183edo: Difference between revisions

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''183edo'' divides the octave into 183 equal parts of 6.557 [[cent]]s each. It is notable as a higher limit system, especially when 7 is left out of the picture. It tempers out the [[~~schisma|~~schisma]], 32805/32768, in the [[~~5-limit|~~5-limit]]. In the [[~~7-limit|~~7-limit]], it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In the [[~~11-limit|~~11-limit]] it tempers out [[540/539]], [[3025/3024]] and [[8019/8000]]; in the [[~~13-limit|~~13-limit]], [[351/350]] and [[676/675]]; in the [[~~17-limit|~~17-limit]] 442/441, 561/560 and 715/714; and in the [[~~19-limit|~~19-limit]] 456/455. It is the [[~~Optimal_patent_val|~~optimal patent val]] for 13-, 17- and 19-limit [[~~Mirkwai_clan|~~mirkat ~~temperament~~]], the 72&183 temperament, and an excellent tuning for the [[rank-~~three_temperament|rank-three~~ temperament]]s [[~~The_Archipelago|~~madagascar and borneo]].

''183edo'' divides the octave into 183 equal parts of 6.557 [[cent]]s each.

== Theory ==

183edo is notable as a higher limit system, especially when 7 is left out of the picture. It tempers out the [[schisma]], 32805/32768, in the [[5-limit]]. In the [[7-limit]], it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In the [[11-limit]], it tempers out [[540/539]], [[3025/3024]] and [[8019/8000]]; in the [[13-limit]], [[351/350]] and [[676/675]]; in the [[17-limit]] 442/441, 561/560 and 715/714; and in the [[19-limit]] 456/455. It is the [[optimal patent val]] for 13-, 17- and 19-limit [[mirkat]] temperament, the 72&183 temperament, and an excellent tuning for the [[rank-3 temperament]]s [[madagascar]] and [[borneo]].

As a no-sevens temperament, it tempers out [[32805/32768]], 5632/5625, [[8019/8000]], [[676/675]], 4425/4424, 6656/6655, [[936/935]], [[1089/1088]], and 1377/1375.

== ~~Just approximation~~ ==

=== Prime harmonics ===

183edo is notable as having especially low error in ''all'' [[prime limit]]s from 11 to 29 for EDOs in the 100 to 200 range, compared using a variety of metrics (prime error punishments), although it has a bad 19 which causes it to fail to be consistent in the [[19-odd-limit]]. It is however a strong no-19's system, being consistent in the no-19's no-35's 29-prime-limited 45-odd-limit add-43. (The prime 43 is added in the set of odd harmonics due to its essentially perfect accuracy. The harmonic 35 is excluded due to the sharpness of 7 compounding and causing inconsistency in ''some'' cases such as for 39/35.) It can also be considered to model the 2.17.29.43 subgroup with extreme accuracy.

[[Category:Equal divisions of the octave]]

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-''183edo'' divides the octave into 183 equal parts of 6.557 [[cent]]s each. It is notable as a higher limit system, especially when 7 is left out of the picture. It tempers out the [[schisma|schisma]], 32805/32768, in the [[5-limit|5-limit]]. In the [[7-limit|7-limit]], it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In the [[11-limit|11-limit]] it tempers out [[540/539]], [[3025/3024]] and [[8019/8000]]; in the [[13-limit|13-limit]], [[351/350]] and [[676/675]]; in the [[17-limit|17-limit]] 442/441, 561/560 and 715/714; and in the [[19-limit|19-limit]] 456/455. It is the [[Optimal_patent_val|optimal patent val]] for 13-, 17- and 19-limit [[Mirkwai_clan|mirkat temperament]], the 72&amp;183 temperament, and an excellent tuning for the [[rank-three_temperament|rank-three temperament]]s [[The_Archipelago|madagascar and borneo]].
+''183edo'' divides the octave into 183 equal parts of 6.557 [[cent]]s each.
+== Theory ==
+edo is notable as a higher limit system, especially when 7 is left out of the picture. It tempers out the [[schisma]], 32805/32768, in the [[5-limit]]. In the [[7-limit]], it tempers out porwell, [[6144/6125]], cataharry, [[19683/19600]] and mirkwai, [[16875/16807]]. In the [[11-limit]], it tempers out [[540/539]], [[3025/3024]] and [[8019/8000]]; in the [[13-limit]], [[351/350]] and [[676/675]]; in the [[17-limit]] 442/441, 561/560 and 715/714; and in the [[19-limit]] 456/455. It is the [[optimal patent val]] for 13-, 17- and 19-limit [[mirkat]] temperament, the 72&amp;183 temperament, and an excellent tuning for the [[rank-3 temperament]]s [[madagascar]] and [[borneo]].
 As a no-sevens temperament, it tempers out [[32805/32768]], 5632/5625, [[8019/8000]], [[676/675]], 4425/4424, 6656/6655, [[936/935]], [[1089/1088]], and 1377/1375.
-== Just approximation ==
+=== Prime harmonics ===
 edo is notable as having especially low error in ''all'' [[prime limit]]s from 11 to 29 for EDOs in the 100 to 200 range, compared using a variety of metrics (prime error punishments), although it has a bad 19 which causes it to fail to be consistent in the [[19-odd-limit]]. It is however a strong no-19's system, being consistent in the no-19's no-35's 29-prime-limited 45-odd-limit add-43. (The prime 43 is added in the set of odd harmonics due to its essentially perfect accuracy. The harmonic 35 is excluded due to the sharpness of 7 compounding and causing inconsistency in ''some'' cases such as for 39/35.) It can also be considered to model the 2.17.29.43 subgroup with extreme accuracy.
-{{Primes in edo|183|columns=10|prec=3}}
+{{Primes in edo|183|columns=10}}
 [[Category:Equal divisions of the octave]]