Xenharmonic Wiki

1106edo: Difference between revisions

← Older edit
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,351 edits
fill in the temperament table
 
(14 intermediate revisions by 4 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1106}}
{{ED intro}}
 
== Theory ==
== Theory ==
1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13 and 17 limits, but even so is distinctly [[consistent]] through the [[17-odd-limit]].  
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 [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441, and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342, and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]].  


It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In higher limits, it supports the 79th-octave temperament [[gold]].
The equal temperament [[tempering out|tempers out]] {{monzo| -53 10 16 }} (kwazy comma) and {{monzo| -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 ===
=== Prime harmonics ===
{{Harmonics in equal|1106}}
{{Harmonics in equal|1106|columns=12}}


=== Divisors ===
=== Subsets and supersets ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.
Since 1106 factors into {{factorization|1106}}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 1753 -1106 }}
| {{Mapping| 1106 1753 }}
| −0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{Monzo| -53 10 16 }}, {{monzo| -13 -46 37 }}
| {{Mapping| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{Mapping| 1106 1753 2568 3105 }}
| −0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| −0.021
| 0.045
| 4.11
|}
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
Line 25: Line 82:
| 231.103
| 231.103
| 8/7
| 8/7
| [[Orga]]
| [[Orga]] (11-limit)
|-
|-
| 1
| 1
Line 34: Line 91:
|-
|-
| 2
| 2
| 401\1106
| 150\1106
| 435.081
| 162.749
| 9/7
| 1125/1024
| [[Crazy]]
|-
| 2
| 401\1106<br>(152\1106)
| 435.081<br>(164.919)
| 9/7<br>(11/10)
| [[Semisupermajor]]
| [[Semisupermajor]]
|-
|-
Line 42: Line 105:
| 479\1106<br>(5\1106)
| 479\1106<br>(5\1106)
| 519.711<br>(5.424)
| 519.711<br>(5.424)
| 27/20<br>(325/324)
| 27/20<br>(5120/5103)
| [[Brahmagupta]]
| [[Brahmagupta]] (7-limit)
|-
|-
| 79
| 79
Line 51: Line 114:
| [[Gold]]
| [[Gold]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
Retrieved from "https://en.xen.wiki/w/1106edo"