29edo: Difference between revisions

{{EDO intro|29}}

! 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]].

=== Standard notation===

 

 

=== Ups and downs ===

Since a sharp raises by three steps, 29edo is a good candidate for [[ups and downs notation]], similar to [[22edo]]. Here, sharps and flats with arrows from [[Helmholtz–Ellis notation]] can be used:

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

! colspan="2" | Tuning Error

| [{{val| 29 46 }}]

| &minus;0.47

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

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

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

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

| [⟨29 46 67 81 100 107 123]]

| [⟨29 46 67 81 100 107 123 131]]

 

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>

| Wesley's comma

| Slendro diesis

| Septimal kleisma, marvel comma

| Undecimal diasecundal comma, telepathma

|+ Table of rank-2 temperaments by generator

! Periods<br> per 8ve

! Generator<br>(Reduced)

! Cents<br>(Reduced)

! Associated Ratio<br>(Reduced)

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.

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

: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>

 

** "All 29" – [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

* [[Arto and Tendo Theory]]

* [[Lumatone mapping for 29edo]]

 

[[Category:3-limit]]

[[Category:Listen]]

[[Category:Subgroup temperaments]]

[[Category:Twentuning]]