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 ==

1600edo is a very strong 37-limit system, being distinctly consistent in the 37-limit with a smaller [[Tenney-~~Euclidean_temperament_measures~~#TE simple badness|relative error]] than anything else with this property until [[4501edo|4501]]. ~~It is also the first division past [[311edo|311]] with a lower 43-limit relative error.~~

1600edo is a very strong 37-limit system, being [[consistency|distinctly consistent]] in the [[37-odd-limit]] with a smaller [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than anything else with this property until [[4501edo|4501]].

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]].

It is also the first division past [[311edo|311]] with a lower [[43-limit]] relative error, being ''almost'' consistent in the [[45-odd-limit]], missing only [[50/39]] and [[39/25]], both of which being off by ''52.6%'' by [[patent val]] mapping, which is still just an error of 0.3945 cents.

===~~Odd~~ harmonics===

In the 7-limit, it supports [[crazy]], it supports In the 11-limit, it supports the rank-3 temperament [[thor]]. In higher limits, it tempers out [[4096/4095]] in the [[13-limit]] (allowing [[schisminic chords]]), [[12376/12375]] in the [[17-limit]], [[6860/6859]] in the 19-limit, and due to being consistent higher than 33-odd-limit it enables the essentially tempered [[flashmic chords]].

===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}}.

=== Prime harmonics ===

{{Harmonics in equal|1600|prec=3|columns=12}}{{Harmonics in equal|1600|columns=12|start=13|prec=3|collapsed=true|title=Approximation of prime harmonics in 1600edo (continued)}}

=== Subsets and supersets ===

Since 1600 factors into {{factorization|1600}}, 1600edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, and 800 }}.

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]]. Similar to the [[Mina]] in the [[27-odd-limit]], All [[45-odd limit]] intervals can be written using integer values of śata, being more in tune than out of tune.

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]].

== 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~~ (¢)

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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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

| −0.0003

| 0.0228

| 3.04

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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

| −0.0157

| 0.0332

| 4.43

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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

| −0.0172

| 0.0329

| 4.39

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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

| −0.0087

| 0.0356

| 4.75

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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

| −0.0163

| 0.0331

| 4.41

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=== Rank-2 temperaments ===

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

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator

! Generator*

! Cents

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

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| 162.75

| 1125/1024

| [[~~Kwazy~~]]

| [[Crazy]]

|-

| 32

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| 17.25

| ?

|[[~~Dike~~]]

| [[Dam]] / [[dike]] / [[polder]]

|-

| 32

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| [[Tetraicosic]]

|}

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if 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.
+{{ED intro}}
 == Theory ==
-edo is a very strong 37-limit system, being distinctly consistent in the 37-limit with a smaller [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] than anything else with this property until [[4501edo|4501]]. It is also the first division past [[311edo|311]] with a lower 43-limit relative error.
+edo is a very strong 37-limit system, being [[consistency|distinctly consistent]] in the [[37-odd-limit]] with a smaller [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than anything else with this property until [[4501edo|4501]].
-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]].
+It is also the first division past [[311edo|311]] with a lower [[43-limit]] relative error, being ''almost'' consistent in the [[45-odd-limit]], missing only [[50/39]] and [[39/25]], both of which being off by ''52.6%'' by [[patent val]] mapping, which is still just an error of 0.3945 cents.
-===Odd harmonics===
-{{Harmonics in equal|1600}}
+In the 7-limit, it supports [[crazy]], it supports In the 11-limit, it supports the rank-3 temperament [[thor]]. In higher limits, it tempers out [[4096/4095]] in the [[13-limit]] (allowing [[schisminic chords]]), [[12376/12375]] in the [[17-limit]], [[6860/6859]] in the 19-limit, and due to being consistent higher than 33-odd-limit it enables the essentially tempered [[flashmic chords]].
-===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}}.
+=== Prime harmonics ===
+{{Harmonics in equal|1600|prec=3|columns=12}}{{Harmonics in equal|1600|columns=12|start=13|prec=3|collapsed=true|title=Approximation of prime harmonics in 1600edo (continued)}}
+=== Subsets and supersets ===
+Since 1600 factors into {{factorization|1600}}, 1600edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, and 800 }}.
+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]]. Similar to the [[Mina]] in the [[27-odd-limit]], All [[45-odd limit]] intervals can be written using integer values of śata, being more in tune than out of tune.
-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]].
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! 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 25: / Line 31: @@
 | 2.3.5
 | {{Monzo| -53 10 16 }}, {{monzo| 26 -75 40 }}
-| [{{val| 1600 2536 3715 }}]
+| {{Mapping| 1600 2536 3715 }}
-| -0.0003
+| −0.0003
 | 0.0228
 | 3.04
@@ Line 32: / Line 38: @@
 | 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
+| −0.0157
 | 0.0332
 | 4.43
@@ Line 39: / Line 45: @@
 | 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
+| −0.0172
 | 0.0329
 | 4.39
@@ Line 46: / Line 52: @@
 | 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
+| −0.0087
 | 0.0356
 | 4.75
@@ Line 53: / Line 59: @@
 | 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
+| −0.0163
 | 0.0331
 | 4.41
@@ Line 61: / Line 67: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
 ! Periods<br>per 8ve
-! Generator
+! Generator*
-! Cents
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 71: / Line 79: @@
 | 162.75
 | 1125/1024
-| [[Kwazy]]
+| [[Crazy]]
 |-
 | 32
@@ Line 77: / Line 85: @@
 | 17.25
 | ?
-|[[Dike]]
+| [[Dam]] / [[dike]] / [[polder]]
 |-
 | 32
@@ Line 97: / Line 105: @@
 | [[Tetraicosic]]
 |}
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct