13edo: Difference between revisions

: This revision was by author [[User:guest|guest]] and made on <tt>2011-07-18 20:05:24 UTC</tt>.<br>

: The original revision id was <tt>241856091</tt>.<br>

|| Degree || Cents || Approximate Ratios* ||

|| 0 || 0 || 1/1 ||

|| 1 || 92.3077 ||   ||

|| 2 || 184.6154 || 10/9, 9/8, 11/10 ||

|| 3 || 276.9231 || 13/11 ||

|| 4 || 369.2308 || 5/4, 16/13, 11/9 ||

|| 5 || 461.5385 || 13/10 ||

|| 6 || 553.84 || 11/8, 18/13 ||

|| 7 || 646.15 || 16/11, 13/9 ||

|| 8 || 738.46 || 20/13 ||

|| 9 || 830.77 || 8/5, 13/8, 18/11 ||

|| 10 || 923.08 || 22/13 ||

|| 11 || 1015.38 || 9/5, 16/9, 20/11 ||

|| 12 || 1107.69 ||   ||

|| 13 || 1200 || 2/1 ||

Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. The simplest MOS scale to support this pentad uses the 2nd degree (~185 cents) as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad).  

         &lt;td&gt;Approximate Ratios*&lt;br /&gt;

         &lt;td&gt;1/1&lt;br /&gt;

         &lt;td&gt;&lt;br /&gt;

         &lt;td&gt;10/9, 9/8, 11/10&lt;br /&gt;

         &lt;td&gt;13/11&lt;br /&gt;

         &lt;td&gt;5/4, 16/13, 11/9&lt;br /&gt;

         &lt;td&gt;13/10&lt;br /&gt;

         &lt;td&gt;11/8, 18/13&lt;br /&gt;

         &lt;td&gt;16/11, 13/9&lt;br /&gt;

         &lt;td&gt;20/13&lt;br /&gt;

         &lt;td&gt;8/5, 13/8, 18/11&lt;br /&gt;

         &lt;td&gt;22/13&lt;br /&gt;

         &lt;td&gt;9/5, 16/9, 20/11&lt;br /&gt;

         &lt;td&gt;&lt;br /&gt;

         &lt;td&gt;2/1&lt;br /&gt;

Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a &amp;quot;stack of 3rds&amp;quot; the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. The simplest MOS scale to support this pentad uses the 2nd degree (~185 cents) as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). &lt;br /&gt;