Wikispaces>PiotrGrochowski |
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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox ET}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | {{ED intro}} |
| : This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-30 14:49:47 UTC</tt>.<br>
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| : The original revision id was <tt>590543392</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">**<span style="font-size: 250%;">218edo</span>**
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| It contains very accurate ratios, here is the collection: 7/4, 11/8, 9/7, 8/7, 9/8, 10/9, 11/10, 17/16
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| **Bold numbers are off within less than 0.1 step size**
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| || fraction: || 3/2 || 4/3 || 5/4 || 8/5 || 5/3 || 6/5 || 7/4 || 8/7 || 10/9 || 9/5 || 9/8 || 16/9 ||
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| || steps in 218edo: || 128 || 90 || 70 || 148 || 161 || 57 || **176** || **42** || 33 || 185 || **37** || **181** ||
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| It's great for 2.9.7.17 subgroup, and good for 2.9.5.7.11.17 subgroup.
| | 218edo is in[[consistent]] to the [[5-odd-limit]], with [[harmonic]] [[3/1|3]] falling about halfway between its steps. However, it contains very accurate ratios, such as [[7/4]], [[9/7]], [[9/8]], [[10/9]], [[11/10]], [[17/16]], and [[19/16]], which are approximated within 0.55-cent deviation (10% the step size). The suggested [[subgroup]]s are therefore 2.9.7.17.19 and 2.9.5.7.11.17.19.23. |
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| Tempers out 640000000000000000/635585924776181463 with patent 5, 7 and 9. This is the difference between three 7/4 ratios and sixteen 10/9 ratios stacked.
| | Commas using the [[13-limit]] patent val: |
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| To handle accurate 3 along with everything mentioned, please explore [[436edo]].</pre></div>
| | ; [[5-limit]]: 20000/19683, 1220703125/1207959552 |
| <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>218edo</title></head><body><strong><span style="font-size: 250%;">218edo</span></strong><br />
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| It contains very accurate ratios, here is the collection: 7/4, 11/8, 9/7, 8/7, 9/8, 10/9, 11/10, 17/16<br />
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| <strong>Bold numbers are off within less than 0.1 step size</strong><br />
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| | ; [[7-limit]]: 4000/3969, 65625/65536, 245/243, 2401/2400 117649/116640 |
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| <table class="wiki_table">
| | ; [[11-limit]]: 4000/3993, 12005/11979, 16384/16335, 4375/4356, 78125/77616, 896/891, 67228/66825, 1375/1372, 6875/6804, 5632/5625, 385/384, 94325/93312, 15488/15435, 75625/75264, 15488/15309, 3388/3375, 1331/1323, 6655/6561, 65219/64800, 43923/43904, 73205/72576, |
| <tr>
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| <td>fraction:<br />
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| </td>
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| <td>3/2<br />
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| </td>
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| <td>4/3<br />
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| </td>
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| <td>5/4<br />
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| </td>
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| <td>8/5<br />
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| </td>
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| <td>5/3<br />
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| </td>
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| <td>6/5<br />
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| </td>
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| <td>7/4<br />
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| </td>
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| <td>8/7<br />
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| </td>
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| <td>10/9<br />
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| </td>
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| <td>9/5<br />
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| </td>
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| <td>9/8<br />
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| </td>
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| <td>16/9<br />
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| </td>
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| </tr>
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| <tr>
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| <td>steps in 218edo:<br />
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| </td>
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| <td>128<br />
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| </td>
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| <td>90<br />
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| </td>
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| <td>70<br />
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| </td>
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| <td>148<br />
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| </td>
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| <td>161<br />
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| </td>
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| <td>57<br />
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| </td>
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| <td><strong>176</strong><br />
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| </td>
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| <td><strong>42</strong><br />
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| </td>
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| <td>33<br />
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| </td>
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| <td>185<br />
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| </td>
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| <td><strong>37</strong><br />
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| </td>
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| <td><strong>181</strong><br />
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| </td>
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| </tr>
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| </table>
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| <br />
