3ed11/9: Difference between revisions

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11 steps of this temperament is an extremely close approximation of 9²:~~13²~~, having only 0.5% relative error. 6 steps is exactly 9²:~~11²~~ (since 3 steps is 9:11), so 9²:~~11²~~:~~13²~~ (81:121:169) is well approximated, which represents the approximate 2:3:4 created by overtones of chimes.<ref>[https://en.wikipedia.org/wiki/Strike_tone#Tuning_a_bell Wikipedia | ''Strike tone'']</ref>

11 steps of this temperament is an extremely close approximation of 92:132, having only 0.5% relative error. 6 steps is exactly 92:112 (since 3 steps is 9:11), so 92:112:132 (81:121:169) is well approximated, which represents the approximate 2:3:4 created by overtones of chimes.<ref>[https://en.wikipedia.org/wiki/Strike_tone#Tuning_a_bell Wikipedia | ''Strike tone'']</ref>

9²:~~11²~~:~~13²~~:~~17²~~:~~23²~~ is also very well approximated, as is ~~19²~~:~~21²~~.

92:112:132:172:232 is also very well approximated, as is 152:192:212.

===Approximation of odd square harmonics relative to 9²===

===Approximation of odd square harmonics relative to 92===

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ratio | steps | relative error | absolute error

1²:9² | -66 | -30.4% | -35.2¢

12:92 | -66 | -30.4% | -35.2¢

3²:9² | -33 | -15.2% | -17.6¢

32:92 | -33 | -15.2% | -17.6¢

5²:9² | -18 | -42.5% | -49.3¢

52:92 | -18 | -42.5% | -49.3¢

7²:9² | -8 | -48.6% | -56.3¢

72:92 | -8 | -48.6% | -56.3¢

9²:9² | 0 | 0% | 0¢

92:92 | 0 | 0% | 0¢

~~11²~~:9² | 6 | 0% | 0¢

112:92 | 6 | 0% | 0¢

~~13²~~:9² | 11 | -0.51% | -0.59¢

132:92 | 11 | -0.51% | -0.59¢

~~15²~~:9² | 15 | -27.4% | -32.1¢

152:92 | 15 | -27.4% | -32.1¢

~~17²~~:9² | 19 | -1.6% | -1.8¢

172:92 | 19 | -1.6% | -1.8¢

~~19²~~:9² | 22 | -34.2% | -39.5¢

192:92 | 22 | -34.2% | -39.5¢

~~21²~~:9² | 25 | -33.4% | -38.7¢

212:92 | 25 | -33.4% | -38.7¢

~~23²~~:9² | 28 | -5.4% | -6.3¢

232:92 | 28 | -5.4% | -6.3¢

[[9ed11/9]] is a possible correction for 15, 19, and 21.

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 {{ED intro}}
-steps of this temperament is an extremely close approximation of 9²:13², having only 0.5% relative error. 6 steps is exactly 9²:11² (since 3 steps is 9:11), so 9²:11²:13² (81:121:169) is well approximated, which represents the approximate 2:3:4 created by overtones of chimes.<ref>[https://en.wikipedia.org/wiki/Strike_tone#Tuning_a_bell Wikipedia | ''Strike tone'']</ref>
+steps of this temperament is an extremely close approximation of 9<sup>2</sup>:13<sup>2</sup>, having only 0.5% relative error. 6 steps is exactly 9<sup>2</sup>:11<sup>2</sup> (since 3 steps is 9:11), so 9<sup>2</sup>:11<sup>2</sup>:13<sup>2</sup> (81:121:169) is well approximated, which represents the approximate 2:3:4 created by overtones of chimes.<ref>[https://en.wikipedia.org/wiki/Strike_tone#Tuning_a_bell Wikipedia | ''Strike tone'']</ref>
-²:11²:13²:17²:23² is also very well approximated, as is 19²:21².
+<sup>2</sup>:11<sup>2</sup>:13<sup>2</sup>:17<sup>2</sup>:23<sup>2</sup> is also very well approximated, as is 15<sup>2</sup>:19<sup>2</sup>:21<sup>2</sup>.
-===Approximation of odd square harmonics relative to 9²===
+===Approximation of odd square harmonics relative to 9<sup>2</sup>===
 {{todo|inline=1|formatting}}
@@ Line 12: / Line 12: @@
 ratio | steps | relative error | absolute error
-²:9² | -66 | -30.4% | -35.2¢
+<sup>2</sup>:9<sup>2</sup> | -66 | -30.4% | -35.2¢
-²:9² | -33 | -15.2% | -17.6¢
+<sup>2</sup>:9<sup>2</sup> | -33 | -15.2% | -17.6¢
-²:9² | -18 | -42.5% | -49.3¢
+<sup>2</sup>:9<sup>2</sup> | -18 | -42.5% | -49.3¢
-²:9² | -8 | -48.6% | -56.3¢
+<sup>2</sup>:9<sup>2</sup> | -8 | -48.6% | -56.3¢
-²:9² | 0 | 0% | 0¢
+<sup>2</sup>:9<sup>2</sup> | 0 | 0% | 0¢
-²:9² | 6 | 0% | 0¢
+<sup>2</sup>:9<sup>2</sup> | 6 | 0% | 0¢
-²:9² | 11 | -0.51% | -0.59¢
+<sup>2</sup>:9<sup>2</sup> | 11 | -0.51% | -0.59¢
-²:9² | 15 | -27.4% | -32.1¢
+<sup>2</sup>:9<sup>2</sup> | 15 | -27.4% | -32.1¢
-²:9² | 19 | -1.6% | -1.8¢
+<sup>2</sup>:9<sup>2</sup> | 19 | -1.6% | -1.8¢
-²:9² | 22 | -34.2% | -39.5¢
+<sup>2</sup>:9<sup>2</sup> | 22 | -34.2% | -39.5¢
-²:9² | 25 | -33.4% | -38.7¢
+<sup>2</sup>:9<sup>2</sup> | 25 | -33.4% | -38.7¢
-²:9² | 28 | -5.4% | -6.3¢
+<sup>2</sup>:9<sup>2</sup> | 28 | -5.4% | -6.3¢
 [[9ed11/9]] is a possible correction for 15, 19, and 21.