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 MasonGreen1 and made on 2016-03-31 14:46:42 UTC.
- The original revision id was 578787405.
- 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-family: 'Times New Roman',Times,serif; font-size: 113%;">43 tone equal temperament</span>= = = **43edo** divides the [[octave]] into 43 [[equal]] parts of 27.907 [[cent]]s 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 Sauveur]] 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]]. Although not [[consistency|consistent]], it performs quite decently in very high limits. It has unambiguous mappings for all prime harmonics up to //64//, with the sole exceptions of 23 and, perhaps, 41. The mappings for composite harmonics can then be derived from those for the primes, giving an almost-complete version of mode 32 of the harmonic series, although the aforementioned lack of consistency will give some unusual results. Indeed, the step size of 43edo is very close to the septimal comma (64:63), while two steps is close to 32:31, and four steps to 16:15. ==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 || || ==Notation of 43edo== Because 43edo is a meantone system, this makes it easier to adapt traditional Western notation to it than to some other tunings. A# and Bb are distinct and the distance between them is one meride. The whole tone is divided into seven merides so this means we can use "third-sharps", "two-thirds-sharps", "third-flats", and "two-thirds-flats" to reach the remaining notes between A and B; notes elsewhere on the scale can be notated similarly. Alternatively, a red-note/blue-note system (similar to that proposed for sixth-tones/[[36edo]]) can be used. Now we have the following sequence of notes, each separated by one meride: A, red A, blue A#, A#, Bb, red Bb, blue B, B. (Note that there are red flats and blue sharps, but no red sharps or blue flats). [[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-family: 'Times New Roman',Times,serif; font-size: 113%;">43 tone equal temperament</span></h1>
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><!-- ws:end:WikiTextHeadingRule:2 --> </h1>
<strong>43edo</strong> divides the <a class="wiki_link" href="/octave">octave</a> into 43 <a class="wiki_link" href="/equal">equal</a> parts of 27.907 <a class="wiki_link" href="/cent">cent</a>s 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 Sauveur</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 />
Although not <a class="wiki_link" href="/consistency">consistent</a>, it performs quite decently in very high limits. It has unambiguous mappings for all prime harmonics up to <em>64</em>, with the sole exceptions of 23 and, perhaps, 41. The mappings for composite harmonics can then be derived from those for the primes, giving an almost-complete version of mode 32 of the harmonic series, although the aforementioned lack of consistency will give some unusual results. Indeed, the step size of 43edo is very close to the septimal comma (64:63), while two steps is close to 32:31, and four steps to 16:15.<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 />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x43 tone equal temperament-Notation of 43edo"></a><!-- ws:end:WikiTextHeadingRule:6 -->Notation of 43edo</h2>
<br />
Because 43edo is a meantone system, this makes it easier to adapt traditional Western notation to it than to some other tunings. A# and Bb are distinct and the distance between them is one meride. The whole tone is divided into seven merides so this means we can use "third-sharps", "two-thirds-sharps", "third-flats", and "two-thirds-flats" to reach the remaining notes between A and B; notes elsewhere on the scale can be notated similarly.<br />
<br />
Alternatively, a red-note/blue-note system (similar to that proposed for sixth-tones/<a class="wiki_link" href="/36edo">36edo</a>) can be used. Now we have the following sequence of notes, each separated by one meride: A, red A, blue A#, A#, Bb, red Bb, blue B, B. (Note that there are red flats and blue sharps, but no red sharps or blue flats).<br />
<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> <em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/43%20edo%20counterpoint.mp3" rel="nofollow">mp3</a></em> Peter Kosmorsky (late 2011) (in meantone)</body></html>