Xenharmonic Wiki

29edo

Revision as of 23:33, 6 March 2023 by Contribution (talk | contribs) (→Logarithmic π and ϕ, Acoustic π and ϕ)
← 28edo 29edo 30edo →
Prime factorization 29 (prime)
Step size 41.3793 ¢ 
Fifth 17\29 (703.448 ¢)
(semiconvergent)
Semitones (A1:m2) 3:2 (124.1 ¢ : 82.76 ¢)
Consistency limit 15
Distinct consistency limit 5

Template:EDO intro

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 slightly sharp, 29edo is a positive temperament – a Superpythagorean instead of a Meantone system.

[File info] [File info]
(Super-)pythagorean diatonic major scale and cadence in 29edo 12edo diatonic major scale and cadence, for comparison

The 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 consistently 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 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 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 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 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 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).

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.

Prime harmonics

Approximation of prime harmonics in 29edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +1.5 -13.9 -17.1 -13.4 -12.9 +19.2 -7.9 -7.6 +4.9 +13.6
Relative (%) +0.0 +3.6 -33.6 -41.3 -32.4 -31.3 +46.4 -19.0 -18.3 +11.9 +32.8
Steps
(reduced) 		29
(0) 		46
(17) 		67
(9) 		81
(23) 		100
(13) 		107
(20) 		119
(3) 		123
(7) 		131
(15) 		141
(25) 		144
(28)

Divisors

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

Intervals

See also: 29edo solfege
Degree Cents Approx. Ratios of the 13-limit Ups and Downs Notation
0 0.000 1/1 P1 unison D
1 41.379 25/24, 33/32, 56/55, 81/80 ^1, vm2 up unison,
downminor 2nd 		^D, vEb
2 82.759 21/20 m2 minor 2nd Eb
3 124.138 16/15, 15/14, 14/13, 13/12 ^m2 upminor 2nd ^Eb
4 165.517 12/11, 11/10 vM2 downmajor 2nd vE
5 206.897 9/8 M2 major 2nd E
6 248.276 8/7, 7/6, 15/13 ^M2, vm3 upmajor 2nd,
downminor 3rd 		^E, vF
·7 289.655 13/11 m3 minor 3rd F
8 331.034 6/5, 11/9 ^m3 upminor 3rd ^F
9 372.414 5/4, 16/13 vM3 downmajor 3rd vF#
10 413.793 14/11 M3 major 3rd F#
11 455.172 9/7, 13/10 ^M3, v4 upmajor 3rd
down 4th 		^F#, vG
·12 496.552 4/3 P4 4th G
13 537.931 11/8, 15/11 ^4 up 4th ^G
14 579.310 7/5, 18/13 vA4, d5 downaug 4th,
dim 5th 		vG#, Ab
15 620.690 10/7, 13/9 A4, ^d5 aug 4th,
updim 5th 		G#, ^Ab
16 662.069 16/11, 22/15 v5 down 5th vA
·17 703.448 3/2 P5 5th A
18 744.828 14/9, 20/13 ^5, vm6 up 5th,
downminor 6th 		^A, vBb
19 786.207 11/7 m6 minor 6th Bb
20 827.586 8/5, 13/8 ^m6 upminor 6th ^Bb
21 868.966 5/3, 18/11 vM6 downmajor 6th vB
·22 910.345 22/13 M6 major 6th B
23 951.724 7/4, 12/7, 26/15 ^M6, vm7 upmajor 6th,
downminor 7th 		^B, vC
24 993.103 16/9 m7 minor 7th C
25 1034.483 11/6, 20/11 ^m7 upminor 7th ^C
26 1075.862 15/8, 28/15, 13/7, 24/13 vM7 downmajor 7th vC#
27 1117.241 40/21 M7 major 7th C#
28 1158.621 48/25, 64/33, 55/28, 160/81 ^M7, v8 upmajor 7th,
down 8ve 		^C#, vD
29 1200.000 2/1 P8 8ve D

