flat of 7/4, and this tells us that (7/4)<sup>8</sup>/(3/2)<sup>11</sup> = 5764801/5668704 must be tempered out also. Taking this, multiplying it by the tetracot comma and taking the fourth root yields [[245/243]], which therefore must be tempered out also. The tetracot comma and 245/243 taken together define 7-limit octacot.


Continuing on, twenty steps of 88 cents gives 1760 cents, which we may compare to the 1751.3 cents of 11/4 and suggests 100/99 being tempered out, and four steps gives 352 cents, which may be compared to the 359.5 cents of 16/13, and suggests 325/324 being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning.
Continuing on, twenty steps of 88 cents gives 1760 cents, which we may compare to the 1751.3 cents of 11/4 and suggests [[100/99]] being tempered out, and four steps gives 352 cents, which may be compared to the 359.5 cents of 16/13, and suggests [[325/324]] being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning.


==The 88cET family==
=== Harmonics ===
[[Gary_Morrison|Gary Morrison]] originally conceived of 88-cent equal tuning (88cET) as composed of steps of exactly 88¢. Nonetheless, composers have recognized a kinship between strict 88cET and some other scales -- in particular, the 41st root of 8 (equivalent to taking three steps of [[41edo]] as a generator with no octaves), the 8th root of 3/2, and the 11th root of 7/4, the latter being a preferred variant of composer and software designer [[X._J._Scott|X. J. Scott]]. These three cousins of strict 88cET have single steps of approximately 87.805¢, 87.744¢, and 88.075¢, respectively. These small differences add up, as can be seen by examining the interval list below.
{{Harmonics in cet|88}}


==Intervals==
== The 88cET family ==
[[Gary Morrison]] originally conceived of 88-cent equal temperament as composed of steps of exactly 88¢. Nonetheless, composers have recognized a kinship between strict 88cET and some other scales – in particular, the 41ed8 (equivalent to taking three steps of [[41edo]] as a generator with no octaves), the 68ed32 (taking every 5 steps of [[68edo]]), the 109ed256 (taking every 8 steps of [[109edo]]), the 150ed2048 (taking every 11 steps of [[150edo]] i.e. the strict 88cET), the [[8edf]], and the 11ed7/4, the latter being a preferred variant of composer and software designer [[X. J. Scott]]. These cousins of strict 88cET have single steps of approximately 87.805¢, 88.235¢, 88.073¢, 88¢, 87.744¢, and 88.075¢, respectively. These small differences add up, as can be seen by examining the interval list below.


