1600edo: Difference between revisions

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{{novelty}}{{stub}}{{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.  
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 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]].  
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}}.


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]].  
=== 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]] (¢)
Line 25: Line 29:
| 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 }}
| -0.0003
| −0.0003
| 0.0228
| 0.0228
| 3.04
| 3.04
Line 32: Line 36:
| 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 }}
| -0.0157
| −0.0157
| 0.0332
| 0.0332
| 4.43
| 4.43
Line 39: Line 43:
| 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 }}
| [{{val| 1600 2536 3715 4492 5535 }}]
| {{Mapping| 1600 2536 3715 4492 5535 }}
| -0.0172
| −0.0172
| 0.0329
| 0.0329
| 4.39
| 4.39
Line 46: Line 50:
| 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
| [{{val| 1600 2536 3715 4492 5535 5921 }}]
| {{Mapping| 1600 2536 3715 4492 5535 5921 }}
| -0.0087
| −0.0087
| 0.0356
| 0.0356
| 4.75
| 4.75
Line 53: Line 57:
| 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
| [{{val| 1600 2536 3715 4492 5535 5921 6540 }}]
| {{Mapping| 1600 2536 3715 4492 5535 5921 6540 }}
| -0.0163
| −0.0163
| 0.0331
| 0.0331
| 4.41
| 4.41
Line 61: Line 65:
=== 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
|-
|-
Line 71: Line 77:
| 162.75
| 162.75
| 1125/1024
| 1125/1024
| [[Kwazy]]
| [[Crazy]]
|-
|-
| 32
| 32
Line 77: Line 83:
| 17.25
| 17.25
| ?
| ?
|[[Dam]] / [[dike]] / [[polder]]
| [[Dam]] / [[dike]] / [[polder]]
|-
|-
| 32
| 32
Line 97: Line 103:
| [[Tetraicosic]]
| [[Tetraicosic]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct

Latest revision as of 11:33, 6 March 2025

← 1599edo 1600edo 1601edo →
Prime factorization 26 × 52
Step size 0.75 ¢ 
Fifth 936\1600 (702 ¢) (→ 117\200)
Semitones (A1:m2) 152:120 (114 ¢ : 90 ¢)
Consistency limit 37
Distinct consistency limit 37

1600 equal divisions of the octave (abbreviated 1600edo or 1600ed2), also called 1600-tone equal temperament (1600tet) or 1600 equal temperament (1600et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1600 equal parts of exactly 0.75 ¢ each. Each step represents a frequency ratio of 21/1600, or the 1600th root of 2.

Theory

1600edo is a very strong 37-limit system, being distinctly consistent in the 37-odd-limit with a smaller relative error than anything else with this property until 4501. It is also the first division past 311 with a lower 43-limit relative error.

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

Approximation of prime harmonics in 1600edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.045 -0.064 +0.174 -0.068 +0.222 +0.045 +0.237 +0.226 +0.173 +0.214
Relative (%) +0.0 +6.0 -8.5 +23.2 -9.1 +29.6 +5.9 +31.6 +30.1 +23.0 +28.6
Steps
(reduced) 		1600
(0) 		2536
(936) 		3715
(515) 		4492
(1292) 		5535
(735) 		5921
(1121) 		6540
(140) 		6797
(397) 		7238
(838) 		7773
(1373) 		7927
(1527)

Subsets and supersets

Since 1600 factors into 26 × 52, 1600edo has subset 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 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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [-53 10 16⟩, [26 -75 40⟩ [⟨1600 2536 3715]] −0.0003 0.0228 3.04
2.3.5.7 4375/4374, [36 -5 0 -10⟩, [-17 5 16 -10⟩ [⟨1600 2536 3715 4492]] −0.0157 0.0332 4.43
2.3.5.7.11 3025/3024, 4375/4374, [24 -1 -5 0 1⟩, [15 1 7 -8 -3⟩ [⟨1600 2536 3715 4492 5535]] −0.0172 0.0329 4.39
2.3.5.7.11.13 3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543 [⟨1600 2536 3715 4492 5535 5921]] −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 [⟨1600 2536 3715 4492 5535 5921 6540]] −0.0163 0.0331 4.41

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
2 217\1600 162.75 1125/1024 Crazy
32 23\1600 17.25 ? Dam / dike / polder
32 121\1600
(21/1600) 		90.75
(15.75) 		48828125/46294416
(?) 		Windrose
32 357\1600
(7\1600) 		267.75
(5.25) 		245/143
(?) 		Germanium
80 629\1600
(9\1600) 		471.75
(6.75) 		130/99
(?) 		Tetraicosic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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