29edo: Difference between revisions

{{EDO intro|29}}

29 is the lowest edo which approximates the [[3/2]] just fifth more accurately than [[12edo]]: 3/2 = 701.955… cents; 17 degrees of 29edo = 703.448… cents. Since the fifth is slightly sharp, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]] – a Parapythagorean instead of a Meantone system.

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 [[Chromatic pairs #Edson|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it.

 

Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[petrmic triad]], a 13-limit [[Dyadic chord|essentially tempered dyadic chord]]. 29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.

 

29edo could be thought of as 12edo's "twin", since the 5-limit error for both is almost exactly the same, but in the opposite direction. 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).

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

| up unison,<br>downminor 2nd

| upmajor 2nd,<br>downminor 3rd

| upmajor 3rd<br>down 4th

| downaug 4th,<br>dim 5th

| aug 4th,<br>updim 5th

| up 5th,<br>downminor 6th

| upmajor 6th,<br>downminor 7th

| upmajor 7th,<br>down 8ve

| {a, b, 0, 1}

| {a, b}, b &lt; -1

| {a, b, -1}

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

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

| {a, b, 1}

| {a, b}, b &gt; 1

| {a, b, 0, -1}

For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]].

{| class="wikitable"

|-

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

== JI approximation ==

=== 15-odd-limit interval mappings ===

{{15-odd-limit|29}}

! rowspan="2" | Optimal<br>8ve Stretch (¢)

! colspan="2" | Tuning Error

| [{{val| 29 46 }}]

| -0.47

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

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

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

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

|2.3.5.7.11.13.19

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

|[⟨29 46 67 81 100 107 123]]

|2.55

|6.16

|2.3.5.7.11.13.19.23

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

|[⟨29 46 67 81 100 107 123 131]]

|2.42

|5.85

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

 

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

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.  

! [[Harmonic limit|Prime<br>Limit]]

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

| [[29-comma|(28 digits)]]

| 43.305

| 29-comma, mystery comma

| 51.120

| 49.166

| 1.9537

| 43.408

| 35.697

| Slendro diesis

| 27.985

| 22.227

| Squalentine

| 21.181

| Gariboh

| 14.516

| Trimyna

| 13.469

| Octagar

| 7.7115

| Septimal kleisma, marvel comma

| 5.7578

| Hemifamity

| [[33554432/33480783|(16 digits)]]

| 3.8041

| 17.399

| 14.367

| 9.6880

| Pentacircle

| 3.9302

| 3.0323

| Wizardharry

| 19.130

| Superleap

|+ Table of rank-2 temperaments by generator

! Periods<br> per 8ve

! Generator<br>(Reduced)

! Cents<br>(Reduced)

! Associated Ratio<br>(Reduced)

 

* [[leapfrog]] diatonic [[5L 2s]] 5552552 (17\29, 1\1)

* [[nautilus]] [[14L 1s]] 222222222222221 (2\29, 1\1)

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

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

Fourteen-note MOSes are worth looking at because taking every other note of them gives a heptatonic, and in many cases diatonic-like, scale. Nautilus[14] is no exception; although the resulting porcupine "diatonic" scale sounds somewhat different from diatonic scales generated from fifths, it can still provide some degree of familiarity. Furthermore, every diatonic chord progression will have at least one loose analogue in nautilus[14], although the chord types might change (for instance, it is possible to have a I-IV-V chord progression where the I is major-odd, and the IV and V are both major-even; the V in this case being on a narrow or "odd" fifth rather than a perfect or "even" fifth).

The fact that the generator size is also a step size means that nautilus makes a good candidate for a [https://en.wikipedia.org/wiki/Generalized_keyboard generalized keyboard]; the fingering of nautilus[14] becomes very simple as a result, perhaps even simpler than with traditional keyboards, despite there being more notes.

Such a scale could be called "[[nicetone]]" as a play on meantone. Since it preserves most of the same 5-limit relationships, nicetone is only slightly xenharmonic (in contrast to [[Superpyth|superpyth]], which is quite blatantly so). The fact that 29edo's superfourth is within a cent of 15:11, and its 13:11 is within half a cent of a just 13:11, are both happy accidents. One just has to make sure that one is using a timbre that allows these higher-limit harmonic relationships to sound apparent and consonant enough to substitute for their simpler counterparts. The nicetone scale is also the cradle of the superdiatonic scales 522352253 and 3243324323 in between the [[leapfrog]] diatonic and chromatic scales.

{{Main|29edo/MOS scales}}

External image: https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-prn1/r90/550502_538613626155939_2005925977_n.jpg<br>

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* [https://www.youtube.com/watch?v=C-FFLXHSO_k ''All 29''] (2023)

* [https://youtu.be/CN4cLOyaVGE ''29edo Negri19 Improvisation No. 1'']] (2024)

* [[Arto and Tendo Theory]]

* [[Lumatone mapping for 29edo]]

 

[[Category:3-limit]]