208edo: Difference between revisions

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The ''208 equal division'' divides the [[Octave|octave]] into 208 equal parts of size 5.769 [[cent|cent]]s each. It tempers out 15625/15552, the kleisma, and is the [[Optimal_patent_val|optimal patent val]] for the kleismic temperament [[Kleismic_family|metakleismic]], and 7, 11 and 13 limit rank three [[Tolermic_family|tolerant]] temperament. It is also the optimal patent val for the rank four [[11-limit|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.

~~208~~ = ~~16 * 13~~, ~~and has divisors 2~~, 4, 8, 16, 13, 26, 52, ~~104~~.

== Theory ==

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.

=~~13-limit transversal~~=

=== Odd harmonics ===

[196/195, 100/99, 91/90, 64/63, 55/54, 49/48, 40/39, 77/75, 36/35, 28/27, 80/77, 25/24, 245/234, 22/21, 21/20, 81/77, 35/33, 52/49, 16/15, 77/72, 15/14, 14/13, 250/231, 13/12, 49/45, 12/11, 35/32, 100/91, 11/10, 54/49, 10/9, 49/44, 39/35, 28/25, 55/49, 9/8, 147/130, 25/22, 91/80, 8/7, 55/48, 147/128, 15/13, 196/169, 64/55, 7/6, 90/77, 75/64, 147/125, 13/11, 77/65, 25/21, 105/88, 117/98, 6/5, 77/64, 40/33, 63/52, 128/105, 11/9, 49/40, 16/13, 154/125, 26/21, 56/45, 96/77, 5/4, 49/39, 44/35, 63/50, 80/63, 14/11, 125/98, 32/25, 77/60, 9/7, 35/27, 100/77, 13/10, 64/49, 55/42, 21/16, 120/91, 33/25, 65/49, 4/3, 147/110, 75/56, 35/26, 66/49, 27/20, 49/36, 15/11, 175/128, 48/35, 11/8, 135/98, 18/13, 245/176, 39/28, 7/5, 108/77, 45/32, 147/104, 64/45, 77/54, 10/7, 56/39, 351/245, 13/9, 196/135, 16/11, 35/24, 143/98, 22/15, 72/49, 40/27, 49/33, 52/35, 112/75, 220/147, 3/2, 98/65, 50/33, 91/60, 32/21, 55/36, 49/32, 20/13, 77/50, 54/35, 14/9, 120/77, 25/16, 196/125, 11/7, 63/40, 100/63, 35/22, 78/49, 8/5, 77/48, 45/28, 21/13, 125/77, 13/8, 49/30, 18/11, 105/64, 104/63, 33/20, 81/49, 5/3, 147/88, 117/70, 42/25, 130/77, 22/13, 245/144, 75/44, 77/45, 12/7, 55/32, 169/98, 26/15, 256/147, 96/55, 7/4, 135/77, 44/25, 260/147, 16/9, 98/55, 25/14, 70/39, 88/49, 9/5, 49/27, 20/11, 91/50, 64/35, 11/6, 90/49, 24/13, 231/125, 13/7, 28/15, 144/77, 15/8, 49/26, 66/35, 91/48, 40/21, 21/11, 245/128, 25/13, 77/40, 27/14, 35/18, 150/77, 39/20, 49/25, 55/28, 63/32, 125/63, 99/50, 195/98, 2]

[[~~Category:Equal divisions of the octave~~|~~###~~]]

=== Subsets and supersets ===

[[Category:~~11-limit~~]]

Since 208 factors into 24 × 13, 208edo has subset edos {{EDOs| 2, 4, 8, 16, 13, 26, 52, and 104 }}.

[[Category:~~13-limit~~]]

[[Category:~~7-limit~~]]

== Regular temperament properties ==

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

|-

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

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

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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

|-

| 2.3.5

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

| {{mapping| 208 330 483 }}

| −0.4301

| 0.5409

| 9.38

|-

| 2.3.5.7

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

| {{mapping| 208 330 483 584 }}

| −0.3586

| 0.4845

| 8.40

|-

| 2.3.5.7.11

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

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

| −0.4330

| 0.4582

| 7.94

|-

| 2.3.5.7.11.13

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

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

| −0.4410

| 0.4187

| 7.26

|}

=== Rank-2 temperaments ===

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

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 47\208

| 271.15

| 1024/875

| [[Quasiorwell]]

|-

| 1

| 55\208

| 317.31

| 6/5

| [[Metakleismic]]

|-

| 4

| 55\208 (3\208)

| 317.31 (17.31)

