296edo: Difference between revisions

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{{Infobox ET

~~| Prime factorization = 23 × 37~~

~~| Step size = 4.05405¢~~

~~| Fifth = 173\296 (701.35¢)~~

~~| Semitones = 27:23 (109.46¢ : 93.24¢)~~

~~| Consistency = 15~~

}}

The '''296 equal divisions of the octave''' ('''296edo'''), or the '''296(-tone) equal temperament''' ('''296tet''', '''296et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 296 parts of about 4.05 [[cent]]s each.

== Theory ==

In the 5-limit, 296et not only tempers out the [[semicomma]] of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its [[optimal patent val]], and tempers out the minortone comma, {{monzo| -16 35 -17 }}. It is also an interesting temperament in higher limits, being distinctly consistent through to the 15-odd-limit. In the 7-limit it tempers out 4375/4374 ([[ragisma]]), 16875/16807 (mirkwai), and 118098/117649 (stearnsma), [[support~~|supporting~~]] 7-limit [[octoid]] ~~temperament~~. In the 11-limit, [[540/539]], 1375/1372, [[3025/3024]], [[4000/3993]], [[6250/6237]] and [[9801/9800]]; in the 13-limit, [[625/624]], [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[6656/6655]], so that it also supports the 11- and 13-limit versions of octoid.

In the 5-limit, 296et not only [[tempering out|tempers out]] the [[semicomma]] of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its [[optimal patent val]], and tempers out the minortone comma, {{monzo| -16 35 -17 }}. It is also an interesting temperament in higher limits, being [[consistency|distinctly consistent]] through to the [[15-odd-limit]]. In the 7-limit it tempers out 4375/4374 ([[ragisma]]), 16875/16807 (mirkwai), and 118098/117649 (stearnsma), [[support]]ing 7-limit [[octoid]] and [[sabric]]. In the 11-limit, [[540/539]], 1375/1372, [[3025/3024]], [[4000/3993]], [[6250/6237]] and [[9801/9800]]; in the 13-limit, [[625/624]], [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[6656/6655]], so that it also supports the 11- and 13-limit versions of octoid. It allows [[swetismic chords]] and [[squbemic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit.

~~296 is divisible by~~ {{~~EDOs~~| ~~2, 4, 8, 37, 74 and 148~~ }}.

=== Prime harmonics ===

=== ~~Prime harmonics~~ ===

=== Subsets and supersets ===

Since 296 factors into {{factorisation|296}}, 296edo has subset edos {{EDOs| 2, 4, 8, 37, 74 and 148 }}.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

|-

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

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

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

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

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

! colspan="2" | Tuning error

|-

Line 29:

Line 25:

| 2.3

| {{monzo| -469 296 }}

| [{{~~val~~| 296 469 }}]

| {{mapping| 296 469 }}

| +0.1904

| 0.1905

Line 36:

Line 32:

| 2.3.5

| 2109375/2097152, {{monzo| -16 35 -17 }}

| [{{~~val~~| 296 469 687 }}]

| {{mapping| 296 469 687 }}

| +0.2962

| 0.2158

Line 43:

Line 39:

| 2.3.5.7

| 4375/4374, 16875/16807, 2100875/2097152

| [{{~~val~~| 296 469 687 831 }}]

| {{mapping| 296 469 687 831 }}

| +0.2138

| 0.2350

Line 50:

Line 46:

| 2.3.5.7.11

| 540/539, 1375/1372, 4000/3993, 2100875/2097152

| [{{~~val~~| 296 469 687 831 1024 }}]

| {{mapping| 296 469 687 831 1024 }}

| +0.1691

| 0.2284

Line 57:

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| 2.3.5.7.11.13

| 540/539, 625/624, 729/728, 1375/1372, 15379/15360

| [{{~~val~~| 296 469 687 831 1024 1095 }}]

| {{mapping| 296 469 687 831 1024 1095 }}

| +0.2012

| 0.2206

Line 65:

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=== Rank-2 temperaments ===

{| 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 per ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

Line 76:

Line 73:

| 182.43

| 10/9

| [[~~Minortone]] / [[mitonic~~]]

| [[Mitonic]]

|-

| 1

Line 82:

Line 79:

| 271.62

| 75/64

| [[~~Orson]] / [[sabric~~]]

| [[Sabric]]

|-

| 1

Line 97:

Line 94:

|-

| 8

| 144\296 (4\296)

| 144\296 (4\296)

| 583.78 (16.22)

| 583.78 (16.22)

| 7/5 (126/125)

| 7/5 (126/125)

| [[Octoid]]

|-

| 37

| 67\296 (3\296)

| 271.62 (12.16)

| 117/100 (?)

| [[Dzelic]]

|}

<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

[[Category:~~Equal divisions of the octave]]~~

[[Category:Sabric]]

~~[[Category:Orson~~]]

