29edo: Difference between revisions

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== Theory ==

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 sharp instead of flat, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]] — a [[Parapyth|~~Parapythagorean~~]] tuning 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 sharp instead of flat, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]] — a [[Parapyth|parapythagorean]] tuning instead of a meantone system.

{| class="wikitable"

| [[File:29edoSuperpythDiatonic.mp3]] [[:File:29edoSuperpythDiatonic.mp3|[File info]]]

| [[File:12edoDiatonic.mp3]] [[:File:12edoDiatonic.mp3|[File info]]]

|-

| ~~(Super-)pythagorean~~ diatonic major scale and cadence in 29edo

| Parapythagorean diatonic major scale and cadence in 29edo

| 12edo diatonic major scale and cadence, for comparison

|}

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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, like how 12 tempers out [[50/49]] but not [[49/48]]; 29 does the opposite. Each also supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively).

A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone.

A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone, since if 2 tones make a 5/4, (4 + 5) * 2/9 tones = 2 tones (9 steps) = 5/4 in 29edo.

=== Prime harmonics ===

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=== 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.

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]].

Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29edo 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 [[7:11:13|1-11/7-13/7 (7:11:13)]] chord, the [[The Archipelago|barbados]] triad [[10:13:15|1-13/10-3/2 (10:13:15)]], the minor barbados triad [[26:30:39|1-15/13-3/2 (26:30:39)]], the [[22:28:33|1-14/11-3/2 (22:28:33)]] triad, the [[22:26:33|1-13/11-3/2 (22:26:33)]] triad, 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.

29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195, 364/363, and 847/845 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~~ ===

Due to 29edo's tone-efficient mapping of 2.3.7/5.11/5.13/5, it makes sense to collapse this subgroup to 29edo. One may then expand the subgroup to the full 13-limit, adding an independent generator to reach primes 5, 7, 11, and 13 in one generator. This is [[mystery]] temperament, which has very low [[badness]] despite so many periods per octave. The 58-note MOS gives scope for harmony, with 29 15-odd-limit otonal chords and 29 utonal chords.

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

=== Interval Flavors ===

29edo has [[Ultramajor and inframinor|inframinor (arto)]], [[Neogothic major and minor|neogothic minor]], [[Submajor_and_supraminor|supraminor]], submajor, neogothic major, and ultramajor (tendo) thirds and sevenths. This is in contrast to systems like [[31edo]], where there are subminor, minor, neutral, major, and supermajor thirds and sevenths. This is due to 29edo representing 2.3.7/5.11/5.13/5 well, and ratios between two primes greater than 3 tend to land between interval categories of intervals in a 2.3.p subgroup. For example, 2.3.5 intervals are major/minor, 2.3.7 intervals are [[Supermajor and subminor|supermajor/subminor]], and 2.3.11 and 2.3.13 intervals are [[Neutral (interval quality)|artoneutral/tendoneutral]]. 31edo, on the other hand, represents 2.3.5.7.11 well, and thus has interval categories represented in 2.3.5, 2.3.7, and 2.3.11. It can also be seen from the fact that the 29&31 temperament, [[tritonic]], maps seconds and thirds to large numbers of generators, so they differ more in tuning between the systems.

=== Subsets and Supersets ===

29edo is the 10th [[prime edo]], following [[23edo]] and coming before [[31edo]]. Its supersets [[58edo]] and [[87edo]] correct many of the higher primes.

== Intervals ==

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! Cents

! Approx. Ratios of the [[13-limit]]

! colspan="3" | [[Ups and downs notation]]

! Chain-of-fifths notation

([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and ^d2)

! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and ^d2)

! colspan="3" |[[SKULO interval names|SKULO interval names and notation]] (K or S = 1)

|-

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| 0.000

| [[1/1]]

| unison

| P1

| unison

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| 1

| 41.379

| [[25/24]], [[33/32]], [[56/55]], [[81/80]]

| [[33/32]], [[40/39]], [[45/44]], [[81/80]], [[64/63]]

| negative diminished 2nd, double diminished 3rd

| ^1, vm2

| up unison, downminor 2nd

| ^D, vEb

| S1, sm2

| K1, S1, sm2

| comma-wide unison, super unison, subminor 2nd

| comma-wide unison, super unison, subminor 2nd

| KD, SD, sEb

|-

| 2

| 82.759

| [[21/20]]

| [[21/20]], [[22/21]], [[135/128]], [[256/243]]

| minor 2nd

| m2

| minor 2nd

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| 124.138

| [[16/15]], [[15/14]], [[14/13]], [[13/12]]

| augmented 1sn

| ^m2

| upminor 2nd

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| 165.517

| [[12/11]], [[11/10]], [[10/9]]

| diminished 3rd

| vM2

| downmajor 2nd

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| 206.897

| [[9/8]]

| major 2nd

| M2

| major 2nd

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| 248.276

| [[8/7]], [[7/6]], [[15/13]]

| double diminished 4th, double augmented 1sn

| ^M2, vm3

| upmajor 2nd, downminor 3rd

| upmajor 2nd, downminor 3rd

| ^E, vF

| SM2, sm3

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| SE, sF

|-

| ·7

| 7

| 289.655

| [[13/11]]

| [[13/11]], [[32/27]]

| minor 3rd

| m3

| minor 3rd

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| 331.034

| [[6/5]], [[11/9]]

| augmented 2nd

| ^m3

| upminor 3rd

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| 372.414

| [[5/4]], [[16/13]]

| diminished 4th

| vM3

| downmajor 3rd

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| 10

| 413.793

| [[14/11]]

| [[14/11]], [[81/64]]

| major 3rd

| M3

| major 3rd

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| 455.172

| [[9/7]], [[13/10]]

| double diminished 5th, double augmented 2nd

| ^M3, v4

| upmajor 3rd down 4th

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| SF#, sG

|-

| ~~·12~~

| 12

| 496.552

| [[4/3]]

| perfect 4th

| P4

| 4th

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| 537.931

| [[11/8]], [[15/11]]

| augmented 3rd

| ^4

| up 4th

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| 579.310

| [[7/5]], [[18/13]]

| diminished 5th

| vA4, d5

| downaug 4th, dim 5th

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| 620.690

| [[10/7]], [[13/9]]

| augmented 4th

| A4, ^d5

| aug 4th, updim 5th

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| 662.069

| [[16/11]], [[22/15]]

| diminished 6th

| v5

| down 5th

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| kA

|-

| ~~·17~~

| 17

| 703.448

| [[3/2]]

| perfect 5th

| P5

| 5th

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| 744.828

| [[14/9]], [[20/13]]

| double augmented 4th, double diminished 7th

| ^5, vm6

| up 5th, downminor 6th

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| 19

| 786.207

| [[11/7]]

| [[11/7]], [[128/81]]

| minor 6th

| m6

| minor 6th

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| 827.586

| [[8/5]], [[13/8]]

| augmented 5th

| ^m6

| upminor 6th

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| 868.966

| [[5/3]], [[18/11]]

| diminished 7th

| vM6

| downmajor 6th

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| kB

|-

| ~~·22~~

| 22

| 910.345

| [[22/13]]

| [[22/13]], [[27/16]]

| major 6th

| M6

| major 6th

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| 951.724

| [[7/4]], [[12/7]], [[26/15]]

| double augmented 5th, double diminished 8ve

| ^M6, vm7

| upmajor 6th, downminor 7th

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| 993.103

| [[16/9]]

| minor 7th

| m7

| minor 7th

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| 1034.483

| [[11/6]], [[20/11]], [[9/5]]

| augmented 6th

| ^m7

| upminor 7th

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| 1075.862

| [[15/8]], [[28/15]], [[13/7]], [[24/13]]

| diminished 8ve

| vM7

| downmajor 7th

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| 27

| 1117.241

| [[40/21]]

| [[40/21]], [[21/11]], [[256/135]], [[243/128]]

| major 7th

| M7

| major 7th

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| 28

| 1158.621

| [[48/25]], [[64/33]], [[55/28]], [[160/81]]

| [[64/33]], [[39/20]], [[88/45]], [[160/81]], [[63/32]]

| diminished 9th, double augmented 6th

| ^M7, v8

| upmajor 7th, down 8ve

| upmajor 7th, down 8ve

| ^C#, vD

| SM7, s8

| supermajor 7th, comma-narrow 8ve, sub 8ve

| supermajor 7th, comma-narrow 8ve, sub 8ve

| SC#, kD, sD

|-

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| 1200.000

| [[2/1]]

| octave

| P8

| 8ve

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

{{dash|C, B♯, D♭, C♯, B𝄪/E𝄫, D, C𝄪/F𝄫, E♭, D♯, F♭, E, D𝄪/G𝄫, F, E♯, G♭, F♯, E𝄪/A𝄫, G, F𝄪, A♭, G♯, B𝄫, A, G𝄪/C𝄫, B♭, A♯, C♭, B, A𝄪/D𝄫, C|s=hair~~|d=long~~}}

