208edo: Difference between revisions

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== Theory ==

~~208et~~ [[tempering out|tempers out]] [[15625/15552]], the kleisma, and is the [[optimal patent val]] for the kleismic temperament [[metakleismic]], and 7-, 11- and 13-limit rank-3 [[tolerant]] temperament. It is also the optimal patent val for the rank-4 [[11-limit]] temperament tempering out [[896/891]], the [[pentacircle]] temperament. Other commas it tempers out include [[2200/2187]] in the 11-limit and [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the 13-limit.

208edo is closely related to [[104edo]], but the mappings for [[harmonic]] [[5/1|5]] differ. As an equal temperament, it [[tempering out|tempers out]] [[15625/15552]], the kleisma, and is the [[optimal patent val]] for the kleismic temperament [[metakleismic]], and 7-, 11- and 13-limit rank-3 [[tolerant]] temperament. It is also the optimal patent val for the rank-4 [[11-limit]] temperament tempering out [[896/891]], the [[pentacircle]] temperament. Other commas it tempers out include [[2200/2187]] in the 11-limit and [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the 13-limit.

=== Odd harmonics ===

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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

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

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

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| 15625/15552, {{monzo| 57 -33 -2 }}

| {{mapping| 208 330 483 }}

| -0.4301

| −0.4301

| 0.5409

| 9.38

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| 2401/2400, 15625/15552, 179200/177147

| {{mapping| 208 330 483 584 }}

| -0.3586

| −0.3586

| 0.4845

| 8.40

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| 896/891, 2200/2187, 2401/2400, 3025/3024

| {{mapping| 208 330 483 584 720 }}

| -0.4330

| −0.4330

| 0.4582

| 7.94

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| 325/324, 352/351, 364/363, 676/675, 2401/2400

| {{mapping| 208 330 483 584 720 770 }}

| -0.4410

| −0.4410

| 0.4187

| 7.26

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=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

| 1

| 47\208

| ~~251~~.15

| 271.15

| 1024/875

| [[Quasiorwell]]

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| 55\208 (3\208)

| 317.31 (17.31)

| 6/5 (81/80)

| 6/5 (126/125)

| [[Quadritikleismic]] (7-limit)

|}

<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

[[Category:Metakleismic]]

[[Category:Tolerant]]

[[Category:Pentacircle]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|208}}
+{{ED intro}}
 == Theory ==
-et [[tempering out|tempers out]] [[15625/15552]], the kleisma, and is the [[optimal patent val]] for the kleismic temperament [[metakleismic]], and 7-, 11- and 13-limit rank-3 [[tolerant]] temperament. It is also the optimal patent val for the rank-4 [[11-limit]] temperament tempering out [[896/891]], the [[pentacircle]] temperament. Other commas it tempers out include [[2200/2187]] in the 11-limit and [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the 13-limit.
+edo is closely related to [[104edo]], but the mappings for [[harmonic]] [[5/1|5]] differ. As an equal temperament, it [[tempering out|tempers out]] [[15625/15552]], the kleisma, and is the [[optimal patent val]] for the kleismic temperament [[metakleismic]], and 7-, 11- and 13-limit rank-3 [[tolerant]] temperament. It is also the optimal patent val for the rank-4 [[11-limit]] temperament tempering out [[896/891]], the [[pentacircle]] temperament. Other commas it tempers out include [[2200/2187]] in the 11-limit and [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the 13-limit.
 === Odd harmonics ===
@@ Line 13: / Line 13: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
 ! [[TE simple badness|Relative]] (%)
@@ Line 25: / Line 26: @@
 | 15625/15552, {{monzo| 57 -33 -2 }}
 | {{mapping| 208 330 483 }}
-| -0.4301
+| −0.4301
 | 0.5409
 | 9.38
@@ Line 32: / Line 33: @@
 | 2401/2400, 15625/15552, 179200/177147
 | {{mapping| 208 330 483 584 }}
-| -0.3586
+| −0.3586
 | 0.4845
 | 8.40
@@ Line 39: / Line 40: @@
 | 896/891, 2200/2187, 2401/2400, 3025/3024
 | {{mapping| 208 330 483 584 720 }}
-| -0.4330
+| −0.4330
 | 0.4582
 | 7.94
@@ Line 46: / Line 47: @@
 | 325/324, 352/351, 364/363, 676/675, 2401/2400
 | {{mapping| 208 330 483 584 720 770 }}
-| -0.4410
+| −0.4410
 | 0.4187
 | 7.26
@@ Line 53: / Line 54: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
 | 1
 | 47\208
-| 251.15
+| 271.15
 | 1024/875
 | [[Quasiorwell]]
@@ Line 75: / Line 77: @@
 | 55\208<br>(3\208)
 | 317.31<br>(17.31)
-| 6/5<br>(81/80)
+| 6/5<br>(126/125)
 | [[Quadritikleismic]] (7-limit)
 |}
+<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
+[[Category:Metakleismic]]
+[[Category:Tolerant]]
+[[Category:Pentacircle]]