161edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
+subsets and supersets
ArrowHead294 (talk | contribs)
15,218 edits
mNo edit summary
Line 13: Line 13:


== 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 54: Line 46:
| 243/242, 351/350, 441/440, 847/845, 3136/3125
| 243/242, 351/350, 441/440, 847/845, 3136/3125
| {{mapping| 161 255 374 452 557 596 }}
| {{mapping| 161 255 374 452 557 596 }}
| -0.046
| &minus;0.046
| 0.449
| 0.449
| 6.03
| 6.03
Line 61: Line 53:
| 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088
| 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088
| {{mapping| 161 255 374 452 557 596 658 }}
| {{mapping| 161 255 374 452 557 596 658 }}
| -0.018
| &minus;0.018
| 0.422
| 0.422
| 5.66
| 5.66
Line 68: Line 60:
| 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512
| 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512
| {{mapping| 161 255 374 452 557 596 658 684 }}
| {{mapping| 161 255 374 452 557 596 658 684 }}
| -0.034
| &minus;0.034
| 0.397
| 0.397
| 5.32
| 5.32
|}
{{comma basis end}}
* 161et has a lower [[TE error|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo|183et]].  
* 161et has a lower [[TE error|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo|183et]].  


=== 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 132: Line 118:
|-
|-
| 7
| 7
| 67\161<br>(2\161)
| 67\161<br />(2\161)
| 499.38<br>(14.91)
| 499.38<br />(14.91)
| 4/3<br>(81/80)
| 4/3<br />(81/80)
| [[Absurdity]]
| [[Absurdity]]
|}
{{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:Mintone]]
[[Category:Mintone]]

Revision as of 04:46, 16 November 2024

← 160edo 161edo 162edo →
Prime factorization 7 × 23
Step size 7.45342 ¢ 
Fifth 94\161 (700.621 ¢)
Semitones (A1:m2) 14:13 (104.3 ¢ : 96.89 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

161et tempers out the würschmidt comma, 393216/390625, in the 5-limit; 3136/3125, 6144/6125 and 2401/2400 in the 7-limit; 243/242, 441/440, 540/539 and 5632/5625 in the 11-limit; and 351/350, 847/845, 1001/1000, 1188/1183, 1575/1573 and 1716/1715 in the 13-limit. It serves as the optimal patent val for the mintone temperament in the 5-, 7-, 11- and 13-limit.

Prime harmonics

In the range of edos from 100 to 200, 161edo is notable as being low in 29-limit relative error.

Approximation of prime harmonics in 161edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.33 +1.26 +0.12 +0.23 +1.71 -0.61 +0.62 -2.19 -1.01 +2.79
Relative (%) +0.0 -17.9 +17.0 +1.6 +3.2 +22.9 -8.2 +8.4 -29.3 -13.5 +37.4
Steps
(reduced) 		161
(0) 		255
(94) 		374
(52) 		452
(130) 		557
(74) 		596
(113) 		658
(14) 		684
(40) 		728
(84) 		782
(138) 		798
(154)

Subsets and supersets

Since 161 factors into 7 × 23, 161edo contains 7edo and 23edo as its subsets.

Regular temperament properties

Template:Comma basis begin |- | 2.3 | [-255 161⟩ | [⟨161 255]] | +0.421 | 0.421 | 5.65 |- | 2.3.5 | 393216/390625, [-17 21 -7⟩ | [⟨161 255 374]] | +0.099 | 0.570 | 7.65 |- | 2.3.5.7 | 2401/2400, 3136/3125, 177147/175000 | [⟨161 255 374 452]] | +0.064 | 0.498 | 6.67 |- | 2.3.5.7.11 | 243/242, 441/440, 3136/3125, 35937/35840 | [⟨161 255 374 452 557]] | +0.037 | 0.448 | 6.01 |- | 2.3.5.7.11.13 | 243/242, 351/350, 441/440, 847/845, 3136/3125 | [⟨161 255 374 452 557 596]] | −0.046 | 0.449 | 6.03 |- | 2.3.5.7.11.13.17 | 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088 | [⟨161 255 374 452 557 596 658]] | −0.018 | 0.422 | 5.66 |- | 2.3.5.7.11.13.17.19 | 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512 | [⟨161 255 374 452 557 596 658 684]] | −0.034 | 0.397 | 5.32 Template:Comma basis end

  • 161et has a lower absolute error than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by 183et.

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 15\161 | 111.80 | 16/15 | Vavoom |- | 1 | 16\161 | 119.25 | 15/14 | Septidiasemi |- | 1 | 17\161 | 126.71 | 14/13 | Mowglic |- | 1 | 25\161 | 186.34 | 10/9 | Mintone |- | 1 | 26\161 | 193.79 | 28/25 | Hemiwürschmidt |- | 1 | 38\161 | 283.23 | 33/28 | Neominor (161f) |- | 1 | 52\161 | 387.58 | 5/4 | Würschmidt (5-limit) |- | 1 | 79\161 | 588.82 | 45/32 | Aufo |- | 7 | 67\161
(2\161) | 499.38
(14.91) | 4/3
(81/80) | Absurdity Template:Rank-2 end {{orf}]

Retrieved from "https://en.xen.wiki/index.php?title=161edo&oldid=165915"