31edo: Difference between revisions

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In mode 16, the most closely-matched harmonics are the composite ones, 21 and 25. Of the other harmonics:

~~<ul><li>~~17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent.~~</li><li>~~19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp. 31edo could be considered a tuning of the 2.5.7.13.17.19 subgroup, on which it is consistent. ~~</li><li>~~23 is about as flat as 11. The chromatic semitone is about half a cent off from 23:22. 31edo could be considered a tuning of the 2.3.5.7.11.23 subgroup, on which it is consistent.~~</li><li>~~27 is quite flat, as it's 3^3 and the error from the meantone fifths accumulates.~~</li><li>~~29 and 31 are both ''very'' sharp, and intervals involving them are unlikely to play any major role.~~</li></ul>~~

* 17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent.

* 19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp. 31edo could be considered a tuning of the 2.5.7.13.17.19 subgroup, on which it is consistent.

* 23 is about as flat as 11. The chromatic semitone is about half a cent off from 23:22. 31edo could be considered a tuning of the 2.3.5.7.11.23 subgroup, on which it is consistent.

* 27 is quite flat, as it's 3^3 and the error from the meantone fifths accumulates.

* 29 and 31 are both ''very'' sharp, and intervals involving them are unlikely to play any major role.

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A large open list of modes (subsets) from 31edo that people have named: [[31edo modes]]. [http://en.wikipedia.org/wiki/Rothenberg_propriety Strictly proper] [[Strictly proper 7-note 31edo scales|7-note 31edo scales]] in the sense of [[David Rothenberg]]. Interesting (to somebody) [[9-note 31edo scales]]. See also [[31edo MOS scales]]. Some of the popular ones:

~~<ul><li>~~31-tone major: 5 5 3 5 5 5 3~~</li><li>~~Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3~~</li><li>~~Harmonic scale 8: 5 5 4 4 4 3 3 3~~</li><li>~~the [[Euler-Fokker genera]] (technically [[JI]] but representable in 31)~~</li></ul>~~

* 31-tone major: 5 5 3 5 5 5 3

* Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3

* Harmonic scale 8: 5 5 4 4 4 3 3 3

* the [[Euler-Fokker genera]] (technically [[JI]] but representable in 31)

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 In mode 16, the most closely-matched harmonics are the composite ones, 21 and 25. Of the other harmonics:
-<ul><li>17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent.</li><li>19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp. 31edo could be considered a tuning of the 2.5.7.13.17.19 subgroup, on which it is consistent. </li><li>23 is about as flat as 11. The chromatic semitone is about half a cent off from 23:22. 31edo could be considered a tuning of the 2.3.5.7.11.23 subgroup, on which it is consistent.</li><li>27 is quite flat, as it's 3^3 and the error from the meantone fifths accumulates.</li><li>29 and 31 are both ''very'' sharp, and intervals involving them are unlikely to play any major role.</li></ul>
+* 17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent.
+* 19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp. 31edo could be considered a tuning of the 2.5.7.13.17.19 subgroup, on which it is consistent.
+* 23 is about as flat as 11. The chromatic semitone is about half a cent off from 23:22. 31edo could be considered a tuning of the 2.3.5.7.11.23 subgroup, on which it is consistent.
+* 27 is quite flat, as it's 3^3 and the error from the meantone fifths accumulates.
+* 29 and 31 are both ''very'' sharp, and intervals involving them are unlikely to play any major role.
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 A large open list of modes (subsets) from 31edo that people have named: [[31edo modes]]. [http://en.wikipedia.org/wiki/Rothenberg_propriety Strictly proper] [[Strictly proper 7-note 31edo scales|7-note 31edo scales]] in the sense of [[David Rothenberg]]. Interesting (to somebody) [[9-note 31edo scales]]. See also [[31edo MOS scales]]. Some of the popular ones:
-<ul><li>31-tone major: 5 5 3 5 5 5 3</li><li>Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3</li><li>Harmonic scale 8: 5 5 4 4 4 3 3 3</li><li>the [[Euler-Fokker genera]] (technically [[JI]] but representable in 31)</li></ul>
+* 31-tone major: 5 5 3 5 5 5 3
+* Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3
+* Harmonic scale 8: 5 5 4 4 4 3 3 3
+* the [[Euler-Fokker genera]] (technically [[JI]] but representable in 31)
 {| class="wikitable"