29edo

29edo as the division of the fourth into 3 and 4 equal parts.

29 equal divisions of the octave, or 29edo, is the tuning system which divides the 2/1 ratio into 29 equal parts of approximately 41.3 cents each. It is notable for providing an accurate approximation of Pythagorean tuning, and for tuning the perfect fourth to a highly divisible interval of 12 steps, allowing 29edo to support structures such as Porcupine and Negri.

General theory

JI approximation

29edo excels at approximating prime 3, but it does not have good approximations to most of the following prime harmonics. However, while 5, 7, 11, and 13 are not tuned well individually, they are all flat by nearly the same amount, and therefore 29edo represents ratios between these primes, such as 13/11 or 7/5, very well. Looking further afield, we find primes 29 and 37 approximated rather well; it should be noted that 32/29 splits 4/3 in three, and 37/32 splits it in two, garnering these two primes important structural roles.

If the patent approximations of 5, 7, 11, and 13 are used, since they are at least rather unambiguous, the accuracy of prime 3 and their shared flatness make 29edo, in fact, consistent to the 15-odd-limit. We find that 19 and 23 share a flat tendency in common with them as well, and if we skip prime 17, the 23-odd-limit is tuned consistently.

Approximation of prime harmonics in 29edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	0.0	+1.5	-13.9	-17.1	-13.4	-12.9	+19.2	-7.9	-7.6	+4.9	+13.6	-3.1
	Relative (%)	0.0	+3.6	-33.6	-41.3	-32.4	-31.3	+46.4	-19.0	-18.3	+11.9	+32.8	-7.4
Steps (reduced)		29 (0)	46 (17)	67 (9)	81 (23)	100 (13)	107 (20)	119 (3)	123 (7)	131 (15)	141 (25)	144 (28)	151 (6)

Intervals and Notation

Due to its accurate tuning of prime 3, 29edo can be notated quite cleanly with the familiar circle of fifths; the whole tone is five steps of 29edo, the diatonic semitone is two steps, and the chromatic semitone is three steps; thus, the sharp and flat represent 3 steps of 29edo.

To represent finer distinctions, up (^) and down (v) accidentals may be used to represent one step of 29edo. This system results in many equivalences in intervals: C𝄪, ^D, vE♭, and F𝄫 are all the same note corresponding to six steps above C.

Diatonic scales

Neomajor, neominor (MOS diatonic)

The MOS Diatonic scale is generated by taking seven adjacent tones from the Circle of Fifths, just as it is in 12edo.  Melodies and chords made using this scale will sound nearly identical to those that can be made using 12edo, with the notable exception of the tritone (which comes in two distinct forms depending on which mode it's found in).

Modes of MOS Diatonic

Gens Up	Step Pattern	Notation	Name
6	5-5-5-2-5-5-2	C - D - E - F♯ - G - A - B - C	Lydian
5	5-5-2-5-5-5-2	C - D - E - F - G - A - B - C	Ionian
4	5-5-2-5-5-2-5	C - D - E - F - G - A - B♭ - C	Mixolydian
3	5-2-5-5-5-2-5	C - D - E♭ - F - G - A - B♭ - C	Dorian
2	5-2-5-5-2-5-5	C - D - E♭ - F - G - A♭ - B♭ - C	Aeolian
1	2-5-5-5-2-5-5	C - D♭ - E♭ - F - G - A♭ - B♭ - C	Phrygian
0	2-5-5-2-5-5-5	C - D♭ - E♭ - F - G♭ - A♭ - B♭ - C	Locrian

Submajor, supraminor (Zarlino diatonic)

TODO: complete section

Ultramajor, inframinor (Omnidiatonic)

TODO: complete section

MOS scales

Chromatic

The Chromatic scale is an extension of the MOS Diatonic scale, which can be found by continuing the sequence along the circle of fifths. Because the circle can be traversed in two possible ways, scales can be extended in an "acute" direction or a "grave" direction.

Modes of Chromatic

Gens Up	Step Pattern	Notation	Name	Notes
11	3-2-3-2-3-2-2-3-2-3-2-2	C - C♯ - D - D♯ - E - E♯ - F♯ - G - G♯ - A - A♯ - B - C	Grave Lydian	Like the seven-note Lydian, lacks a Perfect Fourth over the root.
10	3-2-3-2-2-3-2-3-2-3-2-2	C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - A♯ - B - C	Grave Ionian
9	3-2-3-2-2-3-2-3-2-2-3-2	C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - B♭ - B - C	Grave Mixolydian
8	3-2-2-3-2-3-2-3-2-2-3-2	C - C♯ - D - E♭ - E - F - F♯ - G - G♯ - A - B♭ - B - C	Grave Dorian
7	3-2-2-3-2-3-2-2-3-2-3-2	C - C♯ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C	Grave Aeolian
6	2-3-2-3-2-3-2-2-3-2-3-2	C - D♭ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C	Grave Phrygian	Also accounts for Acute Lydian
5	2-3-2-3-2-2-3-2-3-2-3-2	C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - B - C	Acute Ionian	Also accounts for Grave Locrian
4	2-3-2-3-2-2-3-2-3-2-2-3	C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - C♭ - C	Acute Mixolydian
3	2-3-2-2-3-2-3-2-3-2-2-3	C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - A - B♭ - C♭ - C	Acute Dorian
2	2-3-2-2-3-2-3-2-2-3-2-3	C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C	Acute Aeolian
1	2-2-3-2-3-2-3-2-2-3-2-3	C - D♭ - E𝄫 - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C	Acute Phrygian
0	2-2-3-2-3-2-2-3-2-3-2-3	C - D♭ - E𝄫 - E♭ - F♭ - F - G♭ - A𝄫 - A♭ - B𝄫 - B♭ - C♭ - C	Acute Locrian	Like the seven-note Locrian, lacks a Perfect Fifth over the root.

