214edo: Difference between revisions

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== Theory ==

214edo is (uniquely) consistent through the [[7-odd-limit]]. The patent val for 214edo is {{val| 214 339 497 601 740 792 }}, which [[tempering out|tempers out]] the following commas: 78732/78125 ([[sensipent comma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; 6144/6125 ([[porwell comma]]), 16875/16807 ([[mirkwai comma]]), 321489/320000 (varunisma), and {{monzo| 22 -1 -10 1 }} (quasiorwellisma) in the 7-limit; [[540/539]] ~~and~~ [[~~1375~~/~~1372~~]] in the 11-limit; [[351/350]], [[847/845]], ~~and~~ [[1188/1183]] in the 13-limit. It can be viewed as a 2.13/5 [[subgroup]] temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.

214edo is (uniquely) consistent through the [[7-odd-limit]]. The patent val for 214edo is {{val| 214 339 497 601 740 792 }}, which [[tempering out|tempers out]] the following commas: 78732/78125 ([[sensipent comma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; 6144/6125 ([[porwell comma]]), 16875/16807 ([[mirkwai comma]]), 321489/320000 (varunisma), and {{monzo| 22 -1 -10 1 }} (quasiorwellisma) in the 7-limit; [[540/539]], 1375/1372, [[5632/5625]], in the 11-limit; [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]], and [[4096/4095]] in the 13-limit. It can be viewed as a 2.3.5.13.19.23 [[subgroup]] temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.

=== Prime harmonics ===

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=== Subsets and supersets ===

214 factors into 2 × 107, ~~with~~ [[2edo]] and [[107edo]] as its ~~subset edos~~.

Since 214 factors into 2 × 107, 214edo contains [[2edo]] and [[107edo]] as its subsets.

== Regular temperament properties ==

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|-

| 2.3.5.7.11

| 540/539, 1375/1372, ~~3025~~/~~3024~~, ~~5632~~/~~5625~~

| 540/539, 1375/1372, 5632/5625, 72171/71680

| {{mapping| 214 339 497 601 740 }}

| +0.0897

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|-

| 2.3.5.7.11.13

| 540/539, 847/845~~, 1001/1000~~, 1375/1372, ~~5632~~/~~5625~~

| 351/350, 540/539, 847/845, 1375/1372, 4096/4095

| {{mapping| 214 339 497 601 740 792 }}

| +0.0480

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|-

| 2.3.5.7.11.13.17

| 351/350, 715/714, 936/935, ~~1275/1274, 5544/5525, 5850~~/~~5831~~

| 351/350, 540/539, 715/714, 847/845, 936/935, 4096/4095

| {{mapping| 214 339 497 601 740 792 875 }}

| -0.0144

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| 79\214

| 442.99

| 9/7

| 162/125

| [[~~Sensi~~]]

| [[Sensipent]]

|-

| 1

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| 157.01

| 35/32

| [[Bison]]

| [[Bison]] (214e)

|-

| 2

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is (uniquely) consistent through the [[7-odd-limit]]. The patent val for 214edo is {{val| 214 339 497 601 740 792 }}, which [[tempering out|tempers out]] the following commas: 78732/78125 ([[sensipent comma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; 6144/6125 ([[porwell comma]]), 16875/16807 ([[mirkwai comma]]), 321489/320000 (varunisma), and {{monzo| 22 -1 -10 1 }} (quasiorwellisma) in the 7-limit; [[540/539]] and [[1375/1372]] in the 11-limit; [[351/350]], [[847/845]], and [[1188/1183]] in the 13-limit. It can be viewed as a 2.13/5 [[subgroup]] temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.
+edo is (uniquely) consistent through the [[7-odd-limit]]. The patent val for 214edo is {{val| 214 339 497 601 740 792 }}, which [[tempering out|tempers out]] the following commas: 78732/78125 ([[sensipent comma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; 6144/6125 ([[porwell comma]]), 16875/16807 ([[mirkwai comma]]), 321489/320000 (varunisma), and {{monzo| 22 -1 -10 1 }} (quasiorwellisma) in the 7-limit; [[540/539]], 1375/1372, [[5632/5625]], in the 11-limit; [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]], and [[4096/4095]] in the 13-limit. It can be viewed as a 2.3.5.13.19.23 [[subgroup]] temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.
 === Prime harmonics ===
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
-factors into 2 × 107, with [[2edo]] and [[107edo]] as its subset edos.
+Since 214 factors into 2 × 107, 214edo contains [[2edo]] and [[107edo]] as its subsets.
 == Regular temperament properties ==
@@ Line 44: / Line 44: @@
 |-
 | 2.3.5.7.11
-| 540/539, 1375/1372, 3025/3024, 5632/5625
+| 540/539, 1375/1372, 5632/5625, 72171/71680
 | {{mapping| 214 339 497 601 740 }}
 | +0.0897
@@ Line 51: / Line 51: @@
 |-
 | 2.3.5.7.11.13
-| 540/539, 847/845, 1001/1000, 1375/1372, 5632/5625
+| 351/350, 540/539, 847/845, 1375/1372, 4096/4095
 | {{mapping| 214 339 497 601 740 792 }}
 | +0.0480
@@ Line 58: / Line 58: @@
 |-
 | 2.3.5.7.11.13.17
-| 351/350, 715/714, 936/935, 1275/1274, 5544/5525, 5850/5831
+| 351/350, 540/539, 715/714, 847/845, 936/935, 4096/4095
 | {{mapping| 214 339 497 601 740 792 875 }}
 | -0.0144
@@ Line 83: / Line 83: @@
 | 79\214
 | 442.99
-| 9/7
+| 162/125
-| [[Sensi]]
+| [[Sensipent]]
 |-
 | 1
@@ Line 96: / Line 96: @@
 | 157.01
 | 35/32
-| [[Bison]]
+| [[Bison]] (214e)
 |-
 | 2