1600edo: Difference between revisions
m →Regular temperament properties: cleanup |
→Theory: sata and armodue |
||
| Line 3: | Line 3: | ||
== 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 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. One step of 1600edo is already used as a measure called śata in the context of 16edo [[Armodue theory]]. | ||
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 7-limit, it tempers out the ragisma, 4375/4374. In the 11-limit, it supports the rank-3 temperament [[thor]]. | ||
Revision as of 15:43, 5 March 2023
| ← 1599edo | 1600edo | 1601edo → |
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. One step of 1600edo is already used as a measure called śata in the context of 16edo Armodue theory.
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
| 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 |