3ed11/9: Difference between revisions

← 2ed11/9 3ed11/9 4ed11/9 →
Prime factorization	3 (prime)
Step size	115.803 ¢
Octave	10\3ed11/9 (1158.03 ¢); (semiconvergent)
Twelfth	16\3ed11/9 (1852.84 ¢); (semiconvergent)
Consistency limit	3
Distinct consistency limit	3

← Older edit

VisualWikitext

Latest revision as of 02:45, 6 March 2026

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, as is 15²:19²:21².

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¢

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

↑ Wikipedia | Strike tone

[1] Wikipedia | Strike tone

[1]

@@ Line 2: / Line 2: @@
 {{ED intro}}
-steps of this temperament is an extremely close approximation of 81:121, having only 0.3% relative error. 11 steps is obviously 81:169, so 81:121:169 (9²:11²:13²) 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 is also very well approximated, but 9²:15² has around 25% relative error.
+<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>.
-A simpler name for it is 3ed11/9.
+===Approximation of odd square harmonics relative to 9<sup>2</sup>===
-{{todo|inline=1|format|add values}}
+{{todo|inline=1|formatting}}
-===Approximation of odd square harmonics relative to 9²===
-²:9²
+ratio | steps | relative error | absolute error
-²:9²
+<sup>2</sup>:9<sup>2</sup> | -66 | -30.4% | -35.2¢
-²:9²
+<sup>2</sup>:9<sup>2</sup> | -33 | -15.2% | -17.6¢
-²:9²
+<sup>2</sup>:9<sup>2</sup> | -18 | -42.5% | -49.3¢
-²:9²
+<sup>2</sup>:9<sup>2</sup> | -8 | -48.6% | -56.3¢
-²:9²
+<sup>2</sup>:9<sup>2</sup> | 0 | 0% | 0¢
-²:9²
+<sup>2</sup>:9<sup>2</sup> | 6 | 0% | 0¢
-²:9²
+<sup>2</sup>:9<sup>2</sup> | 11 | -0.51% | -0.59¢
-²:9²
+<sup>2</sup>:9<sup>2</sup> | 15 | -27.4% | -32.1¢
-²:9²
+<sup>2</sup>:9<sup>2</sup> | 19 | -1.6% | -1.8¢
+<sup>2</sup>:9<sup>2</sup> | 22 | -34.2% | -39.5¢
-===Approximation of odd square harmonics===
+<sup>2</sup>:9<sup>2</sup> | 25 | -33.4% | -38.7¢
-²
+<sup>2</sup>:9<sup>2</sup> | 28 | -5.4% | -6.3¢
-²
+[[9ed11/9]] is a possible correction for 15, 19, and 21.
-²
-²
-²
-²
-²
-²
-²

3ed11/9: Difference between revisions

Latest revision as of 02:45, 6 March 2026

Approximation of odd square harmonics relative to 92

Navigation menu

Search

Approximation of odd square harmonics relative to 9²