| 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: @@
-The ''208 equal division'' divides the [[Octave|octave]] into 208 equal parts of size 5.769 [[cent|cent]]s each. It tempers out 15625/15552, the kleisma, and is the [[Optimal_patent_val|optimal patent val]] for the kleismic temperament [[Kleismic_family|metakleismic]], and 7, 11 and 13 limit rank three [[Tolermic_family|tolerant]] temperament. It is also the optimal patent val for the rank four [[11-limit|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.
+{{Infobox ET}}
+{{ED intro}}
-= 16 * 13, and has divisors 2, 4, 8, 16, 13, 26, 52, 104.
+== Theory ==
+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.
-=13-limit transversal=
+=== Odd harmonics ===
-[196/195, 100/99, 91/90, 64/63, 55/54, 49/48, 40/39, 77/75, 36/35, 28/27, 80/77, 25/24, 245/234, 22/21, 21/20, 81/77, 35/33, 52/49, 16/15, 77/72, 15/14, 14/13, 250/231, 13/12, 49/45, 12/11, 35/32, 100/91, 11/10, 54/49, 10/9, 49/44, 39/35, 28/25, 55/49, 9/8, 147/130, 25/22, 91/80, 8/7, 55/48, 147/128, 15/13, 196/169, 64/55, 7/6, 90/77, 75/64, 147/125, 13/11, 77/65, 25/21, 105/88, 117/98, 6/5, 77/64, 40/33, 63/52, 128/105, 11/9, 49/40, 16/13, 154/125, 26/21, 56/45, 96/77, 5/4, 49/39, 44/35, 63/50, 80/63, 14/11, 125/98, 32/25, 77/60, 9/7, 35/27, 100/77, 13/10, 64/49, 55/42, 21/16, 120/91, 33/25, 65/49, 4/3, 147/110, 75/56, 35/26, 66/49, 27/20, 49/36, 15/11, 175/128, 48/35, 11/8, 135/98, 18/13, 245/176, 39/28, 7/5, 108/77, 45/32, 147/104, 64/45, 77/54, 10/7, 56/39, 351/245, 13/9, 196/135, 16/11, 35/24, 143/98, 22/15, 72/49, 40/27, 49/33, 52/35, 112/75, 220/147, 3/2, 98/65, 50/33, 91/60, 32/21, 55/36, 49/32, 20/13, 77/50, 54/35, 14/9, 120/77, 25/16, 196/125, 11/7, 63/40, 100/63, 35/22, 78/49, 8/5, 77/48, 45/28, 21/13, 125/77, 13/8, 49/30, 18/11, 105/64, 104/63, 33/20, 81/49, 5/3, 147/88, 117/70, 42/25, 130/77, 22/13, 245/144, 75/44, 77/45, 12/7, 55/32, 169/98, 26/15, 256/147, 96/55, 7/4, 135/77, 44/25, 260/147, 16/9, 98/55, 25/14, 70/39, 88/49, 9/5, 49/27, 20/11, 91/50, 64/35, 11/6, 90/49, 24/13, 231/125, 13/7, 28/15, 144/77, 15/8, 49/26, 66/35, 91/48, 40/21, 21/11, 245/128, 25/13, 77/40, 27/14, 35/18, 150/77, 39/20, 49/25, 55/28, 63/32, 125/63, 99/50, 195/98, 2]
+{{Harmonics in equal|208}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+=== Subsets and supersets ===
-[[Category:11-limit]]
+Since 208 factors into 2<sup>4</sup> × 13, 208edo has subset edos {{EDOs| 2, 4, 8, 16, 13, 26, 52, and 104 }}.
-[[Category:13-limit]]
-[[Category:7-limit]]
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3.5
+| 15625/15552, {{monzo| 57 -33 -2 }}
+| {{mapping| 208 330 483 }}
+| −0.4301
+| 0.5409
+| 9.38
+|-
+| 2.3.5.7
+| 2401/2400, 15625/15552, 179200/177147
+| {{mapping| 208 330 483 584 }}
+| −0.3586
+| 0.4845
+| 8.40
+|-
+| 2.3.5.7.11
+| 896/891, 2200/2187, 2401/2400, 3025/3024
+| {{mapping| 208 330 483 584 720 }}
+| −0.4330
+| 0.4582
+| 7.94
+|-
+| 2.3.5.7.11.13
+| 325/324, 352/351, 364/363, 676/675, 2401/2400
+| {{mapping| 208 330 483 584 720 770 }}
+| −0.4410
+| 0.4187
+| 7.26
+|}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 1
+| 47\208
+| 271.15
+| 1024/875
+| [[Quasiorwell]]
+|-
+| 1
+| 55\208
+| 317.31
+| 6/5
+| [[Metakleismic]]
+|-
+| 4
+| 55\208<br>(3\208)
+| 317.31<br>(17.31)
+| 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]]