111edo: Difference between revisions

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== Theory ==

111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning.

111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13.

It is also significant for lower limits, especially in terms of what it tempers out; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] rank-4 temperament.

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]].

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, 1375/1372, and notably the [[quartisma]].

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]].

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&~~111~~ temperament, and [[buzzard]], the 58&~~111~~ temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&80 temperament, and [[buzzard]], the 53&58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

~~The prime factorization is 111 = 3 × 37~~.

=== Prime harmonics ===

@@ Line 11: / Line 11: @@
 == Theory ==
-edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning.
+edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13.
-It is also significant for lower limits, especially in terms of what it tempers out; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] rank-4 temperament.
+It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]].
-In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, 1375/1372, and notably the [[quartisma]].
+In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]].
-It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;111 temperament, and [[buzzard]], the 58&amp;111 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.
+It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;80 temperament, and [[buzzard]], the 53&amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.
-The prime factorization is 111 = 3 × 37.
 === Prime harmonics ===