318edo: Difference between revisions
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{{Infobox ET}} | |||
{{ED intro}} | |||
[[Category:Equal divisions of the octave]] | 318edo is [[contorted]] in both the 3-limit and the 5-limit, sharing the same mappings with [[53edo]]. Besides, it shares its representations of the 11th and 17th [[Overtone series|harmonics]] with [[159edo]]. However, compared to 159edo, the [[patent val]]s differ on the mappings for 7, 13, and 19. | ||
In the 5-limit, it tempers out the same commas as 53edo, including the [[32805/32768|schisma (32805/32768)]], the [[15625/15552|kleisma (15625/15552)]], the [[amity comma|amity comma (1600000/1594323)]], the [[semicomma|semicomma (2109375/2097152)]], the [[vulture comma|vulture comma (10485760000/10460353203)]], etc. In the 7-limit it tempers out the stearnsma (118098/117649) and 589824/588245. In the 11-limit it tempers out the swetisma (540/539), the [[4000/3993|wizardharry (4000/3993)]], the [[9801/9800|kalisma (9801/9800)]] and the [[nexuma|nexus comma (1771561/1769472)]]. In the 13-limit, 1575/1573, 2080/2079, it tempers out the [[4096/4095|schismina (4096/4095)]], and the [[cantonisma|cantonisma (10985/10976)]]. | |||
At only slightly more than 3.5{{c}}, the step size of 318edo is really close to being [[unnoticeable comma|unnoticeable]] as is the case with other mega-EDOs in this vicinity, so the steps themselves run a pretty high risk of blending completely into one another. | |||
=== Prime harmonics === | |||
{{harmonics in equal|318}} | |||
=== Subsets and supersets === | |||
318 = 2 × 3 × 53, and has subset edos {{EDOs|1, 2, 3, 6, 53, 106, 159}}. | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | |||
[[Category:53edo]] | |||
[[Category:159edo]] | |||
[[Category:Nexus]] | [[Category:Nexus]] | ||
Latest revision as of 22:50, 20 February 2025
| ← 317edo | 318edo | 319edo → |
318 equal divisions of the octave (abbreviated 318edo or 318ed2), also called 318-tone equal temperament (318tet) or 318 equal temperament (318et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 318 equal parts of about 3.77 ¢ each. Each step represents a frequency ratio of 21/318, or the 318th root of 2.
318edo is contorted in both the 3-limit and the 5-limit, sharing the same mappings with 53edo. Besides, it shares its representations of the 11th and 17th harmonics with 159edo. However, compared to 159edo, the patent vals differ on the mappings for 7, 13, and 19.
In the 5-limit, it tempers out the same commas as 53edo, including the schisma (32805/32768), the kleisma (15625/15552), the amity comma (1600000/1594323), the semicomma (2109375/2097152), the vulture comma (10485760000/10460353203), etc. In the 7-limit it tempers out the stearnsma (118098/117649) and 589824/588245. In the 11-limit it tempers out the swetisma (540/539), the wizardharry (4000/3993), the kalisma (9801/9800) and the nexus comma (1771561/1769472). In the 13-limit, 1575/1573, 2080/2079, it tempers out the schismina (4096/4095), and the cantonisma (10985/10976).
At only slightly more than 3.5 ¢, the step size of 318edo is really close to being unnoticeable as is the case with other mega-EDOs in this vicinity, so the steps themselves run a pretty high risk of blending completely into one another.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.07 | -1.41 | +0.99 | -0.37 | +0.98 | +0.70 | +0.60 | -1.86 | +0.61 | -1.64 |
| Relative (%) | +0.0 | -1.8 | -37.3 | +26.1 | -9.9 | +26.0 | +18.7 | +15.9 | -49.3 | +16.2 | -43.4 | |
| Steps (reduced) |
318 (0) |
504 (186) |
738 (102) |
893 (257) |
1100 (146) |
1177 (223) |
1300 (28) |
1351 (79) |
1438 (166) |
1545 (273) |
1575 (303) | |
Subsets and supersets
318 = 2 × 3 × 53, and has subset edos 1, 2, 3, 6, 53, 106, 159.