240edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 145392267 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 145397585 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2010-05-28 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-28 03:30:26 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>145397585</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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If we round off to the nearest five cents, we end up with a [[Vals and Tuning Space|val]](mapping to primes) for 240edo of <240 380 557 674|. This tempers out the [[http://en.wikipedia.org/wiki/Septimal_kleisma|septimal kleisma]] of 225/224, with the resultant errors (two cents flat for the fifth, a little over a cent flat and sharp, respectively, for the major third and the 7/4) about as low as it is possible to achieve. Retuning 5-limit scales to 240edo is a simple and often very effective way to to make them function as 7-limit scales while retaining very accurate tuning. | If we round off to the nearest five cents, we end up with a [[Vals and Tuning Space|val]](mapping to primes) for 240edo of <240 380 557 674|. This tempers out the [[http://en.wikipedia.org/wiki/Septimal_kleisma|septimal kleisma]] of 225/224, with the resultant errors (two cents flat for the fifth, a little over a cent flat and sharp, respectively, for the major third and the 7/4) about as low as it is possible to achieve. Retuning 5-limit scales to 240edo is a simple and often very effective way to to make them function as 7-limit scales while retaining very accurate tuning. | ||
For higher limits, 240edo tempers out 243/242 in the 11-limit, 351/350 in the 13-limit, and 375/374 in the 17-limit, and adding these to the mix converts marvel temperament into spectacle temperament. This is still a planar temperament, but more complex as two unidecimal neutral thirds of 11/9 make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 the 350 cent interval often employed in 24edo versions of Arabic music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. | For higher limits, 240edo tempers out 243/242 in the 11-limit, 351/350 in the 13-limit, and 375/374 in the 17-limit, and adding these to the mix converts marvel temperament into spectacle temperament. This is still a planar temperament, but more complex as two unidecimal neutral thirds of 11/9 make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 the 350 cent interval often employed in 24edo versions of Arabic music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. We have: | ||
3 ~ 2 (11/9)^2 | |||
5 = 2^2 (5/4) | |||
7 ~ 2 (11/9)^4 (5/4)^2 | |||
11 ~ 2^2 (11/9)^5 | |||
13 ~ 2^3 (11/9)^(-2) (5/4)^4 | |||
17 ~ 2^4 (11/9)^(-3) (5/4)^3 | |||
It should be noted that the exponents of 5/4 above are all positive and go no higher than 4. | |||
13 ~ 2^3 (11/9)^(-2) (5/4)^4 | |||
==Scales== | ==Scales== | ||
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</pre></div> | ! lumma5.scl | ||
! | |||
Carl Lumma's scale, 5-limit just version, TL 19-2-99 | |||
! Also diadie1, prism, Fokker 12-tone just | |||
12 | |||
! | |||
16/15 | |||
9/8 | |||
75/64 | |||
5/4 | |||
4/3 | |||
45/32 | |||
3/2 | |||
8/5 | |||
5/3 | |||
225/128 | |||
15/8 | |||
2/1 | |||
! marvel chords | |||
! [-1, -1, 2]->[-1, 0, -2]||[0, -1, -1]->[0, 0, -1]->[0, 0, 0]->[0, 0, 1]->[0, 0, 2]</pre></div> | |||
<h4>Original HTML content:</h4> | <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>240edo</title></head><body>The 240edo divides the octave into 240 steps of exactly five cents each. Its primary purpose is in tuning marvel temperament and marvel's extension to spectacle temperament.<br /> | <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>240edo</title></head><body>The 240edo divides the octave into 240 steps of exactly five cents each. Its primary purpose is in tuning marvel temperament and marvel's extension to spectacle temperament.<br /> | ||
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If we round off to the nearest five cents, we end up with a <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a>(mapping to primes) for 240edo of &lt;240 380 557 674|. This tempers out the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_kleisma" rel="nofollow">septimal kleisma</a> of 225/224, with the resultant errors (two cents flat for the fifth, a little over a cent flat and sharp, respectively, for the major third and the 7/4) about as low as it is possible to achieve. Retuning 5-limit scales to 240edo is a simple and often very effective way to to make them function as 7-limit scales while retaining very accurate tuning.<br /> | If we round off to the nearest five cents, we end up with a <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a>(mapping to primes) for 240edo of &lt;240 380 557 674|. This tempers out the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_kleisma" rel="nofollow">septimal kleisma</a> of 225/224, with the resultant errors (two cents flat for the fifth, a little over a cent flat and sharp, respectively, for the major third and the 7/4) about as low as it is possible to achieve. Retuning 5-limit scales to 240edo is a simple and often very effective way to to make them function as 7-limit scales while retaining very accurate tuning.<br /> | ||
<br /> | <br /> | ||
For higher limits, 240edo tempers out 243/242 in the 11-limit, 351/350 in the 13-limit, and 375/374 in the 17-limit, and adding these to the mix converts marvel temperament into spectacle temperament. This is still a planar temperament, but more complex as two unidecimal neutral thirds of 11/9 make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 the 350 cent interval often employed in 24edo versions of Arabic music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle.<br /> | For higher limits, 240edo tempers out 243/242 in the 11-limit, 351/350 in the 13-limit, and 375/374 in the 17-limit, and adding these to the mix converts marvel temperament into spectacle temperament. This is still a planar temperament, but more complex as two unidecimal neutral thirds of 11/9 make up a fifth (which is in fact the same fifth as that of 12edo, and the 11/9 the 350 cent interval often employed in 24edo versions of Arabic music.) Musical intervals are therefore generated by octaves, major thirds, and neutral thirds in spectacle. We have:<br /> | ||
<br /> | |||
3 ~ 2 (11/9)^2<br /> | |||
5 = 2^2 (5/4)<br /> | |||
7 ~ 2 (11/9)^4 (5/4)^2<br /> | |||
11 ~ 2^2 (11/9)^5<br /> | |||
13 ~ 2^3 (11/9)^(-2) (5/4)^4<br /> | |||
17 ~ 2^4 (11/9)^(-3) (5/4)^3<br /> | |||
<br /> | |||
It should be noted that the exponents of 5/4 above are all positive and go no higher than 4.<br /> | |||
13 ~ 2^3 (11/9)^(-2) (5/4)^4<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h2> | ||
| Line 99: | Line 138: | ||
1015.<br /> | 1015.<br /> | ||
1085.<br /> | 1085.<br /> | ||
1200.</body></html></pre></div> | 1200.<br /> | ||
<br /> | |||
<br /> | |||
! lumma5.scl<br /> | |||
!<br /> | |||
Carl Lumma's scale, 5-limit just version, TL 19-2-99<br /> | |||
! Also diadie1, prism, Fokker 12-tone just <br /> | |||
12<br /> | |||
!<br /> | |||
16/15<br /> | |||
9/8<br /> | |||
75/64<br /> | |||
5/4<br /> | |||
4/3<br /> | |||
45/32<br /> | |||
3/2<br /> | |||
8/5<br /> | |||
5/3<br /> | |||
225/128<br /> | |||
15/8<br /> | |||
2/1<br /> | |||
! marvel chords<br /> | |||
! [-1, -1, 2]-&gt;[-1, 0, -2]||[0, -1, -1]-&gt;[0, 0, -1]-&gt;[0, 0, 0]-&gt;[0, 0, 1]-&gt;[0, 0, 2]</body></html></pre></div> | |||