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

|}

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, since if 2 tones make a 5/4, (4 + 5) * 2/9 tones = 2 tones (9 steps) = 5/4 in 29edo.

=== 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.

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.

=== 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.

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]].

~~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)~~.

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.

~~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~~.

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.

=== ~~Prime harmonics~~ ===

=== Interval Flavors ===

~~{{Harmonics~~ in ~~equal~~|29|~~columns=~~11}}

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.

=== ~~Divisors~~ ===

=== Subsets and Supersets ===

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

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

== Intervals ==

Line 42:

Line 49:

! Cents

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

! colspan="3" | [[Ups and ~~Downs Notation~~]]

! Chain-of-fifths notation

! 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)

|-

Line 48:

Line 56:

| 0.000

| [[1/1]]

| unison

| P1

| unison

Line 57:

Line 66:

| 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

Line 78:

Line 89:

| 124.138

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

| augmented 1sn

| ^m2

| upminor 2nd

Line 88:

Line 100:

| 165.517

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

| diminished 3rd

| vM2

| downmajor 2nd

Line 98:

Line 111:

| 206.897

| [[9/8]]

| major 2nd

| M2

| major 2nd

Line 108:

Line 122:

| 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

Line 115:

Line 130:

| SE, sF

|-

| ·7

| 7

| 289.655

| [[13/11]]

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

| minor 3rd

| m3

| minor 3rd

Line 128:

Line 144:

| 331.034

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

| augmented 2nd

| ^m3

| upminor 3rd

Line 138:

Line 155:

| 372.414

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

| diminished 4th

| vM3

| downmajor 3rd

Line 147:

Line 165:

| 10

| 413.793

| [[14/11]]

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

| major 3rd

| M3

| major 3rd

Line 158:

Line 177:

| 455.172

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

| double diminished 5th, double augmented 2nd

| ^M3, v4

| upmajor 3rd down 4th

Line 165:

Line 185:

| SF#, sG

|-

| ~~·12~~

| 12

| 496.552

| [[4/3]]

| perfect 4th

| P4

| 4th

Line 178:

Line 199:

| 537.931

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

| augmented 3rd

| ^4

| up 4th

Line 188:

Line 210:

| 579.310

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

| diminished 5th

| vA4, d5

| downaug 4th, dim 5th

Line 198:

Line 221:

| 620.690

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

| augmented 4th

| A4, ^d5

| aug 4th, updim 5th

Line 208:

Line 232:

| 662.069

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

| diminished 6th

| v5

| down 5th

Line 215:

Line 240:

| kA

|-

| ~~·17~~

| 17

| 703.448

| [[3/2]]

| perfect 5th

| P5

| 5th

Line 228:

Line 254:

| 744.828

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

| double augmented 4th, double diminished 7th

| ^5, vm6

| up 5th, downminor 6th

Line 237:

Line 264:

| 19

| 786.207

| [[11/7]]

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

| minor 6th

| m6

| minor 6th

Line 248:

Line 276:

| 827.586

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

| augmented 5th

| ^m6

| upminor 6th

Line 258:

Line 287:

| 868.966

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

| diminished 7th

| vM6

| downmajor 6th

Line 265:

Line 295:

| kB

|-

| ~~·22~~

| 22

| 910.345

| [[22/13]]

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

| major 6th

| M6

| major 6th

Line 278:

Line 309:

| 951.724

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

| double augmented 5th, double diminished 8ve

| ^M6, vm7

| upmajor 6th, downminor 7th

Line 288:

Line 320:

| 993.103

| [[16/9]]

| minor 7th

| m7

| minor 7th

Line 298:

Line 331:

| 1034.483

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

| augmented 6th

| ^m7

| upminor 7th

Line 308:

Line 342:

| 1075.862

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

| diminished 8ve

| vM7

| downmajor 7th

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Line 352:

| 27

| 1117.241

| [[40/21]]

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

| major 7th

| M7

| major 7th

Line 327:

Line 363:

| 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

|-

Line 338:

Line 375:

| 1200.000

| [[2/1]]

| octave

| P8

| 8ve

Line 357:

