217edo: Difference between revisions
Template the intro and odd-limit approximation table; various cleanup |
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== Theory == | == Theory == | ||
217edo is a strong [[19-limit]] system, the smallest | 217edo is a strong [[19-limit]] system, the smallest [[consistency|distinctly consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]]. It shares the same [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s with [[31edo]] (217 = 7 × 31), as well as the [[11/9]] interval (supporting the [[31-comma temperaments #Birds|birds temperament]]). However, compared to 31edo, its [[patent val]] differ on the mappings for [[3/1|3]], [[11/1|11]], [[13/1|13]], [[17/1|17]] and [[19/1|19]] – in fact, this edo has a very accurate 13th harmonic, as well as the [[19/15]] interval. | ||
The equal temperament tempers out the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[escapade comma]], {{monzo| 32 -7 -9 }} in the 5-limit; [[3136/3125]], [[4375/4374]], [[10976/10935]] and 823543/819200 in the 7-limit; [[441/440]], [[4000/3993]], 5632/5625, and [[16384/16335]] in the 11-limit; [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]] and [[4096/4095]] in the 13-limit; 595/594, [[833/832]], [[936/935]], 1156/1155, [[1225/1224]], [[1701/1700]] in the 17-limit; 343/342, 476/475, 969/968, [[1216/1215]], [[1445/1444]], [[1521/1520]] and 1540/1539 in the 19-limit. It allows [[ | The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[escapade comma]], {{monzo| 32 -7 -9 }} in the 5-limit; [[3136/3125]], [[4375/4374]], [[10976/10935]] and [[823543/819200]] in the 7-limit; [[441/440]], [[4000/3993]], [[5632/5625]], and [[16384/16335]] in the 11-limit; [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]] and [[4096/4095]] in the 13-limit; [[595/594]], [[833/832]], [[936/935]], [[1156/1155]], [[1225/1224]], [[1701/1700]] in the 17-limit; [[343/342]], [[476/475]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]] and [[1540/1539]] in the 19-limit. It allows [[minor minthmic chords]], [[werckismic chords]], and [[sinbadmic chords]] in the 13-odd-limit, in addition to [[island chords]] and [[nicolic chords]] in the 15-odd-limit. It provides the [[optimal patent val]] for the 11- and 13-limit [[arch]] and the 11- and 13-limit [[cotoneum]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|217|columns=11}} | {{Harmonics in equal|217|columns=11}} | ||
== JI | == Approximation to JI == | ||
=== Selected just intervals === | === Selected just intervals === | ||
The following table shows how [[23-odd-limit|23-odd-limit intervals]] are represented in 217edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. | The following table shows how [[23-odd-limit|23-odd-limit intervals]] are represented in 217edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. | ||
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| 2.3 | | 2.3 | ||
| {{monzo| 344 -217 }} | | {{monzo| 344 -217 }} | ||
| | | {{mapping| 217 344 }} | ||
| -0.110 | | -0.110 | ||
| 0.1101 | | 0.1101 | ||
| Line 35: | Line 35: | ||
| 2.3.5 | | 2.3.5 | ||
| {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }} | | {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }} | ||
| | | {{mapping| 217 344 504 }} | ||
| -0.186 | | -0.186 | ||
| 0.1398 | | 0.1398 | ||
| Line 41: | Line 41: | ||
|- | |- | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 3136/3125, 4375/4374, | | 3136/3125, 4375/4374, 823543/819200 | ||
| | | {{mapping| 217 344 504 609 }} | ||
| -0.043 | | -0.043 | ||
| 0.2757 | | 0.2757 | ||
| Line 48: | Line 48: | ||
|- | |- | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 441/440, 3136/3125, | | 441/440, 3136/3125, 4000/3993, 4375/4374 | ||
| | | {{mapping| 217 344 504 609 751 }} | ||
| -0.131 | | -0.131 | ||
| 0.3034 | | 0.3034 | ||
| Line 55: | Line 55: | ||
|- | |- | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 364/363, 441/440, 676/675, | | 364/363, 441/440, 676/675, 3136/3125, 4375/4374 | ||
| | | {{mapping| 217 344 504 609 751 803 }} | ||
| -0.111 | | -0.111 | ||
| 0.2808 | | 0.2808 | ||
| Line 62: | Line 62: | ||
|- | |- | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 364/363, 441/440, 595/594, | | 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125 | ||
| | | {{mapping| 217 344 504 609 751 803 887 }} | ||
| -0.099 | | -0.099 | ||
| 0.2616 | | 0.2616 | ||
| Line 69: | Line 69: | ||
|- | |- | ||
| 2.3.5.7.11.13.17.19 | | 2.3.5.7.11.13.17.19 | ||
| 343/342, 364/363, 441/440, | | 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215 | ||
| | | {{mapping| 217 344 504 609 751 803 887 922 }} | ||
| -0.119 | | -0.119 | ||
| 0.2504 | | 0.2504 | ||
| Line 80: | Line 80: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Birds]] | | [[Birds]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Scales == | == Scales == | ||
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* [[Cotoneum29]] | * [[Cotoneum29]] | ||
* [[Cotoneum41]] | * [[Cotoneum41]] | ||
[[Category:Arch]] | [[Category:Arch]] | ||
[[Category:Birds]] | [[Category:Birds]] | ||
[[Category:Cotoneum]] | [[Category:Cotoneum]] | ||