241edo: Difference between revisions
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" Tags: Mobile edit Mobile web edit |
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== Theory == | == Theory == | ||
241edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]]. It has a sharp tendency, with [[prime harmonic]]s 3 through 13 all tuned sharp. As an equal temperament, it [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]], [[19683/19600]] and [[3136/3125]] in the [[7-limit]], [[540/539]], 43923/43904, [[65536/65219]], and [[151263/151250]] in the [[11-limit]], and [[351/350]], [[676/675]], [[729/728]], [[1001/1000]] and [[2080/2079]] in the [[13-limit]]. It provides the [[optimal patent val]] for [[subpental]]. | 241edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]]. It has a sharp tendency, with [[prime harmonic]]s 3 through 13 all tuned sharp. | ||
As an equal temperament, it [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]], [[19683/19600]] and [[3136/3125]] in the [[7-limit]], [[540/539]], [[43923/43904]], [[65536/65219]], and [[151263/151250]] in the [[11-limit]], and [[351/350]], [[676/675]], [[729/728]], [[1001/1000]] and [[2080/2079]] in the [[13-limit]]. It provides the [[optimal patent val]] for [[subpental]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
241edo is the 53rd [[prime edo]]. | 241edo is the 53rd [[prime edo]]. As such, it does not contain any nontrivial subset edos. [[1205edo]], which slices its step in five, is a notable higher-limit system. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
Latest revision as of 13:39, 14 July 2026
| ← 240edo | 241edo | 242edo → |
241 equal divisions of the octave (abbreviated 241edo or 241ed2), also called 241-tone equal temperament (241tet) or 241 equal temperament (241et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 241 equal parts of about 4.98 ¢ each. Each step represents a frequency ratio of 21/241, or the 241st root of 2.
Theory
241edo is distinctly consistent in the 15-odd-limit. It has a sharp tendency, with prime harmonics 3 through 13 all tuned sharp.
As an equal temperament, it tempers out 78732/78125 in the 5-limit, 19683/19600 and 3136/3125 in the 7-limit, 540/539, 43923/43904, 65536/65219, and 151263/151250 in the 11-limit, and 351/350, 676/675, 729/728, 1001/1000 and 2080/2079 in the 13-limit. It provides the optimal patent val for subpental.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.12 | +2.07 | +2.13 | +1.38 | +0.97 | -0.39 | +1.24 | -0.89 | +1.13 | +0.19 |
| Relative (%) | +0.0 | +2.4 | +41.5 | +42.7 | +27.7 | +19.4 | -7.9 | +24.9 | -17.8 | +22.7 | +3.9 | |
| Steps (reduced) |
241 (0) |
382 (141) |
560 (78) |
677 (195) |
834 (111) |
892 (169) |
985 (21) |
1024 (60) |
1090 (126) |
1171 (207) |
1194 (230) | |
Subsets and supersets
241edo is the 53rd prime edo. As such, it does not contain any nontrivial subset edos. 1205edo, which slices its step in five, is a notable higher-limit system.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [382 -241⟩ | [⟨241 382]] | −0.038 | 0.038 | 0.76 |
| 2.3.5 | 78732/78125, [56 -28 -5⟩ | [⟨241 382 560]] | −0.322 | 0.403 | 8.10 |
| 2.3.5.7 | 3136/3125, 19683/19600, 829940/823543 | [⟨241 382 560 677]] | −0.431 | 0.397 | 7.97 |
| 2.3.5.7.11 | 540/539, 3136/3125, 8019/8000, 15488/15435 | [⟨241 382 560 677 834]] | −0.425 | 0.355 | 7.14 |
| 2.3.5.7.11.13 | 351/350, 540/539, 676/675, 3136/3125, 10648/10647 | [⟨241 382 560 677 834 892]] | −0.397 | 0.330 | 6.63 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 20\241 | 99.59 | 200/189 | Quintagar / quinsandric |
| 1 | 50\241 | 248.96 | [-26 18 -1⟩ | Monzismic |
| 1 | 76\241 | 378.42 | 56/45 | Subpental |
| 1 | 89\241 | 443.15 | 162/125 | Sensipent |
| 1 | 100\241 | 497.93 | 4/3 | Gary |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct