Wikispaces>Andrew_Heathwaite |
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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>
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| : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-09-28 16:23:27 UTC</tt>.<br>
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| : The original revision id was <tt>166174095</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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">=88cET=
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| ==Theory== | | == Theory == |
| | 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. |
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| 88 cent equal temperament 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 cents equal temperament are very closely related. | | 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. |
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| 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. |
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| Continuing on, twenty steps of 88 cents gives 1760 cents, whioh 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 suggestes 325/324 being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning.
| | === Harmonics === |
| | {{Harmonics in cet|88}} |
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| ==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. |
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| 88cET is considered a very consonant tuning, and you will find that many of its intervals fall very close to simple ratios in 7- and 11-limit just intonation. It is also extremely close to [[41edo]], which is itself extremely close to the 8th root of 3:2 (a perfect fifth divided into exactly 8 logarithmically equal steps). See chart: | | == 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 |
| | |} |
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| ||~ Degree ||~ 88cET ||~ 41edo ||~ 8th Root ||~ Some Nearby ||
| | == Scales == |
| ||~ ||~ ||~ 3-steps ||~ of 3:2 ||~ JI Intervals ||
| | * [[Symmetrical scales of 88cET]] |
| ||||||||||= **//first octave//** ||
| |
| || 0 || 0 || 0 || 0 || 1/1=0 ||
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| || 1 || 88 || 87.805 || 87.744 || 22/21=80.537, 21/20=84.467, 20/19=88.801, 19/18=93.603 ||
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| || 2 || 176 || 175.610 || 175.489 || [[11_10|11/10]]=165.004, 21/19=173.268, [[10_9|10/9]]=182.404 ||
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| || 3 || 264 || 263.415 || 263.233 || [[7_6|7/6]]=266.871 ||
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| || 4 || 352 || 351.220 || 350.978 || [[11_9|11/9]]= 347.408, 27/22=354.547, 16/13=359.472 ||
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| || 5 || 440 || 439.024 || 438.722 || 32/25=427.373, [[9_7|9/7]]=435.084, 22/17 446.363 ||
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| || 6 || 528 || 526.829 || 526.466 || 19/14=528.687, 49/36=533.742, [[15_11|15/11]]=536.95 ||
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| || 7 || 616 || 614.634 || 614.211 || [[10_7|10/7]]=617.488 ||
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| || 8 || 704 || 702.439 || 701.955 || [[3_2|3/2]]=701.955 ||
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| || 9 || 792 || 790.244 || 789.699 || [[11_7|11/7]]=782.492, 30/19=790.756, 128/81=792.180, 19/12=795.558, 27/17=800.910, [[8_5|8/5]]=813.686 ||
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| || 10 || 880 || 878.049 || 878.444 || [[5_3|5/3]]=884.359 ||
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| || 11 || 968 || 965.854 || 965.188 || [[7_4|7/4]]=968.826 ||
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| || 12 || 1056 || 1053.659 || 1052.933 || [[11_6|11/6]]=1049.363, 35/19=1057.627, 24/13=1061.427 ||
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| || 13 || 1144 || 1141.463 || 1140.677 || 27/14=1137.039, 31/16=1145.036 ||
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| ||||||||||= **//second octave//** ||
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| || 14 || 32 || 29.268 || 28.421 || 65/64=26.841, 64/63=27.264, 63/62=27.700, 58/57=30.109 ||
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| || 15 || 120 || 117.073 || 116.166 || 16/15=111.731, 15/14=119.443, 14/13=128.298 ||
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| || 16 || 208 || 204.878 || 203.910 || 9/8=203.910 ||
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| || 17 || 296 || 292.683 || 291.654 || 13/11=289.210, 32/27=294.135, 19/16=297.513 ||
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| || 18 || 384 || 380.488 || 379.399 || 5/4=386.314 ||
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| || 19 || 472 || 468.293 || 467.143 || 17/13=464.428, 21/16=470.781 ||
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| || 20 || 560 || 556.098 || 554.888 || 11/8=551.318, 18/13=563.382 ||
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| || 21 || 648 || 643.902 || 642.632 || 16/11=648.682 ||
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| || 22 || 736 || 731.707 || 730.376 || 32/21=729.219, 26/17=735.572, 49/32=737.652 ||
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| || 23 || 824 || 819.512 || 818.121 || 8/5=813.686, 45/28=821.398, 21/13=830.253 ||
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| || 24 || 912 || 907.317 || 905.865 || 42/25=898.153, 32/19=902.487, 27/16=905.865, 22/13=910.790, 17/10=918.642 ||
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| || 25 || 1000 || 995.122 || 993.609 || 39/22=991.165, 16/9=996.090, 25/14=1003.802, 34/19=1007.442 ||
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| || 26 || 1088 || 1082.927 || 1081.354 || 28/15=1080.557, 15/8=1088.269 ||
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| || 27 || 1176 || 1170.732 || 1169.098 || 63/32=1172.736, 160/81=1178.494 ||
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| ||||||||||= **//third octave//** ||
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| || 28 || 64 || 58.537 || 56.843 || 33/3253.273, 28/27=62.961, 80/77=66.170, 27/26=65.337 ||
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| || 29 || 152 || 146.341 || 144.587 || 49/45=147.428, 12/11=150.637, 35/32=155.140 ||
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| || 30 || 240 || 234.146 || 232.331 || 8/7=231.174, 23/20=241.961, 15/13=247.741 ||
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| || 31 || 328 || 321.951 || 320.076 || 6/5=315.641, 23/19=330.761 ||
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| || 32 || 416 || 409.756 || 407.820 || 81/64=407.820, 33/26=412.745, 14/11=417.508 ||
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| || 33 || 504 || 497.561 || 495.564 || 85/64=491.269, 4/3=498.045, 75/56=505.757 ||
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| || 34 || 592 || 585.366 || 583.309 || 7/5=582.512, 45/32=590.224, 38/27=591.648 ||
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| || 35 || 680 || 673.171 || 671.053 || 28/19=671.313, 40/27=680.449 ||
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| || 36 || 768 || 760.976 || 758.798 || 17/11=753.637, 99/64=755.228, 14/9=764.916, 39/25=769.855, 25/16=772.627 ||
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| || 37 || 856 || 848.780 || 846.542 || 13/8=840.528, 18/11=852.592 ||
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| || 38 || 944 || 936.585 || 934.286 || 12/7=933.129, 19/11=946.195 ||
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| || 39 || 1032 || 1024.390 || 1022.031 || 9/5=1017.596, 49/27=1031.787, 20/11=1034.996 ||
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| || 40 || 1120 || 1112.195 || 1109.775 || 36/19=1106.397, 243/128=1109.775, 19/10=1111.199, 21/11=1119.463 ||
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| ||||||||||= **//fourth octave//** (near match) ||
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| || 41 || 8 || 0 || 1197.59 || 1/1=0, 2/1=1200 ||
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| ==Compositions== | | == Music == |
| [[http://www.seraph.it/dep/det/88east.mp3|88 East]] by [[Carlo Serafini]] | | ; [[Carlo Serafini]] |
| [[http://www.seraph.it/dep/det/88vocoeast.mp3|88 VocoEast]] by [[Carlo Serafini]]
| | * [http://www.seraph.it/dep/det/88east.mp3 88 East] |
| [[http://www.seraph.it/dep/det/88Bulgarians.mp3|88 Bulgarians]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html|blog entry]])
| | * [http://www.seraph.it/dep/det/88vocoeast.mp3 88 VocoEast] |
| [[http://www.seraph.it/dep/int/88jinglebells.mp3|88 Jingle Bells]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html|blog entry]])</pre></div>
| | * [http://www.seraph.it/dep/det/88Bulgarians.mp3 88 Bulgarians] ([http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html blog entry]) |
| <h4>Original HTML content:</h4>
| | * [http://www.seraph.it/dep/int/88jinglebells.mp3 88 Jingle Bells] ([http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html blog entry]) |
| <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"><html><head><title>88cET</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x88cET"></a><!-- ws:end:WikiTextHeadingRule:0 -->88cET</h1>
