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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
'''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).
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
 
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2008-02-06 09:11:52 UTC</tt>.<br>
== Theory ==
: The original revision id was <tt>16324387</tt>.<br>
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.
: The revision comment was: <tt>88 East added</tt><br>
 
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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.
<h4>Original Wikitext content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">==Compositions==  
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.
[[http://www.seraph.it/XenoTunes4_files/88east.mp3|88 East]] by Carlo Serafini </pre></div>
 
<h4>Original HTML content:</h4>
=== Harmonics ===
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;88cET&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Compositions&lt;/h2&gt;
{{Harmonics in cet|88}}
&lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes4_files/88east.mp3" rel="nofollow"&gt;88 East&lt;/a&gt; by Carlo Serafini&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== 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"
|-
! Degree
! 11ed7/4
! 88cET
! 41ed8
! 8edf
! Solfege <br>syllable
! Some Nearby <br>JI Intervals
|-
! colspan="6" | first octave
!
|-
| 0
| 0
| 0
| 0
| 0
| do
| 1/1=0
|-
| 1
| 88.075
| 88
| 87.805
| 87.744
| rih
| 22/21=80.537, 21/20=84.467, 20/19=88.801, 19/18=93.603
|-
| 2
| 176.15
| 176
| 175.610
| 175.489
| reh
| [[11/10]]=165.004, 21/19=173.268, [[10/9]]=182.404
|-
| 3
| 264.225
| 264
| 263.415
| 263.233
| ma
| [[7/6]]=266.871
|-
| 4
| 352.3
| 352
| 351.220
| 350.978
| mu
| [[11/9]]=347.408, 27/22=354.547, 16/13=359.472
|-
| 5
| 440.375
| 440
| 439.024
| 438.722
| mo
| 32/25=427.373, [[9/7]]=435.084, [[22/17]]=446.363
|-
| 6
| 528.45
| 528
| 526.829
| 526.466
| fih
| [[19/14]]=528.687, 49/36=533.742, [[15/11]]=536.95
|-
| 7
| 616.526
| 616
| 614.634
| 614.211
| se
| [[10/7]]=617.488
|-
| 8
| 704.601
| 704
| 702.439
| 701.955
| sol
| [[3/2]]=701.955
|-
| 9
| 792.676
| 792
| 790.244
| 789.699
| 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
|-
| 10
| 880.751
| 880
| 878.049
| 877.444
| la
| [[5/3]]=884.359
|-
| 11
| 968.826
| 968
| 965.854
| 965.188
| ta
| [[7/4]]=968.826
|-
| 12
| 1056.901
| 1056
| 1053.659
| 1052.933
| tu
| [[11/6]]=1049.363, 35/19=1057.627, 24/13=1061.427
|-
| 13
| 1144.976
| 1144
| 1141.463
| 1140.677
| to
| 27/14=1137.039, 31/16=1145.036
|-
! colspan="6" | second octave
!
|-
| 14
| 33.051
| 32
| 29.268
| 28.421
| di
| 65/64=26.841, 64/63=27.264, 63/62=27.700, 58/57=30.109
|-
| 15
| 121.126
| 120
| 117.073
| 116.166
| ra
| 16/15=111.731, 15/14=119.443, 14/13=128.298
|-
| 16
| 209.201
| 208
| 204.878
| 203.910
| re
| 9/8=203.910
|-
| 17
| 297.276
| 296
| 292.683
| 291.654
| meh
| 13/11=289.210, 32/27=294.135, 19/16=297.513
|-
| 18
| 385.351
| 384
| 380.488
| 379.399
| mi
| 5/4=386.314
|-
| 19
| 473.427
| 472
| 468.293
| 467.143
| fe
| 17/13=464.428, 21/16=470.781
|-
| 20
