324edo: Difference between revisions
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324edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps, making it a [[dual-fifth system]], with the flat fifth being the 700-cent fifth coming from 12edo, and the sharp fifth coming from 162edo. | 324edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps, making it a [[dual-fifth system]], with the flat fifth being the 700-cent fifth coming from 12edo, and the sharp fifth coming from 162edo. | ||
Latest revision as of 17:19, 20 February 2025
| ← 323edo | 324edo | 325edo → |
324 equal divisions of the octave (abbreviated 324edo or 324ed2), also called 324-tone equal temperament (324tet) or 324 equal temperament (324et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 324 equal parts of about 3.7 ¢ each. Each step represents a frequency ratio of 21/324, or the 324th root of 2.
324edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps, making it a dual-fifth system, with the flat fifth being the 700-cent fifth coming from 12edo, and the sharp fifth coming from 162edo.
It is nonetheless an excellent 2.9.15.21.11.13 subgroup tuning.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.75 | -1.13 | +1.54 | -0.21 | +0.53 | +0.21 | +0.62 | -1.25 | -1.22 | -0.41 | +1.36 |
| Relative (%) | +47.2 | -30.5 | +41.7 | -5.6 | +14.4 | +5.8 | +16.7 | -33.8 | -32.9 | -11.1 | +36.6 | |
| Steps (reduced) |
514 (190) |
752 (104) |
910 (262) |
1027 (55) |
1121 (149) |
1199 (227) |
1266 (294) |
1324 (28) |
1376 (80) |
1423 (127) |
1466 (170) | |
Subsets and supersets
Since 324 factors into 22 × 34, 324edo has subset edos 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, and 162. 648edo, which doubles it, gives a possible correction to its harmonic 3.