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}}
}}
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|29}}
{{ED intro}}


== Theory ==
== Theory ==
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| 12edo diatonic major scale and cadence, for comparison
| 12edo diatonic major scale and cadence, for comparison
|}
|}
[[3/1|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.
Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Subgroup temperaments #Edson (2.3.7/5.11/5.13/5 subgroup)|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it.
Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[petrmic triad]], a 13-limit [[Dyadic chord|essentially tempered dyadic chord]]. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.


29edo could be thought of as the "twin" of [[12edo]] in the 5-limit, since 5-limit intervals in 12edo and 29edo are tuned with almost exactly the same absolute errors, but in opposite directions. There are other ways in which they are counterparts (12 tempers out 50:49 but not 49:48; 29 does the opposite). Each supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively).
29edo could be thought of as the "twin" of [[12edo]] in the 5-limit, since 5-limit intervals in 12edo and 29edo are tuned with almost exactly the same absolute errors, but in opposite directions. There are other ways in which they are counterparts (12 tempers out 50:49 but not 49:48; 29 does the opposite). Each supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively).
Line 29: Line 23:


=== Prime harmonics ===
=== Prime harmonics ===
[[3/1|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.
{{Harmonics in equal|29|columns=11}}
{{Harmonics in equal|29|columns=11}}
=== Stacking fifths ===
Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Subgroup temperaments #Edson (2.3.7/5.11/5.13/5 subgroup)|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it.
Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[petrmic triad]], a 13-limit [[Dyadic chord|essentially tempered dyadic chord]].
29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.


=== Divisors ===
=== Divisors ===
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! 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]]
([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and ^d2)
! colspan="3" |[[SKULO interval names|SKULO interval names and notation]] (K or S = 1)
! colspan="3" |[[SKULO interval names|SKULO interval names and notation]] (K or S = 1)
|-
|-
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| downminor
| downminor
| zo
| zo
| {a, b, 0, 1}
| [a, b, 0, 1>
| 7/6, 7/4
| 7/6, 7/4
|-
|-
| minor
| minor
| fourthward wa
| fourthward wa
| {a, b}, b &lt; -1
| [a, b>, b &lt; -1
| 32/27, 16/9
| 32/27, 16/9
|-
|-
| upminor
| upminor
| gu
| gu
| {a, b, -1}
| [a, b, -1>
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| "
| "
| ilo
| ilo
| {a, b, 0, 0, 1}
| [a, b, 0, 0, 1>
| 11/9, 11/6
| 11/9, 11/6
|-
|-
| downmajor
| downmajor
| lu
| lu
| {a, b, 0, 0, -1}
| [a, b, 0, 0, -1>
| 12/11, 18/11
| 12/11, 18/11
|-
|-
| "
| "
| yo
| yo
| {a, b, 1}
| [a, b, 1>
| 5/4, 5/3
| 5/4, 5/3
|-
|-
| major
| major
| fifthward wa
| fifthward wa
| {a, b}, b &gt; 1
| [a, b>, b &gt; 1
| 9/8, 27/16
| 9/8, 27/16
|-
|-
| upmajor
| upmajor
| ru
| ru
| {a, b, 0, -1}
| [a, b, 0, -1>
| 9/7, 12/7
| 9/7, 12/7
|}
|}
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| C upmajor or C up
| C upmajor or C up
|}
|}
For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]].
For a more complete list, see [[Ups and downs notation #Chords and Chord Progressions]].


== Notation ==
== Notation ==
=== Standard notation===
=== Standard notation ===
29edo can be notated three different ways. Using only sharps and flats, the chromatic scale from C is:
29edo can be notated three different ways. Using only sharps and flats, the chromatic scale from C is:


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* C𝄪 = F𝄫
* C𝄪 = F𝄫


=== Sagittal notation ===
=== Ups and downs notation ===
[[Sagittal notation]] is another possibility, as demonstrated by the below example:
Since a sharp raises by three steps, 29edo is a good candidate for [[ups and downs notation]], similar to [[22edo]]. Spoken as up, downsharp, sharp, etc. Note that downsharp (v#) can be respelled as dup (^^).
 
{{Sharpness-sharp3a}}
{| class="wikitable"
|-
| [[File:29edothumb.png|alt=29edothumb.png|29edothumb.png]]
|-
| This example in Sagittal notation shows 29-edo as a fifth-tone system.
|}
 
=== Ups and downs ===
Since a sharp raises by three steps, 29edo is a good candidate for [[ups and downs notation]], similar to [[22edo]]. Here, sharps and flats with arrows from [[Helmholtz–Ellis notation]] can be used:


Here, sharps and flats with arrows from [[Helmholtz–Ellis notation]] are used:
{{Sharpness-sharp3}}
{{Sharpness-sharp3}}


