43edo
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author Andrew_Heathwaite and made on 2013-05-30 12:32:04 UTC.
- The original revision id was 435319322.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
=<span style="color: #027bac; font-size: 103%;">43 tone equal temperament</span>= = = //43edo// divides the octave into 43 equal parts of 27.907 cents each. It is strongly associated with meantone temperament, particularly 1/5 comma meantone, being a good tuning system in the 5, 7, 11, and 13-limit. The version of 11-limit meantone is the one tempering out 99/98, 176/175 and 441/440 sometimes called Huygens. 43-equal has the first good 13-limit meantone available as an equal division of the octave. The baroque, french, ironically hearing and speech impaired acoustician [[@http://en.wikipedia.org/wiki/Joseph_Sauveur|Joseph Saveur]] based his system on 43 equal tones to the octave, calling them "merides". Further information: [[http://tonalsoft.com/enc/m/meride.aspx]] In the 13-limit, we get two versions of meantone equivalent in 43et, one, [[Meantone family#Septimal%20meantone-Unidecimal%20meantone%20aka%20Huygens-Meridetone|meridetone]], tempering out 78/77, the other, [[Meantone family#Septimal%20meantone-Unidecimal%20meantone%20aka%20Huygens-Grosstone|grosstone]], 144/143. Meridetone has generator mapping <0 1 4 10 18 27|, and grosstone <0 1 4 10 18 -16|; 43 supplies the optimal patent val for meridetone. The 43 patent val <43 68 100 121 149 159| maps 5 to 100 steps, allowing the divison of 5 into 20 equal parts, leading to [[Meantone family#Jerome|jerome temperament]], an interesting higher-limit system for which 43 supplies the optimal patent val in the 7, 11, 13, 17, 19 and 23 limits. It also provides the optimal patent val for 11- and 13-limit [[Marvel temperaments#Amavil|amavil temperament]], which is not a meantone temperament. [[Thuja]] temperament is also a possibility, in which five generators, (~11/8)^5 = ~5/1, with [[MOS]] of 15 and 28. 43edo is the 14th [[prime numbers|prime]] edo, following [[41edo]] and coming before [[47edo]]. ==Intervals== || Degrees of 43-EDO || Cents value || Approximate 13-limit Ratios || || 0 || 0 || || || 1 || 27.907 || || || 2 || 55.814 || || || 3 || 83.721 || || || 4 || 111.628 || 17/16, 16/15, 15/14 || || 5 || 139.535 || 12/11, 13/12, 14/13 || || 6 || 167.442 || 11/10 || || 7 || 195.349 || 9/8, 10/9 || || 8 || 223.256 || 8/7 || || 9 || 251.163 || 15/13 || || 10 || 279.07 || 7/6, 13/11 || || 11 || 306.977 || 6/5 || || 12 || 334.884 || 17/14, 39/32 || || 13 || 362.791 || 11/9, 16/13 || || 14 || 390.698 || 5/4 || || 15 || 418.605 || 9/7, 14/11 || || 16 || 446.512 || 13/10 || || 17 || 474.419 || 21/16 || || 18 || 502.326 || 4/3 || || 19 || 530.233 || 15/11 || || 20 || 558.139 || 11/8, 18/13 || || 21 || 586.046 || 7/5 || || 22 || 613.953 || 10/7 || || 23 || 641.86 || 16/11, 13/9 || || 24 || 669.767 || 22/15 || || 25 || 697.674 || 3/2 || || 26 || 725.581 || 32/21 || || 27 || 753.488 || 20/13 || || 28 || 781.395 || 14/9, 11/7 || || 29 || 809.302 || 8/5 || || 30 || 837.209 || 18/11, 13/8 || || 31 || 865.116 || || || 32 || 893.023 || 5/3 || || 33 || 920.93 || 12/7 || || 34 || 948.837 || 26/15 || || 35 || 976.744 || 7/4 || || 36 || 1004.651 || 16/9, 9/5 || || 37 || 1032.558 || 20/11 || || 38 || 1060.465 || 11/6, 24/13, 13/7 || || 39 || 1088.372 || 15/8, 28/15 || || 40 || 1116.279 || || || 41 || 1144.186 || || || 42 || 1172.093 || || [[file:xenharmonic/43 edo counterpoint.mid|43 edo counterpoint.mid]] ////[[http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/43%20edo%20counterpoint.mp3|mp3]]//// Peter Kosmorsky (late 2011) (in meantone)
Original HTML content:
<html><head><title>43edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x43 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #027bac; font-size: 103%;">43 tone equal temperament</span></h1>
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><!-- ws:end:WikiTextHeadingRule:2 --> </h1>
<em>43edo</em> divides the octave into 43 equal parts of 27.907 cents each. It is strongly associated with meantone temperament, particularly 1/5 comma meantone, being a good tuning system in the 5, 7, 11, and 13-limit. The version of 11-limit meantone is the one tempering out 99/98, 176/175 and 441/440 sometimes called Huygens. 43-equal has the first good 13-limit meantone available as an equal division of the octave. The baroque, french, ironically hearing and speech impaired acoustician <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Joseph_Sauveur" rel="nofollow" target="_blank">Joseph Saveur</a> based his system on 43 equal tones to the octave, calling them "merides". Further information: <a class="wiki_link_ext" href="http://tonalsoft.com/enc/m/meride.aspx" rel="nofollow">http://tonalsoft.com/enc/m/meride.aspx</a><br />
<br />
In the 13-limit, we get two versions of meantone equivalent in 43et, one, <a class="wiki_link" href="/Meantone%20family#Septimal%20meantone-Unidecimal%20meantone%20aka%20Huygens-Meridetone">meridetone</a>, tempering out 78/77, the other, <a class="wiki_link" href="/Meantone%20family#Septimal%20meantone-Unidecimal%20meantone%20aka%20Huygens-Grosstone">grosstone</a>, 144/143. Meridetone has generator mapping <0 1 4 10 18 27|, and grosstone <0 1 4 10 18 -16|; 43 supplies the optimal patent val for meridetone.<br />