== Intervals ==
{{todo|cleanup|inline=true}}
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Degree
! Degree
! | 11th root <br>of 7/4
! 11ed7/4
! | 88cET
! 88cET
! | 41st root <br>of 8
! 41ed8
! | [[8edf|8th root <br>of 3/2]]
! 8edf
! | Solfege <br>syllable
! Solfege <br>syllable
! | Some Nearby <br>JI Intervals
! Some Nearby <br>JI Intervals
|-
|-
! colspan="6" | '''''first octave'''''
! colspan="6" | first octave
! |
!  
|-
|-
| | 0
| 0
| | 0
| 0
| | 0
| 0
| | 0
| 0
| | 0
| 0
| | do
| do
| | 1/1=0
| 1/1=0
|-
|-
| | 1
| 1
| | 88.075
| 88.075
| | 88
| 88
| | 87.805
| 87.805
| | 87.744
| 87.744
| | rih
| rih
| | 22/21=80.537, 21/20=84.467, 20/19=88.801, 19/18=93.603
| 22/21=80.537, 21/20=84.467, 20/19=88.801, 19/18=93.603
|-
|-
| | 2
| 2
| | 176.15
| 176.15
| | 176
| 176
| | 175.610
| 175.610
| | 175.489
| 175.489
| | reh
| reh
| | [[11/10]]=165.004, 21/19=173.268, [[10/9]]=182.404
| [[11/10]]=165.004, 21/19=173.268, [[10/9]]=182.404
|-
|-
| | 3
| 3
| | 264.225
| 264.225
| | 264
| 264
| | 263.415
| 263.415
| | 263.233
| 263.233
| | ma
| ma
| | [[7/6]]=266.871
| [[7/6]]=266.871
|-
|-
| | 4
| 4
| | 352.3
| 352.3
| | 352
| 352
| | 351.220
| 351.220
| | 350.978
| 350.978
| | mu
| mu
| | [[11/9]]=347.408, 27/22=354.547, 16/13=359.472
| [[11/9]]=347.408, 27/22=354.547, 16/13=359.472
|-
|-
| | 5
| 5
| | 440.375
| 440.375
| | 440
| 440
| | 439.024
| 439.024
| | 438.722
| 438.722
| | mo
| mo
| | 32/25=427.373, [[9/7]]=435.084, [[22/17]]=446.363
| 32/25=427.373, [[9/7]]=435.084, [[22/17]]=446.363
|-
|-
| | 6
| 6
| | 528.45
| 528.45
| | 528
| 528
| | 526.829
| 526.829
| | 526.466
| 526.466
| | fih
| fih
| | [[19/14]]=528.687, 49/36=533.742, [[15/11]]=536.95
| [[19/14]]=528.687, 49/36=533.742, [[15/11]]=536.95
|-
|-
| | 7
| 7
| | 616.526
| 616.526
| | 616
| 616
| | 614.634
| 614.634
| | 614.211
| 614.211
| | se
| se
| | [[10/7]]=617.488
| [[10/7]]=617.488
|-
|-
| | 8
| 8
| | 704.601
| 704.601
| | 704
| 704
| | 702.439
| 702.439
| | 701.955
| 701.955
| | sol
| sol
| | [[3/2]]=701.955
| [[3/2]]=701.955
|-
|-
| | 9
| 9
| | 792.676
| 792.676
| | 792
| 792
| | 790.244
| 790.244
| | 789.699
| 789.699
| | leh
| leh
| | [[11/7]]=782.492, 30/19=790.756, 128/81=792.180, [[19/12]]=795.558, 27/17=800.910, [[8/5]]=813.686
| [[11/7]]=782.492, 30/19=790.756, 128/81=792.180, [[19/12]]=795.558, 27/17=800.910, [[8/5]]=813.686
|-
|-
| | 10
| 10
| | 880.751
| 880.751
| | 880
| 880
| | 878.049
| 878.049
| | 877.444
| 877.444
| | la
| la
| | [[5/3]]=884.359
| [[5/3]]=884.359
|-
|-
| | 11
| 11
| | 968.826
| 968.826
| | 968
| 968
| | 965.854
| 965.854
| | 965.188
| 965.188
| | ta
| ta
| | [[7/4]]=968.826
| [[7/4]]=968.826
|-
|-
| | 12
| 12
| | 1056.901
| 1056.901
| | 1056
| 1056
| | 1053.659
| 1053.659
| | 1052.933
| 1052.933
| | tu
| tu
| | [[11/6]]=1049.363, 35/19=1057.627, 24/13=1061.427
| [[11/6]]=1049.363, 35/19=1057.627, 24/13=1061.427
|-
|-
| | 13
| 13
| | 1144.976
| 1144.976
| | 1144
| 1144
| | 1141.463
| 1141.463
| | 1140.677
