1200edo: Difference between revisions
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== Theory == | == Theory == | ||
{{ | 1200edo is distinctly [[consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactored]] in the [[5-limit]], having the same mapping as [[600edo]]. Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47. The 47th harmonic is 6666 steps and 666 steps reduced – a funny mathematical coincidence in our decimal system. | ||
1200et tempers out 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 & 1029 temperament, with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200]. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 & 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. | |||
It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721. | |||
=== | === Prime harmonics === | ||
{{Harmonics in equal|1200}} | |||
=== Subsets and supersets === | |||
The divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents. | |||
== 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<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 38: | Line 39: | ||
| 0.0743 | | 0.0743 | ||
| 7.43 | | 7.43 | ||
|} | |} | ||
Revision as of 04:50, 18 April 2023
| ← 1199edo | 1200edo | 1201edo → |
The 1200 equal divisions of the octave (1200edo), or the 1200(-tone) equal temperament (1200tet, 1200et) when viewed from a regular temperament perspective, divides the octave into 1200 equal parts of exactly 1 cent each, and a size close to 1729/1728. It is notable mostly because it is the equal division corresponding to cents.
Theory
1200edo is distinctly consistent through the 11-odd-limit. This means that whole-cent approximations of the 11-odd-limit tonality diamond intervals are conveniently represented through the 11-limit patent val ⟨1200 1902 2786 3369 4151]. It is enfactored in the 5-limit, having the same mapping as 600edo. Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47. The 47th harmonic is 6666 steps and 666 steps reduced – a funny mathematical coincidence in our decimal system.
1200et tempers out 2460375/2458624 and 95703125/95551488 in the 7-limit, supporting the 171 & 1029 temperament, with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200]. It tempers out 9801/9800, 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 & 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200.
It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit POTE tuning of quadritikleismic temperament: ⟨1200 1902 2785 3368]. It also provides the optimal patent val for the 224&752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.045 | -0.314 | +0.174 | -0.318 | +0.472 | +0.045 | +0.487 | -0.274 | +0.423 | -0.036 |
| Relative (%) | +0.0 | +4.5 | -31.4 | +17.4 | -31.8 | +47.2 | +4.5 | +48.7 | -27.4 | +42.3 | -3.6 | |
| Steps (reduced) |
1200 (0) |
1902 (702) |
2786 (386) |
3369 (969) |
4151 (551) |
4441 (841) |
4905 (105) |
5098 (298) |
5428 (628) |
5830 (1030) |
5945 (1145) | |
Subsets and supersets
The divisors of 1200 are 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600. These are all the edos whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7 | 2460375/2458624, 95703125/95551488, [36 -5 0 -10⟩ | [⟨1200 1902 2786 3369]] | +0.0112 | 0.0748 | 7.48 |
| 2.3.5.7.11 | 9801/9800, 234375/234256, 825000/823543, 1771561/1769472 | [⟨1200 1902 2786 3369 4151]] | +0.0273 | 0.0743 | 7.43 |