29edo/Unque's compositional approach

< 29edo

Note: This page is currently under construction, and will be subject to major expansion in the near future. Come back soon!

29edo is far from the most common tuning system advertised to newcomers; instead, the face of microtonality for many beginners are simple enharmonic distinctions (such as those found in 17edo), extended Meantone tunings (such as 31edo), or even further divisions of the familiar 12-tone chromatic scale (such as 24edo). However, I believe that 29edo is one of the best starting places to transition from 12-tone logic into the more expansive xenharmonic colors, and on this page I plan to present not only my experience and approach with 29edo, but my reasoning for why I believe that it should compete with, if not replace, those aforementioned tunings for the role of the introduction to microtonal colors.

Intervals and Notation

29edo's highly accurate 3/2 provides a very familiar Circle of Fifths, which means that the interval categories from the 12-tone chromatic scale remain intact and recognizable, while also introducing a plethora of new categories and finer distinctions that were not present at the broader range of 12edo. Additionally, 29edo introduces one interval to fulfill each of the four interordinal functions, allowing for newcomers to explore the application of these unfamiliar intervals by using them in conjunction with the more familiar diatonic categories.

Intervals of 29edo

Degree	Cents	Category	Notation	Notes
0	0.000	P1	C
1	41.379	A7	B♯	Distinct from the octave. Three major thirds reach this augmented seventh.
2	82.759	m2	D♭
3	124.138	A1	C♯	Distinct from the minor second. This distinction nullifies the familiar enharmonic equivalences.
4	165.517	d3	B𝄪, E𝄫
5	206.897	M2	D
6	248.276	Chthonic	C𝄪, F𝄫	New region in between M2 and m3; two of them make a perfect fourth.
7	289.655	m3	E♭
8	331.034	A2	D♯	Distinct from the minor third, as can be seen in the Harmonic Minor modes.
9	372.414	d4	F♭
10	413.793	M3	E
11	455.172	Naiadic	D𝄪, G𝄫	New region in between M3 and P4; two of them make a major sixth.
12	496.552	P4	F	Just barely flatter than the fourth of 12edo, and closer to justly-tuned 4/3.
13	537.931	A3	E♯
14	579.310	d5	G♭	Distinct from the augmented fourth. Two minor thirds reach this diminished fifth.
15	620.690	A4	F♯	Distinct from the diminished fifth. Three whole tones reach this augmented fourth.
16	662.069	d6	E𝄪, A𝄫
17	703.448	P5	G	Just barely sharper than the fifth of 12edo, and closer to justly-tuned 3/2.
18	744.828	Cocytic	F𝄪	New region between P5 and m6; two of them reduce to a minor third.
19	786.207	m6	A♭
20	827.586	A5	G♯	Distinct from the minor sixth. Two major thirds reach this augmented fifth.
21	868.966	d7	B𝄫	Distinct from the major sixth. Three minor thirds reach this diminished seventh.
22	910.345	M6	A
23	951.724	Ouranic	G𝄪, C𝄫	New region between M6 and m7; two of them reduce to a perfect fifth.
24	993.103	m7	B♭
25	1034.483	A6	A♯	Augmented Sixth chords use this interval, not the typical minor seventh.
26	1075.862	d8	C♭
27	1117.241	M7	B
28	1158.621	d2	A𝄪, D𝄫	Distinct from the octave. Four minor thirds reach this diminished ninth.
29	1200.000	P8	C

As can be seen here, the familiar diatonic categories allow composers to root themselves in established structures, permitting them to fall back onto comprehensible harmony while still allowing for the interordinals and other new colors to be utilized alongside them.

In some cases, using ups and downs notation may be more convenient than the plain circle of fifths. In this notation, the ^ accidental represents raising an interval by a single step of 29edo; 3\29, for instance, may be notated enharmonically as an upminor second rather than an augmented unison in certain scales to avoid accidentals that may be difficult to parse, or alternatively to preserve interval arithmetic.

Interordinal Notations

The four interordinal intervals do not lend themselves well to conventional circle-of-fifths notation; there are several possible ways to interpret these intervals in comparison to the existing ones: each can be seen as a double-augmented interval, an upmajor interval, a downminor interval, or a double-diminished interval, depending on which ordinal class we want to associate it with.

Interordinals

Type	Double-Aug	Upmajor	Downminor	Double-Dim
Chthonic	AA1 (C𝄪)	^M2 (^D)	vm3 (vE♭)	dd4 (F𝄫)
Naiadic	AA2 (D𝄪)	^M3 (^E)	vP4 (vF)	dd5 (G𝄫)
Cocytic	AA4 (F𝄪)	^P5 (^G)	vm6 (vA♭)	dd7 (B𝄫♭)
Ouranic	AA5 (G𝄪)	^M6 (^A)	vm7 (vB♭)	dd1 (C𝄫)

Chords of 29edo

Tertian Triads

One of the most common types of chord formations in Western music is tertian harmony, where triads are formed by stacking two types of thirds. In 29edo, this type of structure can be extended to include several types of diesis-altered thirds that are not present in the diatonic scale, which allows for many new colors that were once unavailable.

