53edo: Difference between revisions
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{{Infobox ET | {{Infobox ET | ||
| Prime factorization = 53 | | Prime factorization = 53 (prime) | ||
| Step size = 22.64151¢ | | Step size = 22.64151¢ | ||
| Fifth = 31\53 | | Fifth = 31\53 (702¢) | ||
| Major 2nd = 9\53 | | Major 2nd = 9\53 (204¢) | ||
| | | Semitones = 5:4 (113¢ : 91¢) | ||
| | | Consistency = 9 | ||
| Monotonicity = 23 | |||
}} | }} | ||
{{Wikipedia| 53 equal temperament }} | {{Wikipedia| 53 equal temperament }} | ||
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For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. | For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. | ||
== Relationship to | == Relationship to 12edo == | ||
Whereas | Whereas 12edo has a circle of twelve 5ths, 53edo has a spiral of twelve 5ths (since 31\53 is on the 7\12 kite in the scale tree). This shows 53edo in a 12edo-friendly format. Excellent for introducing 53edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates. | ||
[[File:53-edo spiral.png|702x702px]] | [[File:53-edo spiral.png|702x702px]] | ||
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One notable property of | One notable property of 53edo is that it offers good approximations for both just and Pythagorean major thirds. | ||
The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so | The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53edo can also be used for 7-limit harmony, tempering out the [[septimal kleisma]], 225/224. | ||
=== 15-odd-limit interval mappings === | === 15-odd-limit interval mappings === | ||
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53et is lower in relative error than any previous equal temperaments in the 3-, 5-, and 13-limit. The next ETs better in these subgroups are 306, 118, and 58, respectively. It is even more prominent in the 2.3.5.7.13.19 and 2.3.5.7.13.19.23 subgroups, and the next ET better in either subgroup is 130. | 53et is lower in relative error than any previous equal temperaments in the 3-, 5-, and 13-limit. The next ETs doing better in these subgroups are 306, 118, and 58, respectively. It is even more prominent in the 2.3.5.7.13.19 and 2.3.5.7.13.19.23 subgroups, and the next ET doing better in either subgroup is 130. | ||
=== Linear temperaments === | === Linear temperaments === | ||
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| 249.06 | | 249.06 | ||
| 15/13 | | 15/13 | ||
| [[Hemischis]] | | [[Hemischis]] / [[hemigari]] | ||
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| 1 | | 1 | ||