111edo: Difference between revisions
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== Theory == | == 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. 111 = | 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 111 = {{factorisation|111}}}}, 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]], 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-limit versions of [[semisept]], the 31 & 80 temperament, and [[buzzard]], the 53 & 58 temperament. [[Gene Ward Smith]]'s trio 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-limit versions of [[semisept]], the 31 & 80 temperament, and [[buzzard]], the 53 & 58 temperament. [[Gene Ward Smith]]'s trio 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. | ||
| Line 35: | Line 35: | ||
| {{monzo| 176 -111 }} | | {{monzo| 176 -111 }} | ||
| {{mapping| 111 176 }} | | {{mapping| 111 176 }} | ||
| | | −0.236 | ||
| 0.236 | | 0.236 | ||
| 2.18 | | 2.18 | ||
| Line 42: | Line 42: | ||
| 78732/78125, 67108864/66430125 | | 78732/78125, 67108864/66430125 | ||
| {{mapping| 111 176 258 }} | | {{mapping| 111 176 258 }} | ||
| | | −0.570 | ||
| 0.510 | | 0.510 | ||
| 4.72 | | 4.72 | ||
| Line 49: | Line 49: | ||
| 1728/1715, 3136/3125, 5120/5103 | | 1728/1715, 3136/3125, 5120/5103 | ||
| {{mapping| 111 176 258 312 }} | | {{mapping| 111 176 258 312 }} | ||
| | | −0.797 | ||
| 0.591 | | 0.591 | ||
| 5.47 | | 5.47 | ||
| Line 56: | Line 56: | ||
| 176/175, 540/539, 1331/1323, 5120/5103 | | 176/175, 540/539, 1331/1323, 5120/5103 | ||
| {{mapping| 111 176 258 312 384 }} | | {{mapping| 111 176 258 312 384 }} | ||
| | | −0.639 | ||
| 0.615 | | 0.615 | ||
| 5.69 | | 5.69 | ||
| Line 63: | Line 63: | ||
| 176/175, 351/350, 540/539, 676/675, 1331/1323 | | 176/175, 351/350, 540/539, 676/675, 1331/1323 | ||
| {{mapping| 111 176 258 312 384 411 }} | | {{mapping| 111 176 258 312 384 411 }} | ||
| | | −0.655 | ||
| 0.562 | | 0.562 | ||
| 5.21 | | 5.21 | ||
| Line 70: | Line 70: | ||
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714 | | 176/175, 256/255, 351/350, 442/441, 540/539, 715/714 | ||
| {{mapping| 111 176 258 312 384 411 454 }} | | {{mapping| 111 176 258 312 384 411 454 }} | ||
| | | −0.672 | ||
| 0.523 | | 0.523 | ||
| 4.84 | | 4.84 | ||
| Line 77: | Line 77: | ||
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475 | | 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475 | ||
| {{mapping| 111 176 258 312 384 411 454 472 }} | | {{mapping| 111 176 258 312 384 411 454 472 }} | ||
| | | −0.740 | ||
| 0.521 | | 0.521 | ||
| 4.83 | | 4.83 | ||
| Line 204: | Line 204: | ||
|- | |- | ||
| 3 | | 3 | ||
| 23\111<br>(14\111) | | 23\111<br />(14\111) | ||
| 248.65<br>(151.35) | | 248.65<br />(151.35) | ||
| 15/13<br>(12/11) | | 15/13<br />(12/11) | ||
| [[Hemimist]] | | [[Hemimist]] | ||
|- | |- | ||
| 3 | | 3 | ||
| 46\111<br>(9\111) | | 46\111<br />(9\111) | ||
| 497.30<br>(97.30) | | 497.30<br />(97.30) | ||
| 4/3<br>(18/17~19/18) | | 4/3<br />(18/17~19/18) | ||
| [[Misty]] | | [[Misty]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||
| Line 223: | Line 223: | ||
== Music == | == Music == | ||
; [[Gene Ward Smith]] | ; [[Gene Ward Smith]] | ||
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) | * ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning | ||
[[Category:Buzzard]] | [[Category:Buzzard]] | ||