210edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|210}}
{{EDO intro|210}}
==Theory==
 
210et tempers out 67108864/66430125 (misty comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 3136/3125, 5120/5103, and 118098/117649 in the 7-limit. It is consistent to the 7-limit, but there is a sharp tendency for harmonics 3, 5, and 7. Using the patent val, it tempers out 176/175, 1375/1372, 8019/8000, and 41503/41472 in the 11-limit; 351/350, 352/351, 847/845, 2197/2187, and 16900/16807 in the 13-limit. Using the 210e val, it tempers out 540/539, 4000/3993, 6912/6875, and 15488/15435 in the 11-limit; 351/350, 364/363, 1001/1000, 2197/2187, and 3584/3575 in the 13-limit.
== Theory ==
===Odd harmonics===
210et [[tempering out|tempers out]] 67108864/66430125 ([[misty comma]]) and 30958682112/30517578125 ([[trisedodge comma]]) in the 5-limit; [[3136/3125]], [[5120/5103]], and 118098/117649 in the 7-limit. It is [[consistent]] to the [[9-odd-limit]], but there is a sharp tendency for [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]].  
 
Using the patent val, it tempers out [[176/175]], 1375/1372, [[8019/8000]], and 41503/41472 in the 11-limit; [[351/350]], [[352/351]], [[847/845]], 2197/2187, and 16900/16807 in the 13-limit. Using the 210e val, it tempers out [[540/539]], [[4000/3993]], 6912/6875, and 15488/15435 in the 11-limit; [[351/350]], [[364/363]], [[1001/1000]], [[2197/2187]], and 3584/3575 in the 13-limit.  
 
=== Odd harmonics ===
{{Harmonics in equal|210}}
{{Harmonics in equal|210}}
===Subsets and supersets===
 
210edo factors into 2 × 3 × 5 × 7, with subset edos {{EDOs|2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105}}.
=== Subsets and supersets ===
==Regular temperament properties==
Since 210 factors into 2 × 3 × 5 × 7, 210edo has subset edos {{EDOs| 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105 }}.
 
== 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<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>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
|{{monzo|111 -70}}
|{{val|210 333}}
| -0.2846
| 0.2845
| 4.98
|-
|-
|2.3.5
| 2.3.5
|{{monzo|26 -12 -3}}, {{monzo|19 10 -15}}
| {{monzo| 26 -12 -3 }}, {{monzo| 19 10 -15 }}
|{{val|210 333 488}}
| {{mapping| 210 333 488 }}
| -0.5138
| -0.5138
| 0.3987
| 0.3987
| 6.98
| 6.98
|-
|-
|2.3.5.7
| 2.3.5.7
|3136/3125, 5120/5103, 118098/117649
| 3136/3125, 5120/5103, 118098/117649
|{{val|210 333 488 590}}
| {{mapping| 210 333 488 590 }}
| -0.6170
| -0.6170
| 0.3888
| 0.3888
| 6.80
| 6.80
|}
|}
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+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
|-
|-
|3
| 3
|123\210<br>(17\210)
| 123\210<br>(17\210)
|702.86<br>(97.14)
| 702.86<br>(97.14)
|3/2<br>(135\128)
| 3/2<br>(135\128)
|[[Misty]]
| [[Misty]]
|-
|-
|5
| 5
|98\210<br>(13\210)
| 98\210<br>(13\210)
|560.00<br>(74.29)
| 560.00<br>(74.29)
|864/625<br>(25/24)
| 864/625<br>(25/24)
|[[Trisedodge]]
| [[Trisedodge]]
|}
|}
 
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 09:13, 3 April 2024

← 209edo 210edo 211edo →
Prime factorization 2 × 3 × 5 × 7
Step size 5.71429 ¢ 
Fifth 123\210 (702.857 ¢) (→ 41\70)
Semitones (A1:m2) 21:15 (120 ¢ : 85.71 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

Theory

210et tempers out 67108864/66430125 (misty comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 3136/3125, 5120/5103, and 118098/117649 in the 7-limit. It is consistent to the 9-odd-limit, but there is a sharp tendency for harmonics 3, 5, and 7.

Using the patent val, it tempers out 176/175, 1375/1372, 8019/8000, and 41503/41472 in the 11-limit; 351/350, 352/351, 847/845, 2197/2187, and 16900/16807 in the 13-limit. Using the 210e val, it tempers out 540/539, 4000/3993, 6912/6875, and 15488/15435 in the 11-limit; 351/350, 364/363, 1001/1000, 2197/2187, and 3584/3575 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 210edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.90 +2.26 +2.60 +1.80 -2.75 -0.53 -2.55 -2.10 -0.37 -2.21 +0.30
Relative (%) +15.8 +39.5 +45.5 +31.6 -48.1 -9.2 -44.7 -36.7 -6.5 -38.7 +5.2
Steps
(reduced) 		333
(123) 		488
(68) 		590
(170) 		666
(36) 		726
(96) 		777
(147) 		820
(190) 		858
(18) 		892
(52) 		922
(82) 		950
(110)

Subsets and supersets

Since 210 factors into 2 × 3 × 5 × 7, 210edo has subset edos 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.5 [26 -12 -3⟩, [19 10 -15⟩ [⟨210 333 488]] -0.5138 0.3987 6.98
2.3.5.7 3136/3125, 5120/5103, 118098/117649 [⟨210 333 488 590]] -0.6170 0.3888 6.80

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
3 123\210
(17\210) 		702.86
(97.14) 		3/2
(135\128) 		Misty
5 98\210
(13\210) 		560.00
(74.29) 		864/625
(25/24) 		Trisedodge

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

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