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

quality color name monzo format examples
downminor zo {a, b, 0, 1} 7/6, 7/4
minor fourthward wa {a, b}, b < -1 32/27, 16/9
upminor gu {a, b, -1} 6/5, 9/5
" ilo {a, b, 0, 0, 1} 11/9, 11/6
downmajor lu {a, b, 0, 0, -1} 12/11, 18/11
" yo {a, b, 1} 5/4, 5/3
major fifthward wa {a, b}, b > 1 9/8, 27/16
upmajor ru {a, b, 0, -1} 9/7, 12/7

All 29edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, yo and ru triads:

color of the 3rd JI chord notes as edosteps notes of C chord written name spoken name
zo 6:7:9 0-6-17 C vEb G Cvm C downminor
gu 10:12:15 0-8-17 C ^Eb G C^m C upminor
yo 4:5:6 0-9-17 C vE G Cv C downmajor or C down
ru 14:18:21 0-11-17 C ^E G C^ C upmajor or C up

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

 
this example in Sagittal notation shows 29-edo as a fifth-tone system.

JI approximation

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 29edo. Prime harmonics are in bold. The following table shows how 15-odd-limit intervals are represented in 29edo. Prime harmonics are in bold.

As 29edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 29edo
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/11, 22/13 0.445 1.1
11/10, 20/11 0.513 1.2
15/13, 26/15 0.535 1.3
13/10, 20/13 0.958 2.3
15/11, 22/15 0.980 2.4
3/2, 4/3 1.493 3.6
9/8, 16/9 2.987 7.2
7/5, 10/7 3.202 7.7
11/7, 14/11 3.715 9.0
13/7, 14/13 4.160 10.1
15/14, 28/15 4.695 11.3
15/8, 16/15 12.407 30.0
13/8, 16/13 12.941 31.3
11/8, 16/11 13.387 32.4
5/4, 8/5 13.900 33.6
13/12, 24/13 14.435 34.9
11/6, 12/11 14.880 36.0
5/3, 6/5 15.393 37.2
13/9, 18/13 15.928 38.5
11/9, 18/11 16.373 39.6
9/5, 10/9 16.886 40.8
7/4, 8/7 17.102 41.3
7/6, 12/7 18.595 44.9
9/7, 14/9 20.088 48.5

Logarithmic π and ϕ, Acoustic π and ϕ

29edo has decent approximations of logarithmic π (4\29), logarithmic ϕ (18\29), acoustic π (19\29) and acoustic ϕ (20\29).[1] These intervals can be better approximated on various lower EDOs, but not all at the same time.

  1. ↑ octave-reduced equivalences

Not until 304 do we find a better EDO in terms of relative error on these intervals.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [46 -29⟩ [⟨29 46]] -0.47 0.47 1.14
2.3.5 250/243, 16875/16384 [⟨29 46 67]] +1.68 3.07 7.41
2.3.5.7 49/48, 225/224, 250/243 [⟨29 46 67 81]] +2.78 3.28 7.91
2.3.5.7.11 49/48, 55/54, 100/99, 225/224 [⟨29 46 67 81 100]] +3.00 2.97 7.15
2.3.5.7.11.13 49/48, 55/54, 100/99, 105/104, 225/224 [⟨29 46 67 81 100 107]] +3.09 2.71 6.54

Commas

29edo tempers out the following commas. This assumes the patent val ⟨29 46 67 81 100 107]. Cent values are rounded to 5 digits.

