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'''206edo''' is the [[EDO|equal division of the octave]] into 206 parts of 5.8252 [[cent]]s each. It is closely related to [[103edo]], but the patent vals differ on the mapping for 3, 11, and 19.
{{Infobox ET}}
{{ED intro}}


It is in[[consistent]] to the 5-limit and higher limit, with four mappings possible for the 19-limit:
== Theory ==
206edo is closely related to [[103edo]], but the [[patent val]]s differ on the mapping for 3, 11, and 19.
 
It is in[[consistent]] to the [[5-odd-limit]] and higher limits, with four mappings possible for the 19-limit:
* {{val| 206 327 478 578 713 762 842 875 }} ([[patent val]])
* {{val| 206 327 478 578 713 762 842 875 }} ([[patent val]])
* {{val| 206 '''326''' 478 578 713 762 842 875 }} (206b)
* {{val| 206 '''326''' 478 578 713 762 842 875 }} (206b)
Line 7: Line 11:
* {{val| 206 327 '''479''' '''579''' 713 '''763''' 842 875 }} (206cdf)  
* {{val| 206 327 '''479''' '''579''' 713 '''763''' 842 875 }} (206cdf)  


Using the patent val, it tempers out  
Using the patent val, it [[tempering out|tempers out]] 48828125/47775744 ([[sycamore comma]]) and 32768000000/31381059609 in the 5-limit; 4000/3969, 84035/82944, and 458752/455625 in the 7-limit; [[385/384]], 2401/2376, 6875/6804, and 9375/9317 in the 11-limit; [[352/351]], [[1575/1573]], 1625/1617, 2197/2178, and [[4096/4095]] in the 13-limit; [[289/288]], [[375/374]], [[442/441]], [[715/714]], and 2000/1989 in the 17-limit; 190/189, 361/360, 665/663, 969/968, 1235/1232, and 1375/1368 in the 19-limit.
* 48828125/47775744 ([[sycamore comma]]) and 32768000000/31381059609 in the 5-limit
 
* 4000/3969, 84035/82944, and 458752/455625 in the 7-limit
Using the 206b val, it tempers out 78732/78125 ([[sensipent comma]]) and 34171875/33554432 ([[ampersand comma]]) in the 5-limit; [[225/224]], [[1029/1024]], and 177147/175000 in the 7-limit; 4375/4356, [[9801/9800]], 15309/15125, and 73728/73205 in the 11-limit; [[351/350]], [[364/363]], [[625/624]], 1701/1690, and 31213/30976 in the 13-limit; [[273/272]], [[833/832]], 850/847, [[1089/1088]], [[1225/1224]], and 1458/1445 in the 17-limit; 210/209, 495/494, 729/722, 1235/1232, and [[1445/1444]] in the 19-limit.
* [[385/384]], 2401/2376, 6875/6804, and 9375/9317 in the 11-limit
 
* [[352/351]], 1575/1573, 1625/1617, 2197/2178, and 4096/4095 in the 13-limit
Using the 206be val, it tempers out [[243/242]], 385/384, 441/440, and 43923/43750 in the 11-limit; 351/350, 625/624, [[847/845]], [[1001/1000]], and [[1573/1568]] in the 13-limit; 273/272, 375/374, 561/560, 715/714, and 833/832 in the 17-limit; 363/361 and 729/722 in the 19-limit.
* 289/288, 375/374, 442/441, 715/714, and 2000/1989 in the 17-limit
* 190/189, 361/360, 665/663, 969/968, 1235/1232, and 1375/1368 in the 19-limit


Using the 206b val, it tempers out  
Using the 206cdf val, it tempers out 2048/2025 ([[diaschisma]]), and {{monzo| 4 -45 29 }} in the 5-limit; [[4375/4374]], 110592/109375, and 235298/234375 in the 7-limit; [[176/175]], [[896/891]], and 1331/1323 in the 11-limit; 640/637, 847/845, 1001/1000, and [[2197/2187]] in the 13-limit; [[136/135]], [[256/255]], 561/560, and 1275/1274 in the 17-limit; 190/189, 476/475, 608/605, 836/833, and 969/968 in the 19-limit.
* 78732/78125 ([[sensipent comma]]) and 34171875/33554432 ([[ampersand]]) in the 5-limit
* 225/224, 1029/1024, and 177147/175000 in the 7-limit
* 4375/4356, [[9801/9800]], 15309/15125, and 73728/73205 in the 11-limit
* 351/350, 364/363, 625/624, 1701/1690, and 31213/30976 in the 13-limit
* 273/272, 833/832, 850/847, 1089/1088, 1225/1224, and 1458/1445 in the 17-limit
* 210/209, 495/494, 729/722, 1235/1232, and 1445/1444 in the 19-limit


Using the 206be val, it tempers out
=== Odd harmonics ===
* [[243/242]], 385/384, 441/440, and 43923/43750 in the 11-limit
{{Harmonics in equal|206}}
* [[351/350]] (ratwolfsma), 625/624, 847/845, 1001/1000, and 1573/1568 in the 13-limit
* 273/272, 375/374, 561/560, 715/714, and 833/832 in the 17-limit
* 363/361 and 729/722 in the 19-limit