Tertian Triads

Chord	Symbol	Notation	Formula	Notes
Major	C maj	C - E - G	10\29 + 7\29	Primary consonance in 5L 2s
Minor	c min	C - E♭ - G	7\29 + 10\29	Primary consonance in 5L 2s
Upmajor	C ^maj	C - ^E - G	11\29 + 6\29
Upminor	c ^min	C - ^E♭ - G	8\29 + 9\29
Downmajor	C vmaj	C - vE - G	9\29 + 8\29
Downminor	c vmin	C - vE♭ - G	6\29 + 11\29
Dietic Major	C maj ^5	C - E - ^G	10\29 + 8\29
Dietic Minor	c min ^5	C - E♭ - ^G	7\29 + 11\29
Dietic Upmajor	C ^maj ^5	C - ^E - ^G	11\29 + 7\29	Primary consonance in Nerevarine
Dietic Upminor	c ^min ^5	C - ^E♭ - ^G	8\29 + 10\29	Primary consonance in Vivecan
Dietic Downmajor	C vmaj ^5	C - vE - ^G	9\29 + 9\29	Primary consonance in 3L 4s
Dietic Downminor	c vmin ^5	C - vE♭ - ^G	6\29 + 12\29
Wolf Major	C maj v5	C - E - vG	10\29 + 6\29
Wolf Minor	c min v5	C - E♭ - vG	7\29 + 9\29
Wolf Upmajor	C ^maj v5	C - ^E - vG	11\29 + 5\29
Wolf Upminor	c ^min v5	C - ^E♭ - vG	8\29 + 8\29	Primary consonance in 4L 3s
Wolf Downmajor	C vmaj v5	C - vE - vG	9\29 + 7\29
Wolf Downminor	c vmin v5	C - vE♭ - vG	6\29 + 10\29
Aug Major	C maj ♯5	C - E - G♯	10\29 + 10\29	Primary consonance in 3L 5s and 5L 1s
Aug Minor	c min ♯5	C - E♭ - G♯	7\29 + 13\29
Aug Upmajor	C ^maj ♯5	C - ^E - G♯	11\29 + 9\29
Aug Upminor	c ^min ♯5	C - ^E♭ - G♯	8\29 + 12\29
Aug Downmajor	C vmaj ♯5	C - vE - G♯	9\29 + 11\29
Aug Downminor	c vmin ♯5	C - vE♭ - G♯	6\29 + 14\29	Upaug thirds are definitely pushing the definition of "thirds"
Dim Major	C maj ♭5	C - E - G♭	10\29 + 4\29	4\29 is a diminished third, so this is still tertian
Dim Minor	c min ♭5	C - E♭ - G♭	7\29 + 7\29	Primary consonance in 4L 5s
Dim Upmajor	C ^maj ♭5	C - ^E - G♭	11\29 + 3\29	Downdim thirds are definitely pushing the definition of "thirds"
Dim Upminor	c ^min ♭5	C - ^E♭ - G♭	8\29 + 6\29
Dim Downmajor	C vmaj ♭5	C - vE - G♭	9\29 + 5\29
Dim Downminor	c vmin ♭5	C - vE♭ - G♭	6\29 + 8\29

Chthonic Triads

If tertian harmony can be considered by taking two intervals that add up to a perfect fifth, then chthonic harmony can be considered by taking two intervals that add up to a perfect fourth. For these purposes, I will consider the "perfect" chthonic (6\29, precisely half of the perfect fourth), the downchthonic (5\29, enharmonically equivalent to a major second), and the upchthonic (7\29, enharmonically equivalent to a minor third).

Chthonic Triads

Chord	Symbol	Notation	Formula	Notes
Chthonic	C ct	C - ^D - F	6\29 + 6\29	Primary consonance in 5L 4s
Upchthonic	C ^ct	C - E♭ - F	7\29 + 5\29
Downchthonic	C vct	C - D - F	5\29 + 7\29	Useful tension in 5L 2s
Wolf Chthonic	C ct ^4	C - ^D - ^F	6\29 + 7\29
Wolf Upchthonic	C ^ct ^4	C - E♭ - ^F	7\29 + 6\29
Wolf Downchthonic	C vct ^4	C - D - ^F	5\29 + 8\29	Useful tension in 4L 3s
Wolf Augchthonic	C #ct ^4	C - D♯ - ^F	8\29 + 5\29
Dietic Chthonic	C ct v4	C - ^D - vF	6\29 + 5\29
Dietic Upchthonic	C ^ct v4	C - E♭ - vF	7\29 + 4\29	Useful tension in 3L 4s
Dietic Downchthonic	C vct v4	C - D - vF	5\29 + 6\29
Dietic Dimchthonic	C ♭ct v4	C - E𝄫 - vF	4\29 + 7\29	Useful tension in 3L 4s

Quartal Inversions

Just like in most Pythagorean-like tunings, the quartal chord and its inversions are very useful as unresolved concordances. These chords are typically used as a functional mediant between a dominant and a tonic, making the tension smoothly transition into the resolution rather than creating a bombastic cadence into the tonic chord.

