217edo: Difference between revisions

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.  

Here below is an algorithmically generated table of no-37 39-odd-limit intervals of 217edo using [[User:Godtone #My Python 3 code|Godtone's code]], with some manually added intervals outside that limit for completeness.

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| [[64/63]]

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| [[36/35]], [[35/34]], [[34/33]]

| [[31/29]], ''[[2187/2048]]''

* 23-limit is not the subgroup it does the best, with the no-23 29- and 31-limit approximated even better.  

* It is also prominent in the 17-limit, with a lower absolute error than any previous equal temperaments, beating [[183edo|183]] and superseded by [[224edo|224]].  

| 16875/16384

| 320/243

<nowiki>*</nowiki> [[normal forms|octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct

217edo can be written in Sagittal using almost the entire Athenian extension (except for {{sagittal| |\ }} {{sagittal| !/ }} {{sagittal| /|| }} {{sagittal| \!! }} since it tempers out [[1240029/1239040]]), by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian. It is identical to [[224edo]]'s Sagittal notation, but it uses the 11/7C for the +6/-6 alteration instead of 55C.<ref name=":1">[[George Secor|George D. Secor]] and [[David Keenan|David C. Keenan]], [https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''], p. 11.</ref>

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| rowspan="2" | {{Sagittal| |( }}

| rowspan="2" | {{Sagittal| )|( }}

| rowspan="2" | {{Sagittal| ~|( }}

| rowspan="2" | {{Sagittal| /| }}

| rowspan="2" | {{Sagittal| |) }}

| rowspan="2" | {{Sagittal| (| }}

| rowspan="2" | {{Sagittal| (|( }}

| rowspan="2" | {{Sagittal| //| }}

| rowspan="2" | {{Sagittal| /|) }}

| rowspan="2" | {{Sagittal| /|\ }}

| {{Sagittal|#}}{{sagittal| \!/ }}

| {{Sagittal|#}}{{sagittal| \!) }}

| {{Sagittal|#}}{{sagittal| \\! }}

| {{Sagittal|#}}{{sagittal| (!( }}

| {{Sagittal|#}}{{sagittal| (! }}

| {{Sagittal|#}}{{sagittal| !) }}

| {{Sagittal|#}}{{sagittal| \! }}

| {{Sagittal|#}}{{sagittal| ~!( }}

| {{Sagittal|#}}{{sagittal| )!( }}

| {{Sagittal|#}}{{sagittal| !( }}

| {{Sagittal|#}}

| {{Sagittal| (|) }}

| {{Sagittal| (|\ }}

| {{Sagittal| )||( }}

| {{Sagittal| ~||( }}

| {{Sagittal| )||~ }}

| {{Sagittal| ||) }}

| {{Sagittal| ||\ }}

| {{Sagittal| (||( }}

| {{Sagittal| //|| }}

| {{Sagittal| /||) }}

| {{Sagittal| /||\ }}

The 5-up (quup) alteration neatly maps to the pythagorean-septimal comma.

|+Ups-and-downs notation

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| ^^

| ^^^

=== 31edo-based notation ===

Since {{nowrap| 217 {{=}} 31 × 7 }}, one ''could'' base the notation on its inherited meantone fifth 126\217 (18\31) instead of its best fifth.

 

This could be useful when [[31edo]] is used as a base tuning, where the whole palette of 217edo is only used to provide subtle inflections of the 31edo pitches, similar to how one might use [[159edo]] to provide subtle corrections of [[53edo]] pitches.

 

! rowspan="2" | Symbol

| rowspan="2" | ^

| ^t

| ^^t

| ^^^t

| vv#

| ^^^>>