| | * [http://www.seraph.it/dep/det/The88thDoor.mp3 The 88th Door] ([http://www.seraph.it/blog_files/927f59ac10125056bcf7871636f246a6-302.html blog entry]) |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x88cET-Theory"></a><!-- ws:end:WikiTextHeadingRule:2 -->Theory</h2>
| |
| <br />
| |
| 88 cent equal temperament uses 88 cents, or 11/150 of an octave, to generate a nonoctave rank one scale. Since 88 cents is an excellent generator for <a class="wiki_link" href="/Tetracot%20family">octacot temperament</a>, it can be viewed as the generator chain of octacot, stripped of octaves. However viewed, octacot and 88 cents equal temperament are very closely related.<br />
| |
| <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)^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. <br />
| |
| <br />
| |
| Continuing on, twenty steps of 88 cents gives 1760 cents, whioh 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 suggestes 325/324 being tempered out. These would give an extended octacot, for which 88 cents would be an excellent generator tuning.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x88cET-Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h2>
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| <br />
| |
| 88cET is considered a very consonant tuning, and you will find that many of its intervals fall very close to simple ratios in 7- and 11-limit just intonation. It is also extremely close to <a class="wiki_link" href="/41edo">41edo</a>, which is itself extremely close to the 8th root of 3:2 (a perfect fifth divided into exactly 8 logarithmically equal steps). See chart:<br />
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| <br />
| |
|
| |
|
| | ; [[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]) |
|
| |
|
| <table class="wiki_table">
| | ; [[Mundoworld]] |
| <tr>
| | * "To Become Water" from ''Mundoworld III'' (2021) – [https://open.spotify.com/track/39gEeGXprXGbAnbq0iyjMF Spotify] | [https://www.youtube.com/watch?v=RBv9c_qlFEk YouTube] |
| <th>Degree<br />
| | * "Mirage Passage" from ''Mirage Passage'' (2024) – [https://open.spotify.com/track/2hAyfHr9XPG96SZPvBNHPP Spotify] | [https://www.youtube.com/watch?v=dWgmmK80I9U YouTube] |
| </th>
| |
| <th>88cET<br />
| |
| </th>
| |
| <th>41edo<br />
| |
| </th>
| |
| <th>8th Root<br />
| |
| </th>
| |
| <th>Some Nearby<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th>3-steps<br />
| |
| </th>
| |
| <th>of 3:2<br />
| |
| </th>
| |
| <th>JI Intervals<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <td colspan="5" style="text-align: center;"><strong><em>first octave</em></strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td>1/1=0<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>88<br />
| |
| </td>
| |
| <td>87.805<br />
| |
| </td>
| |
| <td>87.744<br />
| |
| </td>
| |
| <td>22/21=80.537, 21/20=84.467, 20/19=88.801, 19/18=93.603<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>176<br />
| |
| </td>
| |
| <td>175.610<br />
| |
| </td>
| |
| <td>175.489<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/11_10">11/10</a>=165.004, 21/19=173.268, <a class="wiki_link" href="/10_9">10/9</a>=182.404<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>264<br />
| |
| </td>
| |
| <td>263.415<br />
| |
| </td>
| |
| <td>263.233<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/7_6">7/6</a>=266.871<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>352<br />
| |
| </td>
| |
| <td>351.220<br />
| |
| </td>
| |
| <td>350.978<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/11_9">11/9</a>= 347.408, 27/22=354.547, 16/13=359.472<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>440<br />
| |
| </td>
| |
| <td>439.024<br />
| |
| </td>
| |
| <td>438.722<br />
| |
| </td>
| |
| <td>32/25=427.373, <a class="wiki_link" href="/9_7">9/7</a>=435.084, 22/17 446.363<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>528<br />
| |
| </td>
| |
| <td>526.829<br />
| |
| </td>
| |
| <td>526.466<br />
| |
| </td>
| |
| <td>19/14=528.687, 49/36=533.742, <a class="wiki_link" href="/15_11">15/11</a>=536.95<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>616<br />
| |
| </td>
| |
| <td>614.634<br />
| |
| </td>
| |
| <td>614.211<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/10_7">10/7</a>=617.488<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>704<br />
| |
| </td>
| |
| <td>702.439<br />
| |
| </td>
| |
| <td>701.955<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/3_2">3/2</a>=701.955<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>792<br />
