1106edo: Difference between revisions

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{{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-limit, 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 the 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 ===
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=== Subsets and supersets ===
=== 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"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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|-
|-
| 2.3
| 2.3
| {{monzo| 1753 -1106 }}
| {{Monzo| 1753 -1106 }}
| {{val| 1106 1753 }}
| {{Mapping| 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| -13 -46 37 }}
| {{val| 1106 1753 2568 }}
| {{Mapping| 1106 1753 2568 }}
| +0.001
| +0.001
| 0.019
| 0.019
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| 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 }}
| {{Mapping| 1106 1753 2568 3105 }}
| -0.006
| −0.006
| 0.020
| 0.020
| 1.83
| 1.83
Line 47: Line 48:
| 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 }}
| {{Mapping| 1106 1753 2568 3105 3826 }}
| +0.004
| +0.004
| 0.026
| 0.026
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{val| 1106 1753 2568 3105 3826 4093 }}
| {{Mapping| 1106 1753 2568 3105 3826 4093 }}
| -0.012
| −0.012
| 0.043
| 0.043
| 3.94
| 3.94
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{val| 1106 1753 2568 3105 3826 4093 4521 }}
| {{Mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| -0.021
| −0.021
| 0.045
| 0.045
| 4.11
| 4.11
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=== 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
|-
|-
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| 9/7
| 9/7
| [[Supermajor]]
| [[Supermajor]]
|-
| 2
| 150\1106
| 162.749
| 1125/1024
| [[Crazy]]
|-
|-
| 2
| 2
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| [[Gold]]
| [[Gold]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct

Latest revision as of 11:31, 6 March 2025

← 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

1106 equal divisions of the octave (abbreviated 1106edo or 1106ed2), also called 1106-tone equal temperament (1106tet) or 1106 equal temperament (1106et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1106 equal parts of about 1.08 ¢ each. Each step represents a frequency ratio of 21/1106, or the 1106th root of 2.

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

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⟩, [-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

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
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 Crazy
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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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