181edo: Difference between revisions

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== Theory ==

181edo is only [[consistent]] to the [[7-odd-limit]], though except for [[9/5]], [[23/20]] and their [[octave complement]]s, it is consistent to the [[23-odd-limit]]. Beyond that, it does well on [[prime interval|primes]] [[37/1|37]] and [[43/1|43]], and has unambiguous though not accurate approximations to [[29/1|29]], [[31/1|31]], and [[41/1|41]].

181edo is only [[consistent]] to the [[7-odd-limit]], though except for [[9/5]], [[23/20]] and their [[octave complement]]s, it is consistent to the [[23-odd-limit]]. Beyond that, it does well on [[prime interval|primes]] [[37/1|37]] and [[43/1|43]], and has unambiguous though not accurate approximations to [[29/1|29]], [[31/1|31]], and [[41/1|41]]. However, the composite harmonics [[25/1|25]], [[27/1|27]], [[35/1|35]], and [[39/1|39]] cause inconsistencies, with harmonic 25 itself being inconsistent.

As an equal temperament, 181et [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo| 14 -22 9 }} in the 5-limit; [[2401/2400]], [[5120/5103]], and 390625/387072 in the 7-limit ([[support]]ing the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, [[2200/2187]], and [[4000/3993]] in the 11-limit; [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the 13-limit.

As an equal temperament, 181et [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo| 14 -22 9 }} in the [[5-limit]]; [[2401/2400]], [[5120/5103]], and 390625/387072 in the [[7-limit]] ([[support]]ing the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, [[2200/2187]], and [[4000/3993]] in the [[11-limit]]; and [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the [[13-limit]]. It tempers out [[375/374]], [[595/594]], and [[1275/1274]] in the [[17-limit]], [[400/399]] in the [[19-limit]], and [[300/299]] in the [[23-limit]].

Because its harmonic [[5/1|5]] causes some inconsistencies, and is less accurate than the other harmonics, 181edo can reasonably be treated as a no-5 system, where it is [[purely consistent]]{{idio}} (meaning all harmonics have under 25% [[relative error]]) up to the 23-odd-limit. It tempers out {{Monzo|15 -13 2}} and {{Monzo|-31 -7 15}} in the [[2.3.7 subgroup]]; 26411/26244, [[43923/43904]], and [[131072/130977]] in the [[2.3.7.11 subgroup]]; and [[352/351]], 20449/20412, [[31213/31104]], and 53361/53248 in the 2.3.7.11.13 subgroup. It tempers out [[833/832]] and [[1089/1088]] in the no-5 17-limit, [[343/342]], [[1729/1728]], and [[2432/2431]] in the no-5 19-limit, and [[392/391]] in the no-5 23-limit.

=== Prime harmonics ===

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 == Theory ==
-edo is only [[consistent]] to the [[7-odd-limit]], though except for [[9/5]], [[23/20]] and their [[octave complement]]s, it is consistent to the [[23-odd-limit]]. Beyond that, it does well on [[prime interval|primes]] [[37/1|37]] and [[43/1|43]], and has unambiguous though not accurate approximations to [[29/1|29]], [[31/1|31]], and [[41/1|41]].
+edo is only [[consistent]] to the [[7-odd-limit]], though except for [[9/5]], [[23/20]] and their [[octave complement]]s, it is consistent to the [[23-odd-limit]]. Beyond that, it does well on [[prime interval|primes]] [[37/1|37]] and [[43/1|43]], and has unambiguous though not accurate approximations to [[29/1|29]], [[31/1|31]], and [[41/1|41]]. However, the composite harmonics [[25/1|25]], [[27/1|27]], [[35/1|35]], and [[39/1|39]] cause inconsistencies, with harmonic 25 itself being inconsistent.
-As an equal temperament, 181et [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo| 14 -22 9 }} in the 5-limit; [[2401/2400]], [[5120/5103]], and 390625/387072 in the 7-limit ([[support]]ing the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, [[2200/2187]], and [[4000/3993]] in the 11-limit; [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the 13-limit.
+As an equal temperament, 181et [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo| 14 -22 9 }} in the [[5-limit]]; [[2401/2400]], [[5120/5103]], and 390625/387072 in the [[7-limit]] ([[support]]ing the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, [[2200/2187]], and [[4000/3993]] in the [[11-limit]]; and [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the [[13-limit]]. It tempers out  [[375/374]], [[595/594]], and [[1275/1274]] in the [[17-limit]], [[400/399]] in the [[19-limit]], and [[300/299]] in the [[23-limit]].
+Because its harmonic [[5/1|5]] causes some inconsistencies, and is less accurate than the other harmonics, 181edo can reasonably be treated as a no-5 system, where it is [[purely consistent]]{{idio}} (meaning all harmonics have under 25% [[relative error]]) up to the 23-odd-limit. It tempers out {{Monzo|15 -13 2}} and {{Monzo|-31 -7 15}} in the [[2.3.7 subgroup]]; 26411/26244, [[43923/43904]], and [[131072/130977]] in the [[2.3.7.11 subgroup]]; and [[352/351]], 20449/20412, [[31213/31104]], and 53361/53248 in the 2.3.7.11.13 subgroup. It tempers out [[833/832]] and [[1089/1088]] in the no-5 17-limit, [[343/342]], [[1729/1728]], and [[2432/2431]] in the no-5 19-limit, and [[392/391]] in the no-5 23-limit.
 === Prime harmonics ===