{{dash|C, B♯, D♭, C♯, B𝄪/E𝄫, D, C𝄪/F𝄫, E♭, D♯, F♭, E, D𝄪/G𝄫, F, E♯, G♭, F♯, E𝄪/A𝄫, G, F𝄪, A♭, G♯, B𝄫, A, G𝄪/C𝄫, B♭, A♯, C♭, B, A𝄪/D𝄫, C|s=hair}}

Here, six pairs of enharmonic equivalents exist:

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* C𝄪 = F𝄫

=== ~~Ups and downs~~ notation ===

=== Stein–Zimmermann–Gould notation ===

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 (^^).~~

Since a sharp raises by three steps, 29edo is a good candidate for [[Stein–Zimmermann–Gould notation]], using sharps and flats with arrows similar to [[22edo]]:

~~{{Ups and downs sharpness}}~~

~~Here,~~ sharps and flats with arrows ~~from~~ [[~~Helmholtz–Ellis notation~~]] ~~are used~~:

Note that C♯ is enharmonic to D{{flatup}}, and D♭ is enharmonic to C{{sharpdown}}.

If arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best spelled with double arrows.

=== Kite's ups and downs notation ===

29edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, downsharp, sharp, etc. Note that downsharp (v#) can be respelled as dup (^^).

=== Sagittal notation ===

This notation uses the same sagittal sequence as ~~EDOs~~ [[15edo#Sagittal notation|15]] and [[22edo#Sagittal notation|22]].

This notation uses the same sagittal sequence as edos [[15edo #Sagittal notation|15]] and [[22edo #Sagittal notation|22]].

==== Evo flavor ====

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* [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)

* [https://www.youtube.com/watch?v=VUX9yZiBM6g ''BACH, NEVERENDING CANON, but it has the SHEPARD EFFECT and is tuned into 29edo''] (1742-1749) - rendered by Claudi Meneghin (2025)

; {{W|Nicolaus Bruhns}}

* [https://www.youtube.com/watch?v=me7dHmo3cVs ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)

* [https://www.youtube.com/watch?v=-E-2mszlgWM ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)

; {{W|Kate Bush}}

* [https://www.youtube.com/shorts/QIfqj-8Ojhc ''Army Dreamers'' <nowiki>[short clip]</nowiki>] (1980) - microtonal cover in 29edo by [[Bryan Deister]] (2025)

; {{W|C418}}

* [https://www.youtube.com/shorts/WEu7NzK7u0I ''Cat''] (2011) - microtonal cover in 29edo by [[Bryan Deister]] (2026)

; {{W|Dorian Concept}}

* [https://www.youtube.com/shorts/2NHkGHQ84Qc ''Hide''] (2023/2024) – microtonal cover in 29edo by [[Bryan Deister]] (2025)

; Alan Fennah as "Alternative Radio" (see {{W|Buster (band)|Buster}})

* [https://www.youtube.com/shorts/lOaG5mgYMuM ''Concertina Ballerina''] (1983) – microtonal cover in 29edo by [[Bryan Deister]] (2026)

; {{W|Toby Fox}}

* [https://www.youtube.com/shorts/NYN8EBllJkE ''A Cyber's World''] via ''{{W|Deltarune}} Chapter 2'' (2021) – microtonal cover in 29edo by [[Bryan Deister]] (2023)

* [https://www.youtube.com/watch?v=JOqnRPIOb5o ''Dialtone''] via ''{{W|Deltarune}} Chapter 2'' (2021) – microtonal cover in 29edo by [[Bryan Deister]] (2024)

; {{W|Bart Howard}}

* ''Fly Me to the Moon (29-TET) microtonal cover''] (1954) – microtonal cover in 29edo by ([[Stephen Weigel]] on Lumatone/soft synthesizer and [[Clarissa]] on trumpet) (2026)

** [https://www.youtube.com/watch?v=FFrHIMrAS-E (original performance video)]

** [https://www.youtube.com/watch?v=ZWDCWPOhPAA (transcription)]

; {{W|Kikiyama}} (via {{W|Yume 2kki}})

* [https://www.youtube.com/shorts/UcjQeZot2pk ''Lotus Waters''] (2004) - microtonal cover in 29edo by [[Bryan Deister]] (2025)

; {{W|King Crimson}}

* [https://www.youtube.com/shorts/zWCmzTNddzI ''Discipline''] - microtonal cover in 29edo by [[Bryan Deister]] (2025)

; [https://hsmusic.wiki/artist/james-roach/ James Roach]

