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1600edo: Difference between revisions

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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 ==
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.
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.  


In the 5-limit, it supports [[kwazy]]. In the 7-limit, it tempers out the ragisma, 4375/4374. In the 11-limit, it supports the rank-3 temperament [[thor]].  
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}}
{{Harmonics in equal|1600}}
===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}}.  
=== 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]].
 
== 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" | [[Subgroup]]
! 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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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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 }}
| [{{val| 1600 2536 3715 }}]
| {{Mapping| 1600 2536 3715 }}
| -0.000318
| −0.0003
| 0.022794
| 0.0228
|
| 3.04
|-
|-
| 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 }}
| [{{val| 1600 2536 3715 4492 }}]
| {{Mapping| 1600 2536 3715 4492 }}
| -0.015742
| −0.0157
| 0.033217
| 0.0332
|
| 4.43
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 4375/4374, 184549376/184528125, 7680000000/7672950131
| 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
|
| 4.39
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 91125/91091, 14236560/14235529
| 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
|
| 4.75
|-
|-
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4375/4374, 14875/14872, 154880/154791, 1724800/1724463
| 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 }}
| -0.016332
| −0.0163
|
| 0.0331
|
| 4.41
|}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ 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
| 23\1600
| 17.25
| ?
| [[Dam]] / [[dike]] / [[polder]]
|-
|-
| 32
| 32
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| 245/143<br>(?)
| 245/143<br>(?)
| [[Germanium]]
| [[Germanium]]
|-
| 32
| 23\1600
| 17.25
| ?
| [[Dike]]
|-
|-
| 80
| 80
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| [[Tetraicosic]]
| [[Tetraicosic]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
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