241edo: Difference between revisions

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+prime error table
+RTT table and rank-2 temperaments
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=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|241}}
{{Harmonics in equal|241}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 382 -241 }}
| [{{val| 241 382 }}]
| -0.038
| 0.038
| 0.76
|-
| 2.3.5
| 78732/78125, {{monzo| 56 -28 -5 }}
| [{{val| 241 382 560 }}]
| -0.322
| 0.403
| 8.10
|-
| 2.3.5.7
| 3136/3125, 19683/19600, 829940/823543
| [{{val| 241 382 560 677 }}]
| -0.431
| 0.397
| 7.97
|-
| 2.3.5.7.11
| 540/539, 3136/3125, 8019/8000, 15488/15435
| [{{val| 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
| [{{val| 241 382 560 677 834 892 }}]
| -0.397
| 0.330
| 6.63
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 20\241
| 99.59
| 200/189
| [[Quintagar]] / [[quinsandric]]
|-
| 1
| 50\241
| 248.96
| {{monzo| -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]]
|}


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Prime EDO]]
[[Category:Subpental]]

Revision as of 20:56, 15 March 2022

241edo is the equal division of the octave into 241 parts of 4.9793 cents each. It tempers out 78732/78125 in the 5-limit, 19683/19600 and 3136/3125 in the 7-limit, 65536/65219, 540/539, 43923/43904, 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 the 53rd prime edo.

Prime harmonics

Approximation of prime harmonics in 241edo
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)

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

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		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
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