1200edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
1200edo is notable for being the equal division of the octave whose step size corresponds to exactly 1 [[cent]]. | |||
== Theory == | == Theory == | ||
1200edo is | 1200edo is [[consistency|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 [[enfactoring|enfactored]] in the [[5-limit]], having the same tuning as [[600edo]]. | ||
The equal temperament [[tempering out|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& | 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 & 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721. | ||
Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === 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 thus can be played exactly on any digital audio workstation that offers detuning by cents. | The nontrivial 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 thus can be played exactly on any digital audio workstation that offers detuning by cents. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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| 2.3.5.7 | | 2.3.5.7 | ||
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }} | | 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }} | ||
| | | {{mapping| 1200 1902 2786 3369 }} | ||
| +0.0112 | | +0.0112 | ||
| 0.0748 | | 0.0748 | ||
| Line 35: | Line 39: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472 | | 9801/9800, 234375/234256, 825000/823543, 1771561/1769472 | ||
| | | {{mapping| 1200 1902 2786 3369 4151 }} | ||
| +0.0273 | | +0.0273 | ||
| 0.0743 | | 0.0743 | ||
| 7.43 | | 7.43 | ||
|} | |} | ||
== Music == | |||
; [[Hideya]] | |||
* [https://www.youtube.com/watch?v=FJhmgbuoRHA ''Like scattered blue light''] (2024) | |||
; [[Sevish]] | |||
* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version) | |||
[[Category:Listen]] | |||