Line 395:

| downminor

| zo

| {a, b, 0, 1}

| [a, b, 0, 1>

| 7/6, 7/4

|-

| minor

| fourthward wa

| {a, b}, b < -1

| [a, b>, b < -1

| 32/27, 16/9

|-

| upminor

| gu

| {a, b, -1}

| [a, b, -1>

| 6/5, 9/5

|-

| "

| ilo

| {a, b, 0, 0, 1}

| [a, b, 0, 0, 1>

| 11/9, 11/6

|-

| downmajor

| lu

| {a, b, 0, 0, -1}

| [a, b, 0, 0, -1>

| 12/11, 18/11

|-

| "

| yo

| {a, b, 1}

| [a, b, 1>

| 5/4, 5/3

|-

| major

| fifthward wa

| {a, b}, b > 1

| [a, b>, b > 1

| 9/8, 27/16

|-

| upmajor

| ru

| {a, b, 0, -1}

| [a, b, 0, -1>

| 9/7, 12/7

|}

Line 434:

Line 472:

| 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 ==

=== Standard notation===

=== Standard notation ===

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

{{dash|C, B♯, D♭, C♯, E𝄫, D, C𝄪, E♭, D♯, F♭, E, G𝄫, F, E♯, G♭, F♯, A𝄫, G, F𝄪, A♭, G♯, B𝄫, A, G𝄪, B♭, A♯, C♭, B, A𝄪, 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 450:

Line 488:

* C𝄪 = F𝄫

=== ~~Sagittal~~ notation ===

=== Stein–Zimmermann–Gould 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 [[Stein–Zimmermann–Gould notation]], using sharps and flats with arrows similar to [[22edo]]:

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

~~{| class="wikitable"~~

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

|-

~~| [[File:29edothumb.png|alt=29edothumb.png|29edothumb.png]]~~

|-

~~| This example~~ in ~~Sagittal notation shows 29-edo as a fifth-tone system~~.

|}

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

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

~~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:~~

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 (^^).

~~{{Sharpness-sharp3}}~~

=== Sagittal notation ===

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

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

==== Evo flavor ====

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>

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

==== Revo flavor ====

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 ==

=== Interval mappings ===

Line 489:

Line 544:

| {{monzo| 46 -29 }}

| {{mapping| 29 46 }}

| ~~−0~~.47

| −0.47

| 0.47

| 1.14

Line 495:

Line 550:

| 2.3.5

| 250/243, 16875/16384

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

| {{mapping| 29 46 67 }}

| +1.68

| 3.07

Line 539:

Line 594:

=== 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"

|-

! [[Harmonic limit|Prime limit]]

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

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

! [[Cents]]

Line 739:

Line 794:

| Minor minthma

|}

~~<references />~~

=== Rank-2 temperaments ===

Line 856:

Line 910:

Nicetone scale 5435453 and cadence in 29edo

== Octave stretch or compression ==

29edo's [[prime]]s 5, 7, 11 and 13 are all tuned flat and the 3 has relatively little error, so 29edo can benefit from [[octave stretching]]. Some stretched-octave 29edo tunings include [[116zpi]] or [[equal tuning|96ed10]].

== Scales ==

Line 863:

Line 920:

* [[Leapfrog]] diatonic [[5L 2s]] 5552552 (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]] [[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]] [[9L 1s]] 3333333332 (3\29, 1\1)

* [[Negri]] [[10L 9s]] 2212121212121212121 (3\29 1\1)

* [[Semaphore]] [[4L 1s]] 56666 (6\29, 1\1)

* [[Semaphore]] [[5L 4s]] 551515151 (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]] [[14L 1s]] 222222222222221 (2\29, 1\1)

Line 946:

Line 997:

: Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

</div>

* [[Lumatone mapping for 29edo]]

== Music ==

=== Modern renderings ===

; {{W|Johann Sebastian Bach}}

* [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)

* [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}}

* [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 971:

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/SH5IQOi33Oo ''29edo groove''] (2025)

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

; [[duckapus]]

Line 976:

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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]]

Line 984:

Line 1,066:

* "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)

** "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)

; [[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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Line 1,080:

; [[Budjarn Lambeth]]

* [https://youtu.be/CN4cLOyaVGE ''29edo Porky15 Improvisation No. 1''] (~~2024~~)

* [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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Line 1,095:

; [[Ray Perlner]]

* [https://youtu.be/zvpk7Dnzp_Y ''29 EDO Fugue in Negri 9 Lssssssss "Austro-Hungarian Minor"'']

