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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | : ''Not to be confused with calendar-based scales such as those in [[293edo]], [[400edo]], [[353edo]], or [[Irvic scale|Irvian mode]].'' |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-11-06 07:59:48 UTC</tt>.<br>
| |
| : The original revision id was <tt>379642138</tt>.<br>
| |
| : The revision comment was: <tt></tt><br>
| |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| |
| <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">See [[Hemifamity temperaments#Leapday]].
| |
|
| |
|
| Gencom: [2 3/2; 91/90 133/132 171/170 190/189 833/832 847/845]
| | '''Leapday''' is a [[regular temperament]] for the 7-, 11-, 13-, 17-, and no-19 23-limit. It is based on the [[chain of fifths]], but here, the fifth is tuned slightly sharp of just (approximately 704{{cent}}) so that 6 fifths give [[23/16]], 8 fifths give [[13/8]], 11 fifths give [[11/8]], 15 fifths give [[7/4]], 21 fifths give [[5/4]], and 24 fifths give [[17/16]]. |
| Gencom map: [<1 1 -10 -6 -3 -1 17 6|, <0 1 21 15 11 8 -22 -3|]
| |
|
| |
|
| =Spectrum of Leapday Tunings by Eigenmonzos=
| | Equivalently: |
| ||~ Eigenmonzo ||~ Fifth ||
| | * 5/4, the classical major third, is represented by a triply augmented unison (C–C𝄪♯), |
| || 19/16 || 700.82899 ||
| | * 7/4, the harmonic seventh, is represented by a doubly augmented fifth (C–G𝄪), |
| || 24/19 || 701.11050 ||
| | * 11/8 is represented by an augmented third (C–E♯), |
| || 19/18 || 701.27940 ||
| | * 13/8 is represented by an augmented fifth (C–G♯), |
| || 4/3 || 701.95500 ||
| | * 17/16 is represented by an octave-reduced triply augmented sixth (C–A𝄪♯), and |
| || 15/14 || 702.77754 ||
| | * 23/16 is represented by an augmented fourth (C–F♯). |
| || 7/5 || 702.91463 ||
| |
| || 21/20 || 703.10656 ||
| |
| || 15/11 || 703.35916 ||
| |
| || 15/13 || 703.41008 ||
| |
| || 11/10 || 703.49958 ||
| |
| || 13/10 || 703.52200 ||
| |
| || 13/11 || 703.59676 ||
| |
| || 19/15 || 703.63023 ||
| |
| || 20/19 || 703.70003 ||
| |
| || 26/21 || 703.78166 ||
| |
| || 22/19 || 703.84321 ||
| |
| || 21/19 || 703.85620 ||
| |
| || 22/21 || 703.89259 ||
| |
| || 26/19 || 703.91042 ||
| |
| || 19/14 || 703.96183 ||
| |
| || 16/15 || 704.01221 ||
| |
| || 14/13 || 704.04261 ||
| |
| || 5/4 || 704.11018 (5 limit minimax) ||
| |
| || 18/17 || 704.12311 (19 and 21 limit minimax) ||
| |
| || 17/15 || 704.16621 ||
| |
| || 24/17 || 704.21737 ||
| |
| || 20/17 || 704.21763 ||
| |
| || 6/5 || 704.21794 (7, 15 and 17 limit minimax) ||
| |
| || 17/16 || 704.32021 ||
| |
| || 10/9 || 704.33704 (9, 11 and 13 limit minimax) ||
| |
| || 21/17 || 704.36383 ||
| |
| || 14/11 || 704.37699 ||
| |
| || 21/16 || 704.42381 ||
| |
| || 17/14 || 704.42893 ||
| |
| || 22/17 || 704.43523 ||
| |
| || 17/13 || 704.51908 ||
| |
| || 8/7 || 704.58839 ||
| |
| || 11/8 || 704.66527 ||
| |
| || 7/6 || 704.77649 ||
| |
| || 19/17 || 704.87145 ||
| |
| || 12/11 || 704.93629 ||
| |
| || 9/7 || 704.99353 ||
| |
| || 16/13 || 705.06596 ||
| |
| || 11/9 || 705.26755 ||
| |
| || 13/12 || 705.51038 ||
| |
| || 18/13 || 706.10294 ||
| |
|
| |
|
| </pre></div>
| | As a result, leapday is very much the "opposite" of meantone in many respects, similar to [[superpyth]]: meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of harmonics 7, 11, and 13 are complex, while leapday has the fifth tuned sharp so that intervals of 7, 11, and 13 are relatively simple while intervals of 5 are complex. |
| <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Leapday</title></head><body>See <a class="wiki_link" href="/Hemifamity%20temperaments#Leapday">Hemifamity temperaments</a>.<br />
| |
| <br />
| |
| Gencom: [2 3/2; 91/90 133/132 171/170 190/189 833/832 847/845]<br />
| |
| Gencom map: [&lt;1 1 -10 -6 -3 -1 17 6|, &lt;0 1 21 15 11 8 -22 -3|]<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Spectrum of Leapday Tunings by Eigenmonzos"></a><!-- ws:end:WikiTextHeadingRule:0 -->Spectrum of Leapday Tunings by Eigenmonzos</h1>
| |
|
| |
|
| |
|
| <table class="wiki_table">
| | If ratios of 5 are omitted, the 2.3.7.11.13 [[subgroup]] version of leapday is known as '''leapfrog''', notable as tempering [[parapyth]] (a rank-3 temperament of the 2.3.7.11.13 subgroup) to rank 2 by finding [[~]][[13/8]] at ([[~]][[9/8]])<sup>4</sup>, that is, by tempering out the [[tetris comma]], and is a good combination of simplicity and accuracy, as 5/4 is complex and the canonical mapping for prime 19 is fairly inaccurate. |
| <tr>
| |
| <th>Eigenmonzo<br />
| |
| </th>
| |
| <th>Fifth<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <td>19/16<br />
| |
| </td>
| |
| <td>700.82899<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>24/19<br />
| |
| </td>
| |
| <td>701.11050<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19/18<br />
| |
| </td>
| |
| <td>701.27940<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4/3<br />
| |
| </td>
| |
| <td>701.95500<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15/14<br />
| |
| </td>
| |
| <td>702.77754<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7/5<br />
| |
| </td>
| |
| <td>702.91463<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21/20<br />
| |
| </td>
| |
| <td>703.10656<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15/11<br />
| |
| </td>
| |
| <td>703.35916<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15/13<br />
| |
| </td>
| |
| <td>703.41008<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11/10<br />
| |
| </td>
| |
| <td>703.49958<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13/10<br />
| |
| </td>
| |
| <td>703.52200<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13/11<br />
| |
| </td>
| |
| <td>703.59676<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19/15<br />
| |
| </td>
| |
| <td>703.63023<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20/19<br />
| |
| </td>
| |
| <td>703.70003<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>26/21<br />
| |
| </td>
| |
| <td>703.78166<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22/19<br />
| |
| </td>
| |
| <td>703.84321<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21/19<br />
| |
| </td>
| |
| <td>703.85620<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22/21<br />
| |
| </td>
| |
| <td>703.89259<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>26/19<br />
| |
| </td>
| |
| <td>703.91042<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19/14<br />
| |
| </td>
| |
| <td>703.96183<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16/15<br />
| |
| </td>
| |
| <td>704.01221<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14/13<br />
| |
| </td>
| |
| <td>704.04261<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5/4<br />
| |
| </td>
| |
| <td>704.11018 (5 limit minimax)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18/17<br />
| |
| </td>
| |
| <td>704.12311 (19 and 21 limit minimax)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17/15<br />
| |
| </td>
| |
| <td>704.16621<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>24/17<br />
| |
| </td>
| |
| <td>704.21737<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20/17<br />
| |
| </td>
| |
| <td>704.21763<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6/5<br />
| |
| </td>
| |
| <td>704.21794 (7, 15 and 17 limit minimax)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17/16<br />
| |
| </td>
| |
| <td>704.32021<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10/9<br />
| |
| </td>
| |
| <td>704.33704 (9, 11 and 13 limit minimax)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21/17<br />
| |
| </td>
| |
| <td>704.36383<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14/11<br />
