# 29edo

(Redirected from 29-edo)

# 29 tone equal temperament

29edo divides the 2:1 octave into 29 equal steps of approximately 41.37931 cents. It is the 10th prime edo, following 23edo and coming before 31edo.

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.

 https://en.xen.wiki/w/File:29edoSuperpythDiatonic.mp3 https://en.xen.wiki/w/File:12edoDiatonic.mp3 (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. For those who enjoy the bizarre character of Father temperament, 29edo can also be used to support that temperament, if one imagines 11\29 is approximating both 5/4 and 4/3 (ignoring the better approximations at 10\29 and 12\29, respectively).

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 temperaament 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.

# Intervals and linear temperaments

Degree Cents pions 7mus Approx. ratios of the 15-limit Ups and downs notation Generator for temperaments
0 1/1 P1 unison D
1 41.379 43.862 52.9655 (34.F7316) 25/24~33/32~56/55~81/80 ^1, vm2 up unison,

downminor 2nd

D^, Ebv
2 82.759 87.724 105.931 (69.EE5816) 21/20 m2 minor 2nd Eb Nautilus
3 124.138 131.586 158.897 (9E.E5816) 16/15, 15/14, 14/13, 13/12 ^m2 upminor 2nd Eb^ Negri/Negril
4 165.517 175.448 211.862 (D3.DCB16) 12/11, 11/10 vM2 downmajor 2nd Ev Porcupine/Porky/Coendou
5 206.897 219.31 264.828 (108.D3D16) 9/8 M2 major 2nd E
6 248.276 263.172 317.793 (13D.CB116) 8/7, 7/6, 15/13 ^M2, vm3 upmajor 2nd,

downminor 3rd

E^, Fv Bridgetown/Immunity
289.655 307.0345 370.759 (172.C2316) 13/11 m3 minor 3rd F
8 331.0345 350.897 423.724 (1A7.B9616) 6/5, 11/9 ^m3 upminor 3rd F^
9 372.414 394.759 476.69 (1DC.B0916) 5/4, 16/13 vM3 downmajor 3rd F#v
10 413.793 438.621 529.655 (211.A7C16) 14/11 M3 major 3rd F# Roman
11 455.172 482.483 582.621 (246.9EE16) 9/7, 13/10 ^M3, v4 upmajor 3rd

down 4th

F#^, Gv Ammonite
12· 496.552 526.345 635.586 (27B.96116) 4/3 P4 4th G Cassandra Edson Pepperoni
13 537.931 570.207 688.552 (2B0.8D416) 11/8, 15/11 ^4 up 4th G^ Wilsec
14 579.31 614.069 741.517 (2E5.84716) 7/5, 18/13 vA4, d5 downaug 4th,

dim 5th

G#v, Ab Tritonic
15 620.69 657.931 794.483 (31A.7B916) 10/7, 13/9 A4, ^d5 aug 4th,

updim 5th

G#, Ab^
16 662.069 701.793 847.448 (34F.72C16) 16/11, 22/15 v5 down 5th Av
17· 703.448 745.655 900.414 (384.69F16) 3/2 P5 5th A
18 744.828 789.517 953.379 (3B9.63216) 14/9, 20/13 ^5, vm6 up 5th,

downminor 6th

A^, Bbv
19 786.207 833.379 1006.345 (3EE.58416) 11/7 m6 minor 6th Bb
20 827.586 877.241 1059.31 (423.4F716) 8/5, 13/8 ^m6 upminor 6th Bb^
21 868.9655 921.103 1112.276 (458.46A16) 5/3, 18/11 vM6 downmajor 6th Bv
22· 910.345 964.9655 1165.241 (48D.3DD16) 22/13 M6 major 6th B
23 951.724 1008.828 1218.207 (4C2.34F16) 7/4, 12/7, 26/15 ^M6, vm7 upmajor 6th,