Smitonic

The Smitonic (4L 3s) MOS scale can be found as a circle of supraminor thirds (augmented seconds), or via an "evened out" form of the Harmonic Minor scale. That is, one can derive a mode of smitonic through examining a harmonic minor mode (containing one augmented second, three major seconds, and three minor seconds), narrowing the augmented second by three steps to a major second, and distributing the three steps equally across the minor seconds. The mode names for this scale are given by Ayceman.

Modes of Smitonic

Gens Up	Step Pattern	Notation	Name (Ayceman)	Altered Diatonic Mode
6	5-5-3-5-3-5-3	C - D - E - ^F - ^G - vA - vB - C	Nerevarine	Major Augmented
5	5-3-5-5-3-5-3	C - D - ^E♭ - ^F - ^G - vA - vB - C	Vivecan	Harmonic Minor
4	5-3-5-3-5-5-3	C - D - ^E♭ - ^F - vG - vA - vB - C	Lorkhanic	Lydian ♯2
3	5-3-5-3-5-3-5	C - D - ^E♭ - ^F - vG - vA - B♭ - C	Sothic	Dorian ♯4
2	3-5-5-3-5-3-5	C - ^D♭ - ^E♭ - ^F - vG - vA - B♭ - C	Kagrenacan	Locrian ♯6
1	3-5-3-5-5-3-5	C - ^D♭ - ^E♭ - vF - vG - vA - B♭ - C	Almalexian	Ultralocrian
0	3-5-3-5-3-5-5	C - ^D♭ - ^E♭ - vF - vG - A♭ - B♭ - C	Dagothic	Phrygian Dominant

Gramitonic

The Gramitonic scale takes the role of a diminished scale in 29edo: since four neominor thirds fall short of the octave, the chain of neominor thirds can be extended into this nine-note scale. Note how the four bright modes resemble the pattern of the familiar octatonic scale, with one of the small steps duplicated, and the four darkest modes resemble the rotated variant of that scale; additionally, there is a symmetrical mode that is entirely new to 29edo. The mode names for this scale are given by Lilly Flores.

TODO: add ups and downs

Modes of Gramitonic

Gens Up	Step Pattern	Notation	Name (Flores)
8	6-1-6-1-6-1-6-1-1	C - F𝄫 - E♭ - A𝄫♭ - G♭ - C𝄫♭ - B𝄫 - E𝄫𝄫 - D𝄫 - C	Roi
7	6-1-6-1-6-1-1-6-1	C - F𝄫 - E♭ - A𝄫♭ - G♭ - C𝄫♭ - B𝄫 - A - D𝄫 - C	Steno
6	6-1-6-1-1-6-1-6-1	C - F𝄫 - E♭ - A𝄫♭ - G♭ - F♯ - B𝄫 - A - D𝄫 - C	Limni
5	6-1-1-6-1-6-1-6-1	C - F𝄫 - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - D𝄫 - C	Telma
4	1-6-1-6-1-6-1-6-1	C - B♯ - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - D𝄫 - C	Krini
3	1-6-1-6-1-6-1-1-6	C - B♯ - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - G𝄪 - C	Elos
2	1-6-1-6-1-1-6-1-6	C - B♯ - E♭ - D♯ - G♭ - F♯ - E𝄪 - A - G𝄪 - C	Mychos
1	1-6-1-1-6-1-6-1-6	C - B♯ - E♭ - D♯ - C𝄪♯ - F♯ - E𝄪 - A - G𝄪 - C	Akti
0	1-1-6-1-6-1-6-1-6	C - B♯ - A𝄪♯ - D♯ - C𝄪♯ - F♯ - E𝄪 - A - G𝄪 - C	Dini

Checkertonic

Similarly to the neominor third, the neomajor third of 29edo also does not close at the octave, allowing us to create an 8-note augmented scale. Just like the previous "diminished" scale, notice how the three brightest modes resemble the bright mode of the Tcherepnin scale, with one of the nine steps omitted; the three darkest modes similarly resemble the dark mode of that scale; and the remaining two modes both resemble the symmetrical mode of Tcherepnin. The mode names for this scale are given by R-4981.

