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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>
| | '''28edt''' divides the [[3/1|third harmonic]] into 28 [[equal]] parts of 67.9 [[cent]]s each. It is the simplest [[edt]] which intelligibly doubles the Anti-BP harmonic chain{{clarify}}. |
| : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-20 18:11:30 UTC</tt>.<br>
| |
| : The original revision id was <tt>596295840</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">This is the simplest edt which intelligibly doubles the Anti-BP harmonic chain. It is a schematic temperament, albeit a really strange one due to being a multiple of 4edt and being index-2 in BP.
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| ||~ Degree ||~ cents ||~ notation ||
| | == Theory == |
| || 1/28 || 67.93 || Cq/D@y ||
| |
| || 2/28 || 135.85 || Cp/D@ ||
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| || 3/28 || 203.78 || Cp/Dy ||
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| || 4/28 || 271.71 || D ||
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| || 5/28 || 339.635 || Dp/Ey ||
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| || 6/28 || 407.56 || E ||
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| || 7/28 || 475.49 || Epq/F@y ||
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| || 8/28 || 543.42 || Ep/F@ ||
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| || 9/28 || 611.34 || Eq/Fy ||
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| || 10/28 || 679.27 || F ||
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| || 11/28 || 747.2 || Fq/Gy ||
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| || 12/28 || 815.12 || G ||
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| || 13/28 || 883.05 || Gq/Hy ||
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| || 14/28 || 950.98 || Gp/H@ ||
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| || 15/28 || 1018.9 || Gpq/H@y ||
| |
| || 16/28 || 1086.83 || H ||
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| || 17/28 || 1154.76 || Hq/Jy ||
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| || 18/28 || 1222.685 || J ||
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| || 19/28 || 1290.61 || Jq/A@y ||
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| || 20/28 || 1358.54 || Jp/A@ ||
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| || 21/28 || 1426.47 || Jpq/Ay ||
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| || 22/28 || 1494.39 || A ||
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| || 23/28 || 1562.32 || Aq/B@y ||
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| || 24/28 || 1630.25 || Ap/B@ ||
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| || 25/28 || 1698.17 || Apq/By ||
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| || 26/28 || 1766.1 || B ||
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| || 27/28 || 1834.03 || Bq/Cy ||
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| || 28/28 || 1901.955 || C ||</pre></div>
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| <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>28edt</title></head><body>This is the simplest edt which intelligibly doubles the Anti-BP harmonic chain. It is a schematic temperament, albeit a really strange one due to being a multiple of 4edt and being index-2 in BP.<br />
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| | From a no-twos perspective, 28edt has a nearly-pure 5th harmonic (which tritave-reduces to [[5/3]]), with 41/28 being a continued fraction [[convergent]] to log<sub>3</sub>(5). It poorly approximates harmonic 7, but has an acceptable approximation of harmonic 11, so it makes sense to consider it as a [[3.5.11 subgroup]] system. In the 3.5.11 subgroup, it tempers out the commas [[6655/6561]] and [[4-cent comma|1953125/1948617]]. The former means it supports [[deneb]] temperament, which equates 3 flat [[11/9]]s to [[5/3]], and the latter means it supports [[delta centauri]] temperament, which equates 9 near-just [[5/3]]s tritave-reduced to [[11/9]] (although delta centauri is a [[microtemperament]] and larger EDTs like [[71edt]] offer much better tunings of it). [[56edt]] rectifies the approximation of the 7th harmonic. |
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| <table class="wiki_table">
| | 28edt has a very accurate [[8/1|8th harmonic]], and as such, one step is very close to every third step of [[53edo]], and the tritave-reduced 8th harmonic [[8/3]] and tritave-reduced 8th subharmonic [[9/8]] are equally well-approximated. It tempers out the [[schisma]] in the 8.3.5 subgroup, albeit a really strange tuning due to being a multiple of 4edt and being index-2 in BP. It can also be used for the 3.8.5.11 extension of deneb temperament equating +5 generators to 8/3. |
| <tr>
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| <th>Degree<br />
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| </th>
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| <th>cents<br />
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| </th>
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| <th>notation<br />
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| </th>
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| </tr>
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| <tr>
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| <td>1/28<br />
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| </td>
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| <td>67.93<br />
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| </td>
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| <td>Cq/D@y<br />
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| </td>
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| </tr>
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| <tr>
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| <td>2/28<br />
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| </td>
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| <td>135.85<br />