downminor 7th

B^, Cv
24 993.103 1052.69 1271.172 (4F7.2C216) 16/9 m7 minor 7th C
25 1034.483 1096.552 1324.138 (52C.23516) 11/6, 20/11 ^m7 upminor 7th C^
26 1075.862 1140.414 1377.103 (561.1A816) 15/8, 28/15, 13/7, 24/13 vM7 downmajor 7th C#v
27 1117.241 1184.276 1430.069 (596.11A816) 40/21 M7 major 7th C#
28 1158.621 1228.178 1483.0345 (5CB.08D16) 48/25~64/33~55/28 ~160/81 ^M7, v8 upmajor 7th,

down 8ve

C#^, Dv
29 1200 1272 1536 (60016) 2/1 P8 8ve D

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

quality color 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. 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 Ebv G C.vm C downminor
gu 10:12:15 0-8-17 C Eb^ G C.^m C upminor
yo 4:5:6 0-9-17 C Ev G C.v C downmajor or C dot down
ru 14:18:27 0-11-17 C E^ G C.^ C upmajor or C dot up

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

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

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 29edo (ordered by absolute error).

 Interval, complement Error (abs., in cents) 13/11, 22/13 0.445 11/10, 20/11 0.513 15/13, 26/15 0.535 13/10, 20/13 0.958 15/11, 22/15 0.980 4/3, 3/2 1.493 9/8, 16/9 2.987 7/5, 10/7 3.202 14/11, 11/7 3.715 14/13, 13/7 4.160 15/14, 28/15 4.695 16/15, 15/8 12.407 16/13, 13/8 12.941 11/8, 16/11 13.387 5/4, 8/5 13.900 13/12, 24/13 14.435 12/11, 11/6 14.880 6/5, 5/3 15.393 18/13, 13/9 15.928 11/9, 18/11 16.373 10/9, 9/5 16.886 8/7, 7/4 17.102 7/6, 12/7 18.595 9/7, 14/9 20.088

# Commas

29 EDO tempers out the following commas. (Note: This assumes the val < 29 46 67 81 100 107 |, cent values rounded to 5 digits.)

Comma Monzo Value (Cents) Name 1 Name 2
16875/16384 | -14 3 4 > 51.120 Negri Comma Double Augmentation Diesis
250/243 | 1 -5 3 > 49.166 Maximal Diesis Porcupine Comma
32805/32768 | -15 8 1 > 1.9537 Schisma
525/512 | -9 1 2 1 > 43.408 Avicennma Avicenna's Enharmonic Diesis
49/48 | -4 -1 0 2 > 35.697 Slendro Diesis
686/675 | 1 -3 -2 3 > 27.985 Senga
64827/64000 | -9 3 -3 4 > 22.227 Squalentine
3125/3087 | 0 -2 5 -3 > 21.181 Gariboh
50421/50000 | -4 1 -5 5 > 14.516 Trimyna
4000/3969 | 5 -4 3 -2 > 13.469 Octagar
225/224 | -5 2 2 -1 > 7.7115 Septimal Kleisma Marvel Comma
5120/5103 | 10 -6 1 -1 > 5.7578 Hemifamity
| 25 -14 0 -1 > 3.8041 Garischisma
100/99 | 2 -2 2 0 -1 > 17.399 Ptolemisma
121/120 | -3 -1 -1 0 2 > 14.367 Biyatisma
896/891 | 7 -4 0 1 -1 > 9.6880 Pentacircle
441/440 | -3 2 -1 2 -1 > 3.9302 Werckisma
4000/3993 | 5 -1 3 0 -3 > 3.0323 Wizardharry
9801/9800 | -3 4 -2 -2 2 > 0.17665 Kalisma Gauss' Comma
91/90 | -1 -2 -1 1 0 1 > 19.130 Superleap

A variant of porcupine supported in 29edo is nautilus, which splits the porcupine generator in half (tempering out 50:49 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 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.

Nicetone scale 5435453 and cadence in 29edo

# Music

The Crowning Song by Mats Öljare

Nine Days Later by Mats Öljare

Stranded at Sea by Mats Öljare

## Instruments

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