210edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
ArrowHead294 (talk | contribs)
15,215 edits
mNo edit summary
ArrowHead294 (talk | contribs)
15,215 edits
m Partial undo
Line 3: Line 3:


== Theory ==
== Theory ==
210 = 3 × 70, and 210edo shares its [[3/2|fifth]] with [[70edo]]. It is [[consistent]] to the [[9-odd-limit]], but there is a sharp tendency in the lower [[harmonic]]s. The equal temperament [[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.  
Since {{nowrap|210 {{=}} 3 × 70}}, 210edo shares its [[3/2|fifth]] with [[70edo]]. It is [[consistent]] to the [[9-odd-limit]], but there is a sharp tendency in the lower [[harmonic]]s. The equal temperament [[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.  


Using the 210e val, which does the best, 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. 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, which does the best, 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. 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.  
Line 14: Line 14:


== Regular temperament properties ==
== Regular temperament properties ==
{{comma basis begin}}
{| 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]] (%)
|-
|-
| 2.3.5
| 2.3.5
Line 29: Line 38:
| 0.3888
| 0.3888
| 6.80
| 6.80
{{comma basis end}}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{rank-2 begin}}
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
|-
| 3
| 3
Line 45: Line 61:
| 25/24
| 25/24
| [[Countdown]] (210e)
| [[Countdown]] (210e)
{{rank-2 end}}
|}
{{orf}}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Revision as of 13:01, 16 November 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

Since 210 = 3 × 70, 210edo shares its fifth with 70edo. It is consistent to the 9-odd-limit, but there is a sharp tendency in the lower harmonics. The equal temperament 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.

Using the 210e val, which does the best, 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. 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.

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
(18/17) 		Misty (210gh)
5 13\210 74.29 25/24 Countdown (210e)

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

Retrieved from "https://en.xen.wiki/index.php?title=210edo&oldid=166250"