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| </td>
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| <td>Cp/D@<br />
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| </td>
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| </tr>
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| <tr>
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| <td>3/28<br />
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| </td>
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| <td>203.78<br />
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| </td>
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| <td>Cp/Dy<br />
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| </td>
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| </tr>
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| <tr>
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| <td>4/28<br />
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| </td>
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| <td>271.71<br />
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| </td>
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| <td>D<br />
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| </td>
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| </tr>
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| <tr>
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| <td>5/28<br />
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| </td>
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| <td>339.635<br />
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| </td>
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| <td>Dp/Ey<br />
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| </td>
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| </tr>
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| <tr>
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| <td>6/28<br />
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| </td>
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| <td>407.56<br />
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| </td>
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| <td>E<br />
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| </td>
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| </tr>
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| <tr>
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| <td>7/28<br />
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| </td>
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| <td>475.49<br />
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| </td>
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| <td>Epq/F@y<br />
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| </td>
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| </tr>
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| <tr>
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| <td>8/28<br />
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| </td>
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| <td>543.42<br />
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| </td>
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| <td>Ep/F@<br />
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| </td>
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| </tr>
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| <tr>
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| <td>9/28<br />
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| </td>
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| <td>611.34<br />
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| </td>
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| <td>Eq/Fy<br />
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| </td>
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| </tr>
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| <tr>
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| <td>10/28<br />
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| </td>
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| <td>679.27<br />
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| </td>
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| <td>F<br />
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| </td>
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| </tr>
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| <tr>
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| <td>11/28<br />
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| </td>
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| <td>747.2<br />
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| </td>
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| <td>Fq/Gy<br />
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| </td>
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| </tr>
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| <tr>
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| <td>12/28<br />
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| </td>
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| <td>815.12<br />
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| </td>
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| <td>G<br />
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| </td>
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| </tr>
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| <tr>
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| <td>13/28<br />
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| </td>
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| <td>883.05<br />
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| </td>
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| <td>Gq/Hy<br />
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| </td>
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| </tr>
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| <tr>
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| <td>14/28<br />
