216edo: Difference between revisions

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'''216edo''' is the [[EDO|equal division of the octave]] into 216 parts of 5.5556 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 13-limit: <216 342 502 606 747 799| (patent val), <216 343 502 607 748 800| (216bdef), <216 342 501 606 747 799| (216c), and <216 342 502 607 747 799| (216d). Using the patent val, it tempers out 531441/524288 and 1990656/1953125 in the 5-limit; 126/125, 1029/1024, and 118098/117649 in the 7-limit; 243/242, 3388/3375, 41503/41472, and 43923/43904 in the 11-limit; 676/675, 847/845, 1287/1280, 1701/1690, and 1716/1715 in the 13-limit. Using the 216bdef val, it tempers out 2048/2025 and |1 -46 31> in the 5-limit; 3136/3125, 4000/3969, and 40353607/39858075 in the 7-limit; 2560/2541, 3025/3024, 3388/3375, and 12005/11979 in the 11-limit; 325/324, 364/363, 640/637, and 1716/1715 in the 13-limit. Using the 216c val, it tempers out 15625/15552 and 531441/524288 in the 5-limit; 225/224, 1029/1024, and 4375/4374 in the 7-limit; 243/242, 385/384, 441/440, and 4000/3993 in the 11-limit; 2200/2197 and 2205/2197 in the 13-limit. Using the 216d val, it tempers out 2430/2401, 3136/3125, and 531441/524288 in the 7-limit; 176/175, 243/242, 1375/1372, and 131769/131072 in the 11-limit; 676/675, 1188/1183, 1287/1280, and 3042/3025 in the 13-limit.

[[~~Category~~:~~Edo~~]]

216edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with many mappings possible for the 13-limit. The following four will be discussed here: {{val| 216 342 502 606 747 799 }} ([[patent val]]), {{val| 216 '''343''' 502 '''607''' '''748''' '''800''' }} (216bdef), {{val| 216 342 '''501''' 606 747 799 }} (216c), and {{val| 216 342 502 '''607''' 747 799 }} (216d).

The 216c val is [[Enfactoring|enfactored]] in the 11-limit, and it happens to be of the best accuracy. Like [[72edo|72]], it [[Tempering out|tempers out]] [[15625/15552]] and [[531441/524288]] in the 5-limit; [[225/224]], [[1029/1024]], and [[4375/4374]] in the 7-limit; [[243/242]], [[385/384]], [[441/440]], and [[4000/3993]] in the 11-limit. However, unlike 72, it tempers out [[2200/2197]] and 2205/2197 in the 13-limit, and is an good correction from 72edo's flat [[13/1|13th harmonic]]. This val supports [[phicordial]], a [[weak extension]] of [[miracle]].

The 216bdef val chooses the sharp mapping for each of the harmonics, so it is the opposite of 216c in terms of tuning. It tempers out [[2048/2025]] and {{monzo| 1 -46 31 }} in the 5-limit; [[3136/3125]], [[4000/3969]], and 40353607/39858075 in the 7-limit; [[2560/2541]], [[3025/3024]], [[3388/3375]], and 12005/11979 in the 11-limit; [[325/324]], [[364/363]], [[640/637]], and [[1716/1715]] in the 13-limit.

Using the patent val, it tempers out 531441/524288 and 1990656/1953125 in the 5-limit; [[126/125]], [[1029/1024]], and 118098/117649 in the 7-limit; 243/242, [[3388/3375]], [[41503/41472]], and 43923/43904 in the 11-limit; [[676/675]], [[847/845]], [[1287/1280]], 1701/1690, and [[1716/1715]] in the 13-limit.

Using the 216d val, it tempers out [[2430/2401]], 3136/3125, and 531441/524288 in the 7-limit; [[176/175]], 243/242, 1375/1372, and [[131769/131072]] in the 11-limit; 676/675, [[1188/1183]], 1287/1280, and 3042/3025 in the 13-limit.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 216 factors into 2<sup>3</sup> × 3<sup>3</sup>, 216edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, and 108 }}.

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-'''216edo''' is the [[EDO|equal division of the octave]] into 216 parts of 5.5556 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 13-limit: &lt;216 342 502 606 747 799| (patent val), &lt;216 343 502 607 748 800| (216bdef), &lt;216 342 501 606 747 799| (216c), and &lt;216 342 502 607 747 799| (216d). Using the patent val, it tempers out 531441/524288 and 1990656/1953125 in the 5-limit; 126/125, 1029/1024, and 118098/117649 in the 7-limit; 243/242, 3388/3375, 41503/41472, and 43923/43904 in the 11-limit; 676/675, 847/845, 1287/1280, 1701/1690, and 1716/1715 in the 13-limit. Using the 216bdef val, it tempers out 2048/2025 and |1 -46 31&gt; in the 5-limit; 3136/3125, 4000/3969, and 40353607/39858075 in the 7-limit; 2560/2541, 3025/3024, 3388/3375, and 12005/11979 in the 11-limit; 325/324, 364/363, 640/637, and 1716/1715 in the 13-limit. Using the 216c val, it tempers out 15625/15552 and 531441/524288 in the 5-limit; 225/224, 1029/1024, and 4375/4374 in the 7-limit; 243/242, 385/384, 441/440, and 4000/3993 in the 11-limit; 2200/2197 and 2205/2197 in the 13-limit. Using the 216d val, it tempers out 2430/2401, 3136/3125, and 531441/524288 in the 7-limit; 176/175, 243/242, 1375/1372, and 131769/131072 in the 11-limit; 676/675, 1188/1183, 1287/1280, and 3042/3025 in the 13-limit.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Edo]]
+edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with many mappings possible for the 13-limit. The following four will be discussed here: {{val| 216 342 502 606 747 799 }} ([[patent val]]), {{val| 216 '''343''' 502 '''607''' '''748''' '''800''' }} (216bdef), {{val| 216 342 '''501''' 606 747 799 }} (216c), and {{val| 216 342 502 '''607''' 747 799 }} (216d).
+The 216c val is [[Enfactoring|enfactored]] in the 11-limit, and it happens to be of the best accuracy. Like [[72edo|72]], it [[Tempering out|tempers out]] [[15625/15552]] and [[531441/524288]] in the 5-limit; [[225/224]], [[1029/1024]], and [[4375/4374]] in the 7-limit; [[243/242]], [[385/384]], [[441/440]], and [[4000/3993]] in the 11-limit. However, unlike 72, it tempers out [[2200/2197]] and 2205/2197 in the 13-limit, and is an good correction from 72edo's flat [[13/1|13th harmonic]]. This val supports [[phicordial]], a [[weak extension]] of [[miracle]].
+The 216bdef val chooses the sharp mapping for each of the harmonics, so it is the opposite of 216c in terms of tuning. It tempers out [[2048/2025]] and {{monzo| 1 -46 31 }} in the 5-limit; [[3136/3125]], [[4000/3969]], and 40353607/39858075 in the 7-limit; [[2560/2541]], [[3025/3024]], [[3388/3375]], and 12005/11979 in the 11-limit; [[325/324]], [[364/363]], [[640/637]], and [[1716/1715]] in the 13-limit.
+Using the patent val, it tempers out 531441/524288 and 1990656/1953125 in the 5-limit; [[126/125]], [[1029/1024]], and 118098/117649 in the 7-limit; 243/242, [[3388/3375]], [[41503/41472]], and 43923/43904 in the 11-limit; [[676/675]], [[847/845]], [[1287/1280]], 1701/1690, and [[1716/1715]] in the 13-limit.
+Using the 216d val, it tempers out [[2430/2401]], 3136/3125, and 531441/524288 in the 7-limit; [[176/175]], 243/242, 1375/1372, and [[131769/131072]] in the 11-limit; 676/675, [[1188/1183]], 1287/1280, and 3042/3025 in the 13-limit.
+=== Odd harmonics ===
+{{Harmonics in equal|216}}
+=== Subsets and supersets ===
+Since 216 factors into 2<sup>3</sup> × 3<sup>3</sup>, 216edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, and 108 }}.