* [https://www.youtube.com/shorts/fyPaaW9AyMA ''Pipeorgankind''] (2012) – microtonal cover in 29edo by [[Bryan Deister]] (2024) (the title of the microtonal cover video also includes ''"Homestuck"'', but this appears to be an error)

=== 21st century ===

; [[Charles Loli A.]] ([http://musicool.us/musicool/armonia.htm site]{{dead link}})

* [http://www.microtonalismo.com/el-teclado-29-edo ''Mp3 29EDO - Escala tonal de 17 notas''] {{dead link}}

~~; [[Australopithecine Microtonal Music]]~~

* [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]]

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* [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/shorts/fyPaaW9AyMA ''Homestuck: Pipeorgankind (microtonal cover in 29edo)''] (2024)

* [https://www.youtube.com/shorts/SH5IQOi33Oo ''29edo groove''] (2025)

* [https://www.youtube.com/shorts/~~2NHkGHQ84Qc~~ ''~~Hide - Dorian Concept (microtonal cover in 29edo)''] (2025)~~

* [https://www.youtube.com/shorts/PuaNvxX11II ''an idea in 29edo''] (2026)

* [https://www.youtube.com/shorts/UcjQeZot2pk ''Lotus Waters - Yume 2kki (microtonal cover in 29edo)''] (~~2025~~)

; [[duckapus]]

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; [[E8 Heterotic]]

* [https://~~youtu~~.be/~~_1snAPXErOQ~~?si=~~p2Pucp9aQDW6DMZE~~ ''Glaukos Circuit''] (2019) – chiptune

* [https://www.youtube.com/watch?v=_1snAPXErOQ ''Glaukos Circuit''] (2019) – chiptune

; [[Pedro Laranjeira Finisterra]]

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* [https://www.youtube.com/watch?v=di4qn2VFYbs ''Plane Sonatina No. 1''] (2025)

* [https://www.youtube.com/watch?v=ifvvww20XAU ''Strank Running''] (2025)

; [[groundfault]]

* "The Lake Reflects a Black Sky" from ''A New Dusk'' (2024) – [https://groundfco.bandcamp.com/track/the-lake-reflects-a-black-sky-29-31-20edo Bandcamp] | [https://www.youtube.com/watch?v=1bnEO8vGvbo YouTube (0:00–2:38)] – in part, the rest being in 31edo and 20edo

; [[Igliashon Jones]]

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; [[Budjarn Lambeth]]

* [https://youtu.be/CN4cLOyaVGE ''29edo Porky15 Improvisation''] (2024)

; [[Claudi Meneghin]]

* [https://www.youtube.com/shorts/iAP4MFKyjKk ''Porcupine Canon 3-in-1 on the Lament Bass (29EDO)''] (2026)

; [[NullPointerException Music]]

* [https://www.youtube.com/watch?v=RtbY64I-vYg ''Edolian ~~- Chamber~~''] (2020)

* [https://www.youtube.com/watch?v=RtbY64I-vYg "Chamber"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[Mats Öljare]]

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; [[Chris Vaisvil]]

* [http://micro.soonlabel.com/tuning-survey/daily20111026-bridgetown-14.mp3 ''Route 14 in Bridgetown'']

; [[Randy Wells]] (Australopithecine XEN)

* [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)

; [[Xotla]]