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If arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best spelled with double arrows.
If arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best spelled with double arrows.
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[15edo#Sagittal notation|15]] and [[22edo#Sagittal notation|22]].
==== Evo flavor ====
<imagemap>
File:29-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 511 0 671 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
default [[File:29-EDO_Evo_Sagittal.svg]]
</imagemap>
==== Revo flavor ====
<imagemap>
File:29-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 503 0 663 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
default [[File:29-EDO_Revo_Sagittal.svg]]
</imagemap>


== Approximation to JI ==
== Approximation to JI ==
[[File:29ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|15-odd-limit intervals approximated in 29edo]]
[[File:29ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|15-odd-limit intervals approximated in 29edo]]
=== Interval mappings ===
=== Interval mappings ===
{{Q-odd-limit intervals|29}}
{{Q-odd-limit intervals|29}}
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| {{monzo| 46 -29 }}
| {{monzo| 46 -29 }}
| {{mapping| 29 46 }}
| {{mapping| 29 46 }}
| &minus;0.47
| −0.47
| 0.47
| 0.47
| 1.14
| 1.14
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| 2.3.5
| 2.3.5
| 250/243, 16875/16384
| 250/243, 16875/16384
| [{{val| 29 46 67 }}]
| {{mapping| 29 46 67 }}
| +1.68
| +1.68
| 3.07
| 3.07
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=== Commas ===
=== Commas ===
29edo [[tempers out]] the following [[comma]]s. This assumes the [[patent val]] {{val| 29 46 67 81 100 107 }}. Cent values are rounded to 5 digits.  
29et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[patent val]] {{val| 29 46 67 81 100 107 }}. Cent values are rounded to 5 digits.  


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| 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>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
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| Minor minthma
| Minor minthma
|}
|}
<references />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
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* [[Leapfrog]] diatonic [[5L 2s]] 5552552 (17\29, 1\1)
* [[Leapfrog]] diatonic [[5L 2s]] 5552552 (17\29, 1\1)
* [[Leapfrog]] chromatic [[5L 7s]] 3232323223232322 (17\29, 1\1)
* [[Leapfrog]] chromatic [[5L 7s]] 3232323223232322 (17\29, 1\1)
* [[Leapfrog]] hyperchromatic [[12L 5s]] 21221221221222122122122 (17\29, 1\1)
* [[Porcupine]] [[1L 6s]] 4444445 (4\29, 1\1)
* [[Porcupine]] [[1L 6s]] 4444445 (4\29, 1\1)
* [[Porcupine]] [[7L 1s]] 44444441 (4\29, 1\1)
* [[Porcupine]] [[7L 1s]] 44444441 (4\29, 1\1)
* [[Porcupine]] [[7L 8s]] 313131313131311 (4\29, 1\1)
* [[Porcupine]] [[7L 15s]] 2112112112112112112111 (4\29, 1\1)
* [[Negri]] [[1L 8s]] 333333335 (3\29, 1\1)
* [[Negri]] [[1L 8s]] 333333335 (3\29, 1\1)
* [[Negri]] [[9L 1s]] 3333333332 (3\29, 1\1)
* [[Negri]] [[9L 1s]] 3333333332 (3\29, 1\1)
* [[Negri]] [[10L 9s]] 2212121212121212121 (3\29 1\1)
* [[Semaphore]] [[4L 1s]] 56666 (6\29, 1\1)
* [[Semaphore]] [[4L 1s]] 56666 (6\29, 1\1)
* [[Semaphore]] [[5L 4s]] 551515151 (6\29, 1\1)  
* [[Semaphore]] [[5L 4s]] 551515151 (6\29, 1\1)  
* [[Semaphore]] [[5L 9s]] 41411411411411 (6\29, 1\1)
* [[Semaphore]] [[5L 9s]] 41411411411411 (6\29, 1\1)
* [[Semaphore]] [[5L 14s]] 3113111311131113111 (6\29, 1\1)
* Pathological [[semaphore]] [[5L 19s]] 211121111211112111121111 (6\29, 1\1)
* [[Nautilus]] [[1L 13s]] 22222222222223 (2\29, 1\1)
* [[Nautilus]] [[1L 13s]] 22222222222223 (2\29, 1\1)
* [[Nautilus]] [[14L 1s]] 222222222222221 (2\29, 1\1)
* [[Nautilus]] [[14L 1s]] 222222222222221 (2\29, 1\1)
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: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
</div>
 
* [[Lumatone mapping for 29edo]]
== Music ==
== Music ==
=== Modern renderings ===
=== Modern renderings ===
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* [https://www.youtube.com/watch?v=uGOK7WtVtlM "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=uGOK7WtVtlM "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=jcZaU5PrhvU "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=jcZaU5PrhvU "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=-Sa8IhljHM0 ''BACH - RICERCAR a 6 from the Musical Offering, tuned into 29-EDO'', BWV 1079] (1742-1749) - rendered by Claudi Meneghin (2025)
* [https://www.youtube.com/watch?v=856A7vTqIW8 ''Bach, Art of Fugue: Contrapunctus 11, tuned into 29-edo (harpischord)''] (1740-1746) - rendered by Claudi Meneghin (2025)


