410edo: Difference between revisions

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~~The '''410 equal divisions of the octave''' ('''410edo'''), or the '''410(-tone) equal temperament''' ('''410tet''', '''410et''') when viewed from a [[regular temperament]] perspective, is the [[~~EDO|~~equal division of the octave]] into~~ 410 ~~parts of about 2.93 [[cent]]s each.~~

== Theory ==

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=== ~~Miscellaneous properties~~ ===

=== Divisors ===

Since 410 = 2 × 5 × 41, 410edo has subset edos {{EDOs| 2, 5, 10, 41, 82, and 205 }}. Meanwhile, as every sixth step of [[2460edo]], a step of 410edo is exactly 6 [[mina]]s.

410edo's fifth is on its 240th step, which is a highly composite number. As such, it supports edfs which are divisors of 240. In addition, its perfect fourth is on the 170th step, which while is not highly composite, is still notable to carry a few ed4/3 scales. This can be used to play [[Kartvelian scales]].

== Regular temperament properties ==

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

! rowspan="2" | Subgroup

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

! rowspan="2" | [[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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{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

! Periods per ~~octave~~

! Periods per 8ve

! Generator (~~reduced~~)

! Generator (Reduced)

! Cents (~~reduced~~)

! Cents (Reduced)

! Associated ~~ratio~~

! Associated Ratio

! Temperaments

|-

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| 15/13 (176/175)

| [[Decoid]]

|-

| 41

| 61\410 (1\410)

| 178.54 (2.93)

| 567/512 (352/351)

| [[Hemicountercomp]]

|}

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

== Scales ==

410edo's fifth is on its 240th step, which is a highly composite number. As such, it supports edfs which are divisors of 240. In addition, its perfect fourth is on the 170th step, which while is not highly composite, is still notable to carry a few ed4/3 scales. This can be used to play [[Kartvelian scales]].

* Kartvelian Tetratonic: 120 120 85 85 (simplifies to [[82edo]])

* Kartvelian Decatonic: 48 48 48 48 48 34 34 34 34 34 (simplifies to [[205edo]])

* Kartvelian 22-tonic: 20 20 20 20 20 20 20 20 20 20 20 20 17 17 17 17 17 17 17 17 17~~~~

* Kartvelian 22-tonic: 20 20 20 20 20 20 20 20 20 20 20 20 17 17 17 17 17 17 17 17 17

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

[[Category:Semiluna]]

[[Category:Hemiluna]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''410 equal divisions of the octave''' ('''410edo'''), or the '''410(-tone) equal temperament''' ('''410tet''', '''410et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 410 parts of about 2.93 [[cent]]s each.
+{{EDO intro|410}}
 == Theory ==
@@ Line 12: / Line 12: @@
 {{Harmonics in equal|410|columns=11}}
-=== Miscellaneous properties ===
+=== Divisors ===
 Since 410 = 2 × 5 × 41, 410edo has subset edos {{EDOs| 2, 5, 10, 41, 82, and 205 }}. Meanwhile, as every sixth step of [[2460edo]], a step of 410edo is exactly 6 [[mina]]s.
-edo's fifth is on its 240th step, which is a highly composite number. As such, it supports edfs which are divisors of 240. In addition, its perfect fourth is on the 170th step, which while is not highly composite, is still notable to carry a few ed4/3 scales. This can be used to play [[Kartvelian scales]].
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[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 55: / Line 53: @@
 {| 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<br>(Reduced)
-! Cents<br>(reduced)
+! Cents<br>(Reduced)
-! Associated<br>ratio
+! Associated<br>Ratio
 ! Temperaments
 |-
@@ Line 108: / Line 106: @@
 | 15/13<br>(176/175)
 | [[Decoid]]
+|-
+| 41
+| 61\410<br>(1\410)
+| 178.54<br>(2.93)
+| 567/512<br>(352/351)
+| [[Hemicountercomp]]
 |}
-[[Category:Equal divisions of the octave|###]]
 == Scales ==
+edo's fifth is on its 240th step, which is a highly composite number. As such, it supports edfs which are divisors of 240. In addition, its perfect fourth is on the 170th step, which while is not highly composite, is still notable to carry a few ed4/3 scales. This can be used to play [[Kartvelian scales]].
 * Kartvelian Tetratonic: 120 120 85 85 (simplifies to [[82edo]])
 * Kartvelian Decatonic: 48 48 48 48 48 34 34 34 34 34 (simplifies to [[205edo]])
-* Kartvelian 22-tonic: 20 20 20 20 20 20 20 20 20 20 20 20 17 17 17 17 17 17 17 17 17<!-- 3-digit number -->
+* Kartvelian 22-tonic: 20 20 20 20 20 20 20 20 20 20 20 20 17 17 17 17 17 17 17 17 17
+[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Semiluna]]
 [[Category:Hemiluna]]