1200edo: Difference between revisions
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| Prime factorization = 2<sup>4</sup> × 3 × 5<sup>2</sup> | | Prime factorization = 2<sup>4</sup> × 3 × 5<sup>2</sup> | ||
| Step size = 1¢<sup>by definition</sup> | | Step size = 1¢<sup>by definition</sup> | ||
| Fifth = 702\1200 | | Fifth = 702\1200 (702.00¢) (→ [[200edo|117\200]]) | ||
| | | Semitones = 114:90 (114.00¢ : 90¢) | ||
| Consistency = 11 | |||
}} | }} | ||
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. It is notable mostly because it is the equal division corresponding to cents. | 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. It is notable mostly because it is the equal division corresponding to cents. | ||
==Theory== | == Theory == | ||
{{Harmonics in equal|1200}} | {{Harmonics in equal|1200}} | ||
Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 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, 600}}. These are all the | |||
Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 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, 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. | |||
1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a 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. | 1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a 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. | ||
Revision as of 20:48, 31 March 2022
| ← 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. It is notable mostly because it is the equal division corresponding to cents.
Theory
| 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) | |
Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are 1, 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, 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.
1200edo is uniquely consistent through the 11-limit, which means the intervals of the 11-limit tonality diamond, and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val ⟨1200 1902 2786 3369 4151]. It is contorted in the 5-limit, having the same mapping as 600edo. In the 7-limit, it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by 171edo. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by 494edo. In the 7-limit, it provides a 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.
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 |
| 2.3.7.17.31.41.47 | 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416 | [⟨1200 1902 3369 4905 5945 6429 6666]] | -0.0244 | 0.0351 | 3.51 |