1106edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Expand on theory
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
Line 14: Line 14:


== 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
| {{monzo| 1753 -1106 }}
| {{monzo| 1753 -1106 }}
| {{mapping| 1106 1753 }}
| {{mapping| 1106 1753 }}
| -0.010
| &minus;0.010
| 0.010
| 0.010
| 0.99
| 0.99
Line 41: Line 33:
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{mapping| 1106 1753 2568 3105 }}
| {{mapping| 1106 1753 2568 3105 }}
| -0.006
| &minus;0.006
| 0.020
| 0.020
| 1.83
| 1.83
Line 55: Line 47:
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{mapping| 1106 1753 2568 3105 3826 4093 }}
| {{mapping| 1106 1753 2568 3105 3826 4093 }}
| -0.012
| &minus;0.012
| 0.043
| 0.043
| 3.94
| 3.94
Line 62: Line 54:
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| {{mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| -0.021
| &minus;0.021
| 0.045
| 0.045
| 4.11
| 4.11
|}
{{comma basis end}}


=== 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 96: Line 81:
|-
|-
| 2
| 2
| 401\1106<br>(152\1106)
| 401\1106<br />(152\1106)
| 435.081<br>(164.919)
| 435.081<br />(164.919)
| 9/7<br>(11/10)
| 9/7<br />(11/10)
| [[Semisupermajor]]
| [[Semisupermajor]]
|-
|-
| 7
| 7
| 479\1106<br>(5\1106)
| 479\1106<br />(5\1106)
| 519.711<br>(5.424)
| 519.711<br />(5.424)
| 27/20<br>(5120/5103)
| 27/20<br />(5120/5103)
| [[Brahmagupta]] (7-limit)
| [[Brahmagupta]] (7-limit)
|-
|-
| 79
| 79
| 459\1106<br>(11\1106)
| 459\1106<br />(11\1106)
| 498.011<br>(11.935)
| 498.011<br />(11.935)
| 4/3<br>(?)
| 4/3<br />(?)
| [[Gold]]
| [[Gold]]
|}
{{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}}

Revision as of 04:23, 16 November 2024

← 1105edo 1106edo 1107edo →
Prime factorization 2 × 7 × 79
Step size 1.08499 ¢ 
Fifth 647\1106 (701.989 ¢)
Semitones (A1:m2) 105:83 (113.9 ¢ : 90.05 ¢)
Consistency limit 17
Distinct consistency limit 17

Template:EDO intro

Theory

1106edo is a zeta peak edo. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit relative error being 171, 270, 342, 441 and 612. It is even stronger in the 11-limit; the only ones beating it out now being 270, 342 and 612. It is less strong in the 13- and 17-limit, but even so is distinctly consistent through the 17-odd-limit.

The equal temperament tempers out [-53 10 16⟩ (kwazy comma) and [-13 -46 37⟩ (supermajor comma) in the 5-limit; 4375/4374 and 52734375/52706752 in the 7-limit; 3025/3024 and 9801/9800 in the 11-limit; 4096/4095, 78125/78078, and 105644/105625 in the 13-limit; 2500/2499, 4914/4913, and 8624/8619 in the 17-limit. It notably supports supermajor, brahmagupta, and orga in the 7-limit, and semisupermajor in the 11-limit. In the higher limits, it supports the 79th-octave temperament gold.

Prime harmonics

Approximation of prime harmonics in 1106edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.000 +0.034 -0.057 +0.071 -0.143 +0.340 +0.289 -0.225 -0.065 +0.079 -0.370 +0.374
Relative (%) +0.0 +3.1 -5.2 +6.5 -13.1 +31.4 +26.6 -20.8 -6.0 +7.3 -34.1 +34.5
Steps
(reduced) 		1106
(0) 		1753
(647) 		2568
(356) 		3105
(893) 		3826
(508) 		4093
(775) 		4521
(97) 		4698
(274) 		5003
(579) 		5373
(949) 		5479
(1055) 		5762
(232)

Subsets and supersets

Since 1106 factors into 2 × 7 × 79, it has subset edos 2, 7, 14, 79, 158, and 553.

Regular temperament properties

Template:Comma basis begin |- | 2.3 | [1753 -1106⟩ | [⟨1106 1753]] | −0.010 | 0.010 | 0.99 |- | 2.3.5 | [-53 10 16⟩, [-13 -46 37⟩ | [⟨1106 1753 2568]] | +0.001 | 0.019 | 1.73 |- | 2.3.5.7 | 4375/4374, 52734375/52706752, [46 -14 -3 -6⟩ | [⟨1106 1753 2568 3105]] | −0.006 | 0.020 | 1.83 |- | 2.3.5.7.11 | 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041 | [⟨1106 1753 2568 3105 3826]] | +0.004 | 0.026 | 2.38 |- | 2.3.5.7.11.13 | 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625 | [⟨1106 1753 2568 3105 3826 4093]] | −0.012 | 0.043 | 3.94 |- | 2.3.5.7.11.13.17 | 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619 | [⟨1106 1753 2568 3105 3826 4093 4521]] | −0.021 | 0.045 | 4.11 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 213\1106 | 231.103 | 8/7 | Orga (11-limit) |- | 1 | 401\1106 | 435.081 | 9/7 | Supermajor |- | 2 | 150\1106 | 162.749 | 1125/1024 | Kwazy |- | 2 | 401\1106
(152\1106) | 435.081
(164.919) | 9/7
(11/10) | Semisupermajor |- | 7 | 479\1106
(5\1106) | 519.711
(5.424) | 27/20
(5120/5103) | Brahmagupta (7-limit) |- | 79 | 459\1106
(11\1106) | 498.011
(11.935) | 4/3
(?) | Gold Template:Rank-2 end Template:Orf

Retrieved from "https://en.xen.wiki/index.php?title=1106edo&oldid=165894"