241edo: Difference between revisions

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== 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 ===

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=== 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 ==

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! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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|-

| 2.3

| {{~~monzo~~| 382 -241 }}

| {{Monzo| 382 -241 }}

| {{~~mapping~~| 241 382 }}

| {{Mapping| 241 382 }}

| −0.038

| 0.038

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| 2.3.5

| 78732/78125, {{monzo| 56 -28 -5 }}

| {{~~mapping~~| 241 382 560 }}

| {{Mapping| 241 382 560 }}

| −0.322

| 0.403

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| 2.3.5.7

| 3136/3125, 19683/19600, 829940/823543

| {{~~mapping~~| 241 382 560 677 }}

| {{Mapping| 241 382 560 677 }}

| −0.431

| 0.397

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| 2.3.5.7.11

| 540/539, 3136/3125, 8019/8000, 15488/15435

| {{~~mapping~~| 241 382 560 677 834 }}

| {{Mapping| 241 382 560 677 834 }}

| −0.425

| 0.355

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| 2.3.5.7.11.13

| 351/350, 540/539, 676/675, 3136/3125, 10648/10647

| {{~~mapping~~| 241 382 560 677 834 892 }}

| {{Mapping| 241 382 560 677 834 892 }}

| −0.397

| 0.330

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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Associated ratio*

! Temperaments

|-

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| 50\241

| 248.96

| {{~~monzo~~| -26 18 -1 }}

| {{Monzo| -26 18 -1 }}

| [[Monzismic]]

| [[Monzismic]] (5-limit)

|-

| 1

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| [[Gary]]

|}

<nowiki />* ~~[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and~~ [[normal forms #Minimal-generator form|minimal form]] ~~in parentheses if distinct~~

<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]

[[Category:Subpental]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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]].
+edo 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 ===
@@ Line 9: / Line 11: @@
 === Subsets and supersets ===
-edo is the 53rd [[prime edo]].
+edo 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 ==
@@ Line 17: / Line 19: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 24: / Line 26: @@
 |-
 | 2.3
-| {{monzo| 382 -241 }}
+| {{Monzo| 382 -241 }}
-| {{mapping| 241 382 }}
+| {{Mapping| 241 382 }}
 | −0.038
 | 0.038
@@ Line 32: / Line 34: @@
 | 2.3.5
 | 78732/78125, {{monzo| 56 -28 -5 }}
-| {{mapping| 241 382 560 }}
+| {{Mapping| 241 382 560 }}
 | −0.322
 | 0.403
@@ Line 39: / Line 41: @@
 | 2.3.5.7
 | 3136/3125, 19683/19600, 829940/823543
-| {{mapping| 241 382 560 677 }}
+| {{Mapping| 241 382 560 677 }}
 | −0.431
 | 0.397
@@ Line 46: / Line 48: @@
 | 2.3.5.7.11
 | 540/539, 3136/3125, 8019/8000, 15488/15435
-| {{mapping| 241 382 560 677 834 }}
+| {{Mapping| 241 382 560 677 834 }}
 | −0.425
 | 0.355
@@ Line 53: / Line 55: @@
 | 2.3.5.7.11.13
 | 351/350, 540/539, 676/675, 3136/3125, 10648/10647
-| {{mapping| 241 382 560 677 834 892 }}
+| {{Mapping| 241 382 560 677 834 892 }}
 | −0.397
 | 0.330
@@ Line 63: / Line 65: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br />per 8ve
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br />ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 78: / Line 80: @@
 | 50\241
 | 248.96
-| {{monzo| -26 18 -1 }}
+| {{Monzo| -26 18 -1 }}
-| [[Monzismic]]
+| [[Monzismic]] (5-limit)
 |-
 | 1
@@ Line 99: / Line 101: @@
 | [[Gary]]
 |}
-<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
+<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
 [[Category:Subpental]]