296edo: Difference between revisions

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== 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]]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.

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.

=== Prime harmonics ===

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| 2.3

| {{monzo| -469 296 }}

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

| {{mapping| 296 469 }}

| +0.1904

| 0.1905

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| 2.3.5

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

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

| {{mapping| 296 469 687 }}

| +0.2962

| 0.2158

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| 2.3.5.7

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

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

| {{mapping| 296 469 687 831 }}

| +0.2138

| 0.2350

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| 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

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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

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Dzelic]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:Sabric]]

@@ Line 3: / Line 3: @@
 == 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]]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.
+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.
 === Prime harmonics ===
@@ Line 24: / Line 24: @@
 | 2.3
 | {{monzo| -469 296 }}
-| [{{val| 296 469 }}]
+| {{mapping| 296 469 }}
 | +0.1904
 | 0.1905
@@ Line 31: / Line 31: @@
 | 2.3.5
 | 2109375/2097152, {{monzo| -16 35 -17 }}
-| [{{val| 296 469 687 }}]
+| {{mapping| 296 469 687 }}
 | +0.2962
 | 0.2158
@@ Line 38: / Line 38: @@
 | 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 45: / Line 45: @@
 | 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 52: / Line 52: @@
 | 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 62: / Line 62: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 103: / Line 103: @@
 | [[Dzelic]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 [[Category:Sabric]]