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| </td>
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| <td>950.98<br />
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| </td>
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| <td>Gp/H@<br />
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| </td>
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| </tr>
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| <tr>
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| <td>15/28<br />
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| </td>
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| <td>1018.9<br />
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| </td>
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| <td>Gpq/H@y<br />
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| </td>
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| </tr>
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| <tr>
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| <td>16/28<br />
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| </td>
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| <td>1086.83<br />
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| </td>
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| <td>H<br />
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| </td>
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| </tr>
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| <tr>
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| <td>17/28<br />
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| </td>
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| <td>1154.76<br />
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| </td>
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| <td>Hq/Jy<br />
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| </td>
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| </tr>
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| <tr>
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| <td>18/28<br />
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| </td>
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| <td>1222.685<br />
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| </td>
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| <td>J<br />
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| </td>
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| </tr>
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| <tr>
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| <td>19/28<br />
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| </td>
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| <td>1290.61<br />
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| </td>
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| <td>Jq/A@y<br />
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| </td>
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| </tr>
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| <tr>
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| <td>20/28<br />
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| </td>
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| <td>1358.54<br />
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| </td>
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| <td>Jp/A@<br />
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| </td>
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| </tr>
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| <tr>
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| <td>21/28<br />
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| </td>
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| <td>1426.47<br />
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| </td>
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| <td>Jpq/Ay<br />
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| </td>
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| </tr>
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| <tr>
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| <td>22/28<br />
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| </td>
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| <td>1494.39<br />
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| </td>
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| <td>A<br />
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| </td>
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| </tr>
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| <tr>
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| <td>23/28<br />
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| </td>
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| <td>1562.32<br />
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| </td>
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| <td>Aq/B@y<br />
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| </td>
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| </tr>
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| <tr>
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| <td>24/28<br />
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| </td>
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| <td>1630.25<br />
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| </td>
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| <td>Ap/B@<br />
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| </td>
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| </tr>
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| <tr>
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| <td>25/28<br />
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| </td>
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| <td>1698.17<br />
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| </td>
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| <td>Apq/By<br />
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| </td>
