111edo: Difference between revisions

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== Theory ==

111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]].

111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]].

It is also significant for lower limits, especially in terms of what it [[tempering out|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]].

It is also significant for lower limits, especially in terms of what it [[tempering out|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]].

~~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 further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]].

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 & 80 }} temperament, and [[buzzard]], the {{nowrap| 53 & 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

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 == Theory ==
-edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]].
+edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]].
-It is also significant for lower limits, especially in terms of what it [[tempering out|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]].
+It is also significant for lower limits, especially in terms of what it [[tempering out|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]].
-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 further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]].
 It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 & 80 }} temperament, and [[buzzard]], the {{nowrap| 53 & 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.