836edo: Difference between revisions

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836edo is a strong 11-limit system, having the lowest absolute error and beating [[612edo]].

The equal temperament [[~~Tempering~~ out|tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.

The equal temperament [[tempering out|tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.

Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the [[patent val]], it tempers out [[2200/2197]], [[4096/4095]], 31250/31213 in the 13-limit; and [[1275/1274]], [[2500/2499]], [[2601/2600]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit quasithird. Using the 836f [[val]], it tempers out [[1716/1715]], [[2080/2079]], 15379/15360 in the 13-limit; and [[2431/2430]], 2500/2499, [[4914/4913]], [[5832/5831]], 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.

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=== Subsets and supersets ===

Since 836 factors into ~~22 × 11 × 19~~, 836edo has subset edos {{EDOs| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. 1672edo, which doubles it, provides a good correction for harmonic 13.

Since 836 factors into {{factorization|836}}, 836edo has subset edos {{EDOs| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. 1672edo, which doubles it, provides a good correction for [[harmonic]] [[13/1|13]].

== Regular temperament properties ==

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! Generator*

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Enneadecal]]

|-

|22

| 22

|347\836 (5\836)

| 347\836 (5\836)

|498.09 (7.18)

| 498.09 (7.18)

|4/3 ({{monzo|16 -13 2}})

| 4/3 ({{monzo| 16 -13 2 }})

|[[Major arcana]]

| [[Major arcana]]

|-

| 38

@@ Line 5: / Line 5: @@
 edo is a strong 11-limit system, having the lowest absolute error and beating [[612edo]].
-The equal temperament [[Tempering out|tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.
+The equal temperament [[tempering out|tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.
 Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the [[patent val]], it tempers out [[2200/2197]], [[4096/4095]], 31250/31213 in the 13-limit; and [[1275/1274]], [[2500/2499]], [[2601/2600]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit quasithird. Using the 836f [[val]], it tempers out [[1716/1715]], [[2080/2079]], 15379/15360 in the 13-limit; and [[2431/2430]], 2500/2499, [[4914/4913]], [[5832/5831]], 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
-Since 836 factors into 2<sup>2</sup> × 11 × 19, 836edo has subset edos {{EDOs| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. 1672edo, which doubles it, provides a good correction for harmonic 13.
+Since 836 factors into {{factorization|836}}, 836edo has subset edos {{EDOs| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. 1672edo, which doubles it, provides a good correction for [[harmonic]] [[13/1|13]].
 == Regular temperament properties ==
@@ Line 97: / Line 97: @@
 ! Generator*
 ! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 124: / Line 124: @@
 | [[Enneadecal]]
 |-
-|22
+| 22
-|347\836<br>(5\836)
+| 347\836<br>(5\836)
-|498.09<br>(7.18)
+| 498.09<br>(7.18)
-|4/3<br>({{monzo|16 -13 2}})
+| 4/3<br>({{monzo| 16 -13 2 }})
-|[[Major arcana]]
+| [[Major arcana]]
 |-
 | 38