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29edo: Difference between revisions

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29 is the lowest edo which approximates the [[3/2]] just fifth more accurately than [[12edo]]: 3/2 = 701.955… cents; 17 degrees of 29edo = 703.448… cents. Since the fifth is slightly sharp, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]] – a Superpythagorean instead of a Meantone system.
29 is the lowest edo which approximates the [[3/2]] just fifth more accurately than [[12edo]]: 3/2 = 701.955… cents; 17 degrees of 29edo = 703.448… cents. Since the fifth is slightly sharp, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]] – a Superpythagorean instead of a Meantone system.
{| class="wikitable"
{| class="wikitable"
|[[File:29edoSuperpythDiatonic.mp3]] [[:File:29edoSuperpythDiatonic.mp3|[File info]]]
| [[File:29edoSuperpythDiatonic.mp3]] [[:File:29edoSuperpythDiatonic.mp3|[File info]]]
|[[File:12edoDiatonic.mp3]] [[:File:12edoDiatonic.mp3|[File info]]]
| [[File:12edoDiatonic.mp3]] [[:File:12edoDiatonic.mp3|[File info]]]
|-
|-
|(Super-)pythagorean diatonic major scale and cadence in 29edo
| (Super-)pythagorean diatonic major scale and cadence in 29edo
|12edo diatonic major scale and cadence, for comparison
| 12edo diatonic major scale and cadence, for comparison
|}
|}
The 3 is the only harmonic, of the intelligibly low ones anyway, that 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which [[consistent]]ly represents the [[15-odd-limit]]. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: 250/243 in the [[5-limit]], 49/48 in the [[7-limit]], 55/54 in the [[11-limit]], and 65/64 in the [[13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[19edo]] for [[negri]], as well as an alternative to [[22edo]] or [[15edo]] for [[porcupine]]. 29edo is also an [[oneirotonic]] tuning with generator 11\29, which generates [[ammonite]] temperament.
The 3 is the only harmonic, of the intelligibly low ones anyway, that 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which [[consistent]]ly represents the [[15-odd-limit]]. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: 250/243 in the [[5-limit]], 49/48 in the [[7-limit]], 55/54 in the [[11-limit]], and 65/64 in the [[13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[19edo]] for [[negri]], as well as an alternative to [[22edo]] or [[15edo]] for [[porcupine]]. 29edo is also an [[oneirotonic]] tuning with generator 11\29, which generates [[ammonite]] temperament.
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{{Harmonics in equal|29|columns=11}}
{{Harmonics in equal|29|columns=11}}


=== Miscellany ===
=== Divisors ===
29edo is the 10th [[prime edo]], following [[23edo]] and coming before [[31edo]].
29edo is the 10th [[prime edo]], following [[23edo]] and coming before [[31edo]].


