210edo: Difference between revisions

(3 intermediate revisions by one other user not shown)

Line 1:

== 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.

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 ==

{~~{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

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

| {{mapping| 210 333 488 }}

| ~~−0~~.5138

| −0.5138

| 0.3987

| 6.98

Line 26:

Line 35:

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

| {{mapping| 210 333 488 590 }}

| ~~−0~~.6170

| −0.6170

| 0.3888

| 6.80

~~{{comma basis end}~~}

|}

=== 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

Line 45:

Line 61:

| 25/24

| [[Countdown]] (210e)

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<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 ==
-= 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.
@@ 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 ==
-{{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
 | {{monzo| 26 -12 -3 }}, {{monzo| 19 10 -15 }}
 | {{mapping| 210 333 488 }}
-| &minus;0.5138
+| −0.5138
 | 0.3987
 | 6.98
@@ Line 26: / Line 35: @@
 | 3136/3125, 5120/5103, 118098/117649
 | {{mapping| 210 333 488 590 }}
-| &minus;0.6170
+| −0.6170
 | 0.3888
 | 6.80
-{{comma basis end}}
+|}
 === 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
@@ Line 45: / Line 61: @@
 | 25/24
 | [[Countdown]] (210e)
-{{rank-2 end}}
+|}
-{{orf}}
+<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