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 2<sup>3</sup> × 37
+{{ED intro}}
-| Step size = 4.05405¢
-| Fifth = 173\296 (701.35¢)
-| Semitones = 27:23 (109.46¢ : 93.24¢)
-| Consistency = 15
-}}
-The '''296 equal divisions of the octave''' ('''296edo'''), or the '''296(-tone) equal temperament''' ('''296tet''', '''296et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 296 parts of about 4.05 [[cent]]s each.
 == Theory ==
-In the 5-limit, 296et not only tempers out the [[semicomma]] of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its [[optimal patent val]], and tempers out the minortone comma, {{monzo| -16 35 -17 }}. It is also an interesting temperament in higher limits, being distinctly consistent through to the 15-odd-limit. In the 7-limit it tempers out 4375/4374 ([[ragisma]]), 16875/16807 (mirkwai), and 118098/117649 (stearnsma), [[support|supporting]] 7-limit [[octoid]] temperament. In the 11-limit, [[540/539]], 1375/1372, [[3025/3024]], [[4000/3993]], [[6250/6237]] and [[9801/9800]]; in the 13-limit, [[625/624]], [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[6656/6655]], so that it also supports the 11- and 13-limit versions of octoid.
+In the 5-limit, 296et not only [[tempering out|tempers out]] the [[semicomma]] of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its [[optimal patent val]], and tempers out the minortone comma, {{monzo| -16 35 -17 }}. It is also an interesting temperament in higher limits, being [[consistency|distinctly consistent]] through to the [[15-odd-limit]]. In the 7-limit it tempers out 4375/4374 ([[ragisma]]), 16875/16807 (mirkwai), and 118098/117649 (stearnsma), [[support]]ing 7-limit [[octoid]] and [[sabric]]. In the 11-limit, [[540/539]], 1375/1372, [[3025/3024]], [[4000/3993]], [[6250/6237]] and [[9801/9800]]; in the 13-limit, [[625/624]], [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[6656/6655]], so that it also supports the 11- and 13-limit versions of octoid. It allows [[swetismic chords]] and [[squbemic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit.
-is divisible by {{EDOs| 2, 4, 8, 37, 74 and 148 }}.
+=== Prime harmonics ===
+{{Harmonics in equal|296|columns=11}}
-=== Prime harmonics ===
+=== Subsets and supersets ===
-{{Primes in edo|296}}
+Since 296 factors into {{factorisation|296}}, 296edo has subset edos {{EDOs| 2, 4, 8, 37, 74 and 148 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 29: / Line 25: @@
 | 2.3
 | {{monzo| -469 296 }}
-| [{{val| 296 469 }}]
+| {{mapping| 296 469 }}
 | +0.1904
 | 0.1905
@@ Line 36: / Line 32: @@
 | 2.3.5
 | 2109375/2097152, {{monzo| -16 35 -17 }}
-| [{{val| 296 469 687 }}]
+| {{mapping| 296 469 687 }}
 | +0.2962
 | 0.2158
@@ Line 43: / Line 39: @@
 | 2.3.5.7
 | 4375/4374, 16875/16807, 2100875/2097152
-| [{{val| 296 469 687 831 }}]
+| {{mapping| 296 469 687 831 }}
 | +0.2138
 | 0.2350
@@ Line 50: / Line 46: @@
 | 2.3.5.7.11
 | 540/539, 1375/1372, 4000/3993, 2100875/2097152
-| [{{val| 296 469 687 831 1024 }}]
+| {{mapping| 296 469 687 831 1024 }}
 | +0.1691
 | 0.2284
@@ Line 57: / Line 53: @@
 | 2.3.5.7.11.13
 | 540/539, 625/624, 729/728, 1375/1372, 15379/15360
-| [{{val| 296 469 687 831 1024 1095 }}]
+| {{mapping| 296 469 687 831 1024 1095 }}
 | +0.2012
 | 0.2206
@@ Line 65: / Line 61: @@
 === Rank-2 temperaments ===
 {| 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 octave
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 76: / Line 73: @@
 | 182.43
 | 10/9
-| [[Minortone]] / [[mitonic]]
+| [[Mitonic]]
 |-
 | 1
@@ Line 82: / Line 79: @@
 | 271.62
 | 75/64
-| [[Orson]] / [[sabric]]
+| [[Sabric]]
 |-
 | 1
@@ Line 97: / Line 94: @@
 |-
 | 8
-| 144\296<br>(4\296)
+| 144\296<br />(4\296)
-| 583.78<br>(16.22)
+| 583.78<br />(16.22)
-| 7/5<br>(126/125)
+| 7/5<br />(126/125)
 | [[Octoid]]
+|-
+| 37
+| 67\296<br />(3\296)
+| 271.62<br />(12.16)
+| 117/100<br />(?)
+| [[Dzelic]]
 |}
+<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
-[[Category:Equal divisions of the octave]]
+[[Category:Sabric]]
-[[Category:Orson]]