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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1600}}
{{ED intro}}


== Theory ==
== Theory ==
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]]. 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]].
{{Harmonics in equal|1600}}
 
=== 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 ===
=== 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 }}.  
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]].  
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 ==
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| 26 -75 40 }}
| {{Monzo| -53 10 16 }}, {{monzo| 26 -75 40 }}
| {{mapping| 1600 2536 3715 }}
| {{Mapping| 1600 2536 3715 }}
| −0.0003
| −0.0003
| 0.0228
| 0.0228
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| 2.3.5.7
| 2.3.5.7
| 4375/4374, {{monzo| 36 -5 0 -10 }}, {{monzo| -17 5 16 -10 }}
| 4375/4374, {{monzo| 36 -5 0 -10 }}, {{monzo| -17 5 16 -10 }}
| {{mapping| 1600 2536 3715 4492 }}
| {{Mapping| 1600 2536 3715 4492 }}
| −0.0157
| −0.0157
| 0.0332
| 0.0332
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| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 4375/4374, {{monzo| 24 -1 -5 0 1 }}, {{monzo| 15 1 7 -8 -3 }}
| 3025/3024, 4375/4374, {{monzo| 24 -1 -5 0 1 }}, {{monzo| 15 1 7 -8 -3 }}
| {{mapping| 1600 2536 3715 4492 5535 }}
| {{Mapping| 1600 2536 3715 4492 5535 }}
| −0.0172
| −0.0172
| 0.0329
| 0.0329
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543
| {{mapping| 1600 2536 3715 4492 5535 5921 }}
| {{Mapping| 1600 2536 3715 4492 5535 5921 }}
| −0.0087
| −0.0087
| 0.0356
| 0.0356
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 14875/14872, 63888/63869
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 14875/14872, 63888/63869
| {{mapping| 1600 2536 3715 4492 5535 5921 6540 }}
| {{Mapping| 1600 2536 3715 4492 5535 5921 6540 }}
| −0.0163
| −0.0163
| 0.0331
| 0.0331
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 162.75
| 162.75
| 1125/1024
| 1125/1024
| [[Kwazy]]
| [[Crazy]]
|-
|-
| 32
| 32
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|-
|-
| 32
| 32
| 121\1600<br />(21/1600)
| 121\1600<br>(21/1600)
| 90.75<br />(15.75)
| 90.75<br>(15.75)
| 48828125/46294416<br />(?)
| 48828125/46294416<br>(?)
| [[Windrose]]
| [[Windrose]]
|-
|-
| 32
| 32
| 357\1600<br />(7\1600)
| 357\1600<br>(7\1600)
| 267.75<br />(5.25)
| 267.75<br>(5.25)
| 245/143<br />(?)
| 245/143<br>(?)
| [[Germanium]]
| [[Germanium]]
|-
|-
| 80
| 80
| 629\1600<br />(9\1600)
| 629\1600<br>(9\1600)
| 471.75<br />(6.75)
| 471.75<br>(6.75)
| 130/99<br />(?)
| 130/99<br>(?)
| [[Tetraicosic]]
| [[Tetraicosic]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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
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