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{{Infobox ET | {{Infobox ET}} | ||
{{ED intro}} | |||
}} | |||
{{ | |||
== 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]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|241}} | {{Harmonics in equal|241}} | ||
=== Subsets and supersets === | |||
241edo is the 53rd [[prime edo]]. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | |- | ||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 29: | Line 25: | ||
| 2.3 | | 2.3 | ||
| {{monzo| 382 -241 }} | | {{monzo| 382 -241 }} | ||
| | | {{mapping| 241 382 }} | ||
| | | −0.038 | ||
| 0.038 | | 0.038 | ||
| 0.76 | | 0.76 | ||
| Line 36: | Line 32: | ||
| 2.3.5 | | 2.3.5 | ||
| 78732/78125, {{monzo| 56 -28 -5 }} | | 78732/78125, {{monzo| 56 -28 -5 }} | ||
| | | {{mapping| 241 382 560 }} | ||
| | | −0.322 | ||
| 0.403 | | 0.403 | ||
| 8.10 | | 8.10 | ||
| Line 43: | Line 39: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 3136/3125, 19683/19600, 829940/823543 | | 3136/3125, 19683/19600, 829940/823543 | ||
| | | {{mapping| 241 382 560 677 }} | ||
| | | −0.431 | ||
| 0.397 | | 0.397 | ||
| 7.97 | | 7.97 | ||
| Line 50: | Line 46: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 540/539, 3136/3125, 8019/8000, 15488/15435 | | 540/539, 3136/3125, 8019/8000, 15488/15435 | ||
| | | {{mapping| 241 382 560 677 834 }} | ||
| | | −0.425 | ||
| 0.355 | | 0.355 | ||
| 7.14 | | 7.14 | ||
| Line 57: | Line 53: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 351/350, 540/539, 676/675, 3136/3125, 10648/10647 | | 351/350, 540/539, 676/675, 3136/3125, 10648/10647 | ||
| | | {{mapping| 241 382 560 677 834 892 }} | ||
| | | −0.397 | ||
| 0.330 | | 0.330 | ||
| 6.63 | | 6.63 | ||
| Line 65: | Line 61: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 102: | Line 99: | ||
| [[Gary]] | | [[Gary]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
[[Category:Subpental]] | [[Category:Subpental]] | ||
Latest revision as of 14:23, 20 February 2025
| ← 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.
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