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.  

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 intervals outside that limit for completeness.

|+ 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]]

| 6

| 33.18

| 7

| 38.71

| ''[[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 }}

! Periods<br>per 8ve

! Associated<br>ratio*

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