| | ; [[13-limit]]: 28672/28561, 86240/85683, 20480/20449, 5600/5577, 16807/16731, 25000/24843, 6125/6084, 86625/86528, 68992/68445, 58080/57967, 96800/95823, 847/845, 41503/41067, 33275/33124, 65219/64896, 29575/29403, 4225/4224, 21632/21609, 676/675, 33124/32805, 9295/9261, 46475/45927, 13013/12960, 28561/28512 |
| It's great for 2.9.7.17 subgroup, and good for 2.9.5.7.11.17 subgroup.<br />
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| <br />
| | === Odd harmonics === |
| Tempers out 640000000000000000/635585924776181463 with patent 5, 7 and 9. This is the difference between three 7/4 ratios and sixteen 10/9 ratios stacked.<br />
| | {{Harmonics in equal|218}} |
| <br />
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| To handle accurate 3 along with everything mentioned, please explore <a class="wiki_link" href="/436edo">436edo</a>.</body></html></pre></div>
| | === Subsets and supersets === |
| | Since 218 factors into {{factorization|218}}, 218edo contains [[2edo]] and [[109edo]] as its subsets. [[436edo]], which doubles it, is worth exploring. |
| Prime factorization
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2 × 109
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| Step size
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5.50459 ¢
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| Fifth
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128\218 (704.587 ¢) (→ 64\109)
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| Semitones (A1:m2)
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24:14 (132.1 ¢ : 77.06 ¢)
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| Dual sharp fifth
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128\218 (704.587 ¢) (→ 64\109)
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| Dual flat fifth
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127\218 (699.083 ¢)
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| Dual major 2nd
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37\218 (203.67 ¢)
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| Consistency limit
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3
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| Distinct consistency limit
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3
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218 equal divisions of the octave (abbreviated 218edo or 218ed2), also called 218-tone equal temperament (218tet) or 218 equal temperament (218et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 218 equal parts of about 5.5 ¢ each. Each step represents a frequency ratio of 21/218, or the 218th root of 2.
218edo is inconsistent to the 5-odd-limit, with harmonic 3 falling about halfway between its steps. However, it contains very accurate ratios, such as 7/4, 9/7, 9/8, 10/9, 11/10, 17/16, and 19/16, which are approximated within 0.55-cent deviation (10% the step size). The suggested subgroups are therefore 2.9.7.17.19 and 2.9.5.7.11.17.19.23.
Commas using the 13-limit patent val:
- 5-limit
- 20000/19683, 1220703125/1207959552
- 7-limit
- 4000/3969, 65625/65536, 245/243, 2401/2400 117649/116640
- 11-limit
- 4000/3993, 12005/11979, 16384/16335, 4375/4356, 78125/77616, 896/891, 67228/66825, 1375/1372, 6875/6804, 5632/5625, 385/384, 94325/93312, 15488/15435, 75625/75264, 15488/15309, 3388/3375, 1331/1323, 6655/6561, 65219/64800, 43923/43904, 73205/72576,
- 13-limit
- 28672/28561, 86240/85683, 20480/20449, 5600/5577, 16807/16731, 25000/24843, 6125/6084, 86625/86528, 68992/68445, 58080/57967, 96800/95823, 847/845, 41503/41067, 33275/33124, 65219/64896, 29575/29403, 4225/4224, 21632/21609, 676/675, 33124/32805, 9295/9261, 46475/45927, 13013/12960, 28561/28512
Odd harmonics
Approximation of odd harmonics in 218edo
| Harmonic
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3
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5
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7
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9
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11
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13
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15
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17
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19
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21
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23
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| Error
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Absolute (¢)
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+2.63
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-0.99
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-0.02
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-0.24
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-0.86
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+1.67
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+1.64
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-0.37
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-0.27
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+2.61
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-0.75
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| Relative (%)
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+47.8
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-18.0
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-0.3
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-4.4
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-15.6
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+30.4
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+29.8
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-6.7
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-4.8
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+47.5
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-13.7
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Steps
(reduced)
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346
(128)
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506
(70)
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612
(176)
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691
(37)
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754
(100)
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807
(153)
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852
(198)
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891
(19)
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926
(54)
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958
(86)
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986
(114)
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Subsets and supersets
Since 218 factors into 2 × 109, 218edo contains 2edo and 109edo as its subsets. 436edo, which doubles it, is worth exploring.