228edo: Difference between revisions
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{{Infobox ET | {{Infobox ET}} | ||
{{ED intro}} | |||
It is the first merger of [[12edo]] and [[19edo]], and its step size is the difference between 12edo's and 19edo's fifths. The equal temperament [[tempering out|tempers out]] the [[Pythagorean comma]], 531441/524288, in the 3-limit, and [[225/224]] and [[250047/250000]] in the 7-limit, so that it [[support]]s 7-limit [[compton]] temperament and in fact provides the [[optimal patent val]]. In the 11-limit it tempers out 225/224, [[441/440]], 1375/1372 and 4375/4356, so that it supports 11-limit compton. Aside from the Pythagorean comma, the 12-comma, it tempers out the [[enneadeca]] or 19-tone-comma, and this is reflected in the fact that 228 = 12 × 19. | |||
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=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|228}} | {{Harmonics in equal|228}} | ||
[[Category: | [[Category:Compton]] | ||
Latest revision as of 14:20, 20 February 2025
| ← 227edo | 228edo | 229edo → |
228 equal divisions of the octave (abbreviated 228edo or 228ed2), also called 228-tone equal temperament (228tet) or 228 equal temperament (228et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 228 equal parts of about 5.26 ¢ each. Each step represents a frequency ratio of 21/228, or the 228th root of 2.
It is the first merger of 12edo and 19edo, and its step size is the difference between 12edo's and 19edo's fifths. The equal temperament tempers out the Pythagorean comma, 531441/524288, in the 3-limit, and 225/224 and 250047/250000 in the 7-limit, so that it supports 7-limit compton temperament and in fact provides the optimal patent val. In the 11-limit it tempers out 225/224, 441/440, 1375/1372 and 4375/4356, so that it supports 11-limit compton. Aside from the Pythagorean comma, the 12-comma, it tempers out the enneadeca or 19-tone-comma, and this is reflected in the fact that 228 = 12 × 19.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.96 | -2.10 | -0.40 | +1.35 | +1.31 | +1.58 | +1.20 | +0.31 | +2.49 | -2.36 | -1.96 |
| Relative (%) | -37.1 | -40.0 | -7.7 | +25.7 | +25.0 | +30.0 | +22.9 | +5.8 | +47.3 | -44.8 | -37.2 | |
| Steps (reduced) |
361 (133) |
529 (73) |
640 (184) |
723 (39) |
789 (105) |
844 (160) |
891 (207) |
932 (20) |
969 (57) |
1001 (89) |
1031 (119) | |