<br />
The 43 patent val <43 68 100 121 149 159| maps 5 to 100 steps, allowing the divison of 5 into 20 equal parts, leading to <a class="wiki_link" href="/Meantone%20family#Jerome">jerome temperament</a>, an interesting higher-limit system for which 43 supplies the optimal patent val in the 7, 11, 13, 17, 19 and 23 limits. It also provides the optimal patent val for 11- and 13-limit <a class="wiki_link" href="/Marvel%20temperaments#Amavil">amavil temperament</a>, which is not a meantone temperament. <a class="wiki_link" href="/Thuja">Thuja</a> temperament is also a possibility, in which five generators, (~11/8)^5 = ~5/1, with <a class="wiki_link" href="/MOS">MOS</a> of 15 and 28.<br />
<br />
43edo is the 14th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/41edo">41edo</a> and coming before <a class="wiki_link" href="/47edo">47edo</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x43 tone equal temperament-Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h2>
<br />
<table class="wiki_table">
<tr>
<td>Degrees of 43-EDO<br />
</td>
<td>Cents value<br />
</td>
<td>Approximate 13-limit Ratios<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>27.907<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>55.814<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>83.721<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>111.628<br />
</td>
<td>17/16, 16/15, 15/14<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>139.535<br />
</td>
<td>12/11, 13/12, 14/13<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>167.442<br />
</td>
<td>11/10<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>195.349<br />
</td>
<td>9/8, 10/9<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>223.256<br />
</td>
<td>8/7<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>251.163<br />
</td>
<td>15/13<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>279.07<br />
</td>
<td>7/6, 13/11<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>306.977<br />
</td>
<td>6/5<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>334.884<br />
</td>
<td>17/14, 39/32<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>362.791<br />
</td>
<td>11/9, 16/13<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>390.698<br />
</td>
<td>5/4<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>418.605<br />
</td>
<td>9/7, 14/11<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>446.512<br />
</td>
<td>13/10<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>474.419<br />
</td>
<td>21/16<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>502.326<br />
</td>
<td>4/3<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>530.233<br />
</td>
<td>15/11<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>558.139<br />
</td>
<td>11/8, 18/13<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>586.046<br />
</td>
<td>7/5<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>613.953<br />
</td>
<td>10/7<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>641.86<br />
</td>
<td>16/11, 13/9<br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>669.767<br />
</td>
<td>22/15<br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td>697.674<br />
</td>
<td>3/2<br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>725.581<br />
</td>
<td>32/21<br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>753.488<br />
</td>
<td>20/13<br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>781.395<br />
</td>
<td>14/9, 11/7<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>809.302<br />
</td>
<td>8/5<br />
</td>
</tr>
<tr>
<td>30<br />
</td>
<td>837.209<br />
</td>
<td>18/11, 13/8<br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>865.116<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td>893.023<br />
</td>
<td>5/3<br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td>920.93<br />
</td>
<td>12/7<br />
</td>
</tr>
<tr>
<td>34<br />
</td>
<td>948.837<br />
</td>
<td>26/15<br />
</td>
</tr>
<tr>
<td>35<br />
</td>
<td>976.744<br />
</td>
<td>7/4<br />
</td>
</tr>
<tr>
<td>36<br />
</td>
<td>1004.651<br />
</td>
<td>16/9, 9/5<br />
</td>
</tr>
<tr>
<td>37<br />
</td>
<td>1032.558<br />
</td>
<td>20/11<br />
</td>
</tr>
<tr>
<td>38<br />
</td>
<td>1060.465<br />
</td>
<td>11/6, 24/13, 13/7<br />
</td>
</tr>
<tr>
<td>39<br />
</td>
<td>1088.372<br />
</td>
<td>15/8, 28/15<br />
</td>
</tr>
<tr>
<td>40<br />
</td>
<td>1116.279<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>41<br />
</td>
<td>1144.186<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>42<br />
</td>
<td>1172.093<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<br />
<a href="http://xenharmonic.wikispaces.com/file/view/43%20edo%20counterpoint.mid/311991536/43%20edo%20counterpoint.mid" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/43%20edo%20counterpoint.mid/311991536/43%20edo%20counterpoint.mid');">43 edo counterpoint.mid</a> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/43%20edo%20counterpoint.mp3" rel="nofollow">mp3</a> Peter Kosmorsky (late 2011) (in meantone)</body></html>