111edo: Difference between revisions

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'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent~~|cents~~]].

'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent]]s.

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 ~~temperament~~. 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]] ~~tuning~~ for the corresponding [[11-limit]] rank ~~four~~ temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323~~, 176/175~~, 1375/1372, ~~540/539~~ and [[~~Quartisma|~~117440512/117406179]]. 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 below 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.

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. 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. 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 the [[quartisma]] 117440512/117406179. 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 below 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

The prime factorization is 111 = 3 × 37.

~~<math>111 = 3 \cdot 37</math>~~

Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments.

Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.

{| class="wikitable"

|+Circulating temperaments in 111edo

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-'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cents]].
+'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent]]s.
-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 temperament. 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]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and [[Quartisma|117440512/117406179]]. 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 below 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.
+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. 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. 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 the [[quartisma]] 117440512/117406179. 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 below 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
+The prime factorization is 111 = 3 × 37.
-<math>111 = 3 \cdot 37</math>
 {{Primes in edo|111|columns=11|prec=2}}
-Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments.
+Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
 {| class="wikitable"
 |+Circulating temperaments in 111edo