| 1140.677
| | to
| to
| | 27/14=1137.039, 31/16=1145.036
| 27/14=1137.039, 31/16=1145.036
|-
|-
! colspan="6" | '''''second octave'''''
! colspan="6" | second octave
! |
!  
|-
|-
| | 14
| 14
| | 33.051
| 33.051
| | 32
| 32
| | 29.268
| 29.268
| | 28.421
| 28.421
| | di
| di
| | 65/64=26.841, 64/63=27.264, 63/62=27.700, 58/57=30.109
| 65/64=26.841, 64/63=27.264, 63/62=27.700, 58/57=30.109
|-
|-
| | 15
| 15
| | 121.126
| 121.126
| | 120
| 120
| | 117.073
| 117.073
| | 116.166
| 116.166
| | ra
| ra
| | 16/15=111.731, 15/14=119.443, 14/13=128.298
| 16/15=111.731, 15/14=119.443, 14/13=128.298
|-
|-
| | 16
| 16
| | 209.201
| 209.201
| | 208
| 208
| | 204.878
| 204.878
| | 203.910
| 203.910
| | re
| re
| | 9/8=203.910
| 9/8=203.910
|-
|-
! colspan="6" |''second nonet''
| 17
!
| 297.276
| 296
| 292.683
| 291.654
| meh
| 13/11=289.210, 32/27=294.135, 19/16=297.513
|-
|-
| | 17
| 18
| | 297.276
| 385.351
| | 296
| 384
| | 292.683
| 380.488
| | 291.654
| 379.399
| | meh
| mi
| | 13/11=289.210, 32/27=294.135, 19/16=297.513
| 5/4=386.314
|-
|-
| | 18
| 19
| | 385.351
| 473.427
| | 384
| 472
| | 380.488
| 468.293
| | 379.399
| 467.143
| | mi
| fe
| | 5/4=386.314
| 17/13=464.428, 21/16=470.781
|-
|-
| | 19
| 20
| | 473.427
| 561.502
| | 472
| 560
| | 468.293
| 556.098
| | 467.143
| 554.888
| | fe
| fu
| | 17/13=464.428, 21/16=470.781
| 11/8=551.318, 18/13=563.382
|-
|-
| | 20
| 21
| | 561.502
| 649.577
| | 560
| 648
| | 556.098
| 643.902
| | 554.888
| 642.632
| | fu
| su
| | 11/8=551.318, 18/13=563.382
| 16/11=648.682
|-
|-
| | 21
| 22
| | 649.577
| 737.652
| | 648
| 736
| | 643.902
| 731.707
| | 642.632
| 730.376
| | su
| si
| | 16/11=648.682
| 32/21=729.219, 26/17=735.572, 49/32=737.652
|-
|-
| | 22
| 23
| | 737.652
| 825.727
| | 736
| 824
| | 731.707
| 819.512
| | 730.376
| 818.121
| | si
| le
| | 32/21=729.219, 26/17=735.572, 49/32=737.652
| 8/5=813.686, 45/28=821.398, 21/13=830.253
|-
|-
| | 23
| 24
| | 825.727
| 913.802
| | 824
| 912
| | 819.512
| 907.317
| | 818.121
| 905.865
| | le
| laa
| | 8/5=813.686, 45/28=821.398, 21/13=830.253
| 42/25=898.153, 32/19=902.487, 27/16=905.865, 22/13=910.790, 17/10=918.642
|-
|-
| | 24
| 25
| | 913.802
| 1001.877
| | 912
| 1000
| | 907.317
| 995.122
| | 905.865
| 993.609
| | laa
| teh
| | 42/25=898.153, 32/19=902.487, 27/16=905.865, 22/13=910.790, 17/10=918.642
| 39/22=991.165, 16/9=996.090, 25/14=1003.802, 34/19=1007.442
|-
|-
| | 25
| 26
| | 1001.877
| 1089.952
| | 1000
| 1088
| | 995.122
| 1082.927
| | 993.609
| 1081.354
| | teh
| ti
| | 39/22=991.165, 16/9=996.090, 25/14=1003.802, 34/19=1007.442
| 28/15=1080.557, 15/8=1088.269
|-
|-
| | 26
| 27
| | 1089.952
| 1178.027
| | 1088
| 1176
| | 1082.927
| 1170.732
| | 1081.354
| 1169.098
| | ti
| da
| | 28/15=1080.557, 15/8=1088.269
| 63/32=1172.736, 160/81=1178.494
|-
|-
| | 27
! colspan="6" | third octave
| | 1178.027
!
| | 1176
| | 1170.732
| | 1169.098
| | da
| | 63/32=1172.736, 160/81=1178.494
|-
|-
