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]]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 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.

== 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.

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 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.

The prime factorization is 111 = 3 × 37.

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|23L 65s

|}

== Music ==

* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]

@@ Line 1: / Line 1: @@
 '''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 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.
+== 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.
+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 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.
 The prime factorization is 111 = 3 × 37.
@@ Line 283: / Line 290: @@
 |23L 65s
 |}
 == Music ==
+* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]
-[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]
 [[Category:Theory]]