@@ Line 9: / Line 9: @@
 == Theory ==
-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 sharp instead of flat, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]] &mdash; a [[Parapyth|Parapythagorean]] tuning instead of a meantone system.
+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 sharp instead of flat, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]] &mdash; a [[Parapyth|parapythagorean]] tuning instead of a meantone system.
 {| class="wikitable"
 | [[File:29edoSuperpythDiatonic.mp3]] [[:File:29edoSuperpythDiatonic.mp3|[File info]]]
 | [[File:12edoDiatonic.mp3]] [[:File:12edoDiatonic.mp3|[File info]]]
 |-
-| (Super-)pythagorean diatonic major scale and cadence in 29edo
+| Parapythagorean diatonic major scale and cadence in 29edo
 | 12edo diatonic major scale and cadence, for comparison
 |}
@@ Line 20: / Line 20: @@
 edo 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, like how 12 tempers out [[50/49]] but not [[49/48]]; 29 does the opposite. Each also supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively).
-A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone.
+A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone, since if 2 tones make a 5/4, (4 + 5) * 2/9 tones = 2 tones (9 steps) = 5/4 in 29edo.
 === Prime harmonics ===
@@ Line 27: / Line 27: @@
 === 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.
+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]].
+Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29edo 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 [[7:11:13|1-11/7-13/7 (7:11:13)]] chord, the [[The Archipelago|barbados]] triad [[10:13:15|1-13/10-3/2 (10:13:15)]], the minor barbados triad [[26:30:39|1-15/13-3/2 (26:30:39)]], the [[22:28:33|1-14/11-3/2 (22:28:33)]] triad, the [[22:26:33|1-13/11-3/2 (22:26:33)]] triad, and the [[petrmic triad]], a 13-limit [[Dyadic chord|essentially tempered dyadic chord]].
-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.
+tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195, 364/363, and 847/845 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 ===
+Due to 29edo's tone-efficient mapping of 2.3.7/5.11/5.13/5, it makes sense to collapse this subgroup to 29edo. One may then expand the subgroup to the full 13-limit, adding an independent generator to reach primes 5, 7, 11, and 13 in one generator. This is [[mystery]] temperament, which has very low [[badness]] despite so many periods per octave. The 58-note MOS gives scope for harmony, with 29 15-odd-limit otonal chords and 29 utonal chords.
-edo is the 10th [[prime edo]], following [[23edo]] and coming before [[31edo]].
+=== Interval Flavors ===
+edo has [[Ultramajor and inframinor|inframinor (arto)]], [[Neogothic major and minor|neogothic minor]], [[Submajor_and_supraminor|supraminor]], submajor, neogothic major, and ultramajor (tendo) thirds and sevenths. This is in contrast to systems like [[31edo]], where there are subminor, minor, neutral, major, and supermajor thirds and sevenths. This is due to 29edo representing 2.3.7/5.11/5.13/5 well, and ratios between two primes greater than 3 tend to land between interval categories of intervals in a 2.3.p subgroup. For example, 2.3.5 intervals are major/minor, 2.3.7 intervals are [[Supermajor and subminor|supermajor/subminor]], and 2.3.11 and 2.3.13 intervals are [[Neutral (interval quality)|artoneutral/tendoneutral]]. 31edo, on the other hand, represents 2.3.5.7.11 well, and thus has interval categories represented in 2.3.5, 2.3.7, and 2.3.11. It can also be seen from the fact that the 29&31 temperament, [[tritonic]], maps seconds and thirds to large numbers of generators, so they differ more in tuning between the systems.
