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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | =Basics= |
| : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2014-07-16 08:34:26 UTC</tt>.<br>
| | The ''48 equal division'' divides the [[Octave|octave]] into 48 1/8th tones of precisely 25 [[cent|cent]]s each. Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the [[5-limit|5-limit]], but [[tempering_out|tempers out]] [[2401/2400|2401/2400]] in the [[7-limit|7-limit]], making it a tuning for [[Meantone_family|squares temperament]]. In the [[11-limit|11-limit]] we can add [[99/98|99/98]] and [[121/120|121/120]] to the list, and in the [[13-limit|13-limit]], [[66/65|66/65]]. While [[31edo|31edo]] can also do 13-limit squares, 48 might be preferred for some purposes. |
| : The original revision id was <tt>516359378</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">[[toc|flat]]
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| =Basics= | |
| The //48 equal division// divides the [[octave]] into 48 1/8th tones of precisely 25 [[cent]]s each. Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the [[5-limit]], but [[tempering out|tempers out]] [[2401_2400|2401/2400]] in the [[7-limit]], making it a tuning for [[Meantone family|squares temperament]]. In the [[11-limit]] we can add [[99_98|99/98]] and [[121_120|121/120]] to the list, and in the [[13-limit]], [[66_65|66/65]]. While [[31edo]] can also do 13-limit squares, 48 might be preferred for some purposes. | |
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| =Temperaments= | | =Temperaments= |
| Using its best major third, 48 tempers out 20000/19683, but [[34edo]] does a much better job for this temperament, known as [[Tetracot family|tetracot]]. However in the 7-limit it can be used for [[Jubilismic clan|doublewide temperament]], the 1/2 octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the [[optimal patent val]]. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament [[Jubilismic family|jubilee]], which tempers out 50/49 and 99/98. | | Using its best major third, 48 tempers out 20000/19683, but [[34edo|34edo]] does a much better job for this temperament, known as [[Tetracot_family|tetracot]]. However in the 7-limit it can be used for [[Jubilismic_clan|doublewide temperament]], the 1/2 octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the [[Optimal_patent_val|optimal patent val]]. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament [[Jubilismic_family|jubilee]], which tempers out 50/49 and 99/98. |
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| If 48 is treated as a no-fives system, it still tempers out 99/98 and 243/242 in the 11-limit, leading to a no-fives version of squares for which it does well as a tuning. In the 13 no-fives limit, we can add 144/143 to the list of commas, and we get the no-fives version of 13-limit squares, for which 48 actually defines the [[optimal patent val]]. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425 cent interval serving as both [[9_7|9/7]] and [[14_11|14/11]]. | | If 48 is treated as a no-fives system, it still tempers out 99/98 and 243/242 in the 11-limit, leading to a no-fives version of squares for which it does well as a tuning. In the 13 no-fives limit, we can add 144/143 to the list of commas, and we get the no-fives version of 13-limit squares, for which 48 actually defines the [[Optimal_patent_val|optimal patent val]]. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425 cent interval serving as both [[9/7|9/7]] and [[14/11|14/11]]. |
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| Something close to 48edo is what you get if you cross 16edo with pure fifths, for instance, on a 16-tone guitar. The presence of 12/11 in 16edo allows a string offset of 11/8 to also work for producing perfect fifths. | | Something close to 48edo is what you get if you cross 16edo with pure fifths, for instance, on a 16-tone guitar. The presence of 12/11 in 16edo allows a string offset of 11/8 to also work for producing perfect fifths. |
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| =Compositions= | | =Compositions= |
| [[http://archive.org/download/Quincunx/Quincunx.mp3|Quincunx]] by Jon Lyle Smith
| | [http://archive.org/download/Quincunx/Quincunx.mp3 Quincunx] by Jon Lyle Smith |
| [[http://www.seraph.it/blog_files/a0aafbec9519cfe9600e7a82118da2ee-26.html|Two At Once]] [[http://www.seraph.it/dep/det/twoatonce.mp3|play]] by [[Carlo Serafini]]
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| [[http://www.seraph.it/blog_files/a8435fb03236157e8a60b047e9892594-27.html|Tim's Flutes]] [[http://www.seraph.it/dep/det/Tim%27sFlutes.mp3|play]] by [[Carlo Serafini]]</pre></div>
| | [http://www.seraph.it/blog_files/a0aafbec9519cfe9600e7a82118da2ee-26.html Two At Once] [http://www.seraph.it/dep/det/twoatonce.mp3 play] by [[Carlo_Serafini|Carlo Serafini]] |
| <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>48edo</title></head><body><!-- ws:start:WikiTextTocRule:6:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Basics">Basics</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Temperaments">Temperaments</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->
| | [http://www.seraph.it/blog_files/a8435fb03236157e8a60b047e9892594-27.html Tim's Flutes] [http://www.seraph.it/dep/det/Tim%27sFlutes.mp3 play] by [[Carlo_Serafini|Carlo Serafini]] [[Category:doublewide]] |
