217edo: Difference between revisions

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== Theory ==

217edo is a strong [[19-limit]] system, the smallest [[consistency|distinctly consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]] as well as the no-23 [[31-odd-limit]]. It shares the same [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s with [[31edo]] ({{nowrap| 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]], excelling as a [[2.3.5.13 subgroup|2.3.5.13-subgroup]] temperament. It can be used as a decent approximation of the [[31-limit]], ''almost'' being consistent through the [[31-odd-limit]] except for [[23/14]], [[23/21]], [[29/23]] and their [[octave complement]]s, with errors below the melodic [[just-noticeable difference]]. If one desires even higher consistency and precision, [[311edo]] offers a much better palette.

217edo is a strong [[19-limit]] system, the smallest [[consistency|distinctly consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]] as well as the no-23 [[31-odd-limit]]. It shares the same [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s with [[31edo]] ({{nowrap| 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]], excelling as a [[2.3.5.13 subgroup|2.3.5.13-subgroup]] temperament. It can be used as a decent approximation of the [[31-limit]], ''almost'' being consistent through the [[31-odd-limit]] except for [[23/14]], [[23/21]], [[29/23]] and their [[octave complement]]s, with errors below the melodic [[just-noticeable difference]]. [[224edo]], only a bit bigger, offers a much more accurate [[13-limit]], at the cost of worse higher limits. If one desires even higher consistency and precision, [[311edo]] offers a much better palette.

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]].

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 == Theory ==
-edo is a strong [[19-limit]] system, the smallest [[consistency|distinctly consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]] as well as the no-23 [[31-odd-limit]]. It shares the same [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s with [[31edo]] ({{nowrap| 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]], excelling as a [[2.3.5.13 subgroup|2.3.5.13-subgroup]] temperament. It can be used as a decent approximation of the [[31-limit]], ''almost'' being consistent through the [[31-odd-limit]] except for [[23/14]], [[23/21]], [[29/23]] and their [[octave complement]]s, with errors below the melodic [[just-noticeable difference]]. If one desires even higher consistency and precision, [[311edo]] offers a much better palette.
+edo is a strong [[19-limit]] system, the smallest [[consistency|distinctly consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]] as well as the no-23 [[31-odd-limit]]. It shares the same [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s with [[31edo]] ({{nowrap| 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]], excelling as a [[2.3.5.13 subgroup|2.3.5.13-subgroup]] temperament. It can be used as a decent approximation of the [[31-limit]], ''almost'' being consistent through the [[31-odd-limit]] except for [[23/14]], [[23/21]], [[29/23]] and their [[octave complement]]s, with errors below the melodic [[just-noticeable difference]]. [[224edo]], only a bit bigger, offers a much more accurate [[13-limit]], at the cost of worse higher limits. If one desires even higher consistency and precision, [[311edo]] offers a much better palette.
 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]].