+=== Subsets and Supersets ===
+edo is the 10th [[prime edo]], following [[23edo]] and coming before [[31edo]]. Its supersets [[58edo]] and [[87edo]] correct many of the higher primes.
 == Intervals ==
@@ Line 44: / Line 49: @@
 ! Cents
 ! Approx. Ratios of the [[13-limit]]
-! colspan="3" | [[Ups and downs notation]]
+! Chain-of-fifths notation
-([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and ^d2)
+! colspan="3" | [[Ups and downs notation]]<br>([[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)
 |-
@@ Line 51: / Line 56: @@
 | 0.000
 | [[1/1]]
+| unison
 | P1
 | unison
@@ Line 60: / Line 66: @@
 | 1
 | 41.379
-| [[25/24]], [[33/32]], [[56/55]], [[81/80]]
+| [[33/32]],  [[40/39]],  [[45/44]],<br>[[81/80]], [[64/63]]
+| negative diminished 2nd,<br>double diminished 3rd
 | ^1, vm2
 | up unison,<br />downminor 2nd
 | ^D, vEb
-| S1, sm2
+| K1, S1, sm2
-| comma-wide unison, super unison, subminor 2nd
+| comma-wide unison,<br>super unison, subminor 2nd
 | KD, SD, sEb
 |-
 | 2
 | 82.759
-| [[21/20]]
+| [[21/20]], [[22/21]], [[135/128]], [[256/243]]
+| minor 2nd
 | m2
 | minor 2nd
@@ Line 81: / Line 89: @@
 | 124.138
 | [[16/15]], [[15/14]], [[14/13]], [[13/12]]
+| augmented 1sn
 | ^m2
 | upminor 2nd
@@ Line 91: / Line 100: @@
 | 165.517
 | [[12/11]], [[11/10]], [[10/9]]
+| diminished 3rd
 | vM2
 | downmajor 2nd
@@ Line 101: / Line 111: @@
 | 206.897
 | [[9/8]]
+| major 2nd
 | M2
 | major 2nd
@@ Line 111: / Line 122: @@
 | 248.276
 | [[8/7]], [[7/6]], [[15/13]]
+| double diminished 4th,<br>double augmented 1sn
 | ^M2, vm3
-| upmajor 2nd,<br />downminor 3rd
+| upmajor 2nd,<br>downminor 3rd
 | ^E, vF
 | SM2, sm3
@@ Line 118: / Line 130: @@
 | SE, sF
 |-
-| ·7
+| 7
 | 289.655
-| [[13/11]]
+| [[13/11]], [[32/27]]
+| minor 3rd
 | m3
 | minor 3rd
@@ Line 131: / Line 144: @@
 | 331.034
 | [[6/5]], [[11/9]]
+| augmented 2nd
 | ^m3
 | upminor 3rd
@@ Line 141: / Line 155: @@
 | 372.414
 | [[5/4]], [[16/13]]
+| diminished 4th
 | vM3
 | downmajor 3rd
@@ Line 150: / Line 165: @@
 | 10
 | 413.793
-| [[14/11]]
+| [[14/11]], [[81/64]]
+| major 3rd
 | M3
 | major 3rd
@@ Line 161: / Line 177: @@
 | 455.172
 | [[9/7]], [[13/10]]
+| double diminished 5th,<br>double augmented 2nd
 | ^M3, v4
 | upmajor 3rd<br />down 4th
@@ Line 168: / Line 185: @@
 | SF#, sG
 |-
-| ·12
+| 12
 | 496.552
 | [[4/3]]
+| perfect 4th
 | P4
 | 4th
@@ Line 181: / Line 199: @@
 | 537.931
 | [[11/8]], [[15/11]]
+| augmented 3rd
 | ^4
 | up 4th
@@ Line 191: / Line 210: @@
 | 579.310
 | [[7/5]], [[18/13]]
+| diminished 5th
 | vA4, d5
 | downaug 4th,<br />dim 5th
@@ Line 201: / Line 221: @@
 | 620.690
 | [[10/7]], [[13/9]]
+| augmented 4th
 | A4, ^d5
 | aug 4th,<br />updim 5th
@@ Line 211: / Line 232: @@
 | 662.069
 | [[16/11]], [[22/15]]
+| diminished 6th
 | v5
 | down 5th
@@ Line 218: / Line 240: @@
 | kA
 |-
-| ·17
+| 17
 | 703.448
 | [[3/2]]
+| perfect 5th
 | P5
 | 5th
@@ Line 231: / Line 254: @@
 | 744.828
 | [[14/9]], [[20/13]]
+| double augmented 4th,<br>double diminished 7th
 | ^5, vm6
 | up 5th,<br />downminor 6th
@@ Line 240: / Line 264: @@
 | 19
 | 786.207
-| [[11/7]]
+| [[11/7]], [[128/81]]
+| minor 6th
 | m6
 | minor 6th
@@ Line 251: / Line 276: @@
 | 827.586
 | [[8/5]], [[13/8]]
+| augmented 5th
 | ^m6
 | upminor 6th
@@ Line 261: / Line 287: @@
 | 868.966
 | [[5/3]], [[18/11]]
+| diminished 7th
 | vM6
 | downmajor 6th
@@ Line 268: / Line 295: @@
 | kB
 |-
-| ·22
+| 22
 | 910.345
-| [[22/13]]
+| [[22/13]], [[27/16]]
+| major 6th
 | M6
 | major 6th
@@ Line 281: / Line 309: @@
 | 951.724
 | [[7/4]], [[12/7]], [[26/15]]
+| double augmented 5th,<br>double diminished 8ve
 | ^M6, vm7
 | upmajor 6th,<br />downminor 7th
@@ Line 291: / Line 320: @@