| 561.502
| 560
| 556.098
| 554.888
| fu
| 11/8=551.318, 18/13=563.382
|-
| 21
| 649.577
| 648
| 643.902
| 642.632
| su
| 16/11=648.682
|-
| 22
| 737.652
| 736
| 731.707
| 730.376
| si
| 32/21=729.219, 26/17=735.572, 49/32=737.652
|-
| 23
| 825.727
| 824
| 819.512
| 818.121
| le
| 8/5=813.686, 45/28=821.398, 21/13=830.253
|-
| 24
| 913.802
| 912
| 907.317
| 905.865
| laa
| 42/25=898.153, 32/19=902.487, 27/16=905.865, 22/13=910.790, 17/10=918.642
|-
| 25
| 1001.877
| 1000
| 995.122
| 993.609
| teh
| 39/22=991.165, 16/9=996.090, 25/14=1003.802, 34/19=1007.442
|-
| 26
| 1089.952
| 1088
| 1082.927
| 1081.354
| ti
| 28/15=1080.557, 15/8=1088.269
|-
| 27
| 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
|-
| 29
| 154.177
| 152
| 146.341
| 144.587
| ru
| 49/45=147.428, 12/11=150.637, 35/32=155.140
|-
| 30
| 242.252
| 240
| 234.146
| 232.331
| ri
| 8/7=231.174, 23/20=241.961, 15/13=247.741
|-
| 31
| 330.328
| 328
| 321.951
| 320.076
| me
| 6/5=315.641, 23/19=330.761
|-
| 32
| 418.403
| 416
| 409.756
| 407.820
| maa
| 81/64=407.820, 33/26=412.745, 14/11=417.508
|-
| 33
| 506.478
| 504
| 497.561
| 495.564
| fa
| 85/64=491.269, 4/3=498.045, 75/56=505.757
|-
| 34
| 594.553
| 592
| 585.366
| 583.309
| fi
| 7/5=582.512, 45/32=590.224, 38/27=591.648
|-
| 35
| 682.628
| 680
| 673.171
| 671.053
| sih
| 28/19=671.313, 40/27=680.449
|-
| 36
| 770.703
| 768
| 760.976
| 758.798
| lo
| 17/11=753.637, 99/64=755.228, 14/9=764.916, 39/25=769.855, 25/16=772.627
|-
| 37
| 858.778
| 856
| 848.780
| 846.542
| lu
| 13/8=840.528, 18/11=852.592
|-
| 38
| 946.853
| 944
| 936.585
| 934.286
| li
| 12/7=933.129, 19/11=946.195
|-
| 39
| 1034.928
| 1032
| 1024.390
| 1022.031
| te
| 9/5=1017.596, 49/27=1031.787, 20/11=1034.996
|-
| 40
| 1123.003
| 1120
| 1112.195
| 1109.775
| taa
| 36/19=1106.397, 243/128=1109.775, 19/10=1111.199, 21/11=1119.463
|-
! colspan="6" | fourth octave (near match)
!
|-
| 41
| 11.078
| 8
| 0
| 1197.59
| do
| 1/1=0, 2/1=1200
|}
 
== Scales ==
* [[Symmetrical scales of 88cET]]
 
== 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])
 
; [[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])
 
; [[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]
 
== Further reading ==
* [[Gary Morrison]]’s 2001 [https://soundcloud.com/mr88cet/sets/88cet-lecture-demo-gary-morrison-june-2001 lecture about 88cET]
 
[[Category:Equal-step tuning]]

Latest revision as of 03:18, 28 September 2025

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

Theory

88-cent 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 chords.

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.

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.

Harmonics

Approximation of harmonics in 1ed88c
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +32.0 +34.0 -24.0 +29.7 -22.0 -24.8 +8.0 -19.9 -26.3 -15.3 +10.0
Relative (%) +36.4 +38.7 -27.3 +33.7 -24.9 -28.2 +9.1 -22.6 -29.9 -17.4 +11.4
Step 14 22 27 32 35 38 41 43 45 47 49

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

Scales

Music

Carlo Serafini
Chris Vaisvil
Mundoworld

Further reading

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