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| </tr>
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| <tr>
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| <td>26/28<br />
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| </td>
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| <td>1766.1<br />
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| </td>
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| <td>B<br />
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| </td>
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| </tr>
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| <tr>
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| <td>27/28<br />
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| </td>
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| <td>1834.03<br />
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| </td>
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| <td>Bq/Cy<br />
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| </td>
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| </tr>
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| <tr>
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| <td>28/28<br />
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| </td>
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| <td>1901.955<br />
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| </td>
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| <td>C<br />
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| </td>
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| </tr>
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| </table>
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|
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|
| </body></html></pre></div>
| | Similar to how [[14edt]] can be seen as [[9edo]] with significantly [[stretched and compressed tuning|stretched]] [[octave]]s (~23{{c}}), 28edt can be seen as [[18edo]] with the same stretched octave. Note that using the stretched octave will map 8/1 to an inconsistent approximation rather than the aforementioned accurate 8/1. Considering that [[29edt]] is a compressed 18edo, neither 28edt or 29edt are truly equivalent to 18edo (these two basically correspond to 18edo's dual fifths). |
| | |
| | == Harmonics == |
| | {{Harmonics in equal |
| | | steps = 28 |
| | | num = 3 |
| | | denom = 1 |
| | | intervals = integer |
| | }} |
| | {{Harmonics in equal |
| | | steps = 28 |
| | | num = 3 |
| | | denom = 1 |
| | | start = 12 |
| | | collapsed = 1 |
| | | intervals = integer |
| | }} |
| | |
| | == Intervals == |
| | {| class="wikitable right-all left-4" |
| | |- |
| | ! rowspan="2"| Steps |
| | ! rowspan="2"| [[cent]]s |
| | ! rowspan="2"| [[hekt]]s |
| | ! colspan="2"| Approximate ratios |
| | ! rowspan="2"| notation{{clarify}} |
| | |- |
| | !3.5.11 |
| | !3.8.5.11 |
| | |- |
| | | 1 |
| | | 67.927 |
| | | 46.429 |
| | | [[125/121]] |
| | | [[25/24]] |
| | | Cq/D\\y |
| | |- |
| | | 2 |
| | | 135.854 |
| | | 92.857 |
| | | [[27/25]] |
| | | [[88/81]] |
| | | Cp/D\\ |
| | |- |
| | | 3 |
| | | 203.781 |
| | | 139.286 |
| | | [[135/121]] |
| | | [[9/8]] |
| | | Cpq/Dy |
| | |- |
| | | 4 |
| | | 271.708 |
| | | 185.714 |
| | | [[729/625]] |
| | | [[64/55]], [[75/64]] |
| | | D |
| | |- |
| | | 5 |
| | | 339.635 |
| | | 232.143 |
| | | [[11/9]] |
| | | [[40/33]] |
| | | Dp/Ey |
| | |- |
| | | 6 |
| | | 407.562 |
| | | 278.571 |
| | | [[125/99]] |
| | | [[81/64]] |
| | | E |
| | |- |
| | | 7 |
| | | 475.489 |
| | | 325.000 |
| | | [[33/25]] |
| | | [[72/55]], [[320/243]] |
| | | Epq/F\\y |
| | |- |
| | | 8 |
| | | 543.416 |
| | | 371.429 |
| | | [[15/11]] |
| | | [[11/8]] |
| | | Ep/F\\ |
| | |- |
| | | 9 |
| | | 611.343 |
| | | 417.857 |
| | | [[891/625]] |
| | | [[64/45]] |
| | | Eq/Fy |
| | |- |
| | | 10 |
| | | 679.270 |
| | | 464.286 |
| | | [[81/55]], [[121/81]] |
| | | [[40/27]] |
| | | F |
| | |- |
| | | 11 |
| | | 747.197 |
| | | 510.714 |
| | | [[125/81]] |
| | | [[99/64]] |
| | | Fq/Gy |
| | |- |
| | | 12 |
| | | 815.124 |
| | | 557.143 |
| | | [[121/75]] |
| | | [[8/5]] |
| | | G |
| | |- |
| | | 13 |
| | | 883.051 |
| | | 603.571 |
| | | [[5/3]] |
| | | |
| | | Gq/H\\y |
| | |- |
| | | 14 |
| | | 950.978 |
| | | 650.000 |
| | | [[625/363]], [[1089/625]] |
| | | [[512/297]], [[891/512]] |
| | | Gp/H\\ |
| | |- |
| | | 15 |
| | | 1018.904 |
| | | 696.429 |
| | | [[9/5]] |
| | | |
| | | Gpq/Hy |
| | |- |
| | | 16 |
| | | 1086.831 |
| | | 742.857 |
| | | [[225/121]] |
| | | [[15/8]] |
| | | H |
| | |- |
| | | 17 |
| | | 1154.758 |
| | | 789.286 |
| | | [[243/125]] |
| | | [[64/33]] |
| | | Hq/Jy |
| | |- |
| | | 18 |
| | | 1222.685 |
| | | 835.714 |
| | | [[55/27]], [[243/121]] |
| | | [[81/40]] |
| | | J |
| | |- |
| | | 19 |
| | | 1290.612 |
| | | 882.143 |
| | | [[625/297]] |
| | | [[135/64]] |
| | | Jq/A\\y |
| | |- |
| | | 20 |
| | | 1358.539 |
| | | 928.571 |
| | | [[11/5]] |
| | | [[24/11]] |
| | | Jp/A\\ |
| | |- |
| | | 21 |
| | | 1426.466 |
| | | 975.000 |
| | | [[25/11]] |
| | | [[55/24]], [[729/320]] |
| | | Jpq/Ay |
| | |- |
| | | 22 |
| | | 1494.393 |
| | | 1021.429 |
| | | [[297/125]] |
| | | [[64/27]] |
| | | A |
| | |- |
| | | 23 |
| | | 1562.320 |
| | | 1067.857 |
| | | [[27/11]] |
| | | [[99/40]] |
| | | Aq/B\\y |
| | |- |
| | | 24 |
| | | 1630.247 |
| | | 1114.286 |
| | | [[625/243]] |
| | | [[64/25]], [[165/64]] |
| | | Ap/B\\ |
| | |- |
| | | 25 |
| | | 1698.174 |
| | | 1160.714 |
| | | [[121/45]] |
| | | [[8/3]] |
| | | Apq/By |
| | |- |
| | | 26 |
| | | 1766.101 |
| | | 1207.143 |
| | | [[25/9]] |
| | | [[243/88]] |
| | | B |
| | |- |
| | | 27 |
| | | 1834.028 |
| | | 1253.571 |
| | | [[363/125]] |
| | | [[72/25]] |
| | | Bq/Cy |
| | |- |
| | | 28 |
| | | 1901.955 |
| | | 1300.000 |
| | | [[3/1]] |
| | | |
| | | C |
| | |} |
| | |
| | == Music == |
| | * [http://www.seraph.it/dep/det/NonOctaveMelancholic.mp3 NonOctave Melancholic] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/06940a530481965d791c521ea45012f2-267.html blog entry]) |
| | |
| | {{todo|expand}} |