223edo: Difference between revisions

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~~=<span style="color: #2230ff; font-family: 'Times New Roman',Times,serif; font-size: 113%;">223 equal divisions per octave</span>=~~

'''223edo''' is the [[EDO|equal division of the octave]] into 223 parts of 5.38117 [[cent]]s each. It contains an excellent proportion of Hornbostel Temperament (via 7L2s), between Square root of Pi (184\223), Aureus interval (34/21 in 155\223) and the 6/5 interval (58\223). It is inconsistent to the 5-limit and higher limit, with three mappings possible for the 5-limit: <223 353 518| (optimal patent val), <223 354 518| (223b), and <223 353 517| (223c). Using the optimal patent val, it tempers out the [[Würschmidt family|würschmidt comma]], 393216/390625 and 22876792454961/21990232555520 in the 5-limit; 2401/2400, 3136/3125, and 14348907/14000000 in the 7-limit. Using the 223b val, it tempers out the [[Kleismic family|kleisma]], 15625/15552 and |58, -38, 1> in the 5-limit; 245/243, 3136/3125, and 67108864/66706983 in the 7-limit. Using the 223c val, it tempers out the [[Graviton|gravity comma]], 129140163/128000000 and 35595703125/34359738368 in the 5-limit; 4375/4374, 33075/32768, and 78125/76832 in the 7-limit.

223edo is the ~~48th~~ [[~~prime EDO~~]].

223edo contains an excellent proportion of [[hornbostel]] temperament (via [[7L 2s]]), between square root of π (184\223), Aureus interval (34/21 in 155\223) and the 6/5 interval (58\223). It is in[[consistent]] to the [[5-odd-limit]] and higher limit, with three mappings possible for the 5-limit:

* {{val| 223 353 518 }} ([[patent val]]),

* {{val| 223 '''354''' 518 }} (223b),

* {{val| 223 353 '''517''' }} (223c).

[[~~Category:Edo~~]]

Using the patent val, it tempers out 393216/390625 ([[würschmidt comma]]) and 22876792454961/21990232555520 in the 5-limit; 2401/2400, 3136/3125, and 14348907/14000000 in the 7-limit; 243/242, 441/440, 5632/5625, and 1449459/1433600 in the 11-limit; 847/845, 1188/1183, 1287/1280, and 1573/1568 in the 13-limit.

[[~~Category:Prime EDO~~]]

Using the 223be val, it tempers out 15625/15552 ([[kleisma]]) and {{monzo| 58 -38 1 }} in the 5-limit; 245/243, 3136/3125, and 67108864/66706983 in the 7-limit; 3025/3024, 3388/3375, 4375/4356, and 65536/65219 in the 11-limit; 352/351, 1001/1000, 2197/2178, and 2704/2695 in the 13-limit. Using the 223bef val, it tempers out 196/195, 325/324, 364/363, 625/624, and 49152/49049 in the 13-limit.

Using the 223c val, it tempers out the 129140163/128000000 ([[graviton]]) and 35595703125/34359738368 in the 5-limit; 4375/4374, 33075/32768, and 78125/76832 in the 7-limit; 243/242, 385/384, and 4000/3993 in the 11-limit; 1188/1183, 1573/1568, 1625/1617, 1716/1715, and 3159/3136 in the 13-limit.

Using the 223e val, it tempers out 1944/1925, 2835/2816, and 4000/3993 in the 11-limit; 364/363, 1001/1000, 1701/1690, and 1716/1715 in the 13-limit.

=== Odd harmonics ===

=== Subsets and supersets ===

223edo is the 48th [[prime edo]].

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-=<span style="color: #2230ff; font-family: 'Times New Roman',Times,serif; font-size: 113%;">223 equal divisions per octave</span>=
+{{Infobox ET}}
-'''223edo''' is the [[EDO|equal division of the octave]] into 223 parts of 5.38117 [[cent]]s each. It contains an excellent proportion of Hornbostel Temperament (via 7L2s), between Square root of Pi (184\223), Aureus interval (34/21 in 155\223) and the 6/5 interval (58\223). It is inconsistent to the 5-limit and higher limit, with three mappings possible for the 5-limit: &lt;223 353 518| (optimal patent val), &lt;223 354 518| (223b), and &lt;223 353 517| (223c). Using the optimal patent val, it tempers out the [[Würschmidt family|würschmidt comma]], 393216/390625 and 22876792454961/21990232555520 in the 5-limit; 2401/2400, 3136/3125, and 14348907/14000000 in the 7-limit. Using the 223b val, it tempers out the [[Kleismic family|kleisma]], 15625/15552 and |58, -38, 1&gt; in the 5-limit; 245/243, 3136/3125, and 67108864/66706983 in the 7-limit. Using the 223c val, it tempers out the [[Graviton|gravity comma]], 129140163/128000000 and 35595703125/34359738368 in the 5-limit; 4375/4374, 33075/32768, and 78125/76832 in the 7-limit.
+{{ED intro}}
-edo is the 48th [[prime EDO]].
+edo contains an excellent proportion of [[hornbostel]] temperament (via [[7L 2s]]), between square root of π (184\223), Aureus interval (34/21 in 155\223) and the 6/5 interval (58\223). It is in[[consistent]] to the [[5-odd-limit]] and higher limit, with three mappings possible for the 5-limit:
+* {{val| 223 353 518 }} ([[patent val]]),
+* {{val| 223 '''354''' 518 }} (223b),
+* {{val| 223 353 '''517''' }} (223c).
-[[Category:Edo]]
+Using the patent val, it tempers out 393216/390625 ([[würschmidt comma]]) and 22876792454961/21990232555520 in the 5-limit; 2401/2400, 3136/3125, and 14348907/14000000 in the 7-limit; 243/242, 441/440, 5632/5625, and 1449459/1433600 in the 11-limit; 847/845, 1188/1183, 1287/1280, and 1573/1568 in the 13-limit.
-[[Category:Prime EDO]]
+Using the 223be val, it tempers out 15625/15552 ([[kleisma]]) and {{monzo| 58 -38 1 }} in the 5-limit; 245/243, 3136/3125, and 67108864/66706983 in the 7-limit; 3025/3024, 3388/3375, 4375/4356, and 65536/65219 in the 11-limit; 352/351, 1001/1000, 2197/2178, and 2704/2695 in the 13-limit. Using the 223bef val, it tempers out 196/195, 325/324, 364/363, 625/624, and 49152/49049 in the 13-limit.
+Using the 223c val, it tempers out the 129140163/128000000 ([[graviton]]) and 35595703125/34359738368 in the 5-limit; 4375/4374, 33075/32768, and 78125/76832 in the 7-limit; 243/242, 385/384, and 4000/3993 in the 11-limit; 1188/1183, 1573/1568, 1625/1617, 1716/1715, and 3159/3136 in the 13-limit.
+Using the 223e val, it tempers out 1944/1925, 2835/2816, and 4000/3993 in the 11-limit; 364/363, 1001/1000, 1701/1690, and 1716/1715 in the 13-limit.
+=== Odd harmonics ===
+{{Harmonics in equal|223|columns=11}}
+=== Subsets and supersets ===
+edo is the 48th [[prime edo]].