| |
| </td>
| |
| <td>790.244<br />
| |
| </td>
| |
| <td>789.699<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/11_7">11/7</a>=782.492, 30/19=790.756, 128/81=792.180, 19/12=795.558, 27/17=800.910, <a class="wiki_link" href="/8_5">8/5</a>=813.686<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>880<br />
| |
| </td>
| |
| <td>878.049<br />
| |
| </td>
| |
| <td>878.444<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/5_3">5/3</a>=884.359<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11<br />
| |
| </td>
| |
| <td>968<br />
| |
| </td>
| |
| <td>965.854<br />
| |
| </td>
| |
| <td>965.188<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/7_4">7/4</a>=968.826<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12<br />
| |
| </td>
| |
| <td>1056<br />
| |
| </td>
| |
| <td>1053.659<br />
| |
| </td>
| |
| <td>1052.933<br />
| |
| </td>
| |
| <td><a class="wiki_link" href="/11_6">11/6</a>=1049.363, 35/19=1057.627, 24/13=1061.427<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>1144<br />
| |
| </td>
| |
| <td>1141.463<br />
| |
| </td>
| |
| <td>1140.677<br />
| |
| </td>
| |
| <td>27/14=1137.039, 31/16=1145.036<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td colspan="5" style="text-align: center;"><strong><em>second octave</em></strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14<br />
| |
| </td>
| |
| <td>32<br />
| |
| </td>
| |
| <td>29.268<br />
| |
| </td>
| |
| <td>28.421<br />
| |
| </td>
| |
| <td>65/64=26.841, 64/63=27.264, 63/62=27.700, 58/57=30.109<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>120<br />
| |
| </td>
| |
| <td>117.073<br />
| |
| </td>
| |
| <td>116.166<br />
| |
| </td>
| |
| <td>16/15=111.731, 15/14=119.443, 14/13=128.298<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16<br />
| |
| </td>
| |
| <td>208<br />
| |
| </td>
| |
| <td>204.878<br />
| |
| </td>
| |
| <td>203.910<br />
| |
| </td>
| |
| <td>9/8=203.910<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17<br />
| |
| </td>
| |
| <td>296<br />
| |
| </td>
| |
| <td>292.683<br />
| |
| </td>
| |
| <td>291.654<br />
| |
| </td>
| |
| <td>13/11=289.210, 32/27=294.135, 19/16=297.513<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18<br />
| |
| </td>
| |
| <td>384<br />
| |
| </td>
| |
| <td>380.488<br />
| |
| </td>
| |
| <td>379.399<br />
| |
| </td>
| |
| <td>5/4=386.314<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19<br />
| |
| </td>
| |
| <td>472<br />
| |
| </td>
| |
| <td>468.293<br />
| |
| </td>
| |
| <td>467.143<br />
| |
| </td>
| |
| <td>17/13=464.428, 21/16=470.781<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20<br />
| |
| </td>
| |
| <td>560<br />
| |
| </td>
| |
| <td>556.098<br />
| |
| </td>
| |
| <td>554.888<br />
| |
| </td>
| |
| <td>11/8=551.318, 18/13=563.382<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21<br />
| |
| </td>
| |
| <td>648<br />
| |
| </td>
| |
| <td>643.902<br />
| |
| </td>
| |
| <td>642.632<br />
| |
| </td>
| |
| <td>16/11=648.682<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22<br />
| |
| </td>
| |
| <td>736<br />
| |
| </td>
| |
| <td>731.707<br />
| |
| </td>
| |
| <td>730.376<br />
| |
| </td>
| |
| <td>32/21=729.219, 26/17=735.572, 49/32=737.652<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>23<br />
| |
| </td>
| |
| <td>824<br />
| |
| </td>
| |
| <td>819.512<br />
| |
| </td>
| |
| <td>818.121<br />
| |
| </td>
| |
| <td>8/5=813.686, 45/28=821.398, 21/13=830.253<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>24<br />
| |
| </td>
| |
| <td>912<br />
| |
| </td>
| |
| <td>907.317<br />
| |
| </td>
| |
| <td>905.865<br />
| |
| </td>
| |
| <td>42/25=898.153, 32/19=902.487, 27/16=905.865, 22/13=910.790, 17/10=918.642<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>25<br />
| |
| </td>
| |
| <td>1000<br />
| |
| </td>
| |
| <td>995.122<br />
| |
| </td>
| |
| <td>993.609<br />
| |
| </td>
| |
| <td>39/22=991.165, 16/9=996.090, 25/14=1003.802, 34/19=1007.442<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>26<br />
| |
| </td>
| |
| <td>1088<br />
| |
| </td>
| |
| <td>1082.927<br />
| |
| </td>
| |
| <td>1081.354<br />
| |
| </td>
| |
| <td>28/15=1080.557, 15/8=1088.269<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>27<br />
| |
| </td>
| |
| <td>1176<br />
| |
| </td>
| |
| <td>1170.732<br />
| |
| </td>
| |
| <td>1169.098<br />
| |
| </td>
| |
| <td>63/32=1172.736, 160/81=1178.494<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td colspan="5" style="text-align: center;"><strong><em>third octave</em></strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>28<br />