* [https://www.youtube.com/watch?v=zQFezpL_06o Fugue for 29EDO Piano in Porcupine 7 ssLssss "Zebrian"'']

* [https://www.youtube.com/watch?v=zQFezpL_06o ''Fugue for 29EDO Piano in Porcupine 7 ssLssss "Zebrian"'']

; [[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]]

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Line 1,109:

== 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 ==

[[Category:IMPORTDEBUG - Change External Images]]

[[Category:3-limit record edos|##]]

[[Category:3-limit]]

[[Category:Subgroup temperaments]]

[[Category:~~Listen~~]]

[[Category:Twentuning]]

[[Category:Negri]]

[[Category:Petrmic]]

[[Category:Porcupine]]

[[Category:~~Subgroup temperaments]]~~

[[Category:Listen]]

~~[[Category:Twentuning~~]]

@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|29}}
+{{ED intro}}
 == 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
 |}
+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, since if 2 tones make a 5/4, (4 + 5) * 2/9 tones = 2 tones (9 steps) = 5/4 in 29edo.
+=== 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}}
-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.
+=== 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.
+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]].
-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 (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).
+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.
-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.
+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.
-=== Prime harmonics ===
+=== Interval Flavors ===
-{{Harmonics in equal|29|columns=11}}
+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.
-=== Divisors ===
+=== Subsets and Supersets ===
-edo is the 10th [[prime edo]], following [[23edo]] and coming before [[31edo]].
+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 42: / Line 49: @@
 ! Cents
 ! Approx. Ratios of the [[13-limit]]
-! colspan="3" | [[Ups and Downs Notation]]
+! Chain-of-fifths notation
+! 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 48: / Line 56: @@
 | 0.000
 | [[1/1]]
+| unison
 | P1
 | unison
@@ Line 57: / 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 78: / Line 89: @@
 | 124.138
 | [[16/15]], [[15/14]], [[14/13]], [[13/12]]
+| augmented 1sn
 | ^m2
 | upminor 2nd
@@ Line 88: / Line 100: @@
 | 165.517
 | [[12/11]], [[11/10]], [[10/9]]
+| diminished 3rd
 | vM2
 | downmajor 2nd
@@ Line 98: / Line 111: @@
 | 206.897
 | [[9/8]]
+| major 2nd
 | M2
 | major 2nd
@@ Line 108: / 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 115: / Line 130: @@
 | SE, sF
 |-
-| ·7
+| 7
 | 289.655
-| [[13/11]]
+| [[13/11]], [[32/27]]
+| minor 3rd
 | m3
 | minor 3rd
@@ Line 128: / Line 144: @@
 | 331.034
 | [[6/5]], [[11/9]]
+| augmented 2nd
 | ^m3
 | upminor 3rd
@@ Line 138: / Line 155: @@
 | 372.414
 | [[5/4]], [[16/13]]
+| diminished 4th
 | vM3
 | downmajor 3rd
@@ Line 147: / Line 165: @@
 | 10
 | 413.793
-| [[14/11]]
+| [[14/11]], [[81/64]]
+| major 3rd
 | M3
 | major 3rd
@@ Line 158: / Line 177: @@
 | 455.172
 | [[9/7]], [[13/10]]
+| double diminished 5th,<br>double augmented 2nd
 | ^M3, v4
 | upmajor 3rd<br />down 4th
@@ Line 165: / Line 185: @@
 | SF#, sG
 |-
-| ·12
+| 12
 | 496.552
 | [[4/3]]
+| perfect 4th
 | P4
 | 4th
@@ Line 178: / Line 199: @@
 | 537.931
 | [[11/8]], [[15/11]]
+| augmented 3rd
 | ^4
 | up 4th
@@ Line 188: / Line 210: @@
 | 579.310
 | [[7/5]], [[18/13]]
+| diminished 5th
 | vA4, d5
 | downaug 4th,<br />dim 5th
@@ Line 198: / Line 221: @@
 | 620.690
 | [[10/7]], [[13/9]]
+| augmented 4th
