1000edo: Difference between revisions
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→Regular temperament properties: Added Rank-2 temperaments |
Comma bases; readability |
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== Theory == | == Theory == | ||
1000edo is related to 200edo, but the [[patent val]]s differ on the mapping for 5 and 7. In the [[5-limit]], it tempers out luna comma | 1000edo is related to 200edo, but the [[patent val]]s differ on the mapping for 5 and 7. In the [[5-limit]], it tempers out {{monzo| 38 -2 -15 }} (luna comma) and {{monzo| -17 62 -35 }} (senior comma). In the [[7-limit]], it tempers out [[4375/4374]], 201768035/201326592, and 165288374272/164794921875, leading to the [[lunatic]] temperament and [[seniority]] temperament. It also tempers out [[3025/3024]], [[9801/9800]], and 391314/390625 in the [[11-limit]]; [[1001/1000]], [[4225/4224]], 4459/4455, and [[10648/10647]] in the [[13-limit]], leading to the [[deca]] temperament and [[donar]] temperament. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 41: | Line 41: | ||
|- | |- | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 4375/4374, | | 4375/4374, 201768035/201326592, {{monzo| 12 -3 -14 9 }} | ||
| {{mapping| 1000 1585 2322 2807 }} | | {{mapping| 1000 1585 2322 2807 }} | ||
| +0.0215 | | +0.0215 | ||
| Line 48: | Line 48: | ||
|- | |- | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 3025/3024, 4375/4374, | | 3025/3024, 4375/4374, 391314/390625, {{monzo| -32 13 1 2 1 }} | ||
| {{mapping| 1000 1585 2322 2807 3459 }} | | {{mapping| 1000 1585 2322 2807 3459 }} | ||
| +0.0472 | | +0.0472 | ||
| Line 55: | Line 55: | ||
|- | |- | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 1001/1000, 3025/3024, | | 1001/1000, 3025/3024, 4225/4224, 4375/4374, 708883245/708837376 | ||
| {{mapping| 1000 1585 2322 2807 3459 3700 }} | | {{mapping| 1000 1585 2322 2807 3459 3700 }} | ||
| +0.0631 | | +0.0631 | ||
| Line 61: | Line 61: | ||
| 7.14 | | 7.14 | ||
|} | |} | ||
=== 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 | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|161\1000 | | 161\1000 | ||
|193.200 | | 193.200 | ||
|262144/234375 | | 262144/234375 | ||
|[[ | | [[Lunatic]] (7-limit) | ||
|- | |- | ||
|1 | | 1 | ||
|269\1000 | | 269\1000 | ||
|322.800 | | 322.800 | ||
| | | 3087/2560 | ||
|[[ | | [[Seniority]] | ||
|- | |- | ||
|4 | | 4 | ||
|317\1000<br>(67\1000) | | 317\1000<br>(67\1000) | ||
|380.400<br>(80.400) | | 380.400<br>(80.400) | ||
|5103/4096<br>(22/21) | | 5103/4096<br>(22/21) | ||
|[[Quasithird]] | | [[Quasithird]] | ||
|- | |- | ||
|10 | | 10 | ||
|263\1000<br>(37\1000) | | 263\1000<br>(37\1000) | ||
|315.600<br>(44.400) | | 315.600<br>(44.400) | ||
|6/5<br>(15/14) | | 6/5<br>(15/14) | ||
|[[Deca]] | | [[Deca]] | ||
|} | |} | ||
Revision as of 07:08, 6 October 2023
| ← 999edo | 1000edo | 1001edo → |
1000edo is notable mostly because it is the equal division corresponding to millioctaves.
Theory
1000edo is related to 200edo, but the patent vals differ on the mapping for 5 and 7. In the 5-limit, it tempers out [38 -2 -15⟩ (luna comma) and [-17 62 -35⟩ (senior comma). In the 7-limit, it tempers out 4375/4374, 201768035/201326592, and 165288374272/164794921875, leading to the lunatic temperament and seniority temperament. It also tempers out 3025/3024, 9801/9800, and 391314/390625 in the 11-limit; 1001/1000, 4225/4224, 4459/4455, and 10648/10647 in the 13-limit, leading to the deca temperament and donar temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.045 | +0.086 | -0.426 | -0.518 | -0.528 | -0.555 | +0.087 | +0.526 | +0.023 | -0.236 |
| Relative (%) | +0.0 | +3.7 | +7.2 | -35.5 | -43.2 | -44.0 | -46.3 | +7.2 | +43.8 | +1.9 | -19.6 | |
| Steps (reduced) |
1000 (0) |
1585 (585) |
2322 (322) |
2807 (807) |
3459 (459) |
3700 (700) |
4087 (87) |
4248 (248) |
4524 (524) |
4858 (858) |
4954 (954) | |
Subsets and supersets
1000edo carries the interval size measure millioctave and has subset edos 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500.
2000edo, which doubles 1000edo, is consistent in the 29-odd-limit and thus provides good corrections for the 2.7.11.13.17.23 subgroup.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [317 -200⟩ | [⟨1000 1585]] | -0.0142 | 0.0142 | 1.18 |
| 2.3.5 | [38 -2 -15⟩, [55 -64 20⟩ | [⟨1000 1585 2322]] | -0.0219 | 0.0159 | 1.33 |
| 2.3.5.7 | 4375/4374, 201768035/201326592, [12 -3 -14 9⟩ | [⟨1000 1585 2322 2807]] | +0.0215 | 0.0764 | 6.37 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 391314/390625, [-32 13 1 2 1⟩ | [⟨1000 1585 2322 2807 3459]] | +0.0472 | 0.0854 | 7.12 |
| 2.3.5.7.11.13 | 1001/1000, 3025/3024, 4225/4224, 4375/4374, 708883245/708837376 | [⟨1000 1585 2322 2807 3459 3700]] | +0.0631 | 0.0857 | 7.14 |
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 161\1000 | 193.200 | 262144/234375 | Lunatic (7-limit) |
| 1 | 269\1000 | 322.800 | 3087/2560 | Seniority |
| 4 | 317\1000 (67\1000) |
380.400 (80.400) |
5103/4096 (22/21) |
Quasithird |
| 10 | 263\1000 (37\1000) |
315.600 (44.400) |
6/5 (15/14) |
Deca |