210edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|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
| {{monzo| 26 -12 -3 }}, {{monzo| 19 10 -15 }}
| {{monzo| 26 -12 -3 }}, {{monzo| 19 10 -15 }}
| {{mapping| 210 333 488 }}
| {{mapping| 210 333 488 }}
| -0.5138
| &minus;0.5138
| 0.3987
| 0.3987
| 6.98
| 6.98
Line 34: Line 26:
| 3136/3125, 5120/5103, 118098/117649
| 3136/3125, 5120/5103, 118098/117649
| {{mapping| 210 333 488 590 }}
| {{mapping| 210 333 488 590 }}
| -0.6170
| &minus;0.6170
| 0.3888
| 0.3888
| 6.80
| 6.80
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! 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>(18/17)
| 3/2<br />(18/17)
| [[Misty]] (210gh)
| [[Misty]] (210gh)
|-
|-
Line 59: Line 45:
| 25/24
| 25/24
| [[Countdown]] (210e)
| [[Countdown]] (210e)
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 02:50, 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

210 = 3 × 70, and 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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 3 | 123\210
(17\210) | 702.86
(97.14) | 3/2
(18/17) | Misty (210gh) |- | 5 | 13\210 | 74.29 | 25/24 | Countdown (210e) Template:Rank-2 end Template:Orf

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