282edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
→Regular temperament properties: 23-limit notability
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
Line 16: Line 16:


== 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]] (%)
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 32 -7 -9 }}, {{monzo| -7 22 -12 }}
| {{monzo| 32 -7 -9 }}, {{monzo| -7 22 -12 }}
| {{mapping| 282 447 655 }}
| {{mapping| 282 447 655 }}
| -0.1684
| &minus;0.1684
| 0.1671
| 0.1671
| 3.93
| 3.93
Line 36: Line 28:
| 6144/6125, 118098/117649, 250047/250000
| 6144/6125, 118098/117649, 250047/250000
| {{mapping| 282 447 655 792 }}
| {{mapping| 282 447 655 792 }}
| -0.2498
| &minus;0.2498
| 0.2020
| 0.2020
| 4.75
| 4.75
Line 43: Line 35:
| 540/539, 4000/3993, 5632/5625, 137781/137500
| 540/539, 4000/3993, 5632/5625, 137781/137500
| {{mapping| 282 447 655 792 976 }}
| {{mapping| 282 447 655 792 976 }}
| -0.3081
| &minus;0.3081
| 0.2151
| 0.2151
| 5.06
| 5.06
Line 50: Line 42:
| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575
| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575
| {{mapping| 282 447 655 792 976 1044 }}
| {{mapping| 282 447 655 792 976 1044 }}
| -0.3480
| &minus;0.3480
| 0.2156
| 0.2156
| 5.07
| 5.07
Line 57: Line 49:
| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197
| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197
| {{mapping| 282 447 655 792 976 1044 1153 }}
| {{mapping| 282 447 655 792 976 1044 1153 }}
| -0.3481
| &minus;0.3481
| 0.1996
| 0.1996
| 4.69
| 4.69
Line 64: Line 56:
| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573
| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573
| {{mapping| 282 447 655 792 976 1044 1153 1198 }}
| {{mapping| 282 447 655 792 976 1044 1153 1198 }}
| -0.3152
| &minus;0.3152
| 0.2061
| 0.2061
| 4.84
| 4.84
Line 71: Line 63:
| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287
| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287
| {{mapping| 282 447 655 792 976 1044 1153 1198 1276 }}
| {{mapping| 282 447 655 792 976 1044 1153 1198 1276 }}
| -0.3173
| &minus;0.3173
| 0.1944
| 0.1944
| 4.57
| 4.57
|}
{{comma basis end}}
* 282et has a lower relative error than any previous equal temperaments in the 23-limit, past [[270edo|270]] and before [[311edo|311]].  
* 282et has a lower relative error than any previous equal temperaments in the 23-limit, past [[270edo|270]] and before [[311edo|311]].  


=== 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 123: Line 109:
|-
|-
| 6
| 6
| 51\282<br>(4\282)
| 51\282<br />(4\282)
| 217.02<br>(17.02)
| 217.02<br />(17.02)
| 17/15<br>(105/104)
| 17/15<br />(105/104)
| [[Stearnscape]]
| [[Stearnscape]]
|-
|-
| 6
| 6
| 80\282<br>(14\282)
| 80\282<br />(14\282)
| 340.43<br>(59.57)
| 340.43<br />(59.57)
| 162/133<br>(88/85)
| 162/133<br />(88/85)
| [[Semiseptichrome]]
| [[Semiseptichrome]]
|-
|-
| 6
| 6
| 117\282<br>(23\282)
| 117\282<br />(23\282)
| 497.87<br>(97.87)
| 497.87<br />(97.87)
| 4/3<br>(128/121)
| 4/3<br />(128/121)
| [[Sextile]]
| [[Sextile]]
|}
{{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:Jupiter]]
[[Category:Septisuperfourth]]
[[Category:Septisuperfourth]]
[[Category:Jupiter]]

Revision as of 05:19, 16 November 2024

← 281edo 282edo 283edo →
Prime factorization 2 × 3 × 47
Step size 4.25532 ¢ 
Fifth 165\282 (702.128 ¢) (→ 55\94)
Semitones (A1:m2) 27:21 (114.9 ¢ : 89.36 ¢)
Consistency limit 29
Distinct consistency limit 23

Template:EDO intro

Theory

282edo is the smallest edo distinctly consistent through to the 23-odd-limit, and also the smallest consistent to the 29-odd-limit. It shares the same 3rd, 7th, and 13th harmonics with 94edo (282 = 3 × 94), as well as 11/10 and 20/17 (supporting the garistearn temperament). It has a distinct sharp tendency for odd harmonics up to 29.

