836edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
Line 60: Line 52:
| 0.0337
| 0.0337
| 2.35
| 2.35
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5.7.11.13
| 2.3.5.7.11.13
| style="border-top: double;" | 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213
| 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213
| style="border-top: double;" | {{mapping| 836 1325 1941 2347 2892 3094 }} (836)
| {{mapping| 836 1325 1941 2347 2892 3094 }} (836)
| style="border-top: double;" | -0.0085
| &minus;0.0085
| style="border-top: double;" | 0.0785
| 0.0785
| style="border-top: double;" | 5.47
| 5.47
|-
|-
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374
| 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374
| {{mapping| 836 1325 1941 2347 2892 3094 3417 }} (836)
| {{mapping| 836 1325 1941 2347 2892 3094 3417 }} (836)
| -0.0014
| &minus;0.0014
| 0.0747
| 0.0747
| 5.21
| 5.21
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5.7.11.13
| 2.3.5.7.11.13
| style="border-top: double;" | 1716/1715, 2080/2079, 3025/3024, 15379/15360, 234375/234256
| 1716/1715, 2080/2079, 3025/3024, 15379/15360, 234375/234256
| style="border-top: double;" | {{mapping| 836 1325 1941 2347 2892 3093 }} (836f)
| {{mapping| 836 1325 1941 2347 2892 3093 }} (836f)
| style="border-top: double;" | +0.0561
| +0.0561
| style="border-top: double;" | 0.0805
| 0.0805
| style="border-top: double;" | 5.60
| 5.60
|-
|-
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
Line 88: Line 80:
| 0.0747
| 0.0747
| 5.20
| 5.20
|}
{{comma basis end}}
* 836et is notable in the 11-limit with a lower absolute error than any previous equal temperaments, past [[612edo|612]] and before [[1084edo|1084]].
* 836et is notable in the 11-limit with a lower absolute error than any previous equal temperaments, past [[612edo|612]] and before [[1084edo|1084]].


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 113: Line 99:
|-
|-
| 2
| 2
| 265\836<br>(56\836)
| 265\836<br />(56\836)
| 380.38<br>(80.38)
| 380.38<br />(80.38)
| 81/65<br>(22/21)
| 81/65<br />(22/21)
| [[Quasithird]] (836)
| [[Quasithird]] (836)
|-
|-
| 19
| 19
| 347\836<br>(5\836)
| 347\836<br />(5\836)
| 498.09<br>(7.18)
| 498.09<br />(7.18)
| 4/3<br>(225/224)
| 4/3<br />(225/224)
| [[Enneadecal]]
| [[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
| 38
| 347\836<br>(5\836)
| 347\836<br />(5\836)
| 498.09<br>(7.18)
| 498.09<br />(7.18)
| 4/3<br>(225/224)
| 4/3<br />(225/224)
| [[Hemienneadecal]]
| [[Hemienneadecal]]
|-
|-
| 44
| 44
| 347\836<br>(5\836)
| 347\836<br />(5\836)
| 498.09<br>(7.18)
| 498.09<br />(7.18)
| 4/3<br>(18375/18304)
| 4/3<br />(18375/18304)
| [[Ruthenium]]
| [[Ruthenium]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


[[Category:Quasithird]]
[[Category:Quasithird]]

Revision as of 05:30, 16 November 2024

← 835edo 836edo 837edo →
Prime factorization 22 × 11 × 19
Step size 1.43541 ¢ 
Fifth 489\836 (701.914 ¢)
Semitones (A1:m2) 79:63 (113.4 ¢ : 90.43 ¢)
Consistency limit 11
Distinct consistency limit 11

Template:EDO intro

Theory

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

The equal temperament 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.

Prime harmonics

Approximation of prime harmonics in 836edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.041 -0.189 +0.074 -0.122 +0.621 -0.171 -0.384 +0.434 -0.391 +0.419
Relative (%) +0.0 -2.9 -13.2 +5.1 -8.5 +43.2 -11.9 -26.7 +30.2 -27.2 +29.2
Steps
(reduced) 		836
(0) 		1325
(489) 		1941
(269) 		2347
(675) 		2892
(384) 		3094
(586) 		3417
(73) 		3551
(207) 		3782
(438) 		4061
(717) 		4142
(798)

Subsets and supersets

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

Regular temperament properties

Template:Comma basis begin |- | 2.3 | [-1325 836⟩ | [⟨836 1325]] | +0.0130 | 0.0130 | 0.90 |- | 2.3.5 | [-14 -19 19⟩, [-69 45 -1⟩ | [⟨836 1325 1941]] | +0.0358 | 0.0340 | 2.37 |- | 2.3.5.7 | 4375/4374, 703125/702464, [41 -4 2 -14⟩ | [⟨836 1325 1941 2347]] | +0.0203 | 0.0399 | 2.78 |- | 2.3.5.7.11 | 3025/3024, 4375/4374, 234375/234256, [22 -4 2 -6 -1⟩ | [⟨836 1325 1941 2347 2892]] | +0.0233 | 0.0362 | 2.52 |- | 2.3.5.7.11.17 | 2500/2499, 3025/3024, 4375/4374, 57375/57344, 108086/108045 | [⟨836 1325 1941 2347 2892 3417]] | +0.0264 | 0.0337 | 2.35 |- style="border-top: double;" | 2.3.5.7.11.13 | 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213 | [⟨836 1325 1941 2347 2892 3094]] (836) | −0.0085 | 0.0785 | 5.47 |- | 2.3.5.7.11.13.17 | 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374 | [⟨836 1325 1941 2347 2892 3094 3417]] (836) | −0.0014 | 0.0747 | 5.21 |- style="border-top: double;" | 2.3.5.7.11.13 | 1716/1715, 2080/2079, 3025/3024, 15379/15360, 234375/234256 | [⟨836 1325 1941 2347 2892 3093]] (836f) | +0.0561 | 0.0805 | 5.60 |- | 2.3.5.7.11.13.17 | 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4914/4913, 11271/11264 | [⟨836 1325 1941 2347 2892 3093 3417]] (836f) | +0.0541 | 0.0747 | 5.20 Template:Comma basis end

  • 836et is notable in the 11-limit with a lower absolute error than any previous equal temperaments, past 612 and before 1084.

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 347\836 | 498.09 | 4/3 | Counterschismic |- | 2 | 161\836 | 231.10 | 8/7 | Orga (836f) |- | 2 | 265\836
(56\836) | 380.38
(80.38) | 81/65
(22/21) | Quasithird (836) |- | 19 | 347\836
(5\836) | 498.09
(7.18) | 4/3
(225/224) | Enneadecal |- | 22 | 347\836
(5\836) | 498.09
(7.18) | 4/3
([16 -13 2⟩) | Major arcana |- | 38 | 347\836
(5\836) | 498.09
(7.18) | 4/3
(225/224) | Hemienneadecal |- | 44 | 347\836
(5\836) | 498.09
(7.18) | 4/3
(18375/18304) | Ruthenium Template:Rank-2 end Template:Orf

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