! colspan="6" | '''''third octave'''''
| 28
! |  
| 66.102
| 64
| 58.537
| 56.843
| ro
| 33/3253.273, 28/27=62.961, 80/77=66.170, 27/26=65.337
|-
|-
| | 28
| 29
| | 66.102
| 154.177
| | 64
| 152
| | 58.537
| 146.341
| | 56.843
| 144.587
| | ro
| ru
| | 33/3253.273, 28/27=62.961, 80/77=66.170, 27/26=65.337
| 49/45=147.428, 12/11=150.637, 35/32=155.140
|-
|-
| | 29
| 30
| | 154.177
| 242.252
| | 152
| 240
| | 146.341
| 234.146
| | 144.587
| 232.331
| | ru
| ri
| | 49/45=147.428, 12/11=150.637, 35/32=155.140
| 8/7=231.174, 23/20=241.961, 15/13=247.741
|-
|-
| | 30
| 31
| | 242.252
| 330.328
| | 240
| 328
| | 234.146
| 321.951
| | 232.331
| 320.076
| | ri
| me
| | 8/7=231.174, 23/20=241.961, 15/13=247.741
| 6/5=315.641, 23/19=330.761
|-
|-
| | 31
| 32
| | 330.328
| 418.403
| | 328
| 416
| | 321.951
| 409.756
| | 320.076
| 407.820
| | me
| maa
| | 6/5=315.641, 23/19=330.761
| 81/64=407.820, 33/26=412.745, 14/11=417.508
|-
|-
| | 32
| 33
| | 418.403
| 506.478
| | 416
| 504
| | 409.756
| 497.561
| | 407.820
| 495.564
| | maa
| fa
| | 81/64=407.820, 33/26=412.745, 14/11=417.508
| 85/64=491.269, 4/3=498.045, 75/56=505.757
|-
|-
! colspan="6" |''third nonet''
| 34
!
| 594.553
| 592
| 585.366
| 583.309
| fi
| 7/5=582.512, 45/32=590.224, 38/27=591.648
|-
|-
| | 33
| 35
| | 506.478
| 682.628
| | 504
| 680
| | 497.561
| 673.171
| | 495.564
| 671.053
| | fa
| sih
| | 85/64=491.269, 4/3=498.045, 75/56=505.757
| 28/19=671.313, 40/27=680.449
|-
|-
| | 34
| 36
| | 594.553
| 770.703
| | 592
| 768
| | 585.366
| 760.976
| | 583.309
| 758.798
| | fi
| lo
| | 7/5=582.512, 45/32=590.224, 38/27=591.648
| 17/11=753.637, 99/64=755.228, 14/9=764.916, 39/25=769.855, 25/16=772.627
|-
|-
| | 35
| 37
| | 682.628
| 858.778
| | 680
| 856
| | 673.171
| 848.780
| | 671.053
| 846.542
| | sih
| lu
| | 28/19=671.313, 40/27=680.449
| 13/8=840.528, 18/11=852.592
|-
|-
| | 36
| 38
| | 770.703
| 946.853
| | 768
| 944
| | 760.976
| 936.585
| | 758.798
| 934.286
| | lo
| li
| | 17/11=753.637, 99/64=755.228, 14/9=764.916, 39/25=769.855, 25/16=772.627
| 12/7=933.129, 19/11=946.195
|-
|-
| | 37
| 39
| | 858.778
| 1034.928
| | 856
| 1032
| | 848.780
| 1024.390
| | 846.542
| 1022.031
| | lu
| te
| | 13/8=840.528, 18/11=852.592
| 9/5=1017.596, 49/27=1031.787, 20/11=1034.996
|-
|-
| | 38
| 40
| | 946.853
| 1123.003
| | 944
| 1120
| | 936.585
| 1112.195
| | 934.286
| 1109.775
| | li
| taa
| | 12/7=933.129, 19/11=946.195
| 36/19=1106.397, 243/128=1109.775, 19/10=1111.199, 21/11=1119.463
|-
|-
| | 39
! colspan="6" | fourth octave (near match)
| | 1034.928
!
| | 1032
| | 1024.390
| | 1022.031
| | te
| | 9/5=1017.596, 49/27=1031.787, 20/11=1034.996
|-
|-
| | 40
| 41
| | 1123.003
| 11.078
| | 1120
| 8
| | 1112.195
| 0
| | 1109.775
| 1197.59
| | taa
| do
| | 36/19=1106.397, 243/128=1109.775, 19/10=1111.199, 21/11=1119.463
| 1/1=0, 2/1=1200
|-
! colspan="6" | '''''fourth octave''''' (near match)
! |
|-
| | 41
| | 11.078
| | 8
| | 0
| | 1197.59
| | do
| | 1/1=0, 2/1=1200
|}
|}


