29edo: Difference between revisions

Line 479:

! 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]] (¢)

Line 487:

| 2.3

| {{monzo| 46 -29 }}

| [{{~~val~~| 29 46 }}]

| {{mapping| 29 46 }}

| −0.47

| 0.47

Line 501:

| 2.3.5.7

| 49/48, 225/224, 250/243

| [{{~~val~~| 29 46 67 81 }}]

| {{mapping| 29 46 67 81 }}

| +2.78

| 3.28

Line 508:

| 2.3.5.7.11

| 49/48, 55/54, 100/99, 225/224

| [{{~~val~~| 29 46 67 81 100 }}]

| {{mapping| 29 46 67 81 100 }}

| +3.00

| 2.97

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| 2.3.5.7.11.13

| 49/48, 55/54, 100/99, 105/104, 225/224

| [{{~~val~~| 29 46 67 81 100 107 }}]

| {{mapping| 29 46 67 81 100 107 }}

| +3.09

| 2.71

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| 2.3.5.7.11.13.19

| 49/48, 55/54, 65/64, 77/76, 100/99, 105/104

| ~~[⟨29~~ 46 67 81 100 107 123]]

| {{mapping| 29 46 67 81 100 107 123 }}

| +2.91

| 2.55

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| 2.3.5.7.11.13.19.23

| 49/48, 55/54, 65/64, 70/69, 77/76, 100/99, 105/104

| ~~[⟨29~~ 46 67 81 100 107 123 131]]

| {{mapping| 29 46 67 81 100 107 123 131 }}

| +2.76

| 2.42

| 5.85

|}

* 29et (29g val) has a lower relative error than any previous equal temperament in the [[23-limit]]. The next equal temperament doing better in this subgroup is [[46edo|46]].

29et (29g val) has a lower relative error than any previous equal temperament in the [[23-limit]]. The next equal temperament doing better in this subgroup is [[46edo|46]].

* 29et does well in the no-17 [[19-limit]] and no-17 23-limit, being consistent to the no-17 [[23-odd-limit]]. However, [[15edo]] is lower in relative error in both these subgroups than 29.

29et does well in the no-17 [[19-limit]] and no-17 23-limit, being consistent to the no-17 [[23-odd-limit]]. However, [[15edo]] is lower in relative error in both these subgroups than 29.

=== Commas ===

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Line 542:

{| class="commatable wikitable center-all left-3 right-4 left-6"

|-

! [[Harmonic limit|Prime ~~Limit~~]]

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

! [[Monzo]]

Line 605:

Line 603:

| 35.70

| Zozo

| ~~Slendro~~ diesis

| Semaphoresma, slendro diesis

|-

| 7

Line 748:

Line 746:

|+ Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio~~ (Reduced)~~

! Associated Ratio*

! Temperament

|-

Line 838:

Line 836:

| [[Edson]]

|}

<nowiki/>* [[Normal lists|octave-reduced form]], reduced to the first half-octave

=== The Tetradecatonic System ===

A variant of porcupine [[support~~|supported~~]] in 29edo is [[nautilus]], which splits the porcupine generator in half (tempering out 49:48 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine.

A variant of porcupine [[support]]ed in 29edo is [[nautilus]], which splits the porcupine generator in half (tempering out 49:48 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine.

The MOS nautilus[14] contains both "even" tetrads (approximating 4:5:6:7 or its inverse) as well as "odd" tetrads (approximating the "Bohlen-Pierce-like" chord 9:11:13:15, or its inverse). Both types are recognizable and consonant, if somewhat heavily tempered. Moreover, one of the four types of tetrads may be built on '''each''' scale degree of nautilus[14], thus there are as many chords as there are notes, so nautilus[14] has a "circulating" quality to it with as much freedom of modulation as possible. To be exact, there are 4 "major-even", 4 "minor-even", 3 "major-odd", and 3 "minor-odd" chords.

