894edo: Difference between revisions
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== Theory == | == Theory == | ||
894edo is [[consistent]] to the [[17-odd-limit]], and except for [[19/13]], [[19/17]] and their [[octave complement]]s, it is consistent to the [[31-odd-limit]], thus making it a | 894edo is [[consistent]] to the [[17-odd-limit]], and except for [[19/13]], [[19/17]], [[25/13]], [[25/17]] and their [[octave complement]]s, it is consistent to the [[31-odd-limit]], thus making it a reasonable [[31-limit]] system. Its [[prime interval|primes]] [[5/1|5]], [[17/1|17]], [[19/1|19]], and [[23/1|23]] come from [[149edo]], its [[11/1|11]] from [[298edo]], and its [[7/1|7]] and [[13/1|13]] from [[447edo]]. | ||
As an equal temperament, it [[tempering out|tempers out]] {{monzo| 23 6 -14 }} ([[vishnu comma]]) in the [[5-limit]]; 250047/250000 ([[landscape comma]]) in the [[7-limit]]; [[9801/9800]], [[131072/130977]], [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], 34398/34375 in the [[13-limit]]; [[1156/1155]], [[2431/2430]], [[2601/2600]], [[11016/11011]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[3250/3249]] in the [[19-limit]]; and [[1863/1862]], [[2300/2299]] among others in the [[23-limit]]. | As an equal temperament, it [[tempering out|tempers out]] {{monzo| 23 6 -14 }} ([[vishnu comma]]) in the [[5-limit]]; 250047/250000 ([[landscape comma]]) in the [[7-limit]]; [[9801/9800]], [[131072/130977]], [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], 34398/34375 in the [[13-limit]]; [[1156/1155]], [[2431/2430]], [[2601/2600]], [[11016/11011]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[3250/3249]] in the [[19-limit]]; and [[1863/1862]], [[2300/2299]] among others in the [[23-limit]]. | ||
Latest revision as of 15:19, 18 June 2026
| ← 893edo | 894edo | 895edo → |
894 equal divisions of the octave (abbreviated 894edo or 894ed2), also called 894-tone equal temperament (894tet) or 894 equal temperament (894et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 894 equal parts of about 1.34 ¢ each. Each step represents a frequency ratio of 21/894, or the 894th root of 2.
Theory
894edo is consistent to the 17-odd-limit, and except for 19/13, 19/17, 25/13, 25/17 and their octave complements, it is consistent to the 31-odd-limit, thus making it a reasonable 31-limit system. Its primes 5, 17, 19, and 23 come from 149edo, its 11 from 298edo, and its 7 and 13 from 447edo.
As an equal temperament, it tempers out [23 6 -14⟩ (vishnu comma) in the 5-limit; 250047/250000 (landscape comma) in the 7-limit; 9801/9800, 131072/130977, 151263/151250 in the 11-limit; 1716/1715, 2080/2079, 4096/4095, 4225/4224, 34398/34375 in the 13-limit; 1156/1155, 2431/2430, 2601/2600, 11016/11011 in the 17-limit; 1521/1520, 2376/2375, 3250/3249 in the 19-limit; and 1863/1862, 2300/2299 among others in the 23-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.058 | +0.263 | +0.302 | +0.360 | -0.259 | -0.257 | +0.474 | -0.086 | -0.047 | -0.069 |
| Relative (%) | +0.0 | +4.4 | +19.6 | +22.5 | +26.8 | -19.3 | -19.2 | +35.3 | -6.4 | -3.5 | -5.2 | |
| Steps (reduced) |
894 (0) |
1417 (523) |
2076 (288) |
2510 (722) |
3093 (411) |
3308 (626) |
3654 (78) |
3798 (222) |
4044 (468) |
4343 (767) |
4429 (853) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.337 | +0.468 | -0.108 | +0.265 | +0.321 | -0.111 | -0.106 | -0.112 | +0.169 | +0.398 | +0.564 |
| Relative (%) | -25.1 | +34.9 | -8.1 | +19.8 | +23.9 | -8.3 | -7.9 | -8.4 | +12.6 | +29.7 | +42.0 | |
| Steps (reduced) |
4657 (187) |
4790 (320) |
4851 (381) |
4966 (496) |
5121 (651) |
5259 (789) |
5302 (832) |
5423 (59) |
5498 (134) |
5534 (170) |
5636 (272) | |
Subsets and supersets
Since 894 factors into primes as 2 × 3 × 149, 894edo contains subset edos 2, 3, 6, 149, 298, and 447.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1417 -894⟩ | [⟨894 1417]] | −0.0184 | 0.0184 | 1.37 |
| 2.3.5 | [23 6 -14⟩, [93 -66 5⟩ | [⟨894 1417 2076]] | −0.0501 | 0.0473 | 3.52 |
| 2.3.5.7 | 250047/250000, 43046721/43025920, 134217728/133984375 | [⟨894 1417 2076 2510]] | −0.0644 | 0.0479 | 3.57 |
| 2.3.5.7.11 | 9801/9800, 131072/130977, 151263/151250, 1771875/1771561 | [⟨894 1417 2076 2510 3093]] | −0.0724 | 0.0456 | 3.40 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 4096/4095, 34398/34375, 1574640/1574573 | [⟨894 1417 2076 2510 3093 3308]] | −0.0486 | 0.0675 | 5.03 |
| 2.3.5.7.11.13.17 | 1156/1155, 1716/1715, 2080/2079, 2431/2430, 4096/4095, 34398/34375 | [⟨894 1417 2076 2510 3093 3308 3654]] | −0.0327 | 0.0737 | 5.49 |
| 2.3.5.7.11.13.17.19 | 1156/1155, 1521/1520, 1716/1715, 2080/2079, 2376/2375, 2431/2430, 3250/3249 | [⟨894 1417 2076 2510 3093 3308 3654 3798]] | −0.0425 | 0.0737 | 5.49 |
| 2.3.5.7.11.13.17.19.23 | 1156/1155, 1521/1520, 1716/1715, 1863/1862, 2080/2079, 2376/2375, 2431/2430, 3250/3249 | [⟨894 1417 2076 2510 3093 3308 3654 3798 4044]] | −0.0357 | 0.0721 | 5.37 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 2 | 53\894 | 71.14 | 25/24 | Vishnu (5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct