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{{Infobox ET}}
{{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 ==
== Theory ==
{{Harmonics in equal|1600}}
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]].
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. 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.


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}}.  
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.  


In the 5-limit, it supports [[kwazy]].
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]].  


In the 7-limit, it tempers out the ragisma, 4375/4374.
=== 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)}}


In the 11-limit, it supports the rank-3 temperament [[thor]].  
=== 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.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| 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]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|2.3.5
! [[TE error|Absolute]] (¢)
|{{Monzo|-53, 10, 16}}, {{Monzo|26, -75, 40}}
! [[TE simple badness|Relative]] (%)
|[{{val|1600 2536 3715}}]
| -0.000318
|0.022794
|
|-
|-
|2.3.5.7
| 2.3.5
|4375/4374, {{Monzo|36, -5, 0, -10}}, {{Monzo|-17, 5, 16, -10}}
| {{Monzo| -53 10 16 }}, {{monzo| 26 -75 40 }}
|[{{val|1600 2536 3715 4492}}]
| {{Mapping| 1600 2536 3715 }}
| -0.015742
| −0.0003
|0.033217
| 0.0228
|
| 3.04
|-
|-
|2.3.5.7.11
| 2.3.5.7
|3025/3024, 4375/4374, 184549376/184528125, 7680000000/7672950131
| 4375/4374, {{monzo| 36 -5 0 -10 }}, {{monzo| -17 5 16 -10 }}
|[{{val|1600 2536 3715 4492 5535}}]
| {{Mapping| 1600 2536 3715 4492 }}
|?
| −0.0157
|?
| 0.0332
|
| 4.43
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11
|3025/3024, 4096/4095, 4375/4374, 91125/91091, 14236560/14235529
| 3025/3024, 4375/4374, {{monzo| 24 -1 -5 0 1 }}, {{monzo| 15 1 7 -8 -3 }}
|[{{val|1600 2536 3715 4492 5535 5921}}]
| {{Mapping| 1600 2536 3715 4492 5535 }}
|?
| −0.0172
|?
| 0.0329
|
| 4.39
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13
|2500/2499, 3025/3024, 4375/4374, 14875/14872, 154880/154791, 1724800/1724463
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543
|[{{val|1600 2536 3715 4492 5535 5921 6540}}]
| {{Mapping| 1600 2536 3715 4492 5535 5921 }}
| -0.016332
| −0.0087
|
| 0.0356
|
| 4.75
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 14875/14872, 63888/63869
| {{Mapping| 1600 2536 3715 4492 5535 5921 6540 }}
| −0.0163
| 0.0331
| 4.41
|}
|}
[[Category:Equal divisions of the octave|####]]


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Periods<br>per 8ve
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
!Generator
!Cents
!Associated<br>ratio
!Temperaments
|-
|-
|2
! Periods<br>per 8ve
|217\1600
! Generator*
|162.75
! Cents*
|1125/1024
! Associated<br>ratio*
|[[Kwazy]]
! Temperaments
|-
|-
|32
| 2
|121\1600<br>(21/1600)
| 217\1600
|90.75<br>(15.75)
| 162.75
|48828125/46294416<br>(?)
| 1125/1024
|[[Windrose]]
| [[Crazy]]
|-
|-
|32
| 32
|357\1600<br>(7\1600)
| 23\1600
|267.75<br>(5.25)
| 17.25
|245/143<br>(?)
| ?
|[[Germanium]]
| [[Dam]] / [[dike]] / [[polder]]
|-
|-
|32
| 32
|23\1600
| 121\1600<br>(21/1600)
|17.25
| 90.75<br>(15.75)
|?
| 48828125/46294416<br>(?)
|[[Dike]]
| [[Windrose]]
|-
|-
|80
| 32
|629\1600<br>(9\1600)
| 357\1600<br>(7\1600)
|471.75<br>(6.75)
| 267.75<br>(5.25)
|130/99<br>(?)
| 245/143<br>(?)
|[[Tetraicosic]]
| [[Germanium]]
|}<!-- 4-digit number -->
|-
| 80
| 629\1600<br>(9\1600)
| 471.75<br>(6.75)
| 130/99<br>(?)
| [[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
Retrieved from "https://en.xen.wiki/w/1600edo"