The equal temperament tempers out 6144/6125 (porwell), 118098/117649 (stearnsma), and 250047/250000 (landscape comma) in the 7-limit, and 540/539 and 5632/5625 in the 11-limit, so that it provides the optimal patent val for the jupiter temperament; it also tempers out 4000/3993 and 234375/234256, providing the optimal patent val for septisuperfourth temperament. In the 13-limit, it tempers out 729/728, 1575/1573, 1716/1715, 2080/2079, and 10648/10647.

It allows essentially tempered chords including swetismic chords, squbemic chords, and petrmic chords in the 13-odd-limit, in addition to nicolic chords in the 15-odd-limit.

Prime harmonics

Approximation of prime harmonics in 282edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.17 +0.92 +1.39 +1.87 +2.03 +1.43 +0.36 +1.51 +0.21 -0.35
Relative (%) +0.0 +4.1 +21.6 +32.6 +44.0 +47.6 +33.5 +8.4 +35.6 +4.9 -8.3
Steps
(reduced) 		282
(0) 		447
(165) 		655
(91) 		792
(228) 		976
(130) 		1044
(198) 		1153
(25) 		1198
(70) 		1276
(148) 		1370
(242) 		1397
(269)

Subsets and supersets

Since 282 factors into 2 × 3 × 47, 282edo has subset edos 2, 3, 47, 94, and 141.

Regular temperament properties

Template:Comma basis begin |- | 2.3.5 | [32 -7 -9⟩, [-7 22 -12⟩ | [⟨282 447 655]] | −0.1684 | 0.1671 | 3.93 |- | 2.3.5.7 | 6144/6125, 118098/117649, 250047/250000 | [⟨282 447 655 792]] | −0.2498 | 0.2020 | 4.75 |- | 2.3.5.7.11 | 540/539, 4000/3993, 5632/5625, 137781/137500 | [⟨282 447 655 792 976]] | −0.3081 | 0.2151 | 5.06 |- | 2.3.5.7.11.13 | 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575 | [⟨282 447 655 792 976 1044]] | −0.3480 | 0.2156 | 5.07 |- | 2.3.5.7.11.13.17 | 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197 | [⟨282 447 655 792 976 1044 1153]] | −0.3481 | 0.1996 | 4.69 |- | 2.3.5.7.11.13.17.19 | 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573 | [⟨282 447 655 792 976 1044 1153 1198]] | −0.3152 | 0.2061 | 4.84 |- | 2.3.5.7.11.13.17.19.23 | 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287 | [⟨282 447 655 792 976 1044 1153 1198 1276]] | −0.3173 | 0.1944 | 4.57 Template:Comma basis end

  • 282et has a lower relative error than any previous equal temperaments in the 23-limit, past 270 and before 311.

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 13\282 | 55.32 | 33/32 | Escapade |- | 1 | 133\282 | 565.96 | 4096/2835 | Trident (282ef) |- | 2 | 13\282 | 55.32 | 33/32 | Septisuperfourth |- | 2 | 43\282 | 182.98 | 10/9 | Unidecmic |- | 3 | 33\282 | 140.43 | 243/224 | Septichrome |- | 3 | 37\282 | 157.45 | 35/32 | Nessafof (7-limit) |- | 6 | 51\282
(4\282) | 217.02
(17.02) | 17/15
(105/104) | Stearnscape |- | 6 | 80\282
(14\282) | 340.43
(59.57) | 162/133
(88/85) | Semiseptichrome |- | 6 | 117\282
(23\282) | 497.87
(97.87) | 4/3
(128/121) | Sextile Template:Rank-2 end Template:Orf

Retrieved from "https://en.xen.wiki/index.php?title=282edo&oldid=165946"