@@ Line 479: / Line 479: @@
 ! 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]] (¢)
@@ Line 487: / Line 487: @@
 | 2.3
 | {{monzo| 46 -29 }}
-| [{{val| 29 46 }}]
+| {{mapping| 29 46 }}
 | &minus;0.47
 | 0.47
@@ Line 501: / Line 501: @@
 | 2.3.5.7
 | 49/48, 225/224, 250/243
-| [{{val| 29 46 67 81 }}]
+| {{mapping| 29 46 67 81 }}
 | +2.78
 | 3.28
@@ Line 508: / Line 508: @@
 | 2.3.5.7.11
 | 49/48, 55/54, 100/99, 225/224
-| [{{val| 29 46 67 81 100 }}]
+| {{mapping| 29 46 67 81 100 }}
 | +3.00
 | 2.97
@@ Line 515: / Line 515: @@
 | 2.3.5.7.11.13
 | 49/48, 55/54, 100/99, 105/104, 225/224
-| [{{val| 29 46 67 81 100 107 }}]
+| {{mapping| 29 46 67 81 100 107 }}
 | +3.09
 | 2.71
@@ Line 522: / Line 522: @@
 | 2.3.5.7.11.13.19
 | 49/48, 55/54, 65/64, 77/76, 100/99, 105/104
-| [⟨29 46 67 81 100 107 123]]
+| {{mapping| 29 46 67 81 100 107 123 }}
 | +2.91
 | 2.55
@@ Line 529: / Line 529: @@
 | 2.3.5.7.11.13.19.23
 | 49/48, 55/54, 65/64, 70/69, 77/76, 100/99, 105/104
-| [⟨29 46 67 81 100 107 123 131]]
+| {{mapping| 29 46 67 81 100 107 123 131 }}
 | +2.76
 | 2.42
 | 5.85
 |}
+* 29et (29g val) has a lower relative error than any previous equal temperament in the [[23-limit]]. The next equal temperament doing better in this subgroup is [[46edo|46]].
-et (29g val) has a lower relative error than any previous equal temperament in the [[23-limit]].  The next equal temperament doing better in this subgroup is [[46edo|46]].
+* 29et does well in the no-17 [[19-limit]] and no-17 23-limit, being consistent to the no-17 [[23-odd-limit]]. However, [[15edo]] is lower in relative error in both these subgroups than 29.
-et does well in the no-17 [[19-limit]] and no-17 23-limit, being consistent to the no-17 [[23-odd-limit]]. However, [[15edo]] is lower in relative error in both these subgroups than 29.
 === Commas ===
@@ Line 544: / Line 542: @@
 {| class="commatable wikitable center-all left-3 right-4 left-6"
 |-
-! [[Harmonic limit|Prime<br>Limit]]
+! [[Harmonic limit|Prime<br>limit]]
 ! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
 ! [[Monzo]]
@@ Line 605: / Line 603: @@
 | 35.70
 | Zozo
-| Slendro diesis
+| Semaphoresma, slendro diesis
 |-
 | 7
@@ Line 748: / Line 746: @@
 |+ Table of rank-2 temperaments by generator
 |-
-! Periods<br> per 8ve
+! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated Ratio<br>(Reduced)
+! Associated Ratio*
 ! Temperament
 |-
@@ Line 838: / Line 836: @@
 | [[Edson]]
 |}
+<nowiki/>* [[Normal lists|octave-reduced form]], reduced to the first half-octave
 === The Tetradecatonic System ===
-A variant of porcupine [[support|supported]] in 29edo is [[nautilus]], which splits the porcupine generator in half (tempering out 49:48 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine.
+A variant of porcupine [[support]]ed in 29edo is [[nautilus]], which splits the porcupine generator in half (tempering out 49:48 in the process), thus resulting in a different mapping for 7 than standard porcupine. Nautilus also extends to the 13-limit much more easily than does standard porcupine.
 The MOS nautilus[14] contains both "even" tetrads (approximating 4:5:6:7 or its inverse) as well as "odd" tetrads (approximating the "Bohlen-Pierce-like" chord 9:11:13:15, or its inverse). Both types are recognizable and consonant, if somewhat heavily tempered. Moreover, one of the four types of tetrads may be built on '''each''' scale degree of nautilus[14], thus there are as many chords as there are notes, so nautilus[14] has a "circulating" quality to it with as much freedom of modulation as possible. To be exact, there are 4 "major-even", 4 "minor-even", 3 "major-odd", and 3 "minor-odd" chords.