29edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 215339230 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 215349154 - 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:xenwolf|xenwolf]] and made on <tt>2011-03-30 03:10:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>215349154</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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29edo divides the 2:1 octave into 29 equal steps of approximately 41.37931 cents. | 29edo divides the 2:1 octave into 29 equal steps of approximately 41.37931 cents. | ||
29 is the lowest edo which approximates the 3:2 just fifth more accurately than 12edo: 3/2 = 701.955... cents; 17 degrees of 29edo = 703.448... cents. Since the fifth is slightly sharp, 29edo is a [[positive temperament]] -- a Superpythagorean instead of a Meantone system. | 29 is the lowest edo which approximates the [[3_2|3:2]] just fifth more accurately than [[12edo]]: 3/2 = 701.955... cents; 17 degrees of 29edo = 703.448... cents. Since the fifth is slightly sharp, 29edo is a [[positive temperament]] -- a Superpythagorean instead of a Meantone system. | ||
The 3 is the only harmonic, of the intelligibly low ones anyway, that 29-edo approximates, and it does so quite well. Hence one possible use for 29edo is as an equally tempered pythagorean scale. However, it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 chord, the [[The Archipelago|barbados triad]] 1-13/10-3/2, the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 triad and the 1-13/11-3/2 triad. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas. | The 3 is the only harmonic, of the intelligibly low ones anyway, that 29-edo approximates, and it does so quite well. Hence one possible use for 29edo is as an equally tempered pythagorean scale. However, it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 chord, the [[The Archipelago|barbados triad]] 1-13/10-3/2, the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 triad and the 1-13/11-3/2 triad. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas. | ||
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29edo divides the 2:1 octave into 29 equal steps of approximately 41.37931 cents.<br /> | 29edo divides the 2:1 octave into 29 equal steps of approximately 41.37931 cents.<br /> | ||
<br /> | <br /> | ||
29 is the lowest edo which approximates the 3:2 just fifth more accurately than 12edo: 3/2 = 701.955... cents; 17 degrees of 29edo = 703.448... cents. Since the fifth is slightly sharp, 29edo is a <a class="wiki_link" href="/positive%20temperament">positive temperament</a> -- a Superpythagorean instead of a Meantone system. <br /> | 29 is the lowest edo which approximates the <a class="wiki_link" href="/3_2">3:2</a> just fifth more accurately than <a class="wiki_link" href="/12edo">12edo</a>: 3/2 = 701.955... cents; 17 degrees of 29edo = 703.448... cents. Since the fifth is slightly sharp, 29edo is a <a class="wiki_link" href="/positive%20temperament">positive temperament</a> -- a Superpythagorean instead of a Meantone system. <br /> | ||
<br /> | <br /> | ||
The 3 is the only harmonic, of the intelligibly low ones anyway, that 29-edo approximates, and it does so quite well. Hence one possible use for 29edo is as an equally tempered pythagorean scale. However, it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 chord, the <a class="wiki_link" href="/The%20Archipelago">barbados triad</a> 1-13/10-3/2, the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 triad and the 1-13/11-3/2 triad. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.<br /> | The 3 is the only harmonic, of the intelligibly low ones anyway, that 29-edo approximates, and it does so quite well. Hence one possible use for 29edo is as an equally tempered pythagorean scale. However, it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 chord, the <a class="wiki_link" href="/The%20Archipelago">barbados triad</a> 1-13/10-3/2, the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 triad and the 1-13/11-3/2 triad. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.<br /> | ||