| |
| </td>
| |
| <td>704.37699<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21/16<br />
| |
| </td>
| |
| <td>704.42381<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17/14<br />
| |
| </td>
| |
| <td>704.42893<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22/17<br />
| |
| </td>
| |
| <td>704.43523<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17/13<br />
| |
| </td>
| |
| <td>704.51908<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8/7<br />
| |
| </td>
| |
| <td>704.58839<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11/8<br />
| |
| </td>
| |
| <td>704.66527<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7/6<br />
| |
| </td>
| |
| <td>704.77649<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19/17<br />
| |
| </td>
| |
| <td>704.87145<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12/11<br />
| |
| </td>
| |
| <td>704.93629<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9/7<br />
| |
| </td>
| |
| <td>704.99353<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16/13<br />
| |
| </td>
| |
| <td>705.06596<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11/9<br />
| |
| </td>
| |
| <td>705.26755<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13/12<br />
| |
| </td>
| |
| <td>705.51038<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18/13<br />
| |
| </td>
| |
| <td>706.10294<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| </body></html></pre></div>
| | Leapday was named by [[Herman Miller]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10589.html Yahoo! Tuning Group (Archive) | ''Some 13-limit temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10604.html Yahoo! Tuning Group (Archive) | ''24 13-limit temperaments supported by 46'']</ref>. |
| | |
| | See [[Hemifamity temperaments #Leapday]] or [[No-fives subgroup temperaments #Leapfrog]] for more technical data. |
| | |
| | == Interval chain == |
| | In the following table, odd harmonics 1–23 are in '''bold'''. |
| | |
| | {| class="wikitable center-1 right-2" |
| | |- |
| | ! rowspan="2" | # |
| | ! rowspan="2" | Cents* |
| | ! colspan="2" | Approximate ratios |
| | |- |
| | ! 13-limit |
| | ! Additional ratios<br />of 17 and 23 |
| | |- |
| | | 0 |
| | | 0.0 |
| | | '''1/1''' |
| | | |
| | |- |
| | | 1 |
| | | 704.3 |
| | | '''3/2''' |
| | | |
| | |- |
| | | 2 |
| | | 208.6 |
| | | '''9/8''' |
| | | 17/15, 26/23 |
| | |- |
| | | 3 |
| | | 912.9 |
| | | 22/13, 27/16 |
| | | 17/10 |
| | |- |
| | | 4 |
| | | 417.2 |
| | | 14/11, 33/26 |
| | | 23/18 |
| | |- |
| | | 5 |
| | | 1121.5 |
| | | 21/11, 40/21 |
| | | 23/12, 44/23 |
| | |- |
| | | 6 |
| | | 625.8 |
| | | 10/7, 13/9 |
| | | '''23/16''' |
| | |- |
| | | 7 |
| | | 130.0 |
| | | 13/12, 14/13, 15/14 |
| | | |
| | |- |
| | | 8 |
| | | 834.3 |
| | | '''13/8''', 21/13 |
| | | 34/21 |
| | |- |
| | | 9 |
| | | 338.6 |
| | | 11/9, 39/32, 40/33 |
| | | 17/14, 28/23 |
| | |- |
| | | 10 |
| | | 1042.9 |
| | | 11/6, 20/11 |
| | | 42/23 |
| | |- |
| | | 11 |
| | | 547.2 |
| | | '''11/8''', 15/11 |
| | | |
| | |- |
| | | 12 |
| | | 51.5 |
| | | 28/27, 33/32, 40/39, 45/44 |
| | | 34/33, 35/34 |
| | |- |
| | | 13 |
| | | 755.8 |
| | | 14/9, 20/13 |
| | | 17/11 |
| | |- |
| | | 14 |
| | | 260.1 |
| | | 7/6, 15/13 |
| | | |
| | |- |
| | | 15 |
| | | 964.4 |
| | | '''7/4''' |
| | | 40/23 |
| | |- |
| | | 16 |
| | | 468.7 |
| | | '''21/16''' |
| | | 17/13, 30/23 |
| | |- |
| | | 17 |
| | | 1173.0 |
| | | 63/32, 160/81 |
| | | 45/23, 51/26 |
| | |- |
| | | 18 |
| | | 677.3 |
| | | 40/27 |
| | | 34/23 |
| | |- |
| | | 19 |
| | | 181.6 |
| | | 10/9 |
| | | |
| | |- |
| | | 20 |
| | | 885.8 |
| | | 5/3 |
| | | |
| | |- |
| | | 21 |
| | | 390.1 |
| | | '''5/4''' |
| | | |
| | |- |
| | | 22 |
| | | 1094.4 |
| | | '''15/8''' |
| | | 17/9 |
| | |- |
| | | 23 |
| | | 598.7 |
| | | 45/32 |
| | | 17/12 |
