1600edo: Difference between revisions

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Line 21: Line 21:
! 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
|-
|-
Line 28: Line 28:
|-
|-
| 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
Line 36: 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 }}
| {{mapping| 1600 2536 3715 4492 }}
| {{Mapping| 1600 2536 3715 4492 }}
| −0.0157
| −0.0157
| 0.0332
| 0.0332
Line 43: 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 }}
| {{mapping| 1600 2536 3715 4492 5535 }}
| {{Mapping| 1600 2536 3715 4492 5535 }}
| −0.0172
| −0.0172
| 0.0329
| 0.0329
Line 50: 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
| {{mapping| 1600 2536 3715 4492 5535 5921 }}
| {{Mapping| 1600 2536 3715 4492 5535 5921 }}
| −0.0087
| −0.0087
| 0.0356
| 0.0356
Line 57: 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
| {{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
|-
|-
Line 77: Line 77:
| 162.75
| 162.75
| 1125/1024
| 1125/1024
| [[Kwazy]]
| [[Crazy]]
|-
|-
| 32
| 32
Line 86: Line 86:
|-
|-
| 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 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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