2118edo: Difference between revisions

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{{Infobox ET

~~|Prime factorization = 2 × 3 × 353~~

~~|Fifth = 1239\2118 (→[[706edo|413/706]])~~

~~|Major 2nd = 360\2118~~

~~|Semitones = 201:159~~

~~|Consistency=11~~}}

== Theory ==

Primes approximated with less than 1 standard deviation in 2118edo are: 2, 3, 5, 7, 11, 19, 23, 29, 31, 43. Overall, it offers excellent double-13's 31-limit harmony, as both mappings of 13 (2118 and 2118f vals) have useful interpretations.

2118edo provides a 43-limit approximation of [[secor]] with [[46/43]] (206 steps), however this reduces to 103\1059, meaning that it is a compound of two circles of such secor. In addition, it offers a 205-step generator "meantone secor" which is described by a {{nowrap|31 & 2118}} temperament, also in the 2.3.5.7.11.23.43 subgroup, and also offers a meantone fifth. The comma basis for the "meantone secor" temperament is 5376/5375, 9317/9315, 25921/25920, 151263/151250, and 10551296/10546875.

=== Prime harmonics ===

Primes with less than 1 standard deviation in 2118edo are: 2, 3, 5, 7, 11, 19, 23, 29, 31, 43. Overall, it offers excellent double-13s 31-limit harmony, as both mappings of 13 (2118 and 2118f vals) have useful interpretations.

2118edo provides a 43-limit approximation of [[secor]] with [[46/43]] (206 steps), however this reduces to 103\1059, meaning that it is a compound of two circles of such secor. In addition, it offers a 205-step generator "meantone secor" which is described by a 31 & 2118 temperament, also in the 2.3.5.7.11.23.43 subgroup, and ~~also offers a meantone fifth. The comma basis for the "meantone secor" temperament is 5376/5375, 9317/9315, 25921/25920, 151263/151250, and 10551296/10546875.~~

=== Subsets and supersets ===

2118edo is 6 times the [[353edo]], meaning it can be used to play a compound of 6 chains of the [[Hemimean clan #Rectified hebrew|rectified hebrew]] temperament.

2118edo is 6 times the [[353edo]], meaning it can be used to play a compound of 6 chains of the [[Hemimean clan#Rectified ~~Hebrew~~|~~Rectified Hebrew~~]] temperament.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

~~! rowspan="2" |Subgroup~~

~~! rowspan="2" |[[Comma list]]~~

~~! rowspan="2" |[[Mapping]]~~

~~! rowspan="2" |Optimal~~

~~8ve stretch (¢)~~

~~! colspan="2" |Tuning error~~

|-

![[~~TE error~~|~~Absolute~~]] ~~(¢)~~

! rowspan="2" | [[Subgroup]]

![[~~TE simple badness|Relative~~]] (%)

! rowspan="2" | [[Comma list|Comma List]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve Stretch (¢)

! colspan="2" | Tuning Error

|-

|2.3.5

! [[TE error|Absolute]] (¢)

|{{monzo|38 -2 15}}, {{monzo| -11 130 -84}}

! [[TE simple badness|Relative]] (%)

|[{{~~val~~|2118 3357 4918}}]

|-

| -0.0186

| 2.3.5

|0.0156

| {{monzo| 38 -2 15 }}, {{monzo| -11 130 -84 }}

| {{mapping| 2118 3357 4918 }}

| −0.0186

| 0.0156

|

|-

|2.3.5.7

| 2.3.5.7

|250047/250000,{{monzo|-1 -18 -3 13}}, {{monzo|38 -2 -15 0}}

| 250047/250000, {{monzo| -1 -18 -3 13 }}, {{monzo| 38 -2 -15 }}

|[{{~~val~~|2118 3357 4918 5946}}]

| {{mapping| 2118 3357 4918 5946 }}

| -0.0150

| −0.0150

|0.0148

| 0.0148

|

|-

|2.3.5.7.11

| 2.3.5.7.11

|9801/9800, 250047/250000, {{monzo|25 1 -4 0 -5}}, {{monzo|16 -7 -9 2 3}}

| 9801/9800, 250047/250000, {{monzo| 25 1 -4 0 -5 }}, {{monzo| 16 -7 -9 2 3 }}

|[{{~~val~~|2118 3357 4918 5946 7927}}]

