210edo: Difference between revisions

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== Theory ==

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.

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.

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=== Subsets and supersets ===

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 }}.

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

== Regular temperament properties ==

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| {{monzo| 26 -12 -3 }}, {{monzo| 19 10 -15 }}

| {{mapping| 210 333 488 }}

| ~~−0~~.5138

| −0.5138

| 0.3987

| 6.98

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| 3136/3125, 5120/5103, 118098/117649

| {{mapping| 210 333 488 590 }}

| ~~−0~~.6170

| −0.6170

| 0.3888

| 6.80

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| [[Countdown]] (210e)

|}

<nowiki />* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if ~~it is~~ distinct

<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|210}}
+{{ED intro}}
 == Theory ==
-Since {{nowrap|210 {{=}} 3 &times; 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.
+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.
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-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 }}.
+Since 210 factors into {{factorisation|210}}, 210edo has subset edos {{EDOs| 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105 }}.
 == Regular temperament properties ==
@@ Line 28: / Line 28: @@
 | {{monzo| 26 -12 -3 }}, {{monzo| 19 10 -15 }}
 | {{mapping| 210 333 488 }}
-| &minus;0.5138
+| −0.5138
 | 0.3987
 | 6.98
@@ Line 35: / Line 35: @@
 | 3136/3125, 5120/5103, 118098/117649
 | {{mapping| 210 333 488 590 }}
-| &minus;0.6170
+| −0.6170
 | 0.3888
 | 6.80
@@ Line 62: / Line 62: @@
 | [[Countdown]] (210e)
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct