1600edo: Difference between revisions

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~~The '''~~1600 equal divisions of the octave''' ('''1600edo'''), or the '''1600-tone equal temperament''' ('''1600tet'''), '''1600 equal temperament''' ('''1600et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1600 [[equal]] parts of exactly 750 [[cent|millicents]] each.

== Theory ==

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In the 5-limit, it supports [[kwazy]]. In the 11-limit, it supports the rank-3 temperament [[thor]]. In higher limits, it tempers out [[12376/12375]] in the 17-limit and due to being consistent higher than 33-odd-limit it enables the essentially tempered [[flashmic chords]].

===Odd harmonics===

=== Odd harmonics ===

===Subsets and supersets===

=== Subsets and supersets ===

1600's divisors are {{EDOs|1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, 800}}.

One step of it is the [[relative cent]] for [[16edo|16]]. ~~It's~~ high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for interval size measure. One step of 1600edo is already used as a measure called ''śata'' in the context of 16edo [[Armodue theory]].

One step of it is the [[relative cent]] for [[16edo|16]]. Its high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for interval size measure. One step of 1600edo is already used as a measure called ''śata'' in the context of 16edo [[Armodue theory]].

== Regular temperament properties ==

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

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

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

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

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

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

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| 2.3.5

| {{Monzo| -53 10 16 }}, {{monzo| 26 -75 40 }}

| [{{~~val~~| 1600 2536 3715 }}]

| {{mapping| 1600 2536 3715 }}

| -0.0003

| 0.0228

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| 2.3.5.7

| 4375/4374, {{monzo| 36 -5 0 -10 }}, {{monzo| -17 5 16 -10 }}

| [{{~~val~~| 1600 2536 3715 4492 }}]

| {{mapping| 1600 2536 3715 4492 }}

| -0.0157

| 0.0332

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| 2.3.5.7.11

| 3025/3024, 4375/4374, {{monzo| 24 -1 -5 0 1 }}, {{monzo| 15 1 7 -8 -3 }}

| [{{~~val~~| 1600 2536 3715 4492 5535 }}]

| {{mapping| 1600 2536 3715 4492 5535 }}

| -0.0172

| 0.0329

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| 2.3.5.7.11.13

| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543

| [{{~~val~~| 1600 2536 3715 4492 5535 5921 }}]

| {{mapping| 1600 2536 3715 4492 5535 5921 }}

| -0.0087

| 0.0356

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| 2.3.5.7.11.13.17

| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 14875/14872, 63888/63869

| [{{~~val~~| 1600 2536 3715 4492 5535 5921 6540 }}]

| {{mapping| 1600 2536 3715 4492 5535 5921 6540 }}

| -0.0163

| 0.0331

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

! Periods<br>per 8ve

! Generator

! Generator*

! Cents

! Cents*

! Associated<br>Ratio

! Temperaments

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| [[Tetraicosic]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''1600 equal divisions of the octave''' ('''1600edo'''), or the '''1600-tone equal temperament''' ('''1600tet'''), '''1600 equal temperament''' ('''1600et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1600 [[equal]] parts of exactly 750 [[cent|millicents]] each.
+{{EDO intro|1600}}
 == Theory ==
@@ Line 6: / Line 6: @@
 In the 5-limit, it supports [[kwazy]]. In the 11-limit, it supports the rank-3 temperament [[thor]]. In higher limits, it tempers out [[12376/12375]] in the 17-limit and due to being consistent higher than 33-odd-limit it enables the essentially tempered [[flashmic chords]].
-===Odd harmonics===
+=== Odd harmonics ===
 {{Harmonics in equal|1600}}
-===Subsets and supersets===
+=== Subsets and supersets ===
 's divisors are {{EDOs|1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, 800}}.
-One step of it is the [[relative cent]] for [[16edo|16]]. It's high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for interval size measure. One step of 1600edo is already used as a measure called ''śata'' in the context of 16edo [[Armodue theory]].
+One step of it is the [[relative cent]] for [[16edo|16]]. Its high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for interval size measure. One step of 1600edo is already used as a measure called ''śata'' in the context of 16edo [[Armodue theory]].
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
 ! rowspan="2" | Optimal<br>8ve Stretch (¢)
@@ Line 25: / Line 28: @@
 | 2.3.5
 | {{Monzo| -53 10 16 }}, {{monzo| 26 -75 40 }}
-| [{{val| 1600 2536 3715 }}]
+| {{mapping| 1600 2536 3715 }}
 | -0.0003
 | 0.0228
@@ Line 32: / Line 35: @@
 | 2.3.5.7
 | 4375/4374, {{monzo| 36 -5 0 -10 }}, {{monzo| -17 5 16 -10 }}
-| [{{val| 1600 2536 3715 4492 }}]
+| {{mapping| 1600 2536 3715 4492 }}
 | -0.0157
 | 0.0332
@@ Line 39: / Line 42: @@
 | 2.3.5.7.11
 | 3025/3024, 4375/4374, {{monzo| 24 -1 -5 0 1 }}, {{monzo| 15 1 7 -8 -3 }}
-| [{{val| 1600 2536 3715 4492 5535 }}]
+| {{mapping| 1600 2536 3715 4492 5535 }}
 | -0.0172
 | 0.0329
@@ Line 46: / Line 49: @@
 | 2.3.5.7.11.13
 | 3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543
-| [{{val| 1600 2536 3715 4492 5535 5921 }}]
+| {{mapping| 1600 2536 3715 4492 5535 5921 }}
 | -0.0087
 | 0.0356
@@ Line 53: / Line 56: @@
 | 2.3.5.7.11.13.17
 | 2500/2499, 3025/3024, 4096/4095, 4375/4374, 14875/14872, 63888/63869
-| [{{val| 1600 2536 3715 4492 5535 5921 6540 }}]
+| {{mapping| 1600 2536 3715 4492 5535 5921 6540 }}
 | -0.0163
 | 0.0331
@@ Line 62: / Line 65: @@
 {| class="wikitable center-all left-5"
 ! Periods<br>per 8ve
-! Generator
+! Generator*
-! Cents
+! Cents*
 ! Associated<br>Ratio
 ! Temperaments
@@ Line 97: / Line 100: @@
 | [[Tetraicosic]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct