223edo: Difference between revisions
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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). | |||
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 {{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 === | |||
223edo is the 48th [[prime edo]]. | |||
Latest revision as of 16:34, 4 March 2025
| ← 222edo | 223edo | 224edo → |
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 5.38 ¢ 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
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.40 | +1.13 | -0.22 | +0.57 | -2.44 | -1.07 | -1.27 | +2.67 | -1.55 | -2.62 | +1.32 |
| Relative (%) | -44.7 | +21.0 | -4.0 | +10.7 | -45.3 | -19.8 | -23.7 | +49.6 | -28.8 | -48.7 | +24.6 | |
| Steps (reduced) |
353 (130) |
518 (72) |
626 (180) |
707 (38) |
771 (102) |
825 (156) |
871 (202) |
912 (20) |
947 (55) |
979 (87) |
1009 (117) | |
Subsets and supersets
223edo is the 48th prime edo.