2129edo: Difference between revisions

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== Theory ==

2129edo is only [[consistent]] to the [[5-odd-limit]], where it tempers out the [[schisma]]. Otherwise its poor approximation to both [[harmonic]]s [[3/1|3]] and [[5/1|5]] commends itself to a 2.9.15.7.11.13.… [[subgroup]] interpretation.

2129edo is only [[consistent]] to the [[5-odd-limit]], where it tempers out the [[schisma]]. Otherwise its poor approximation to both [[harmonic]]s [[3/1|3]] and [[5/1|5]] commends itself to a 2.9.15.7.11.13.… [[subgroup]] interpretation. However, its representation of [[5/3]] and its octave complement [[6/5]] are extremely accurate, due to being a continued fraction convergent to their logarithms.

=== Odd harmonics ===

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== Regular temperament properties ==

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

|-

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

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

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

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

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

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

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

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| {{monzo| -6749 2129 }}

| {{mapping| 2129 6749 }}

| -0.0204

| −0.0204

| 0.0204

| 3.62

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| {{monzo| 37 29 -33 }}, {{monzo| 209 -61 -4 }}

| {{mapping| 2129 6749 8318 }}

| -0.0247

| −0.0247

| 0.0177

| 3.14

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| 24414062500/24407490807, 13841287201/13839609375, 2199023255552/2197176384375

| {{mapping| 2129 6749 8318 5977 }}

| -0.0256

| −0.0256

| 0.0154

| 2.73

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| 9800/9801, 5767168/5764801, 104857600/104825259, 13841287201/13839609375

| {{mapping| 2129 6749 8318 5977 7365 }}

| -0.0162

| −0.0162

| 0.0232

| 4.12

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| 10648/10647, 9801/9800, 196625/196608, 36924979/36905625, 304117528/303807105

| {{mapping| 2129 6749 8318 5977 7365 7878 }}

| -0.0075

| −0.0075

| 0.0288

| 5.11

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| 2431/2430, 10648/10647, 9801/9800, 845325/845152, 297440/297381, 11275335/11275264, 15980544/15978655

| {{mapping| 2129 6749 8318 5977 7365 7878 8702 }}

| -0.0024

| −0.0024

| 0.0295

| 5.2

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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 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| 498.262

| 4/3

| [[Helmholtz]]

| [[Helmholtz (temperament)|Helmholtz]]

|}

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

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

== Scales ==

* [[Hemischis29]]

== Music ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|2129}}
+{{ED intro}}
 == Theory ==
-edo is only [[consistent]] to the [[5-odd-limit]], where it tempers out the [[schisma]]. Otherwise its poor approximation to both [[harmonic]]s [[3/1|3]] and [[5/1|5]] commends itself to a 2.9.15.7.11.13.… [[subgroup]] interpretation.
+edo is only [[consistent]] to the [[5-odd-limit]], where it tempers out the [[schisma]]. Otherwise its poor approximation to both [[harmonic]]s [[3/1|3]] and [[5/1|5]] commends itself to a 2.9.15.7.11.13.… [[subgroup]] interpretation. However, its representation of [[5/3]] and its octave complement [[6/5]] are extremely accurate, due to being a continued fraction convergent to their logarithms.
 === Odd harmonics ===
@@ Line 13: / Line 13: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
 ! [[TE simple badness|Relative]] (%)
@@ Line 25: / Line 26: @@
 | {{monzo| -6749 2129 }}
 | {{mapping| 2129 6749 }}
-| -0.0204
+| −0.0204
 | 0.0204
 | 3.62
@@ Line 32: / Line 33: @@
 | {{monzo| 37 29 -33 }}, {{monzo| 209 -61 -4 }}
 | {{mapping| 2129 6749 8318 }}
-| -0.0247
+| −0.0247
 | 0.0177
 | 3.14
@@ Line 39: / Line 40: @@
 | 24414062500/24407490807, 13841287201/13839609375, 2199023255552/2197176384375
 | {{mapping| 2129 6749 8318 5977 }}
-| -0.0256
+| −0.0256
 | 0.0154
 | 2.73
@@ Line 46: / Line 47: @@
 | 9800/9801, 5767168/5764801, 104857600/104825259, 13841287201/13839609375
 | {{mapping| 2129 6749 8318 5977 7365 }}
-| -0.0162
+| −0.0162
 | 0.0232
 | 4.12
@@ Line 53: / Line 54: @@
 | 10648/10647, 9801/9800, 196625/196608, 36924979/36905625, 304117528/303807105
 | {{mapping| 2129 6749 8318 5977 7365 7878 }}
-| -0.0075
+| −0.0075
 | 0.0288
 | 5.11
@@ Line 60: / Line 61: @@
 | 2431/2430, 10648/10647, 9801/9800, 845325/845152, 297440/297381, 11275335/11275264, 15980544/15978655
 | {{mapping| 2129 6749 8318 5977 7365 7878 8702 }}
-| -0.0024
+| −0.0024
 | 0.0295
 | 5.2
@@ Line 67: / Line 68: @@
 === 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 8ve
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 78: / Line 80: @@
 | 498.262
 | 4/3
-| [[Helmholtz]]
+| [[Helmholtz (temperament)|Helmholtz]]
 |}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Scales ==
+* [[Hemischis29]]
 == Music ==