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29 tone equal temperament | Intervals and linear temperaments | Commas | The Tetradecatonic System | Nicetone | Scales | Music

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.

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

List of 29et rank two temperaments by badness

Degree
Cents
Approx. ratiosof the 15-limit
Ups and downs notation
Generator for temperaments
0
0
1/1
P1
unison
D

1
41.379
25/24~33/32~56/55~81/80
^1, vm2
up unison,
downminor 2nd
D^, Ebv

2
82.759
21/20
m2
minor 2nd
Eb
Nautilus
3
124.138
16/15, 15/14, 14/13, 13/12
^m2
upminor 2nd
Eb^
Negri/Negril
4
165.517
12/11, 11/10
vM2
downmajor 2nd
Ev
Porcupine/Porky/Coendou
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^, Fv
Bridgetown/Immunity

289.655
13/11
m3
minor 3rd
F

8
331.035
6/5, 11/9
^m3
upminor 3rd
F^

9
372.414
5/4, 16/13
vM3
downmajor 3rd
F#v

10
413.793
14/11
M3
major 3rd
F#
Roman
11
455.172
9/7, 13/10
^M3, v4
upmajor 3rd
down 4th
F#^, Gv
Ammonite
12·
496.552
4/3
P4
4th
G
Cassandra Edson Pepperoni
13
537.931
11/8, 15/11
^4
up 4th
G^
Wilsec
14
579.310
7/5, 18/13
vA4, d5
downaug 4th,
dim 5th
G#v, Ab
Tritonic
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
Av

17·
703.448
3/2
P5
5th
A

18
744.828
14/9, 20/13
^5, vm6
up 5th,
downminor 6th
A^, Bbv

19
786.207
11/7
m6
minor 6th
Bb

20
827.586
8/5, 13/8
^m6
upminor 6th
Bb^

21
868.965
5/3, 18/11
vM6
downmajor 6th
Bv

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^, Cv

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
C#v

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#^, Dv

29
1200
2/1
P8
8ve
D

See also: 29edo solfege

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Ups and Downs Notation - Chord names in other EDOs.
29edothumb.png
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


The Tetradecatonic System


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.

Scales

bridgetown9
bridgetown14
Escala Tonal de 17 tonos - Charles Loli

Music

Mp3 29EDO - Escala tonal de 17 notasby Charles Loli A.
Paint in the Water 29 by Igliashon Jones
Nautilus Reverie by Igliashon Calvin Jones-Coolidge
Howling of the Holy by Igliashon Jones
Route 14 in Bridgetown by Chris Vaisvil
The Crowning Song by Mats Öljare
Nine Days Later by Mats Öljare
Stranded at Sea by Mats Öljare

Instruments

Guitar 29EDO

Bass 29EDO