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! Degree
! Degree
! Cents
! Cents
! Approx. ratios of the [[13-limit]]
! Approx. Ratios of the [[13-limit]]
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! Generator for temperaments
|-
|-
| 0
| 0
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| unison
| unison
| D
| D
|
|-
|-
| 1
| 1
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| up unison,<br>downminor 2nd
| up unison,<br>downminor 2nd
| ^D, vEb
| ^D, vEb
|
|-
|-
| 2
| 2
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| minor 2nd
| minor 2nd
| Eb
| Eb
| [[Nautilus]]
|-
|-
| 3
| 3
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| upminor 2nd
| upminor 2nd
| ^Eb
| ^Eb
| [[Negri]]/[[Negril]]
|-
|-
| 4
| 4
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| downmajor 2nd
| downmajor 2nd
| vE
| vE
| [[Porcupine]]/[[Porky]]/[[Coendou]]
|-
|-
| 5
| 5
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| major 2nd
| major 2nd
| E
| E
| [[Chromatic_pairs#Baldy|Baldy]]
|-
|-
| 6
| 6
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| upmajor 2nd,<br>downminor 3rd
| upmajor 2nd,<br>downminor 3rd
| ^E, vF
| ^E, vF
| [[Chromatic_pairs#Bridgetown|Bridgetown]]/[[Immunity]]
|-
|-
| ·7
| ·7
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| minor 3rd
| minor 3rd
| F
| F
| [[Chromatic_pairs#Gariberttet|Gariberttet]]
|-
|-
| 8
| 8
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| upminor 3rd
| upminor 3rd
| ^F
| ^F
| [[Dicot_family#Flat|Flat]]
|-
|-
| 9
| 9
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| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
| [[Chromatic_pairs#Sephiroth|Sephiroth]]
|-
|-
| 10
| 10
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| major 3rd
| major 3rd
| F#
| F#
| [[Roman]]
|-
|-
| 11
| 11
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| upmajor 3rd<br>down 4th
| upmajor 3rd<br>down 4th
| ^F#, vG
| ^F#, vG
| [[Ammonite]]
|-
|-
| ·12
| ·12
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| 4th
| 4th
| G
| G
| [[Cassandra]] [[Chromatic_pairs#Edson|Edson]] [[Chromatic_pairs#Pepperoni|Pepperoni]]
|-
|-
| 13
| 13
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| up 4th
| up 4th
| ^G
| ^G
| [[Wilsec]]
|-
|-
| 14
| 14
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| downaug 4th,<br>dim 5th
| downaug 4th,<br>dim 5th
| vG#, Ab
| vG#, Ab
| [[Tritonic]]
|-
|-
| 15
| 15
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| aug 4th,<br>updim 5th
| aug 4th,<br>updim 5th
| G#, ^Ab
| G#, ^Ab
|
|-
|-
| 16
| 16
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| down 5th
| down 5th
| vA
| vA
|
|-
|-
| ·17
| ·17
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| 5th
| 5th
| A
| A
|
|-
|-
| 18
| 18
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| up 5th,<br>downminor 6th
| up 5th,<br>downminor 6th
| ^A, vBb
| ^A, vBb
|
|-
|-
| 19
| 19
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| minor 6th
| minor 6th
| Bb
| Bb
|
|-
|-
| 20
| 20
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| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
|
|-
|-
| 21
| 21
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| downmajor 6th
| downmajor 6th
| vB
| vB
|
|-
|-
| ·22
| ·22
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| major 6th
| major 6th
| B
| B
|
|-
|-
| 23
| 23
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| upmajor 6th,<br>downminor 7th
| upmajor 6th,<br>downminor 7th
| ^B, vC
| ^B, vC
|
|-
|-
| 24
| 24
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| minor 7th
| minor 7th
| C
| C
|
|-
|-
| 25
| 25
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| upminor 7th
| upminor 7th
| ^C
| ^C
|
|-
|-
| 26
| 26
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| downmajor 7th
| downmajor 7th
| vC#
| vC#
|
|-
|-
| 27
| 27
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| major 7th
| major 7th
| C#
| C#
|
|-
|-
| 28
| 28
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| upmajor 7th,<br>down 8ve
| upmajor 7th,<br>down 8ve
| ^C#, vD
| ^C#, vD
|
|-
|-
| 29
| 29
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| 8ve
| 8ve
| D
| D
|
|}
|}


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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! 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 error|Absolute]] (¢)
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<references/>
<references/>


=== Linear temperaments ===
=== Rank-2 temperaments ===
* [[List of 29et rank two temperaments by badness]]
* [[List of 29et rank two temperaments by badness]]
{| class="wikitable center-all left-5"
|+ Table of rank-2 temperaments by generator
|-
! Periods<br> per 8ve
! Generator<br>(Reduced)
! Cents<br>(Reduced)
! Associated Ratio<br>(Reduced)
! Temperament
|-
| 1
| 2\29
| 82.8
| 21/20
| [[Nautilus]]
|-
| 1
| 3\29
| 124.1
| 14/13
| [[Negri]] / [[negril]] / [[negroni]]
|-
| 1
| 4\29
| 165.5
| 11/10
| [[Porky]] / [[coendou]]
|-
| 1
| 5\29
| 206.9
| 9/8
| [[Baldy]]
|-
| 1
| 6\29
| 248.3
| 15/13
| [[Immunity]] / [[immune]]<br>[[Hemigari]]
|-
| 1
| 7\29
| 289.7
| 13/11
| [[Gariberttet]]
|-
| 1
| 9\29
| 372.4
| 5/4
| [[Sephiroth]]
|-
| 1
| 10\29
| 413.8
| 9/7
| [[Roman]]
|-
| 1
| 11\29
| 455.2
| 13/10
| [[Ammonite]]
|-
| 1
| 12\29
| 496.6
| 4/3
| [[Garibaldi]] / [[andromeda]]<br>[[Leapday]]
|-
| 1
| 13\29
| 537.9
| 15/11
| [[Wilsec]]
|-
| 1
| 14\29
| 579.3
| 7/5
| [[Tritonic]]
|}


Important MOSes include:  
Important MOSes include:  
Retrieved from "https://en.xen.wiki/w/29edo"