==Scales==
== Scales ==
<ul><li>[[symmetrical_scales_of_88cET|symmetrical scales of 88cET]]</li></ul>
* [[Symmetrical scales of 88cET]]
 
==Compositions==
<span style=""><span style="">[http://www.seraph.it/dep/det/88east.mp3 88 East]</span> </span> by [[Carlo_Serafini|Carlo Serafini]]


<span style=""><span style="">[http://www.seraph.it/dep/det/88vocoeast.mp3 88 VocoEast]</span> </span> by [[Carlo_Serafini|Carlo Serafini]]
== Music ==
; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/88east.mp3 88 East]
* [http://www.seraph.it/dep/det/88vocoeast.mp3 88 VocoEast]
* [http://www.seraph.it/dep/det/88Bulgarians.mp3 88 Bulgarians] ([http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html blog entry])
* [http://www.seraph.it/dep/int/88jinglebells.mp3 88 Jingle Bells] ([http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html blog entry])
* [http://www.seraph.it/dep/det/The88thDoor.mp3 The 88th Door] ([http://www.seraph.it/blog_files/927f59ac10125056bcf7871636f246a6-302.html blog entry])


<span style=""><span style="">[http://www.seraph.it/dep/det/88Bulgarians.mp3 88 Bulgarians]</span> </span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html blog entry])
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/88cent_nonoctave/STE-004_88_cent_guitar.mp3 88 cent guitar improvisation]
* [http://micro.soonlabel.com/88cent_nonoctave/Prelude_in_88_Cent_Tuning.mp3 A Simple Prelude for 88 Cent Piano] ([http://micro.soonlabel.com/88cent_nonoctave/A_Simple_Prelude_in_88_Cent_Tuning.pdf scordata])


<span style=""><span style="">[http://www.seraph.it/dep/int/88jinglebells.mp3 88 Jingle Bells]</span> </span> by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html blog entry])
; [[Mundoworld]]
* "To Become Water" from ''Mundoworld III'' (2021) – [https://open.spotify.com/track/39gEeGXprXGbAnbq0iyjMF Spotify] | [https://www.youtube.com/watch?v=RBv9c_qlFEk YouTube]
* "Mirage Passage" from ''Mirage Passage'' (2024) – [https://open.spotify.com/track/2hAyfHr9XPG96SZPvBNHPP Spotify] | [https://www.youtube.com/watch?v=dWgmmK80I9U YouTube]


<span style=""><span style="">[http://micro.soonlabel.com/88cent_nonoctave/STE-004_88_cent_guitar.mp3 88 cent guitar improvisation]</span></span> by [http://www.chrisvaisvil.com Chris Vaisvil]
== Further reading ==
* [[Gary Morrison]]’s 2001 [https://soundcloud.com/mr88cet/sets/88cet-lecture-demo-gary-morrison-june-2001 lecture about 88cET]


<span style=""><span style="">[http://micro.soonlabel.com/88cent_nonoctave/Prelude_in_88_Cent_Tuning.mp3 A Simple Prelude for 88 Cent Piano]</span></span> by [http://chrisvaisvil.com/?p=951 Chris Vaisvil] ([http://micro.soonlabel.com/88cent_nonoctave/A_Simple_Prelude_in_88_Cent_Tuning.pdf scordata])
[[Category:Equal-step tuning]]
[[Category:Equal-step tuning]]
[[Category:Edonoi]]
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