1619edo: Difference between revisions
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== Theory == | == Theory == | ||
1619edo is excellent in the 13-limit | 1619edo is excellent in the 13-limit, where it tempers out [[4225/4224]], [[4375/4374]], [[6656/6655]], 78125/78078, and 117649/117612. It also notably tempers out [[123201/123200]]. It supports [[vidar]], which has the comma basis 4225/4224, 4375/4374, and 6656/6655, and other unnamed expansions of the [[ragismic]] temperament such as the 270 & 441 & 1619, tempering out 4225/4224, 4375/4374, 655473/655360, or the 72 & 270 & 494 & 1619 temperament tempering out 6656/6655 and 2912000/2910897. | ||
1619edo has 7/6 on 360th step, a highly divisible number, 27/25 on 180th, and 33/32 on 72nd as a consequence of tempering out the commas. This means that 72ed33/32 is virtually equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the 832 & 1619 temperament, which tempers out 196625/196608, 200000/199927, 2912000/2910897, and 3764768/3764475. | |||
1619edo has 7/6 on 360th step, a highly divisible number, 27/25 on 180th, and 33/32 on 72nd as a consequence of tempering out the commas. This means that 72ed33/32 is equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the temperament | |||
1619edo is the 256th [[Prime edo]]. | 1619edo is the 256th [[Prime edo]]. | ||
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! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
| {{monzo| -2566 1619 }} | | {{monzo| -2566 1619 }} | ||
| [{{val| 1619 2566 }}] | | [{{val| 1619 2566 }}] | ||
| Line 65: | Line 63: | ||
| 0.032 | | 0.032 | ||
| 4.2 | | 4.2 | ||
|} | |} | ||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
Revision as of 18:22, 12 April 2022
| ← 1618edo | 1619edo | 1620edo → |
1619edo divides the octave into parts of about 0.741 cents each.
Theory
1619edo is excellent in the 13-limit, where it tempers out 4225/4224, 4375/4374, 6656/6655, 78125/78078, and 117649/117612. It also notably tempers out 123201/123200. It supports vidar, which has the comma basis 4225/4224, 4375/4374, and 6656/6655, and other unnamed expansions of the ragismic temperament such as the 270 & 441 & 1619, tempering out 4225/4224, 4375/4374, 655473/655360, or the 72 & 270 & 494 & 1619 temperament tempering out 6656/6655 and 2912000/2910897.
1619edo has 7/6 on 360th step, a highly divisible number, 27/25 on 180th, and 33/32 on 72nd as a consequence of tempering out the commas. This means that 72ed33/32 is virtually equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the 832 & 1619 temperament, which tempers out 196625/196608, 200000/199927, 2912000/2910897, and 3764768/3764475.
1619edo is the 256th Prime edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.040 | -0.149 | -0.080 | +0.134 | -0.009 | +0.295 | -0.293 | +0.262 | -0.053 |
| Relative (%) | +0.0 | -5.4 | -20.2 | -10.8 | +18.0 | -1.2 | +39.8 | -39.5 | +35.3 | -7.1 | |
| Steps (reduced) |
1619 (0) |
2566 (947) |
3759 (521) |
4545 (1307) |
5601 (744) |
5991 (1134) |
6618 (142) |
6877 (401) |
7324 (848) |
7865 (1389) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-2566 1619⟩ | [⟨1619 2566]] | 0.013 | 0.013 | 1.7 |
| 2.3.5 | [-69 45 -1⟩, [-82 -1 36⟩ | [⟨1619 2566 3759]] | 0.030 | 0.026 | 3.5 |
| 2.3.5.7 | 4375/4374, [-6 3 9 -7⟩, [-67 14 6 11⟩ | [⟨1619 2566 3759 4545]] | 0.030 | 0.023 | 3.1 |
| 2.3.5.7.11 | 117649/117612, 151263/151250, 759375/758912, 117440512/117406179 | [⟨1619 2566 3759 4545 5601]] | 0.016 | 0.034 | 4.0 |
| 2.3.5.7.11.13 | 4225/4224, 43940/43923, 151263/151250, 91125/91091, 123201/123200 | [⟨1619 2566 3759 4545 5601 5991]] | 0.013 | 0.032 | 4.2 |