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 is also very well approximated | 9²:11²:13²:17²:23² is also very well approximated. | ||
===Approximation of odd square harmonics relative to 9²=== | ===Approximation of odd square harmonics relative to 9²=== | ||
{{todo|inline=1|formatting}} | |||
ratio | steps | relative error | absolute error | |||
1²:9² | -66 | -30.4% | -35.2¢ | |||
3²:9² | -33 | -15.2% | -17.6¢ | |||
5²:9² | -18 | -42.5% | -49.3¢ | |||
7²:9² | -8 | -48.6% | -56.3¢ | |||
9²:9² | 0 | 0% | 0¢ | |||
11²:9² | 6 | 0% | 0¢ | |||
9² | 13²:9² | 11 | -0.51% | -0.59¢ | ||
15²:9² | 15 | -27.4% | -32.1¢ | |||
17²:9² | 19 | -1.6% | -1.8¢ | |||
19²:9² | 22 | -34.2% | -39.5¢ | |||
21²:9² | 25 | -33.4% | -38.7¢ | |||
23²:9² | 28 | -5.4% | -6.3¢ | |||
Revision as of 22:37, 19 July 2025
| ← 2ed11/9 | 3ed11/9 | 4ed11/9 → |
(semiconvergent)
(semiconvergent)
3 equal divisions of 11/9 (abbreviated 3ed11/9) is a nonoctave tuning system that divides the interval of 11/9 into 3 equal parts of about 116 ¢ each. Each step represents a frequency ratio of (11/9)1/3, or the cube root of 11/9.
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.[1]
9²:11²:13²:17²:23² is also very well approximated.
Approximation of odd square harmonics relative to 9²
|
Todo: formatting |
ratio | steps | relative error | absolute error
1²:9² | -66 | -30.4% | -35.2¢
3²:9² | -33 | -15.2% | -17.6¢
5²:9² | -18 | -42.5% | -49.3¢
7²:9² | -8 | -48.6% | -56.3¢
9²:9² | 0 | 0% | 0¢
11²:9² | 6 | 0% | 0¢
13²:9² | 11 | -0.51% | -0.59¢
15²:9² | 15 | -27.4% | -32.1¢
17²:9² | 19 | -1.6% | -1.8¢
19²:9² | 22 | -34.2% | -39.5¢
21²:9² | 25 | -33.4% | -38.7¢
23²:9² | 28 | -5.4% | -6.3¢