210edo: Difference between revisions

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==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 ===

===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"

! 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 8ve Stretch (¢)

! rowspan="2" | Optimal 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

| 0.3987

| 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

| 0.3888

| 6.80

|}

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperaments

|-

|3

| 3

|123\210 (17\210)

| 123\210 (17\210)

|702.86 (97.14)

| 702.86 (97.14)

|3/2 (135\128)

| 3/2 (135\128)

|[[Misty]]

| [[Misty]]

|-

|5

| 5

|98\210 (13\210)

| 98\210 (13\210)

|560.00 (74.29)

| 560.00 (74.29)

|864/625 (25/24)

| 864/625 (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|~~###~~]] ~~~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
 {{EDO intro|210}}
-==Theory==
-et 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===
+et [[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}}
-===Subsets and supersets===
-edo 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"
-! 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
 | 0.3987
 | 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
 | 0.3888
 | 6.80
 |}
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! 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>(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 -->