223edo

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← 222edo223edo224edo →
Prime factorization 223 (prime)
Step size 5.38117¢
Fifth 130\223 (699.552¢)
Semitones (A1:m2) 18:19 (96.86¢ : 102.2¢)
Dual sharp fifth 131\223 (704.933¢)
Dual flat fifth 130\223 (699.552¢)
Dual major 2nd 38\223 (204.484¢)
Consistency limit 3
Distinct consistency limit 3

223 equal divisions of the octave (abbreviated 223edo or 223ed2), also called 223-tone equal temperament (223tet) or 223 equal temperament (223et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 223 equal parts of about 5.381 ¢ each. Each step represents a frequency ratio of 21/223, or the 223rd root of 2.

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 inconsistent to the 5-odd-limit and higher limit, with three mappings possible for the 5-limit:

  • 223 353 518] (patent val),
  • 223 354 518] (223b),
  • 223 353 517] (223c).

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.

Using the 223be val, it tempers out 15625/15552 (kleisma) and [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

Approximation of odd harmonics in 223edo
Harmonic 3 5 7 9 11 13 15 17 19 21
Error absolute (¢) -2.40 +1.13 -0.22 +0.57 -2.44 -1.07 -1.27 +2.67 -1.55 -2.62
relative (%) -45 +21 -4 +11 -45 -20 -24 +50 -29 -49
Steps
(reduced)
353
(130)
518
(72)
626
(180)
707
(38)
771
(102)
825
(156)
871
(202)
912
(20)
947
(55)
979
(87)

Subsets and supersets

223edo is the 48th prime edo.