3ed11/9: Difference between revisions
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11 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> | |||
9<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 9<sup>2</sup>=== | |||
{{todo|inline=1|formatting}} | |||
ratio | steps | relative error | absolute error | |||
1<sup>2</sup>:9<sup>2</sup> | -66 | -30.4% | -35.2¢ | |||
3<sup>2</sup>:9<sup>2</sup> | -33 | -15.2% | -17.6¢ | |||
5<sup>2</sup>:9<sup>2</sup> | -18 | -42.5% | -49.3¢ | |||
7<sup>2</sup>:9<sup>2</sup> | -8 | -48.6% | -56.3¢ | |||
9<sup>2</sup>:9<sup>2</sup> | 0 | 0% | 0¢ | |||
11<sup>2</sup>:9<sup>2</sup> | 6 | 0% | 0¢ | |||
13<sup>2</sup>:9<sup>2</sup> | 11 | -0.51% | -0.59¢ | |||
15<sup>2</sup>:9<sup>2</sup> | 15 | -27.4% | -32.1¢ | |||
17<sup>2</sup>:9<sup>2</sup> | 19 | -1.6% | -1.8¢ | |||
19<sup>2</sup>:9<sup>2</sup> | 22 | -34.2% | -39.5¢ | |||
21<sup>2</sup>:9<sup>2</sup> | 25 | -33.4% | -38.7¢ | |||
23<sup>2</sup>:9<sup>2</sup> | 28 | -5.4% | -6.3¢ | |||
[[9ed11/9]] is a possible correction for 15, 19, and 21. | |||