| <!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:0 -->Basics</h1>
| | [[Category:edo]] |
| The <em>48 equal division</em> divides the <a class="wiki_link" href="/octave">octave</a> into 48 1/8th tones of precisely 25 <a class="wiki_link" href="/cent">cent</a>s each. Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the <a class="wiki_link" href="/5-limit">5-limit</a>, but <a class="wiki_link" href="/tempering%20out">tempers out</a> <a class="wiki_link" href="/2401_2400">2401/2400</a> in the <a class="wiki_link" href="/7-limit">7-limit</a>, making it a tuning for <a class="wiki_link" href="/Meantone%20family">squares temperament</a>. In the <a class="wiki_link" href="/11-limit">11-limit</a> we can add <a class="wiki_link" href="/99_98">99/98</a> and <a class="wiki_link" href="/121_120">121/120</a> to the list, and in the <a class="wiki_link" href="/13-limit">13-limit</a>, <a class="wiki_link" href="/66_65">66/65</a>. While <a class="wiki_link" href="/31edo">31edo</a> can also do 13-limit squares, 48 might be preferred for some purposes.<br />
| | [[Category:jubilee]] |
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| | [[Category:listen]] |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Temperaments</h1>
| | [[Category:squares]] |
| Using its best major third, 48 tempers out 20000/19683, but <a class="wiki_link" href="/34edo">34edo</a> does a much better job for this temperament, known as <a class="wiki_link" href="/Tetracot%20family">tetracot</a>. However in the 7-limit it can be used for <a class="wiki_link" href="/Jubilismic%20clan">doublewide temperament</a>, the 1/2 octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a>. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament <a class="wiki_link" href="/Jubilismic%20family">jubilee</a>, which tempers out 50/49 and 99/98.<br />
| | [[Category:subgroup]] |
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| If 48 is treated as a no-fives system, it still tempers out 99/98 and 243/242 in the 11-limit, leading to a no-fives version of squares for which it does well as a tuning. In the 13 no-fives limit, we can add 144/143 to the list of commas, and we get the no-fives version of 13-limit squares, for which 48 actually defines the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a>. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425 cent interval serving as both <a class="wiki_link" href="/9_7">9/7</a> and <a class="wiki_link" href="/14_11">14/11</a>.<br />
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| Something close to 48edo is what you get if you cross 16edo with pure fifths, for instance, on a 16-tone guitar. The presence of 12/11 in 16edo allows a string offset of 11/8 to also work for producing perfect fifths.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:4 -->Compositions</h1>
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| <a class="wiki_link_ext" href="http://archive.org/download/Quincunx/Quincunx.mp3" rel="nofollow">Quincunx</a> by Jon Lyle Smith<br />
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| <a class="wiki_link_ext" href="http://www.seraph.it/blog_files/a0aafbec9519cfe9600e7a82118da2ee-26.html" rel="nofollow">Two At Once</a> <a class="wiki_link_ext" href="http://www.seraph.it/dep/det/twoatonce.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a><br />
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| <a class="wiki_link_ext" href="http://www.seraph.it/blog_files/a8435fb03236157e8a60b047e9892594-27.html" rel="nofollow">Tim's Flutes</a> <a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Tim%27sFlutes.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a></body></html></pre></div>
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Basics
The 48 equal division divides the octave into 48 1/8th tones of precisely 25 cents each. Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the 5-limit, but tempers out 2401/2400 in the 7-limit, making it a tuning for squares temperament. In the 11-limit we can add 99/98 and 121/120 to the list, and in the 13-limit, 66/65. While 31edo can also do 13-limit squares, 48 might be preferred for some purposes.
Temperaments
Using its best major third, 48 tempers out 20000/19683, but 34edo does a much better job for this temperament, known as tetracot. However in the 7-limit it can be used for doublewide temperament, the 1/2 octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the optimal patent val. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament jubilee, which tempers out 50/49 and 99/98.
If 48 is treated as a no-fives system, it still tempers out 99/98 and 243/242 in the 11-limit, leading to a no-fives version of squares for which it does well as a tuning. In the 13 no-fives limit, we can add 144/143 to the list of commas, and we get the no-fives version of 13-limit squares, for which 48 actually defines the optimal patent val. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425 cent interval serving as both 9/7 and 14/11.
Something close to 48edo is what you get if you cross 16edo with pure fifths, for instance, on a 16-tone guitar. The presence of 12/11 in 16edo allows a string offset of 11/8 to also work for producing perfect fifths.
Compositions
Quincunx by Jon Lyle Smith
Two At Once play by Carlo Serafini
Tim's Flutes play by Carlo Serafini