 | A4, ^d5
 | aug 4th,<br />updim 5th
@@ Line 208: / Line 232: @@
 | 662.069
 | [[16/11]], [[22/15]]
+| diminished 6th
 | v5
 | down 5th
@@ Line 215: / Line 240: @@
 | kA
 |-
-| ·17
+| 17
 | 703.448
 | [[3/2]]
+| perfect 5th
 | P5
 | 5th
@@ Line 228: / Line 254: @@
 | 744.828
 | [[14/9]], [[20/13]]
+| double augmented 4th,<br>double diminished 7th
 | ^5, vm6
 | up 5th,<br />downminor 6th
@@ Line 237: / Line 264: @@
 | 19
 | 786.207
-| [[11/7]]
+| [[11/7]], [[128/81]]
+| minor 6th
 | m6
 | minor 6th
@@ Line 248: / Line 276: @@
 | 827.586
 | [[8/5]], [[13/8]]
+| augmented 5th
 | ^m6
 | upminor 6th
@@ Line 258: / Line 287: @@
 | 868.966
 | [[5/3]], [[18/11]]
+| diminished 7th
 | vM6
 | downmajor 6th
@@ Line 265: / Line 295: @@
 | kB
 |-
-| ·22
+| 22
 | 910.345
-| [[22/13]]
+| [[22/13]], [[27/16]]
+| major 6th
 | M6
 | major 6th
@@ Line 278: / 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 288: / Line 320: @@
 | 993.103
 | [[16/9]]
+| minor 7th
 | m7
 | minor 7th
@@ Line 298: / Line 331: @@
 | 1034.483
 | [[11/6]], [[20/11]], [[9/5]]
+| augmented 6th
 | ^m7
 | upminor 7th
@@ Line 308: / Line 342: @@
 | 1075.862
 | [[15/8]], [[28/15]], [[13/7]], [[24/13]]
+| diminished 8ve
 | vM7
 | downmajor 7th
@@ Line 317: / Line 352: @@
 | 27
 | 1117.241
-| [[40/21]]
+| [[40/21]], [[21/11]], [[256/135]], [[243/128]]
+| major 7th
 | M7
 | major 7th
@@ Line 327: / 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 338: / Line 375: @@
 | 1200.000
 | [[2/1]]
+| octave
 | P8
 | 8ve
@@ Line 357: / Line 395: @@
 | downminor
 | zo
-| {a, b, 0, 1}
+| [a, b, 0, 1>
 | 7/6, 7/4
 |-
 | minor
 | fourthward wa
-| {a, b}, b &lt; -1
+| [a, b>, b &lt; -1
 | 32/27, 16/9
 |-
 | upminor
 | gu
-| {a, b, -1}
+| [a, b, -1>
 | 6/5, 9/5
 |-
 | "
 | ilo
-| {a, b, 0, 0, 1}
+| [a, b, 0, 0, 1>
 | 11/9, 11/6
 |-
 | downmajor
 | lu
-| {a, b, 0, 0, -1}
+| [a, b, 0, 0, -1>
 | 12/11, 18/11
 |-
 | "
 | yo
-| {a, b, 1}
+| [a, b, 1>
 | 5/4, 5/3
 |-
 | major
 | fifthward wa
-| {a, b}, b &gt; 1
+| [a, b>, b &gt; 1
 | 9/8, 27/16
 |-
 | upmajor
 | ru
-| {a, b, 0, -1}
+| [a, b, 0, -1>
 | 9/7, 12/7
 |}
@@ Line 434: / Line 472: @@
 | 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 ==
-=== Standard notation===
+=== Standard notation ===
 edo can be notated three different ways. Using only sharps and flats, the chromatic scale from C is:
-{{dash|C, B♯, D♭, C♯, E𝄫, D, C𝄪, E♭, D♯, F♭, E, G𝄫, F, E♯, G♭, F♯, A𝄫, G, F𝄪, A♭, G♯, B𝄫, A, G𝄪, B♭, A♯, C♭, B, A𝄪, 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 450: / Line 488: @@
 * C𝄪 = F𝄫
-=== Sagittal notation ===
+=== Stein–Zimmermann–Gould 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 [[Stein–Zimmermann–Gould notation]], using sharps and flats with arrows similar to [[22edo]]:
+{{Sharpness-sharp3-szg}}
+Note that C♯ is enharmonic to D{{flatup}}, and D♭ is enharmonic to C{{sharpdown}}.
-{| class="wikitable"
+If arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best spelled with double arrows.
-|-
-| [[File:29edothumb.png|alt=29edothumb.png|29edothumb.png]]
-|-
-| This example in Sagittal notation shows 29-edo as a fifth-tone system.
-|}
-=== Ups and downs ===
+=== Kite's ups and downs notation ===
-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:
+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}}
-{{Sharpness-sharp3}}
+=== Sagittal notation ===
+This notation uses the same sagittal sequence as edos [[15edo #Sagittal notation|15]] and [[22edo #Sagittal notation|22]].
-Note that C♯ is enharmonic to D{{flatup}}, and D♭ is enharmonic to C{{sharpdown}}.
