231edo: Difference between revisions

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== Theory ==

In the 5-limit, 231et tempers out the [[kleisma]], 15625/15552, and in the 7-limit [[1029/1024]], so that it [[support]]s the [[tritikleismic]] temperament, and in fact provides the [[optimal patent val]]. In the 11-limit it tempers out [[385/384]], [[441/440]] and [[4000/3993]], leading to 11-limit tritikleismic for which it also gives the optimal patent val.

231 years is the number of years in a 41 out of 231 leap week cycle, which corresponds to a 41 & 149 temperament tempering out 132055/131072, 166375/165888, and 2460375/2458624. This type of solar calendar leap rule scale may actually be of more use to harmony, since a 41 note subset mimics [[41edo]], a rather useful edo harmonically, and it preserves the simple commas mentioned above.

=== Odd harmonics ===

In the 5-limit it tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it [[support]]s [[Kleismic_family#Tritikleismic|tritikleismic temperament]], and in fact provides the [[optimal patent val]]. In the 11-limit it tempers out 385/384, 441/440 and 4000/3993, leading to 11-limit tritikleismic for which it also gives the optimal patent val.

231 years is the number of years in a 41 out of 231 leap week cycle, which corresponds to a 41 & 149 temperament tempering out 132055/131072, 166375/165888, and 2460375/2458624. This type of solar calendar leap rule scale may actually be of more use to harmony, since a 41 note subset mimics [[41edo]], a rather useful EDO harmonically, and it preserves the simple commas mentioned above - [http://x31eq.com/cgi-bin/rt.cgi?ets=41%26231&limit=11 see here.]

== Regular temperament properties ==

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|-

| 2.3.5

| 15625/15552, [-64, 36~~, 3⟩~~

| 15625/15552, {{monzo| -64 36 3 }}

| [{{val|231 366 536}}]

| [{{val| 231 366 536 }}]

| 0.410

| 0.334

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| 2.3.5.7

| 1029/1024, 15625/15552, 823543/820125

| [{{val|231 366 536 648}}]

| [{{val| 231 366 536 648 }}]

| 0.539

| 0.365

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|-

| 2.3.5.7.11

| 385/384, 441/440, ~~14700~~/~~14641~~, ~~2460375~~/~~2458624~~

| 385/384, 441/440, 4000/3993, 823543/820125

| [{{val|231 366 536 648 799}}]

| [{{val| 231 366 536 648 799 }}]

| 0.469

| 0.354

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|}

== Rank ~~two~~ temperaments ~~by generator~~ ==

== Rank-2 temperaments ==

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

! Periods per octave

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| 62\231

| 322.08

|

| 135/112

| Dee leap week

|-

| 3

| 61\231 (16\231)

| 61\231 (16\231)

| 316.88 (83.12)

| 316.88 (83.12)

| 6/5

| Tritrikleismic

|}

[[Category:~~tritikleismic~~]]

[[Category:Equal divisions of the octave]]

[[Category:Tritikleismic]]

@@ Line 2: / Line 2: @@
 == Theory ==
+In the 5-limit, 231et tempers out the [[kleisma]], 15625/15552, and in the 7-limit [[1029/1024]], so that it [[support]]s the [[tritikleismic]] temperament, and in fact provides the [[optimal patent val]]. In the 11-limit it tempers out [[385/384]], [[441/440]] and [[4000/3993]], leading to 11-limit tritikleismic for which it also gives the optimal patent val.
+years is the number of years in a 41 out of 231 leap week cycle, which corresponds to a 41 & 149 temperament tempering out 132055/131072, 166375/165888, and 2460375/2458624. This type of solar calendar leap rule scale may actually be of more use to harmony, since a 41 note subset mimics [[41edo]], a rather useful edo harmonically, and it preserves the simple commas mentioned above.
+=== Odd harmonics ===
 {{Harmonics in equal|231}}
-In the 5-limit it tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it [[support]]s [[Kleismic_family#Tritikleismic|tritikleismic temperament]], and in fact provides the [[optimal patent val]]. In the 11-limit it tempers out 385/384, 441/440 and 4000/3993, leading to 11-limit tritikleismic for which it also gives the optimal patent val.
-years is the number of years in a 41 out of 231 leap week cycle, which corresponds to a 41 & 149 temperament tempering out 132055/131072, 166375/165888, and 2460375/2458624. This type of solar calendar leap rule scale may actually be of more use to harmony, since a 41 note subset mimics [[41edo]], a rather useful EDO harmonically, and it preserves the simple commas mentioned above - [http://x31eq.com/cgi-bin/rt.cgi?ets=41%26231&limit=11 see here.]
 == Regular temperament properties ==
@@ Line 19: / Line 21: @@
 |-
 | 2.3.5
-| 15625/15552, [-64, 36, 3⟩
+| 15625/15552, {{monzo| -64 36 3 }}
-| [{{val|231 366 536}}]
+| [{{val| 231 366 536 }}]
 | 0.410
 | 0.334
@@ Line 27: / Line 29: @@
 | 2.3.5.7
 | 1029/1024, 15625/15552, 823543/820125
-| [{{val|231 366 536 648}}]
+| [{{val| 231 366 536 648 }}]
 | 0.539
 | 0.365
@@ Line 33: / Line 35: @@
 |-
 | 2.3.5.7.11
-| 385/384, 441/440, 14700/14641, 2460375/2458624
+| 385/384, 441/440, 4000/3993, 823543/820125
-| [{{val|231 366 536 648 799}}]
+| [{{val| 231 366 536 648 799 }}]
 | 0.469
 | 0.354
@@ Line 40: / Line 42: @@
 |}
-== Rank two temperaments by generator ==
+== Rank-2 temperaments ==
 {| class="wikitable center-all left-5"
 ! Periods <br> per octave
@@ Line 51: / Line 53: @@
 | 62\231
 | 322.08
-|
+| 135/112
 | Dee leap week
 |-
 | 3
-| 61\231 <br> (16\231)
+| 61\231<br>(16\231)
-| 316.88 <br> (83.12)
+| 316.88<br>(83.12)
 | 6/5
 | Tritrikleismic
 |}
-[[Category:tritikleismic]]
+[[Category:Equal divisions of the octave]]
+[[Category:Tritikleismic]]