Prime
Limit 		Ratio[1] Monzo Cents Color name Name(s)
3 (28 digits) [46 -29⟩ 43.305 Wa-29 29-comma, mystery comma
5 16875/16384 [-14 3 4⟩ 51.120 Laquadyo Negri comma, double augmentation diesis
5 250/243 [1 -5 3⟩ 49.166 Triyo Porcupine comma, maximal diesis
5 32805/32768 [-15 8 1⟩ 1.9537 Layo Schisma
7 525/512 [-9 1 2 1⟩ 43.408 Lazoyoyo Avicennma, Avicenna's enharmonic diesis
7 49/48 [-4 -1 0 2⟩ 35.697 Zozo Slendro diesis
7 686/675 [1 -3 -2 3⟩ 27.985 Trizo-agugu Senga
7 64827/64000 [-9 3 -3 4⟩ 22.227 Laquadzo-atrigu Squalentine
7 3125/3087 [0 -2 5 -3⟩ 21.181 Triru-aquinyo Gariboh
7 50421/50000 [-4 1 -5 5⟩ 14.516 Quinzogu Trimyna
7 4000/3969 [5 -4 3 -2⟩ 13.469 Rurutriyo Octagar
7 225/224 [-5 2 2 -1⟩ 7.7115 Ruyoyo Septimal kleisma, marvel comma
7 5120/5103 [10 -6 1 -1⟩ 5.7578 Saruyo Hemifamity
7 (16 digits) [25 -14 0 -1⟩ 3.8041 Sasaru Garischisma
11 100/99 [2 -2 2 0 -1⟩ 17.399 Luyoyo Ptolemisma
11 121/120 [-3 -1 -1 0 2⟩ 14.367 Lologu Biyatisma
11 896/891 [7 -4 0 1 -1⟩ 9.6880 Saluzo Pentacircle
11 441/440 [-3 2 -1 2 -1⟩ 3.9302 Luzozogu Werckisma
11 4000/3993 [5 -1 3 0 -3⟩ 3.0323 Trithuyo Wizardharry
13 91/90 [-1 -2 -1 1 0 1⟩ 19.130 Thozogu Superleap
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator
(Reduced) 		Cents
(Reduced) 		Associated Ratio
(Reduced) 		Temperament
1 2\29 82.8 21/20 Nautilus
1 3\29 124.1 14/13 Negri / negril / negroni
1 4\29 165.5 11/10 Porky / coendou
1 5\29 206.9 9/8 Baldy
1 6\29 248.3 15/13 Immunity / immune
Hemigari
1 7\29 289.7 13/11 Gariberttet
1 9\29 372.4 5/4 Sephiroth
1 10\29 413.8 9/7 Roman
1 11\29 455.2 13/10 Ammonite
1 12\29 496.6 4/3 Garibaldi / andromeda
Leapday
1 13\29 537.9 15/11 Wilsec
1 14\29 579.3 7/5 Tritonic

Important MOSes include:

The Tetradecatonic System

A variant of porcupine 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.

Nautilus[14] scale (Lsssssssssssss) in 29edo

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 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.

If one can tolerate the tuning error (which is roughly equal to that of 12edo, albeit in the opposite direction for the 5- and 7-limits), this tetradecatonic scale is worth exploring. 29edo is often neglected since it falls so close to the much more popular and well-studied 31edo, but 29 does have its own advantages, and this is one of them.

Nicetone

29edo is not a meantone system, but it could nonetheless be used as a basis for common-practice music if one considers the superfourth as a consonant, alternative type of fourth, and the 11:13:16 as an alternative type of consonant "doubly minor" triad. We can then use a diatonic scale such as 5435453 (which resembles Didymus' 5-limit JI diatonic scale, but with the syntonic comma being exaggerated in size). This scale has a very similar harmonic structure to a meantone diatonic scale, except that one of its minor triads is doubly-minor.

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, 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.

Nicetone scale 5435453 and cadence in 29edo

Common Nicetone tunings
Tuning L:m:s Good Just Approximations Other comments Degrees
D vE F G vA vB
9/8 5/4 4/3 3/2 5/3 15/8
Just 1.825:1.6325:1 Just 9/8, 5/4 and 4/3 203.91 386.314 498.045 701.955 884.359 1088.269
15edo 3:2:1 240 400 480 720 880 1120
18edo 4:2:1 Wolf fourth and fifth 266.667 466.667 733.333 866.667 1133.333
20edo 4:3:1 240 420 480 720 900 1140
21edo 5:2:1 Wolf fourth and fifth 285.714 400 457.143 742.857 857.143 1142.857
22edo 4:3:2 Also has diatonic MOS 218.182 381.182 490.909 709.091 872.727 1090.909
23edo 5:3:1 14/11 Wolf fourth and fifth 260.87 417.391 469.565 730.435 886.9565 1147.826
24edo 6:2:1 Also has neutral diatonic MOS 300 400 450 750 850 1050
25edo 5:3:2