 | 993.103
 | [[16/9]]
+| minor 7th
 | m7
 | minor 7th
@@ Line 301: / Line 331: @@
 | 1034.483
 | [[11/6]], [[20/11]], [[9/5]]
+| augmented 6th
 | ^m7
 | upminor 7th
@@ Line 311: / Line 342: @@
 | 1075.862
 | [[15/8]], [[28/15]], [[13/7]], [[24/13]]
+| diminished 8ve
 | vM7
 | downmajor 7th
@@ Line 320: / Line 352: @@
 | 27
 | 1117.241
-| [[40/21]]
+| [[40/21]], [[21/11]], [[256/135]], [[243/128]]
+| major 7th
 | M7
 | major 7th
@@ Line 330: / Line 363: @@
 | 28
 | 1158.621
-| [[48/25]], [[64/33]], [[55/28]], [[160/81]]
+| [[64/33]], [[39/20]], [[88/45]],<br>[[160/81]], [[63/32]]
+| diminished 9th,<br>double augmented 6th
 | ^M7, v8
-| upmajor 7th,<br />down 8ve
+| upmajor 7th,<br>down 8ve
 | ^C#, vD
 | SM7, s8
-| supermajor 7th, comma-narrow 8ve, sub 8ve
+| supermajor 7th,<br>comma-narrow 8ve, sub 8ve
 | SC#, kD, sD
 |-
@@ Line 341: / Line 375: @@
 | 1200.000
 | [[2/1]]
+| octave
 | P8
 | 8ve
@@ Line 443: / Line 478: @@
 edo can be notated three different ways. Using only sharps and flats, the chromatic scale from C is:
-{{dash|C, B♯, D♭, C♯, B𝄪/E𝄫, D, C𝄪/F𝄫, E♭, D♯, F♭, E, D𝄪/G𝄫, F, E♯, G♭, F♯, E𝄪/A𝄫, G, F𝄪, A♭, G♯, B𝄫, A, G𝄪/C𝄫, B♭, A♯, C♭, B, A𝄪/D𝄫, C|s=hair|d=long}}
+{{dash|C, B♯, D♭, C♯, B𝄪/E𝄫, D, C𝄪/F𝄫, E♭, D♯, F♭, E, D𝄪/G𝄫, F, E♯, G♭, F♯, E𝄪/A𝄫, G, F𝄪, A♭, G♯, B𝄫, A, G𝄪/C𝄫, B♭, A♯, C♭, B, A𝄪/D𝄫, C|s=hair}}
 Here, six pairs of enharmonic equivalents exist:
@@ Line 453: / Line 488: @@
 * C𝄪 = F𝄫
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
-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 (^^).
+Since a sharp raises by three steps, 29edo is a good candidate for [[Stein–Zimmermann–Gould notation]], using sharps and flats with arrows similar to [[22edo]]:
-{{Ups and downs sharpness}}
+{{Sharpness-sharp3-szg}}
-Here, sharps and flats with arrows from [[Helmholtz–Ellis notation]] are used:
-{{Sharpness-sharp3}}
 Note that C♯ is enharmonic to D{{flatup}}, and D♭ is enharmonic to C{{sharpdown}}.
 If arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best spelled with double arrows.
+=== Kite's ups and downs notation ===
+edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, downsharp, sharp, etc. Note that downsharp (v#) can be respelled as dup (^^).
+{{Ups and downs sharpness}}
 === Sagittal notation ===
-This notation uses the same sagittal sequence as EDOs [[15edo#Sagittal notation|15]] and [[22edo#Sagittal notation|22]].
+This notation uses the same sagittal sequence as edos [[15edo #Sagittal notation|15]] and [[22edo #Sagittal notation|22]].
 ==== Evo flavor ====
@@ Line 969: / Line 1,005: @@
 * [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)
+* [https://www.youtube.com/watch?v=VUX9yZiBM6g ''BACH, NEVERENDING CANON, but it has the SHEPARD EFFECT and is tuned into 29edo''] (1742-1749) - rendered by Claudi Meneghin (2025)
 ; {{W|Nicolaus Bruhns}}
 * [https://www.youtube.com/watch?v=me7dHmo3cVs ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
 * [https://www.youtube.com/watch?v=-E-2mszlgWM ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
+; {{W|Kate Bush}}
+* [https://www.youtube.com/shorts/QIfqj-8Ojhc ''Army Dreamers'' <nowiki>[short clip]</nowiki>] (1980) - microtonal cover in 29edo by [[Bryan Deister]] (2025)
+; {{W|C418}}
+* [https://www.youtube.com/shorts/WEu7NzK7u0I ''Cat''] (2011) - microtonal cover in 29edo by [[Bryan Deister]] (2026)
+; {{W|Dorian Concept}}
+* [https://www.youtube.com/shorts/2NHkGHQ84Qc ''Hide''] (2023/2024) – microtonal cover in 29edo by [[Bryan Deister]] (2025)
+; Alan Fennah as "Alternative Radio" (see {{W|Buster (band)|Buster}})
+* [https://www.youtube.com/shorts/lOaG5mgYMuM ''Concertina Ballerina''] (1983) – microtonal cover in 29edo by [[Bryan Deister]] (2026)
