1600edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 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
| &minus;0.0003
| 0.0228
| 0.0228
| 3.04
| 3.04
Line 36: Line 28:
| 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
| &minus;0.0157
| 0.0332
| 0.0332
| 4.43
| 4.43
Line 43: Line 35:
| 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
| &minus;0.0172
| 0.0329
| 0.0329
| 4.39
| 4.39
Line 50: Line 42:
| 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
| &minus;0.0087
| 0.0356
| 0.0356
| 4.75
| 4.75
Line 57: Line 49:
| 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
| &minus;0.0163
| 0.0331
| 0.0331
| 4.41
| 4.41
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 2
| 2
Line 85: Line 70:
|-
|-
| 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]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 04:45, 16 November 2024

← 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

Template:EDO intro

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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 2 | 217\1600 | 162.75 | 1125/1024 | Kwazy |- | 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 Template:Rank-2 end Template:Orf

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