| |
| </td>
| |
| <td>64<br />
| |
| </td>
| |
| <td>58.537<br />
| |
| </td>
| |
| <td>56.843<br />
| |
| </td>
| |
| <td>33/3253.273, 28/27=62.961, 80/77=66.170, 27/26=65.337<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>29<br />
| |
| </td>
| |
| <td>152<br />
| |
| </td>
| |
| <td>146.341<br />
| |
| </td>
| |
| <td>144.587<br />
| |
| </td>
| |
| <td>49/45=147.428, 12/11=150.637, 35/32=155.140<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>30<br />
| |
| </td>
| |
| <td>240<br />
| |
| </td>
| |
| <td>234.146<br />
| |
| </td>
| |
| <td>232.331<br />
| |
| </td>
| |
| <td>8/7=231.174, 23/20=241.961, 15/13=247.741<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>31<br />
| |
| </td>
| |
| <td>328<br />
| |
| </td>
| |
| <td>321.951<br />
| |
| </td>
| |
| <td>320.076<br />
| |
| </td>
| |
| <td>6/5=315.641, 23/19=330.761<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>32<br />
| |
| </td>
| |
| <td>416<br />
| |
| </td>
| |
| <td>409.756<br />
| |
| </td>
| |
| <td>407.820<br />
| |
| </td>
| |
| <td>81/64=407.820, 33/26=412.745, 14/11=417.508<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>33<br />
| |
| </td>
| |
| <td>504<br />
| |
| </td>
| |
| <td>497.561<br />
| |
| </td>
| |
| <td>495.564<br />
| |
| </td>
| |
| <td>85/64=491.269, 4/3=498.045, 75/56=505.757<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>34<br />
| |
| </td>
| |
| <td>592<br />
| |
| </td>
| |
| <td>585.366<br />
| |
| </td>
| |
| <td>583.309<br />
| |
| </td>
| |
| <td>7/5=582.512, 45/32=590.224, 38/27=591.648<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>35<br />
| |
| </td>
| |
| <td>680<br />
| |
| </td>
| |
| <td>673.171<br />
| |
| </td>
| |
| <td>671.053<br />
| |
| </td>
| |
| <td>28/19=671.313, 40/27=680.449<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>36<br />
| |
| </td>
| |
| <td>768<br />
| |
| </td>
| |
| <td>760.976<br />
| |
| </td>
| |
| <td>758.798<br />
| |
| </td>
| |
| <td>17/11=753.637, 99/64=755.228, 14/9=764.916, 39/25=769.855, 25/16=772.627<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>37<br />
| |
| </td>
| |
| <td>856<br />
| |
| </td>
| |
| <td>848.780<br />
| |
| </td>
| |
| <td>846.542<br />
| |
| </td>
| |
| <td>13/8=840.528, 18/11=852.592<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>38<br />
| |
| </td>
| |
| <td>944<br />
| |
| </td>
| |
| <td>936.585<br />
| |
| </td>
| |
| <td>934.286<br />
| |
| </td>
| |
| <td>12/7=933.129, 19/11=946.195<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>39<br />
| |
| </td>
| |
| <td>1032<br />
| |
| </td>
| |
| <td>1024.390<br />
| |
| </td>
| |
| <td>1022.031<br />
| |
| </td>
| |
| <td>9/5=1017.596, 49/27=1031.787, 20/11=1034.996<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>40<br />
| |
| </td>
| |
| <td>1120<br />
| |
| </td>
| |
| <td>1112.195<br />
| |
| </td>
| |
| <td>1109.775<br />
| |
| </td>
| |
| <td>36/19=1106.397, 243/128=1109.775, 19/10=1111.199, 21/11=1119.463<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td colspan="5" style="text-align: center;"><strong><em>fourth octave</em></strong> (near match)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>41<br />
| |
| </td>
| |
| <td>8<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td>1197.59<br />
| |
| </td>
| |
| <td>1/1=0, 2/1=1200<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | == Further reading == |
| <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x88cET-Compositions"></a><!-- ws:end:WikiTextHeadingRule:6 -->Compositions</h2>
| | * [[Gary Morrison]]’s 2001 [https://soundcloud.com/mr88cet/sets/88cet-lecture-demo-gary-morrison-june-2001 lecture about 88cET] |
| <a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88east.mp3" rel="nofollow">88 East</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a><br />
| | |
| <a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88vocoeast.mp3" rel="nofollow">88 VocoEast</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a><br />
| | [[Category:Equal-step tuning]] |
| <a class="wiki_link_ext" href="http://www.seraph.it/dep/det/88Bulgarians.mp3" rel="nofollow">88 Bulgarians</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/9660ca3450a996ea8b55713cbf36151f-15.html" rel="nofollow">blog entry</a>)<br />
| |
| <a class="wiki_link_ext" href="http://www.seraph.it/dep/int/88jinglebells.mp3" rel="nofollow">88 Jingle Bells</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/495ec175ce56cf38cb399d1cd24db164-17.html" rel="nofollow">blog entry</a>)</body></html></pre></div>
| |