1600edo: Difference between revisions

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== 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
! rowspan="2" | Optimal<br>8ve Stretch (¢)
8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" |Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[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 }}
|[{{val|1600 2536 3715}}]
| [{{val| 1600 2536 3715 }}]
| -0.000318
| -0.000318
|0.022794
| 0.022794
|
|
|-
|-
|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}}]
| [{{val| 1600 2536 3715 4492 }}]
| -0.015742
| -0.015742
|0.033217
| 0.033217
|
|
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|3025/3024, 4375/4374, 184549376/184528125, 7680000000/7672950131
| 3025/3024, 4375/4374, 184549376/184528125, 7680000000/7672950131
|[{{val|1600 2536 3715 4492 5535}}]
| [{{val| 1600 2536 3715 4492 5535 }}]
|?
|  
|?
|  
|
|
|-
|-
|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, 91125/91091, 14236560/14235529
|[{{val|1600 2536 3715 4492 5535 5921}}]
| [{{val| 1600 2536 3715 4492 5535 5921 }}]
|?
|  
|?
|  
|
|
|-
|-
|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, 4375/4374, 14875/14872, 154880/154791, 1724800/1724463
|[{{val|1600 2536 3715 4492 5535 5921 6540}}]
| [{{val| 1600 2536 3715 4492 5535 5921 6540 }}]
| -0.016332
| -0.016332
|
|
|
|
|}
|}
[[Category:Equal divisions of the octave|####]]


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Periods<br>per 8ve
! Periods<br>per 8ve
!Generator
! Generator
!Cents
! Cents
!Associated<br>ratio
! Associated<br>Ratio
!Temperaments
! Temperaments
|-
|-
|2
| 2
|217\1600
| 217\1600
|162.75
| 162.75
|1125/1024
| 1125/1024
|[[Kwazy]]
| [[Kwazy]]
|-
|-
|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]]
|-
|-
|32
| 32
|23\1600
| 23\1600
|17.25
| 17.25
|?
| ?
|[[Dike]]
| [[Dike]]
|-
|-
|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]]
|}<!-- 4-digit number -->
|}

Revision as of 08:51, 24 February 2023

← 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

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 millicents each.

Theory

1600edo is a very strong 37-limit system, being distinctly consistent in the 37-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. One step of it is the relative cent for 16. It's high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for interval size measure.

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.

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

1600's divisors are 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, 800.

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]] -0.000318 0.022794
2.3.5.7 4375/4374, [36 -5 0 -10⟩, [-17 5 16 -10⟩ [⟨1600 2536 3715 4492]] -0.015742 0.033217
2.3.5.7.11 3025/3024, 4375/4374, 184549376/184528125, 7680000000/7672950131 [⟨1600 2536 3715 4492 5535]]
2.3.5.7.11.13 3025/3024, 4096/4095, 4375/4374, 91125/91091, 14236560/14235529 [⟨1600 2536 3715 4492 5535 5921]]
2.3.5.7.11.13.17 2500/2499, 3025/3024, 4375/4374, 14875/14872, 154880/154791, 1724800/1724463 [⟨1600 2536 3715 4492 5535 5921 6540]] -0.016332

Rank-2 temperaments

Periods
per 8ve 		Generator Cents Associated
Ratio 		Temperaments
2 217\1600 162.75 1125/1024 Kwazy
32 121\1600
(21/1600) 		90.75
(15.75) 		48828125/46294416
(?) 		Windrose
32 357\1600
(7\1600) 		267.75
(5.25) 		245/143
(?) 		Germanium
32 23\1600 17.25 ? Dike
80 629\1600
(9\1600) 		471.75
(6.75) 		130/99
(?) 		Tetraicosic
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