EDOs to ETs: Difference between revisions
Wikispaces>igliashon **Imported revision 242341681 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 242361441 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-21 22:06:37 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>242361441</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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|| Cents || 0 || 120 || 240 || 360 || 480 || 600 || 720 || 840 || 960 || 1080 || 1200 || | || Cents || 0 || 120 || 240 || 360 || 480 || 600 || 720 || 840 || 960 || 1080 || 1200 || | ||
|| Ratio || 1/1 || 15/14 || 8/7 || 5/4 || 4/3 || 7/5 || 3/2 || 8/5 || 7/4 || 13/7 || 2/1 || | || Ratio || 1/1 || 15/14 || 8/7 || 5/4 || 4/3 || 7/5 || 3/2 || 8/5 || 7/4 || 13/7 || 2/1 || | ||
These ratios generate the 2.3.5.7.13 subgroup. | |||
This is step 1, actually, because we have only one ratio mapped to each degree. In step 2, we add more by finding the intervals between the original intervals, as well as the intervals that appear by multiplying ratios, and end up with something like this: | This is step 1, actually, because we have only one ratio mapped to each degree. In step 2, we add more by finding the intervals between the original intervals, as well as the intervals that appear by multiplying ratios, and end up with something like this: | ||
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40/21 || 2/1 || | 40/21 || 2/1 || | ||
From this list, we can then figure out the commas that are tempered out simply by finding the differences between all the ratios that are mapped to the same degree. One drawback to this approach is that if the ratios chosen in step 1 have are not approximate with very high accuracy, when we get to step 2 we may still end up conflating ratios which bear little sonic resemblance. You can see in the above chart that 3\10 approximates 5/4, 6/5, and 16/13, and when tuned Just, all three of these ratios are quite sonically distinct. | From this list, we can then figure out the commas that are tempered out simply by finding the differences between all the ratios that are mapped to the same degree. Alternatively, we can use the subgroup val for the generated subgroup, in this case 2.3.5.7.13, and from that find a basis for the commas as 25/24, 28/27, 40/39 and 50/49. One drawback to this entire approach is that if the ratios chosen in step 1 have are not approximate with very high accuracy, when we get to step 2 we may still end up conflating ratios which bear little sonic resemblance. You can see in the above chart that 3\10 approximates 5/4, 6/5, and 16/13, and when tuned Just, all three of these ratios are quite sonically distinct. | ||
Another approach is to ignore individual dyads and instead concentrate on the largest and lowest subgroup of the harmonic series that can be well-approximated in the EDO. This approach has not yet been mathematically formalized, so deciding which subgroup of the harmonic series is the "best fit" for an EDO is currently somewhat a matter of personal discretion. One interpretation of 10-EDO based on this method, however, would be to treat it as a 2.7.13.15 subgroup, as 10-EDO provides very convincing matches to all of those harmonics. This leads to the following mapping: | Another approach is to ignore individual dyads and instead concentrate on the largest and lowest subgroup of the harmonic series that can be well-approximated in the EDO. This approach has not yet been mathematically formalized, so deciding which subgroup of the harmonic series is the "best fit" for an EDO is currently somewhat a matter of personal discretion. One interpretation of 10-EDO based on this method, however, would be to treat it as a 2.7.13.15 subgroup, as 10-EDO provides very convincing matches to all of those harmonics. This leads to the following mapping: | ||
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</table> | </table> | ||
<br /> | |||
These ratios generate the 2.3.5.7.13 subgroup.<br /> | |||
<br /> | <br /> | ||
This is step 1, actually, because we have only one ratio mapped to each degree. In step 2, we add more by finding the intervals between the original intervals, as well as the intervals that appear by multiplying ratios, and end up with something like this:<br /> | This is step 1, actually, because we have only one ratio mapped to each degree. In step 2, we add more by finding the intervals between the original intervals, as well as the intervals that appear by multiplying ratios, and end up with something like this:<br /> | ||
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<br /> | <br /> | ||
From this list, we can then figure out the commas that are tempered out simply by finding the differences between all the ratios that are mapped to the same degree. One drawback to this approach is that if the ratios chosen in step 1 have are not approximate with very high accuracy, when we get to step 2 we may still end up conflating ratios which bear little sonic resemblance. You can see in the above chart that 3\10 approximates 5/4, 6/5, and 16/13, and when tuned Just, all three of these ratios are quite sonically distinct.<br /> | From this list, we can then figure out the commas that are tempered out simply by finding the differences between all the ratios that are mapped to the same degree. Alternatively, we can use the subgroup val for the generated subgroup, in this case 2.3.5.7.13, and from that find a basis for the commas as 25/24, 28/27, 40/39 and 50/49. One drawback to this entire approach is that if the ratios chosen in step 1 have are not approximate with very high accuracy, when we get to step 2 we may still end up conflating ratios which bear little sonic resemblance. You can see in the above chart that 3\10 approximates 5/4, 6/5, and 16/13, and when tuned Just, all three of these ratios are quite sonically distinct.<br /> | ||
<br /> | <br /> | ||
Another approach is to ignore individual dyads and instead concentrate on the largest and lowest subgroup of the harmonic series that can be well-approximated in the EDO. This approach has not yet been mathematically formalized, so deciding which subgroup of the harmonic series is the &quot;best fit&quot; for an EDO is currently somewhat a matter of personal discretion. One interpretation of 10-EDO based on this method, however, would be to treat it as a 2.7.13.15 subgroup, as 10-EDO provides very convincing matches to all of those harmonics. This leads to the following mapping:<br /> | Another approach is to ignore individual dyads and instead concentrate on the largest and lowest subgroup of the harmonic series that can be well-approximated in the EDO. This approach has not yet been mathematically formalized, so deciding which subgroup of the harmonic series is the &quot;best fit&quot; for an EDO is currently somewhat a matter of personal discretion. One interpretation of 10-EDO based on this method, however, would be to treat it as a 2.7.13.15 subgroup, as 10-EDO provides very convincing matches to all of those harmonics. This leads to the following mapping:<br /> | ||