414edo: Difference between revisions

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~~The '''414 equal divisions of the octave''' ('''414edo'''), or the '''414(-tone) equal temperament''' ('''414tet''', '''414et''') when viewed from a [[regular temperament]] perspective, is the [[~~EDO|~~equal division of the octave]] into~~ 414 ~~parts of about 2.90 [[cent]]s each.~~

== Theory ==

414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for 5. It is [[consistent]] to the [[17-odd-limit]], tempering out {{monzo| -36 11 8 }} (submajor comma) and {{monzo|1 -27 18}} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], 1275/1274, and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].

414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1|5]]. It is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the harmonics, making for a good full 17-limit system. The equal temperament [[tempering out|tempers out]] {{monzo| -36 11 8 }} (submajor comma) and {{monzo| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].

=== Prime harmonics ===

=== Subsets and supersets ===

Since 414 factors into {{factorization|414}}, 414edo has subset edos {{EDOs| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.

== Regular temperament properties ==

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

! rowspan="2" | Subgroup

! rowspan="2" | [Subgroup]]

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

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

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

! rowspan="2" | Optimal 8ve ~~stretch~~ (¢)

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

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

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

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| 2.3.5

| {{monzo| -36 11 8 }}, {{monzo| 1 -27 18 }}

| [{{~~val~~| 414 656 961 }}]

| {{mapping| 414 656 961 }}

| +0.2222

| 0.1575

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| 2.3.5.7

| 2401/2400, 4375/4374, {{monzo| -36 11 8 }}

| [{{~~val~~| 414 656 961 1162 }}]

| {{mapping| 414 656 961 1162 }}

| +0.2299

| 0.1371

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| 2.3.5.7.11

| 2401/2400, 3025/3024, 4375/4374, 1366875/1362944

| [{{~~val~~| 414 656 961 1162 1432 }}]

| {{mapping| 414 656 961 1162 1432 }}

| +0.2182

| 0.1248

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| 2.3.5.7.11.13

| 625/624, 729/728, 1575/1573, 2200/2197, 2401/2400

| [{{~~val~~| 414 656 961 1162 1432 1532 }}]

| {{mapping| 414 656 961 1162 1432 1532 }}

| +0.1795

| 0.1431

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| 2.3.5.7.11.13.17

| 625/624, 729/728, 833/832, 1089/1088, 1225/1224, 2200/2197

| [{{~~val~~| 414 656 961 1162 1432 1532 1692 }}]

| {{mapping| 414 656 961 1162 1432 1532 1692 }}

| +0.1751

| 0.1329

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

|+Table of rank-2 temperaments by generator

! Periods per ~~octave~~

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperaments

|-

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| [[Semihemiennealimmal]]

|}

<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}}
-The '''414 equal divisions of the octave''' ('''414edo'''), or the '''414(-tone) equal temperament''' ('''414tet''', '''414et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 414 parts of about 2.90 [[cent]]s each.
+{{EDO intro|414}}
 == Theory ==
-edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for 5. It is [[consistent]] to the [[17-odd-limit]], tempering out {{monzo| -36 11 8 }} (submajor comma) and {{monzo|1 -27 18}} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], 1275/1274, and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].
+edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1|5]]. It is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the harmonics, making for a good full 17-limit system. The equal temperament [[tempering out|tempers out]] {{monzo| -36 11 8 }} (submajor comma) and {{monzo| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].
 === Prime harmonics ===
 {{Harmonics in equal|414}}
+=== Subsets and supersets ===
+Since 414 factors into {{factorization|414}}, 414edo has subset edos {{EDOs| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+! rowspan="2" | [Subgroup]]
-! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Comma list|Comma List]]
 ! 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]] (¢)
@@ Line 21: / Line 24: @@
 | 2.3.5
 | {{monzo| -36 11 8 }}, {{monzo| 1 -27 18 }}
-| [{{val| 414 656 961 }}]
+| {{mapping| 414 656 961 }}
 | +0.2222
 | 0.1575
@@ Line 28: / Line 31: @@
 | 2.3.5.7
 | 2401/2400, 4375/4374, {{monzo| -36 11 8 }}
-| [{{val| 414 656 961 1162 }}]
+| {{mapping| 414 656 961 1162 }}
 | +0.2299
 | 0.1371
@@ Line 35: / Line 38: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 4375/4374, 1366875/1362944
-| [{{val| 414 656 961 1162 1432 }}]
+| {{mapping| 414 656 961 1162 1432 }}
 | +0.2182
 | 0.1248
@@ Line 42: / Line 45: @@
 | 2.3.5.7.11.13
 | 625/624, 729/728, 1575/1573, 2200/2197, 2401/2400
-| [{{val| 414 656 961 1162 1432 1532 }}]
+| {{mapping| 414 656 961 1162 1432 1532 }}
 | +0.1795
 | 0.1431
@@ Line 49: / Line 52: @@
 | 2.3.5.7.11.13.17
 | 625/624, 729/728, 833/832, 1089/1088, 1225/1224, 2200/2197
-| [{{val| 414 656 961 1162 1432 1532 1692 }}]
+| {{mapping| 414 656 961 1162 1432 1532 1692 }}
 | +0.1751
 | 0.1329
@@ Line 58: / Line 61: @@
 {| class="wikitable center-all left-5"
 |+Table of rank-2 temperaments by generator
-! Periods<br>per octave
+! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 94: / Line 97: @@
 | [[Semihemiennealimmal]]
 |}
+<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 -->