1600edo: Difference between revisions
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Adopt template: EDO intro; cleanup; clarify the title row of the rank-2 temp table |
Adopt template: Factorization; misc. cleanup |
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== Theory == | == Theory == | ||
1600edo is a very strong 37-limit system, being distinctly consistent in the 37-limit with a smaller [[Tenney- | 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]]. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
1600 | 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]]. | 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]]. | ||
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|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{ | | {{monzo| -53 10 16 }}, {{monzo| 26 -75 40 }} | ||
| {{mapping| 1600 2536 3715 }} | | {{mapping| 1600 2536 3715 }} | ||
| -0.0003 | | -0.0003 | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+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 80: | Line 82: | ||
| 17.25 | | 17.25 | ||
| ? | | ? | ||
|[[Dam]] / [[dike]] / [[polder]] | | [[Dam]] / [[dike]] / [[polder]] | ||
|- | |- | ||
| 32 | | 32 | ||
Revision as of 13:59, 30 October 2023
| ← 1599edo | 1600edo | 1601edo → |
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
| 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
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 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 |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct