29edo

Theory

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.

(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

Degree	Cents	Approx. ratios of the 15-limit	Ups and downs notation			Generator for temperaments
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	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	vE	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, vF	Bridgetown/Immunity
7·	289.655	13/11	m3	minor 3rd	F
8	331.0345	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#	Roman
11	455.172	9/7, 13/10	^M3, v4	upmajor 3rd down 4th	^F#, vG	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	579.31	7/5, 18/13	vA4, d5	downaug 4th, dim 5th	vG#, Ab	Tritonic
15	620.69	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	868.9655	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	2/1	P8	8ve	D

See also: 29edo solfege

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 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	14:18:27	0-11-17	C ^E G	C^	C upmajor or C 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.)

Ratio	Monzo	Cents	Color Name	Name 1	Name 2
16875/16384	| -14 3 4 >	51.120	Laquadyo	Negri Comma	Double Augmentation Diesis
250/243	| 1 -5 3 >	49.166	Triyo	Maximal Diesis	Porcupine Comma
32805/32768	| -15 8 1 >	1.9537	Layo	Schisma
525/512	| -9 1 2 1 >	43.408	Lazoyoyo	Avicennma	Avicenna's Enharmonic Diesis
49/48	| -4 -1 0 2 >	35.697	Zozo	Slendro Diesis
686/675	| 1 -3 -2 3 >	27.985	Trizo-agugu	Senga
64827/64000	| -9 3 -3 4 >	22.227	Laquadzo-atrigu	Squalentine
3125/3087	| 0 -2 5 -3 >	21.181	Triru-aquinyo	Gariboh
50421/50000	| -4 1 -5 5 >	14.516	Quinzogu	Trimyna
4000/3969	| 5 -4 3 -2 >	13.469	Rurutriyo	Octagar
225/224	| -5 2 2 -1 >	7.7115	Ruyoyo	Septimal Kleisma	Marvel Comma
5120/5103	| 10 -6 1 -1 >	5.7578	Saruyo	Hemifamity
	| 25 -14 0 -1 >	3.8041	Sasaru	Garischisma
100/99	| 2 -2 2 0 -1 >	17.399	Luyoyo	Ptolemisma
121/120	| -3 -1 -1 0 2 >	14.367	Lologu	Biyatisma
896/891	| 7 -4 0 1 -1 >	9.6880	Saluzo	Pentacircle
441/440	| -3 2 -1 2 -1 >	3.9302	Luzozogu	Werckisma
4000/3993	| 5 -1 3 0 -3 >	3.0323	Trithuyo	Wizardharry
9801/9800	| -3 4 -2 -2 2 >	0.17665	Bilorugu	Kalisma	Gauss' Comma
91/90	| -1 -2 -1 1 0 1 >	19.130	Thozogu	Superleap

Linear temperaments

List of 29et rank two temperaments by badness

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.

Nicetone scale 5435453 and cadence in 29edo

Scales

bridgetown9

bridgetown14

Escala Tonal de 17 tonos - Charles Loli

Instruments

Guitar 29EDO from Peru - Charles Loli and Antonio Huamani

Bass 29EDO from Peru - Charles Loli and Antonio Huamani

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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