1106edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1106}}
{{EDO intro|1106}}
== 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 [[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-limit, but even so is 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]].
It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[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}}


=== 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 2 × 7 × 79, 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"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal
! rowspan="2" | Optimal<br>8ve Stretch (¢)
8ve Stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" |Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|1753 -1106}}
| {{monzo| 1753 -1106 }}
|{{val|1106 1753}}
| {{val| 1106 1753 }}
| -0.010
| -0.010
|0.010
| 0.010
|0.99
| 0.99
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-53 10 16}}, {{monzo|40 -56 21}}
| {{monzo| -53 10 16 }}, {{monzo| 40 -56 21 }}
|{{val|1106 1753 2568}}
| {{val| 1106 1753 2568 }}
| +0.001
| +0.001
|0.019
| 0.019
|1.73
| 1.73
|-
|-
|2.3.5.7
| 2.3.5.7
|4375/4374, 52734375/52706752, {{monzo|46 -14 -3 -6}}
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
|{{val|1106 1753 2568 3105}}
| {{val| 1106 1753 2568 3105 }}
|<nowiki>-0.006</nowiki>
| -0.006
|0.020
| 0.020
|1.83
| 1.83
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
|{{val|1106 1753 2568 3105 3826}}
| {{val| 1106 1753 2568 3105 3826 }}
| +0.004
| +0.004
|0.026
| 0.026
|2.38
| 2.38
|-
|-
|2.3.5.7.11.13
|2.3.5.7.11.13
|3025/3024, 4096/4095, 4375/4374, 456533/456300, 928125/927472
| 3025/3024, 4096/4095, 4375/4374, 456533/456300, 928125/927472
|{{val|1106 1753 2568 3105 3826 4093}}
| {{val| 1106 1753 2568 3105 3826 4093 }}
|<nowiki>-0.012</nowiki>
| -0.012
|0.043
| 0.043
|3.94
| 3.94
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|2500/2499, 3025/3024, 4096/4095, 8624/8619, 9801/9800, 14875/14572
| 2500/2499, 3025/3024, 4096/4095, 8624/8619, 9801/9800, 14875/14572
|{{val|1106 1753 2568 3105 3826 4093 4521}}
| {{val| 1106 1753 2568 3105 3826 4093 4521 }}
|<nowiki>-0.021</nowiki>
| -0.021
|0.045
| 0.045
|4.11
| 4.11
|}
|}
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"

Revision as of 01:29, 6 July 2023

← 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.

It notably supports supermajor, brahmagupta, and orga in the 7-limit, and notably 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
Error Absolute (¢) +0.000 +0.034 -0.057 +0.071 -0.143 +0.340 +0.289 -0.225 -0.065 +0.079 -0.370
Relative (%) +0.0 +3.1 -5.2 +6.5 -13.1 +31.4 +26.6 -20.8 -6.0 +7.3 -34.1
Steps
(reduced) 		1106
(0) 		1753
(647) 		2568
(356) 		3105
(893) 		3826
(508) 		4093
(775) 		4521
(97) 		4698
(274) 		5003
(579) 		5373
(949) 		5479
(1055)

Subsets and supersets

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

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [1753 -1106⟩ ⟨1106 1753] -0.010 0.010 0.99
2.3.5 [-53 10 16⟩, [40 -56 21⟩ ⟨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, 456533/456300, 928125/927472 ⟨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, 8624/8619, 9801/9800, 14875/14572 ⟨1106 1753 2568 3105 3826 4093 4521] -0.021 0.045 4.11

Rank-2 temperaments

Periods
per 8ve 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 213\1106 231.103 8/7 Orga
1 401\1106 435.081 9/7 Supermajor
2 401\1106 435.081 9/7 Semisupermajor
7 479\1106
(5\1106) 		519.711
(5.424) 		27/20
(325/324) 		Brahmagupta
79 459\1106
(11\1106) 		498.011
(11.935) 		4/3
(?) 		Gold
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