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]]. [[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]].

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 useful intervals outside that limit. Intervals in italics are inconsistently mapped.

|+ style="font-size: 105%; white-space: nowrap;" | Table of 217edo intervals

! #

! Cents

! Marks

! Approximate intervals

| 0

| 0

| P1

| 1

| 5.53

| 2

| 11.06

| 3

| 16.59

| 4

| 22.12

| [[81/80]]

| 5

| 27.65

| [[64/63]], ''[[531441/524288]]''

| 6

| 33.18

| 7

| 38.71

| [[36/35]], [[35/34]], [[34/33]], [[1053/1024]]

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

| ''[[23/21]]'', [[34/31]]

| ''[[33/29]]''

| ''[[33/28]]'', [[46/39]], [[13/11]]

| [[17/14]], ''[[28/23]]''

| [[44/35]], [[39/31]], [[34/27]], ''[[29/23]]''

| ''[[46/29]]'', [[27/17]], [[62/39]], [[35/22]]

| ''[[23/14]]'', [[28/17]]

| [[22/13]], [[39/23]], ''[[56/33]]''

| ''[[58/33]]''

| [[31/17]], ''[[42/23]]''

| 217

| 1200.

| P8

| [[2/1]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

| {{Monzo| 344 -217 }}

| {{Mapping| 217 344 }}

| {{Monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }}

| {{Mapping| 217 344 504 }}

| {{Mapping| 217 344 504 609 }}

| {{Mapping| 217 344 504 609 751 }}

| {{Mapping| 217 344 504 609 751 803 }}

| {{Mapping| 217 344 504 609 751 803 887 }}

| {{Mapping| 217 344 504 609 751 803 887 922 }}

| {{Mapping| 217 344 504 609 751 803 887 922 982 }}

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

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notable in the 17-limit, with a lower absolute error than any previous equal temperaments, beating [[183edo|183]] and superseded by [[224edo|224]].  

! Periods<br>per 8ve

! Associated<br>ratio*

| 33/32

| 25/19

| 94\217<br>(1\217)

| 519.82<br>(5.53)

| 27/20<br>(325/324)

| 90\217<br>(1\217)

| 497.70<br>(5.53)

| 4/3<br>(243/242)

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

217edo can be written in Sagittal using ''almost'' the entire Athenian extension 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>

! 1

! 2

! 3

! 4

! 5

! 6

! 7

! 8

! 9

! 10

! 11

! 12

! 13

! 14

! 15

! 16

! 17

! 18

! 19

! 20

! 21

| rowspan="2" | <big>{{Sagittal| |( }}</big>

| rowspan="2" | <big>{{Sagittal| )|( }}</big>

| rowspan="2" | <big>{{Sagittal| ~|( }}</big>

| rowspan="2" | <big>{{Sagittal| /| }}</big>

| rowspan="2" | <big>{{Sagittal| |) }}</big>

| rowspan="2" | <big>{{Sagittal| (| }}</big>

| rowspan="2" | <big>{{Sagittal| (|( }}</big>

| rowspan="2" | <big>{{Sagittal| //| }}</big>

| rowspan="2" | <big>{{Sagittal| /|) }}</big>

| rowspan="2" | <big>{{Sagittal| /|\ }}</big>

| <small>{{Sagittal|#}}{{sagittal| \!/ }}</small>

| <small>{{Sagittal|#}}{{sagittal| \!) }}</small>

| <small>{{Sagittal|#}}{{sagittal| \\! }}</small>

| <small>{{Sagittal|#}}{{sagittal| (!( }}</small>

| <small>{{Sagittal|#}}{{sagittal| (! }}</small>

| <small>{{Sagittal|#}}{{sagittal| !) }}</small>

| <small>{{Sagittal|#}}{{sagittal| \! }}</small>

| <small>{{Sagittal|#}}{{sagittal| ~!( }}</small>

| <small>{{Sagittal|#}}{{sagittal| )!( }}</small>

| <small>{{Sagittal|#}}{{sagittal| !( }}</small>

| <small>{{Sagittal|#}}</small>

| <big>{{Sagittal| (|) }}</big>

| <big>{{Sagittal| (|\ }}</big>

| <big>{{Sagittal| )||( }}</big>

| <big>{{Sagittal| ~||( }}</big>

| <big>{{Sagittal| )||~ }}</big>

| <big>{{Sagittal| ||) }}</big>

| <big>{{Sagittal| ||\ }}</big>

| <big>{{Sagittal| (||( }}</big>

| <big>{{Sagittal| //|| }}</big>

| <big>{{Sagittal| /||) }}</big>

| <big>{{Sagittal| /||\ }}</big>

The 5-up (quup) alteration neatly maps to the pythagorean-septimal comma.{{Ups and downs sharpness|217|false}}

 

 

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. It also can be written with SZ half-sharps and up/down. The main drawback of this notation is that 3/2 is no longer P5 but '''^P5''' , so that ~4:5:6 is from C written as C-E-^G, compared to C-^<E-G to the patent val chain-of-fifths ups-and-downs notation.

|+Alternative 31edo-based notation

|-

| rowspan="2" | ^

| rowspan="2" | ^^

| rowspan="2" | ^^^

=== 7edo-based whitewood notation ===

Since {{nowrap| 217 {{=}} 7 × 31}}, one ''could'' use the inherited [[Whitewood family|whitewood]] fifth 124\217, ditch sharps and flats, and instead use ups and downs to represent pitch deviations from the 7edo nominals. Since 31=15*2+1, all pitches can be notated with ups, downs, quips and quids. ~4:5:6 becomes C-^^^>E-^^^G.

! Sharp

Symbol

| rowspan="2" |h

| ^

|^^

|^^^

vv>

| vv>>>

^^^>>