+==== 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>
-If arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best spelled with double arrows.
+==== 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 ==
 [[File:29ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|15-odd-limit intervals approximated in 29edo]]
 === Interval mappings ===
 {{Q-odd-limit intervals|29}}
@@ Line 489: / Line 544: @@
 | {{monzo| 46 -29 }}
 | {{mapping| 29 46 }}
-| &minus;0.47
+| −0.47
 | 0.47
 | 1.14
@@ Line 495: / Line 550: @@
 | 2.3.5
 | 250/243, 16875/16384
-| [{{val| 29 46 67 }}]
+| {{mapping| 29 46 67 }}
 | +1.68
 | 3.07
@@ Line 539: / Line 594: @@
 === Commas ===
-edo [[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.
+et [[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"
 |-
 ! [[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]]
 ! [[Cents]]
@@ Line 739: / Line 794: @@
 | Minor minthma
 |}
-<references />
 === Rank-2 temperaments ===
@@ Line 856: / Line 910: @@
 Nicetone scale 5435453 and cadence in 29edo
+== Octave stretch or compression ==
+edo's [[prime]]s 5, 7, 11 and 13 are all tuned flat and the 3 has relatively little error, so 29edo can benefit from [[octave stretching]]. Some stretched-octave 29edo tunings include [[116zpi]] or [[equal tuning|96ed10]].
 == Scales ==
@@ Line 863: / Line 920: @@
 * [[Leapfrog]] diatonic [[5L 2s]] 5552552 (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]] [[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]] [[9L 1s]] 3333333332 (3\29, 1\1)
-* [[Negri]] [[10L 9s]] 2212121212121212121 (3\29 1\1)
 * [[Semaphore]] [[4L 1s]] 56666 (6\29, 1\1)
 * [[Semaphore]] [[5L 4s]] 551515151 (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]] [[14L 1s]] 222222222222221 (2\29, 1\1)
@@ Line 946: / Line 997: @@
 : <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
 </div>
+* [[Lumatone mapping for 29edo]]
 == Music ==
 === Modern renderings ===
 ; {{W|Johann Sebastian Bach}}
-* [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)
+* [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}}
+* [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 971: / 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/SH5IQOi33Oo ''29edo groove''] (2025)
+* [https://www.youtube.com/shorts/PuaNvxX11II ''an idea in 29edo''] (2026)
 ; [[duckapus]]
@@ Line 976: / 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 984: / Line 1,066: @@
 * "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)
-** "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)
+; [[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 993: / Line 1,080: @@
 ; [[Budjarn Lambeth]]
-* [https://youtu.be/CN4cLOyaVGE ''29edo Porky15 Improvisation No. 1''] (2024)
+* [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 1,005: / Line 1,095: @@
 ; [[Ray Perlner]]
 * [https://youtu.be/zvpk7Dnzp_Y ''29 EDO Fugue in Negri 9 Lssssssss "Austro-Hungarian Minor"'']
-* [https://www.youtube.com/watch?v=zQFezpL_06o Fugue for 29EDO Piano in Porcupine 7 ssLssss "Zebrian"'']
+* [https://www.youtube.com/watch?v=zQFezpL_06o ''Fugue for 29EDO Piano in Porcupine 7 ssLssss "Zebrian"'']
 ; [[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 1,015: / Line 1,109: @@
 == 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:3-limit record edos|##]] <!-- 2-digit number -->
-[[Category:3-limit]]
+[[Category:Subgroup temperaments]]
-[[Category:Listen]]
+[[Category:Twentuning]]
 [[Category:Negri]]
 [[Category:Petrmic]]
 [[Category:Porcupine]]
-[[Category:Subgroup temperaments]]
+[[Category:Listen]]
-[[Category:Twentuning]]