| {{mapping| 2118 3357 4918 5946 7927 }}

|~~<nowiki>-0~~.0096~~</nowiki>~~

| −0.0096

|0.0172

| 0.0172

|

|}

~~[[Category:Equal divisions of the octave|####]] ~~

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-|Prime factorization = 2 × 3 × 353
+{{ED intro}}
-|Fifth = 1239\2118 (→[[706edo|413/706]])
-|Major 2nd = 360\2118
-|Semitones = 201:159
-|Consistency=11}}
-{{EDO intro|2118}}
 == Theory ==
+Primes approximated with less than 1 standard deviation in 2118edo are: 2, 3, 5, 7, 11, 19, 23, 29, 31, 43. Overall, it offers excellent double-13's 31-limit harmony, as both mappings of 13 (2118 and 2118f vals) have useful interpretations.
+edo provides a 43-limit approximation of [[secor]] with [[46/43]] (206 steps), however this reduces to 103\1059, meaning that it is a compound of two circles of such secor. In addition, it offers a 205-step generator "meantone secor" which is described by a {{nowrap|31 &amp; 2118}} temperament, also in the 2.3.5.7.11.23.43 subgroup, and also offers a meantone fifth. The comma basis for the "meantone secor" temperament is 5376/5375, 9317/9315, 25921/25920, 151263/151250, and 10551296/10546875.
+=== Prime harmonics ===
 {{Harmonics in equal|2118}}
-Primes with less than 1 standard deviation in 2118edo are: 2, 3, 5, 7, 11, 19, 23, 29, 31, 43. Overall, it offers excellent double-13s 31-limit harmony, as both mappings of 13 (2118 and 2118f vals) have useful interpretations.
-edo provides a 43-limit approximation of [[secor]] with [[46/43]] (206 steps), however this reduces to 103\1059, meaning that it is a compound of two circles of such secor. In addition, it offers a 205-step generator "meantone secor" which is described by a 31 & 2118 temperament, also in the 2.3.5.7.11.23.43 subgroup, and also offers a meantone fifth. The comma basis for the "meantone secor" temperament is 5376/5375, 9317/9315, 25921/25920, 151263/151250, and 10551296/10546875.
+=== Subsets and supersets ===
+edo is 6 times the [[353edo]], meaning it can be used to play a compound of 6 chains of the [[Hemimean clan #Rectified hebrew|rectified hebrew]] temperament.
-edo is 6 times the [[353edo]], meaning it can be used to play a compound of 6 chains of the [[Hemimean clan#Rectified Hebrew|Rectified Hebrew]] temperament.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |Subgroup
-! rowspan="2" |[[Comma list]]
-! rowspan="2" |[[Mapping]]
-! rowspan="2" |Optimal
-ve stretch (¢)
-! colspan="2" |Tuning error
 |-
-![[TE error|Absolute]] (¢)
+! rowspan="2" | [[Subgroup]]
-![[TE simple badness|Relative]] (%)
+! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! colspan="2" | Tuning Error
 |-
-|2.3.5
+! [[TE error|Absolute]] (¢)
-|{{monzo|38 -2 15}}, {{monzo| -11 130 -84}}
+! [[TE simple badness|Relative]] (%)
-|[{{val|2118 3357 4918}}]
+|-
-| -0.0186
+| 2.3.5
-|0.0156
+| {{monzo| 38 -2 15 }}, {{monzo| -11 130 -84 }}
+| {{mapping| 2118 3357 4918 }}
+| −0.0186
+| 0.0156
 |
 |-
-|2.3.5.7
+| 2.3.5.7
-|250047/250000,{{monzo|-1 -18 -3 13}}, {{monzo|38 -2 -15 0}}
+| 250047/250000, {{monzo| -1 -18 -3 13 }}, {{monzo| 38 -2 -15 }}
-|[{{val|2118 3357 4918 5946}}]
+| {{mapping| 2118 3357 4918 5946 }}
-| -0.0150
+| −0.0150
-|0.0148
+| 0.0148
 |
 |-
-|2.3.5.7.11
+| 2.3.5.7.11
-|9801/9800, 250047/250000, {{monzo|25 1 -4 0 -5}}, {{monzo|16 -7 -9 2 3}}
+| 9801/9800, 250047/250000, {{monzo| 25 1 -4 0 -5 }}, {{monzo| 16 -7 -9 2 3 }}
-|[{{val|2118 3357 4918 5946 7927}}]
+| {{mapping| 2118 3357 4918 5946 7927 }}
-|<nowiki>-0.0096</nowiki>
+| −0.0096
-|0.0172
+| 0.0172
 |
 |}
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->