240edo
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2010-05-28 03:46:51 UTC.
- The original revision id was 145398915.
- 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:
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. 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. 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== Here are some examples of scales retuned to 240edo and hence exhibiting marvel temperament. ! duodene.scl ! Ellis's Duodene : genus [33355] 12 ! 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1 ! duodene240.scl ! Ellis's Duodene : genus [33355] retuned to 240edo 12 ! 115. 200. 315. 385. 500. 585. 700. 815. 885. 1015. 1085. 1200. ! 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 ! lumma5_240.scl ! Carl Lumma's scale aka diadie1, 240edo version 12 ! 115. 200. 270. 385. 500. 585. 700. 815. 885. 970. 1085. 1200. ! marvel chords ! [-1, -1, 2]->[-1, 0, -2]||[0, -1, -1]->[0, 0, -1]->[0, 0, 0]->[0, 0, 1]->[0, 0, 2] ! pum14.scl pum14 scale 14 ! 25/24 16/15 10/9 75/64 5/4 4/3 64/45 3/2 25/16 8/5 5/3 16/9 15/8 2 ! pum14_240.scl pum14 in 240edo 14 ! 70. 115. 185. 270. 385. 500. 615. 700. 770. 815. 885. 1000. 1085. 1200. ! tetrads [[0, -1, 0], [0, -1, 1], [1, -1, 1], [1, -1, 2], ! [0, 0, 2], [0, -1, -2], [0, 0, 1], [0, -1, -1]]
Original HTML content:
<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 /> <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 <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 /> 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 /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h2> <br /> Here are some examples of scales retuned to 240edo and hence exhibiting marvel temperament.<br /> <br /> ! duodene.scl<br /> !<br /> Ellis's Duodene : genus [33355]<br /> 12<br /> !<br /> 16/15<br /> 9/8<br /> 6/5<br /> 5/4<br /> 4/3<br /> 45/32<br /> 3/2<br /> 8/5<br /> 5/3<br /> 9/5<br /> 15/8<br /> 2/1<br /> <br /> ! duodene240.scl<br /> !<br /> Ellis's Duodene : genus [33355] retuned to 240edo<br /> 12<br /> !<br /> 115.<br /> 200.<br /> 315.<br /> 385.<br /> 500.<br /> 585.<br /> 700.<br /> 815.<br /> 885.<br /> 1015.<br /> 1085.<br /> 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 /> <br /> <br /> ! lumma5_240.scl<br /> !<br /> Carl Lumma's scale aka diadie1, 240edo version<br /> 12<br /> !<br /> 115.<br /> 200.<br /> 270.<br /> 385.<br /> 500.<br /> 585.<br /> 700.<br /> 815.<br /> 885.<br /> 970.<br /> 1085.<br /> 1200.<br /> ! marvel chords<br /> ! [-1, -1, 2]->[-1, 0, -2]||[0, -1, -1]->[0, 0, -1]->[0, 0, 0]->[0, 0, 1]->[0, 0, 2]<br /> <br /> ! pum14.scl<br /> pum14 scale<br /> 14<br /> !<br /> 25/24<br /> 16/15<br /> 10/9<br /> 75/64<br /> 5/4<br /> 4/3<br /> 64/45<br /> 3/2<br /> 25/16<br /> 8/5<br /> 5/3<br /> 16/9<br /> 15/8<br /> 2<br /> <br /> ! pum14_240.scl<br /> pum14 in 240edo<br /> 14<br /> !<br /> 70.<br /> 115.<br /> 185.<br /> 270.<br /> 385.<br /> 500.<br /> 615.<br /> 700.<br /> 770.<br /> 815.<br /> 885.<br /> 1000.<br /> 1085.<br /> 1200.<br /> ! tetrads [[0, -1, 0], [0, -1, 1], [1, -1, 1], [1, -1, 2], ! [0, 0, 2], [0, -1, -2], [0, 0, 1], [0, -1, -1]]</body></html>