Using the 206cdf val, it tempers out
=== Subsets and supersets ===
* 2048/2025 ([[diaschisma]]),  and {{Monzo| 4 -45 29 }}; in the 5-limit
206edo contains 2edo and 103edo as subsets. 412edo, which doubles it, provides an excellent correction to the approximation of harmonic 3.
* 4375/4374, 110592/109375, and 235298/234375 in the 7-limit
* [[176/175]], 896/891, and 1331/1323 in the 11-limit
* 640/637, 847/845, 1001/1000, and 2197/2187 in the 13-limit
* 136/135, 256/255, 561/560, and 1275/1274 in the 17-limit
* 190/189, 476/475, 608/605, 836/833, and 969/968 in the 19-limit


== Scales ==
== Scales ==
* [[skwares8]]
* [[skwares8]]
* [[skwares11]]
* [[skwares11]]
* [[skwares14]]
* [[skwares14]]
[[Category:Theory]]
[[Category:Equal divisions of the octave]]
{{todo|improve readability}}

Latest revision as of 14:15, 20 February 2025

← 205edo 206edo 207edo →
Prime factorization 2 × 103
Step size 5.82524 ¢ 
Fifth 121\206 (704.854 ¢)
Semitones (A1:m2) 23:13 (134 ¢ : 75.73 ¢)
Dual sharp fifth 121\206 (704.854 ¢)
Dual flat fifth 120\206 (699.029 ¢) (→ 60\103)
Dual major 2nd 35\206 (203.883 ¢)
(semiconvergent)
Consistency limit 3
Distinct consistency limit 3

206 equal divisions of the octave (abbreviated 206edo or 206ed2), also called 206-tone equal temperament (206tet) or 206 equal temperament (206et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 206 equal parts of about 5.83 ¢ each. Each step represents a frequency ratio of 21/206, or the 206th root of 2.

Theory

206edo is closely related to 103edo, but the patent vals differ on the mapping for 3, 11, and 19.

It is inconsistent to the 5-odd-limit and higher limits, with four mappings possible for the 19-limit:

  • ⟨206 327 478 578 713 762 842 875] (patent val)
  • ⟨206 326 478 578 713 762 842 875] (206b)
  • ⟨206 326 478 578 712 762 842 875] (206be)
  • ⟨206 327 479 579 713 763 842 875] (206cdf)

Using the patent val, it tempers out 48828125/47775744 (sycamore comma) and 32768000000/31381059609 in the 5-limit; 4000/3969, 84035/82944, and 458752/455625 in the 7-limit; 385/384, 2401/2376, 6875/6804, and 9375/9317 in the 11-limit; 352/351, 1575/1573, 1625/1617, 2197/2178, and 4096/4095 in the 13-limit; 289/288, 375/374, 442/441, 715/714, and 2000/1989 in the 17-limit; 190/189, 361/360, 665/663, 969/968, 1235/1232, and 1375/1368 in the 19-limit.

Using the 206b val, it tempers out 78732/78125 (sensipent comma) and 34171875/33554432 (ampersand comma) in the 5-limit; 225/224, 1029/1024, and 177147/175000 in the 7-limit; 4375/4356, 9801/9800, 15309/15125, and 73728/73205 in the 11-limit; 351/350, 364/363, 625/624, 1701/1690, and 31213/30976 in the 13-limit; 273/272, 833/832, 850/847, 1089/1088, 1225/1224, and 1458/1445 in the 17-limit; 210/209, 495/494, 729/722, 1235/1232, and 1445/1444 in the 19-limit.

Using the 206be val, it tempers out 243/242, 385/384, 441/440, and 43923/43750 in the 11-limit; 351/350, 625/624, 847/845, 1001/1000, and 1573/1568 in the 13-limit; 273/272, 375/374, 561/560, 715/714, and 833/832 in the 17-limit; 363/361 and 729/722 in the 19-limit.

Using the 206cdf val, it tempers out 2048/2025 (diaschisma), and [4 -45 29⟩ in the 5-limit; 4375/4374, 110592/109375, and 235298/234375 in the 7-limit; 176/175, 896/891, and 1331/1323 in the 11-limit; 640/637, 847/845, 1001/1000, and 2197/2187 in the 13-limit; 136/135, 256/255, 561/560, and 1275/1274 in the 17-limit; 190/189, 476/475, 608/605, 836/833, and 969/968 in the 19-limit.

Odd harmonics

Approximation of odd harmonics in 206edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +2.90 -1.85 -1.84 -0.03 +2.08 -1.69 +1.05 -0.10 -0.43 +1.06 +0.85
Relative (%) +49.8 -31.7 -31.5 -0.5 +35.7 -29.1 +18.1 -1.7 -7.3 +18.3 +14.6
Steps
(reduced) 		327
(121) 		478
(66) 		578
(166) 		653
(35) 		713
(95) 		762
(144) 		805
(187) 		842
(18) 		875
(51) 		905
(81) 		932
(108)

Subsets and supersets

206edo contains 2edo and 103edo as subsets. 412edo, which doubles it, provides an excellent correction to the approximation of harmonic 3.

Scales

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