| | |} |
| | <nowiki />* In 13-limit CTE tuning |
| | |
| | == Tunings == |
| | === Tuning spectrum === |
| | This spectrum assumes 19-limit leapday. |
| | |
| | {| class="wikitable center-all left-4" |
| | ! Edo<br>generator |
| | ! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]* |
| | ! Generator (¢) |
| | ! Comments |
| | |- |
| | | |
| | | 19/16 |
| | | 700.829 |
| | | |
| | |- |
| | | |
| | | 19/12 |
| | | 701.110 |
| | | |
| | |- |
| | | |
| | | 19/18 |
| | | 701.279 |
| | | |
| | |- |
| | | |
| | | 3/2 |
| | | 701.955 |
| | | |
| | |- |
| | | 24\41 |
| | | |
| | | 702.439 |
| | | 41cc… val, lower bound of 5-odd-limit diamond monotone |
| | |- |
| | | |
| | | 15/14 |
| | | 702.778 |
| | | |
| | |- |
| | | |
| | | 7/5 |
| | | 702.915 |
| | | |
| | |- |
| | | |
| | | 21/20 |
| | | 703.107 |
| | | |
| | |- |
| | | |
| | | 15/11 |
| | | 703.359 |
| | | |
| | |- |
| | | |
| | | 15/13 |
| | | 703.410 |
| | | |
| | |- |
| | | 17\29 |
| | | |
| | | 703.448 |
| | | 29g val, lower bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone |
| | |- |
| | | |
| | | 11/10 |
| | | 703.500 |
| | | |
| | |- |
| | | |
| | | 13/10 |
| | | 703.522 |
| | | |
| | |- |
| | | |
| | | 13/11 |
| | | 703.597 |
| | | |
| | |- |
| | | |
| | | 19/15 |
| | | 703.630 |
| | | |
| | |- |
| | | |
| | | 19/10 |
| | | 703.700 |
| | | |
| | |- |
| | | |
| | | 21/13 |
| | | 703.782 |
| | | |
| | |- |
| | | |
| | | 19/11 |
| | | 703.843 |
| | | |
| | |- |
| | | |
| | | 21/19 |
| | | 703.856 |
| | | |
| | |- |
| | | |
| | | 21/11 |
| | | 703.893 |
| | | |
| | |- |
| | | |
| | | 19/13 |
| | | 703.910 |
| | | |
| | |- |
| | | |
| | | 19/14 |
| | | 703.962 |
| | | |
| | |- |
| | | |
| | | 19/17 |
| | | 703.979 |
| | | 19- and 21-odd-limit minimax |
| | |- |
| | | 44\75 |
| | | |
| | | 704.000 |
| | | 75dfgh val |
| | |- |
| | | |
| | | 15/8 |
| | | 704.012 |
| | | |
| | |- |
| | | |
| | | 17/14 |
| | | 704.014 |
| | | |
| | |- |
| | | |
| | | 17/13 |
| | | 704.027 |
| | | |
| | |- |
| | | |
| | | 13/7 |
| | | 704.043 |
| | | |
| | |- |
| | | |
| | | 5/4 |
| | | 704.110 |
| | | 5-odd-limit minimax |
| | |- |
| | | |
| | | 17/11 |
| | | 704.126 |
| | | |
| | |- |
| | | 71\121 |
| | | |
| | | 704.132 |
| | | 121defgh val |
| | |- |
| | | |
| | | 5/3 |
| | | 704.218 |
| | | 7-, 15- and 17-odd-limit minimax |
| | |- |
| | | |
| | | 21/17 |
| | | 704.272 |
| | | |
| | |- |
| | | |
| | | 9/5 |
| | | 704.337 |
| | | 9-, 11- and 13-odd-limit minimax |
| | |- |
| | | 27\46 |
| | | |
| | | 704.348 |
| | | |
| | |- |
| | | |
| | | 17/16 |
| | | 704.373 |
| | | |
| | |- |
| | | |
| | | 11/7 |
| | | 704.377 |
| | | |
| | |- |
| | | |
| | | 21/16 |
| | | 704.424 |
| | | |
| | |- |
| | | |
| | | 17/12 |
| | | 704.478 |
| | | |
| | |- |
| | | |
| | | 7/4 |
| | | 704.588 |
| | | |
| | |- |
| | | |
| | | 17/9 |
| | | 704.593 |
| | | |
| | |- |
| | | |
| | | 11/8 |
| | | 704.665 |
| | | |
| | |- |
| | | 37\63 |
| | | |
| | | 704.762 |
| | | 63ch val |
| | |- |
| | | |
| | | 7/6 |
| | | 704.776 |
| | | |
| | |- |
| | | |
| | | 11/6 |
| | | 704.936 |
| | | |
| | |- |
| | | |
| | | 9/7 |
| | | 704.994 |
| | | |
| | |- |
| | | |
| | | 13/8 |
| | | 705.066 |
| | | |
| | |- |
| | | |
| | | 11/9 |
| | | 705.268 |
| | | |
| | |- |
| | | |
| | | 13/12 |
| | | 705.510 |
| | | |
| | |- |
| | | 10\17 |
| | | |
| | | 705.882 |
| | | 17cg val, upper bound of 5-, 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone |
| | |- |
| | | |
| | | 13/9 |
| | | 706.103 |
| | | |
| | |- |
| | | |
| | | 17/10 |
| | | 706.214 |
| | | |
| | |- |
| | | |
| | | 17/15 |
| | | 708.343 |
| | | |
| | |} |
| | <nowiki/>* Besides the octave |
| | |
| | == References and external links == |
| | <references/> |
| | |
| | [[Category:Leapday| ]] <!-- Main article --> |
| | [[Category:Rank-2 temperaments]] |
| | [[Category:Sengic temperaments]] |
| | [[Category:Hemifamity temperaments]] |