5:4:1

240 384

432

480 720 864

912

1104
26edo 6:3:1 Also has diatonic MOS 276.923 415.385 461.5385 738.4615 876.923 1153.846
27edo 5:4:2

7:2:1

Also has diatonic MOS 222.222

311.111

400 488.889

444.444

711.111

755.556

888.889

844.444

1111.111

1155.556

28edo 6:3:2

6:4:1

257.143 385.714

428.571

471.429 728.571 857.143

900

1114.286

1157.143

29edo 5:4:3

7:3:1

Gentle fifth

Also has diatonic MOS

206.897

289.655

372.414

413.793

496.551

455.172

703.449

745.828

868.9655 1075.862

1158.721

30edo 6:5:1

8:2:1

240

320

440

400

480

440

720

760

920

840

1160
31edo 7:3:2

7:4:1

Also has diatonic MOS 270.968 386.314

425.8065

464.516 735.484 851.613

890.323

1122.581

1161.29

32edo 6:4:3

6:5:2

8:3:1

Also has diatonic MOS 225

300

375

412.5

487.5

450

712.5

750

862.5

900

1087.5

1125

1162.5

33edo 7:4:2

7:5:1

13/11 Also has diatonic MOS 254.5455 400

436.364

472.727 727.272 872.727

909.091

1127.273

1163.636

34edo 6:5:3

8:3:2

8:4:1

25/24

50/49

Gentle fifth

Also has neutral diatonic MOS

211.765

282.353

388.235

423.529

494.118

458.8235

705.882

741.1765

882.353

847.059

1094.118

1129.412

1164.706

35edo 7:4:3

7:5:2

7:6:1

33/26 240 377.143

411.429

445.714

480 720 857.143

891.429

925.714

1097.143

1131.429

1165.714

36edo 6:5:4 27/25 Also has Porcupine MOS 200 366.667 500 700 866.667 1066.667
37edo 7:5:3

7:6:2

Has 37edo just major triad 227.027 389.189 486.4865 713.5135 875.676 1102.703
38edo 8:4:3

8:5:2

8:6:1

6/5, 33/26 and 14/13 or 28/27 Has wolf major and minor triads

Also has neutral diatonic MOS

252.632 378.947

410.526

442.105

473.684 726.318 852.632

884.2105

915.7895

1105.263

1136.842

1168.421

39edo 7:5:4

7:6:3

Also has diatonic MOS 215.385 369.231

400

492.308 707.692 861.5385

892.308

1076.923

1107.

40edo 8:5:3

8:7:1

Golden Nicetone

Also has diatonic MOS

240 390

450

480 720 870

930

1110

1170

41edo 7:6:4 Also has diatonic MOS 204.878 380.488 497.561 702.439 878.049 1082.927
42edo 8:5:4

8:6:3

8:7:2

6/5 Also has diatonic MOS 228.571 371.429

400

428.571

485.714 714.286 857.143

885.714

914.286

1085.714

1114.286

1142.857

43edo 7:6:5 Also has diatonic MOS 195.349 362.791 502.326 697.674 865.116 1060.465
44edo 8:7:3 Also has neutral diatonic MOS 218.182 409.091 490.909 709.091 900 1118.182
46edo 8:6:5

8:7:4

10/9 Gentle fifth

Also has diatonic MOS

208.696 365.217

391.304

495.652 704.348 860.87

886.9565

1069.565

1095.652

48edo 8:7:5 200 375 500 700 875 1075
50edo 8:7:6 Also has diatonic MOS 192 360 504 696 864 1056

Scales

Well temperaments

Instruments

Guitar 29EDO from Peru - Charles Loli and Antonio Huamani

 

Bass 29EDO from Peru - Charles Loli and Antonio Huamani

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

WARNING: MediaWiki doesn't have very good support for external images.
Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

Music

Charles Loli A. (site)
Igliashon Jones
Chris Vaisvil
Ray Perlner
Mats Öljare

See also

Retrieved from "https://en.xen.wiki/index.php?title=29edo&oldid=104852"