TODO: add ups and downs

Modes of Checkertonic

Gens Up	Step Pattern	Notation	Name (R-4981)
7	8-1-8-1-1-8-1-1	C - G𝄫♭ - F♭ - C𝄫♭ - B𝄫♭ - A♭ - E𝄫♭ - D𝄫 - C	King
6	8-1-1-8-1-8-1-1	C - G𝄫♭ - F♭ - E - B𝄫♭ - A♭ - E𝄫𝄫 - D𝄫 - C	Queen
5	8-1-1-8-1-1-8-1	C - G𝄫♭ - F♭ - E - B𝄫♭ - A♭ - G♯ - D𝄫 - C	Marshall
4	1-8-1-8-1-1-8-1	C - B♯ - F♭ - E - B𝄫♭ - A♭ - G♯ - D𝄫 - C	Cardinal
3	1-8-1-1-8-1-8-1	C - B♯ - F♭ - E - D𝄪 - A♭ - G♯ - D𝄫 - C	Rook
2	1-8-1-1-8-1-1-8	C - B♯ - F♭ - E - D𝄪 - A♭ - G♯ - F𝄪♯ - C	Bishop
1	1-1-8-1-8-1-1-8	C - B♯ - A𝄪♯ - E - D𝄪 - A♭ - G♯ - F𝄪♯ - C	Knight
0	1-1-8-1-1-8-1-8	C - B♯ - A𝄪♯ - E - D𝄪 - C𝄪𝄪 - G♯ - F𝄪♯ - C	Pawn

Machinoid

Just like the thirds, we can notice that the whole tones in 29edo do not close at the octave; instead, we see that six whole tones reach an augmented seventh, which exceeds the size of the octave by an edostep. However, the octave can still be closed by employing one diminished third (equivalent to a downmajor second) to act as a "wolf" version of the whole tone; this leads to Machinoid, a whole tone scale that has six distinct modes. The mode names for this scale are given by Lilly Flores.

Modes of Machinoid

Gens Up	Step Pattern	Notation	Name (Flores)
5	5-5-5-5-5-4	C - D - E - F♯ - G♯ - A♯ - C	Erev
4	5-5-5-5-4-5	C - D - E - F♯ - G♯ - B♭ -C	Oplen
3	5-5-5-4-5-5	C - D - E - F♯ - A♭ - B♭ - C	Layla
2	5-5-4-5-5-5	C - D - E - G♭ - A♭ - B♭ - C	Shemesh
1	5-4-5-5-5-5	C - D - F♭ - G♭ - A♭ - B♭ - C	Boqer
0	4-5-5-5-5-5	C - E𝄫 - F♭ - G♭ - A♭ - B♭ - C	Tsohorayim

Multiples

As 29edo's chain of fifths is quite accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 29edo. The temperament they share is called Mystery, preserving 29edo's 2.3.7/5.11/5.13/5 subgroup.

58edo

58edo splits the octave and perfect fifth in half, and provides a greatly improved 17th harmonic, along with improved mappings for 5, 11, 13, and especially 7, that all share a strong sharp tendency. It is a good tuning for temperaments such as Diaschismic, based on the half-octave, and Hemififths, based on the neutral third. It is also rather strong for the purposes of interval categorization schemes such as ADIN due to the fact that it has both neutral and interordinal intervals.

Approximation of prime harmonics in 58edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	0.0	+1.5	+6.8	+3.6	+7.3	+7.7	-1.5	-7.9	-7.6	+4.9	-7.1	-3.1
	Relative (%)	0.0	+7.2	+32.8	+17.3	+35.3	+37.4	-7.3	-38.0	-36.7	+23.7	-34.3	-14.8
Steps (reduced)		58 (0)	92 (34)	135 (19)	163 (47)	201 (27)	215 (41)	237 (5)	246 (14)	262 (30)	282 (50)	287 (55)	302 (12)

87edo

87edo splits the octave and perfect fifth into three. As 29edo's primes 5, 11, and 13 are close to 1/3 of a step off, 87edo tunes them near just, especially prime 5. Prime 7 (and especially the interval 9/7) receives the most damage out of the 13-limit, but it is dented flatwards precisely enough to provide an essentially ideal tuning for Rodan while still maintaining 15-odd-limit consistency. Other important structures that 87edo supports include Kleismic and Didacus.

Approximation of prime harmonics in 87edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	0.0	+1.5	-0.1	-3.3	+0.4	+0.9	+5.4	+5.9	+6.2	+4.9	-0.2	-3.1
	Relative (%)	0.0	+10.8	-0.8	-24.0	+2.9	+6.2	+39.1	+43.0	+45.0	+35.6	-1.5	-22.2
Steps (reduced)		87 (0)	138 (51)	202 (28)	244 (70)	301 (40)	322 (61)	356 (8)	370 (22)	394 (46)	423 (75)	431 (83)	453 (18)