810edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro}} | {{EDO intro}} | ||
== Theory == | |||
810 = 270 × 3, and 810edo has three copies of [[270edo]] in the 13-limit (and the 2.3.5.7.11.13.19 [[subgroup]]). It makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, only [[consistent]] to the [[9-odd-limit]]. [[11/9]], [[13/12]], [[13/9]], [[13/10]], and their [[octave complement]]s are all mapped inconsistently in this edo. | |||
As an equal temperament, it [[tempering out|tempers out]] [[4914/4913]] in the 17-limit; and [[2024/2023]], [[2737/2736]], and [[3520/3519]] in the 23-limit. Although it does quite well in these limits, it is way less efficient as [[270edo]]'s or [[540edo]]'s mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied essentially tempered chords are worth the load of all the extra notes. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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Since 810 factors into {{factorization|810}}, 810edo has subset edos {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405 }}. | Since 810 factors into {{factorization|810}}, 810edo has subset edos {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405 }}. | ||
== Regular temperament properties == | |||
{{ | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list|Comma List]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve Stretch (¢) | |||
! colspan="2" | Tuning Error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095, 4914/4913 | |||
| {{mapping| 810 1284 1881 2274 2802 2997 3311 }} | |||
| -0.0281 | |||
| 0.1025 | |||
| 6.92 | |||
|- | |||
| 2.3.5.7.11.13.17.19 | |||
| 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728, 4914/4913 | |||
| {{mapping| 810 1284 1881 2274 2802 2997 3311 3441 }} | |||
| -0.0324 | |||
| 0.0966 | |||
| 6.52 | |||
|- | |||
| 2.3.5.7.11.13.17.19.23 | |||
| 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728, 2024/2023, 2737/2736 | |||
| {{mapping| 810 1284 1881 2274 2802 2997 3311 3441 3664 }} | |||
| -0.0257 | |||
| 0.0930 | |||
| 6.28 | |||
|} | |||