; {{W|Nicolaus Bruhns}}
; {{W|Nicolaus Bruhns}}
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; [[Australopithecine Microtonal Music]]
; [[Australopithecine Microtonal Music]]
* [https://www.youtube.com/watch?v=yvCVAxyU5ZU ''Toy Shoppe''] (2024)
* [https://www.youtube.com/watch?v=yvCVAxyU5ZU ''Toy Shoppe''] (2024)
* [https://www.youtube.com/watch?v=3pAU6_QunmU ''The Sea of Swirly Twirly Gumdrops''] (2024)


; [[User:CellularAutomaton|CellularAutomaton]]
; [[User:CellularAutomaton|CellularAutomaton]]
Line 970: Line 985:
* [https://www.youtube.com/watch?v=HGQ2b6v0TWE ''Glass Animals - Life Itself''] (2023)
* [https://www.youtube.com/watch?v=HGQ2b6v0TWE ''Glass Animals - Life Itself''] (2023)
* [https://www.youtube.com/watch?v=ktk0VWbUbDg ''microtonal improvisation in 29edo''] (2023)
* [https://www.youtube.com/watch?v=ktk0VWbUbDg ''microtonal improvisation in 29edo''] (2023)
* [https://www.youtube.com/shorts/fyPaaW9AyMA ''Homestuck: Pipeorgankind (microtonal cover in 29edo)''] (2024)
* [https://www.youtube.com/shorts/SH5IQOi33Oo ''29edo groove''] (2025)


; [[duckapus]]
; [[duckapus]]
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* "Chill Bells" from ''Melancholie'' (2023) [https://open.spotify.com/track/30Ik57efXmIae1YSgTgdIE Spotify] | [https://francium223.bandcamp.com/track/chill-bells Bandcamp] | [https://www.youtube.com/watch?v=Qbz2a4PMnjY YouTube]
* "Chill Bells" from ''Melancholie'' (2023) [https://open.spotify.com/track/30Ik57efXmIae1YSgTgdIE Spotify] | [https://francium223.bandcamp.com/track/chill-bells Bandcamp] | [https://www.youtube.com/watch?v=Qbz2a4PMnjY YouTube]
* from ''XenRhythms'' (2024)  
* from ''XenRhythms'' (2024)  
** "All 29" – [https://francium223.bandcamp.com/track/all-29 Bandcamp] | [https://www.youtube.com/watch?v=C-FFLXHSO_k YouTube]
** "All 29" – [https://open.spotify.com/track/7I0fLgRPKdqCK51PUnt4Oe Spotify] | [https://francium223.bandcamp.com/track/all-29 Bandcamp] | [https://www.youtube.com/watch?v=C-FFLXHSO_k YouTube]
** "Do Not Immerse Yourself In Fire Or Water" – [https://francium223.bandcamp.com/track/do-not-immerse-yourself-in-fire-or-water Bandcamp] | [https://www.youtube.com/watch?v=NHDvNQbBHV0 YouTube] – immunity[14] in 29edo tuning
** "Do Not Immerse Yourself In Fire Or Water" – [https://open.spotify.com/track/5FUmlRP1JjLsH99vXbJhXH Spotify] | [https://francium223.bandcamp.com/track/do-not-immerse-yourself-in-fire-or-water Bandcamp] | [https://www.youtube.com/watch?v=NHDvNQbBHV0 YouTube] – in Immunity[14], 29edo tuning
* [https://www.youtube.com/watch?v=di4qn2VFYbs ''Plane Sonatina No. 1''] (2025)
* [https://www.youtube.com/watch?v=ifvvww20XAU ''Strank Running''] (2025)


; [[Igliashon Jones]]
; [[Igliashon Jones]]
Line 1,014: Line 1,033:


== See also ==
== See also ==
* [[Arto and Tendo Theory]]
* [[User:Unque/29edo Composition Theory|Unque's approach]]
* [[Lumatone mapping for 29edo]]
* [[Extraclassical tonality]]
* [[67ed5]] – octave-stretched 29edo, improves harmonics 5.7.11.13.17 but damages 2.3
 
== Notes ==
<references group="note" />
 
== References ==
<references />


[[Category:IMPORTDEBUG - Change External Images]]
[[Category:IMPORTDEBUG - Change External Images]]
 
[[Category:3-limit record edos|##]] <!-- 2-digit number -->
[[Category:3-limit]]
[[Category:Subgroup temperaments]]
[[Category:Listen]]
[[Category:Twentuning]]
[[Category:Negri]]
[[Category:Negri]]
[[Category:Petrmic]]
[[Category:Petrmic]]
[[Category:Porcupine]]
[[Category:Porcupine]]
[[Category:Subgroup temperaments]]
[[Category:Listen]]
[[Category:Twentuning]]
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