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.

It [[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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| 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213

| {{mapping| 836 1325 1941 2347 2892 3094 }} (836)

| ~~−0~~.0085

| −0.0085

| 0.0785

| 5.47

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| 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374

| {{mapping| 836 1325 1941 2347 2892 3094 3417 }} (836)

| ~~−0~~.0014

| −0.0014

| 0.0747

| 5.21

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| [[Ruthenium]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if ~~it is~~ distinct

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Quasithird]]

@@ 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.
+It [[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 65: / Line 65: @@
 | 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213
 | {{mapping| 836 1325 1941 2347 2892 3094 }} (836)
-| &minus;0.0085
+| −0.0085
 | 0.0785
 | 5.47
@@ Line 72: / Line 72: @@
 | 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374
 | {{mapping| 836 1325 1941 2347 2892 3094 3417 }} (836)
-| &minus;0.0014
+| −0.0014
 | 0.0747
 | 5.21
@@ Line 144: / Line 144: @@
 | [[Ruthenium]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:Quasithird]]