+; {{W|Toby Fox}}
+* [https://www.youtube.com/shorts/NYN8EBllJkE ''A Cyber's World''] via ''{{W|Deltarune}} Chapter 2'' (2021) – microtonal cover in 29edo by [[Bryan Deister]] (2023)
+* [https://www.youtube.com/watch?v=JOqnRPIOb5o ''Dialtone''] via ''{{W|Deltarune}} Chapter 2'' (2021) – microtonal cover in 29edo by [[Bryan Deister]] (2024)
+; {{W|Bart Howard}}
+* ''Fly Me to the Moon (29-TET) microtonal cover''] (1954) – microtonal cover in 29edo by ([[Stephen Weigel]] on Lumatone/soft synthesizer and [[Clarissa]] on trumpet) (2026)
+** [https://www.youtube.com/watch?v=FFrHIMrAS-E (original performance video)]
+** [https://www.youtube.com/watch?v=ZWDCWPOhPAA (transcription)]
+; {{W|Kikiyama}} (via {{W|Yume 2kki}})
+* [https://www.youtube.com/shorts/UcjQeZot2pk ''Lotus Waters''] (2004) - microtonal cover in 29edo by [[Bryan Deister]] (2025)
+; {{W|King Crimson}}
+* [https://www.youtube.com/shorts/zWCmzTNddzI ''Discipline''] - microtonal cover in 29edo by [[Bryan Deister]] (2025)
+; [https://hsmusic.wiki/artist/james-roach/ James Roach]
+* [https://www.youtube.com/shorts/fyPaaW9AyMA ''Pipeorgankind''] (2012) – microtonal cover in 29edo by [[Bryan Deister]] (2024) (the title of the microtonal cover video also includes ''"Homestuck"'', but this appears to be an error)
 === 21st century ===
 ; [[Charles Loli A.]] ([http://musicool.us/musicool/armonia.htm site]{{dead link}})
 * [http://www.microtonalismo.com/el-teclado-29-edo ''Mp3 29EDO - Escala tonal de 17 notas''] {{dead link}}
-; [[Australopithecine Microtonal Music]]
-* [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]]
@@ Line 988: / Line 1,051: @@
 * [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/shorts/fyPaaW9AyMA ''Homestuck: Pipeorgankind (microtonal cover in 29edo)''] (2024)
 * [https://www.youtube.com/shorts/SH5IQOi33Oo ''29edo groove''] (2025)
-* [https://www.youtube.com/shorts/2NHkGHQ84Qc ''Hide - Dorian Concept (microtonal cover in 29edo)''] (2025)
+* [https://www.youtube.com/shorts/PuaNvxX11II ''an idea in 29edo''] (2026)
-* [https://www.youtube.com/shorts/UcjQeZot2pk ''Lotus Waters - Yume 2kki (microtonal cover in 29edo)''] (2025)
 ; [[duckapus]]
@@ Line 997: / Line 1,058: @@
 ; [[E8 Heterotic]]
-* [https://youtu.be/_1snAPXErOQ?si=p2Pucp9aQDW6DMZE ''Glaukos Circuit''] (2019) – chiptune
+* [https://www.youtube.com/watch?v=_1snAPXErOQ ''Glaukos Circuit''] (2019) – chiptune
 ; [[Pedro Laranjeira Finisterra]]
@@ Line 1,009: / Line 1,070: @@
 * [https://www.youtube.com/watch?v=di4qn2VFYbs ''Plane Sonatina No. 1''] (2025)
 * [https://www.youtube.com/watch?v=ifvvww20XAU ''Strank Running''] (2025)
+; [[groundfault]]
+* "The Lake Reflects a Black Sky" from ''A New Dusk'' (2024) – [https://groundfco.bandcamp.com/track/the-lake-reflects-a-black-sky-29-31-20edo Bandcamp] | [https://www.youtube.com/watch?v=1bnEO8vGvbo YouTube (0:00–2:38)] – in part, the rest being in 31edo and 20edo
 ; [[Igliashon Jones]]
@@ Line 1,017: / Line 1,081: @@
 ; [[Budjarn Lambeth]]
 * [https://youtu.be/CN4cLOyaVGE ''29edo Porky15 Improvisation''] (2024)
+; [[Claudi Meneghin]]
+* [https://www.youtube.com/shorts/iAP4MFKyjKk ''Porcupine Canon 3-in-1 on the Lament Bass (29EDO)''] (2026)
 ; [[NullPointerException Music]]
-* [https://www.youtube.com/watch?v=RtbY64I-vYg ''Edolian - Chamber''] (2020)
+* [https://www.youtube.com/watch?v=RtbY64I-vYg "Chamber"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)
 ; [[Mats Öljare]]
@@ Line 1,032: / Line 1,099: @@
 ; [[Chris Vaisvil]]
 * [http://micro.soonlabel.com/tuning-survey/daily20111026-bridgetown-14.mp3 ''Route 14 in Bridgetown'']
+; [[Randy Wells]] (Australopithecine XEN)
+* [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)
 ; [[Xotla]]