Quartal Inversions

Gens Up	Symbol	Notation	Formula	Resolves to
2	C4	C - F - B♭	12\29 + 12\29	F (^)maj, B♭ (v)min
1	C sus4	C - F - G	12\29 + 5\29	C (^)maj, F (v)min
0	C sus2	C - D - G	5\29 + 12\29	G (^)maj, C (v)min

Additionally, color can be created by replacing one of the fourths with an upfourth or downfourth:

Quartal Alterations

Symbol	Formula	Rooted	First Inversion	Second Inversion
C^4	13\29 + 12\29	C - ^F - ^B♭	C - F - vG	C - vD - G
C4 ^7	12\29 + 13\29	C - F - ^B♭	C - ^F - G	C - vD - vG
Cv4	11\29 + 12\29	C - vF - vB♭	C - F - ^G	C - ^D - G
C4 v7	12\29 + 11\29	C - F - vB♭	C - vF - G	C - ^D - ^G

Scales of 29edo

5L 2s

The 5L 2s 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.

Modes of 5L 2s

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

5L 7s

The 5L 7s scale is an extension of 5L 2s created by continuing the generator sequence. Because the Circle of Fifths is bidirectional, the seven modes can be extended either by continuing the sequence upwards or downwards; those created by going up the chain are called grave modes, and those extended by going down the chain are called acute modes.

Modes of 5L 7s

Gens Up	Step Pattern	Notation	Name	Notes
11	LsLsLssLsLss	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	LsLssLsLsLss	C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - A♯ - B - C	Grave Ionian
9	LsLssLsLssLs	C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - B♭ - B - C	Grave Mixolydian
8	LssLsLsLssLs	C - C♯ - D - E♭ - E - F - F♯ - G - G♯ - A - B♭ - B - C	Grave Dorian
7	LssLsLssLsLs	C - C♯ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C	Grave Aeolian
6	sLsLsLssLsLs	C - D♭ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C	Grave Phrygian	Also accounts for Acute Lydian
5	sLsLssLsLsLs	C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - B - C	Acute Ionian	Also accounts for Grave Locrian
4	sLsLssLsLssL	C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - C♭ - C	Acute Mixolydian
3	sLssLsLsLssL	C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - A - B♭ - C♭ - C	Acute Dorian
2	sLssLsLssLsL	C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C	Acute Aeolian
1	ssLsLsLssLsL	C - D♭ - E𝄫 - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C	Acute Phrygian
0	ssLsLssLsLsL	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.

4L 3s

The 4L 3s scale can be thought of as an alteration of the Harmonic Minor scale, which is unique to 29edo. If we notice that the augmented second is precisely three steps larger than a major second, we can distribute this error amongst the three semitones that occur in the scale, which reduces the scale to a maximum variety of two. We may also notice that this scale's pattern creates a circle of augmented seconds, which can be used to quantify the brightness of the seven modes.

The mode names for this scale are given by Ayceman.

Modes of 4L 3s

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

4L 5s

The 4L 5s scale takes the role of a diminished scale in 29edo. Since four minor thirds fall short of the octave, the chain of minor thirds can be extended into this enneatonic form. 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.

Modes of 4L 5s

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

3L 5s

Similarly to the minor third, the major third of 29edo also does not close at the octave, allowing us to create an octatonic augmented scale. Just like the 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.

Modes of 3L 5s

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

3L 4s

The first truly unheard-of scale that 29edo pulls off is its approximation of the neutral scale by stacking the downmajor third seven times. Like 4L 3s, this scale uses harmony based on upfifths and downfifths rather than normal perfect fifths, which makes its harmony more distant from familiar structures. Just like 5L 3s, it can be compared to the Tcherepnin scale, and as such it relies on augmented triads as its source of harmony; however, this scale pattern removes two of the nine Tcherepnin steps rather than three, reducing it to a more standard heptatonic form.

The modes names for this scale are given by Andrew Heathwaite. They can also be named by comparing two diatonic modes.

Modes of 3L 4s

Gens Up	Step Pattern	Notation	Name (Heathwaite)	Mixed Diatonic
6	LsLsLss	C - vD♯ - vE - ^F♯ - ^G - A♯ - B - C	Dril	Dorian / Lydian
5	LsLssLs	C - vD♯ - vE - ^F♯ - ^G - ^A♭ - B - C	Gil	Lydian / Aeolian
4	LssLsLs	C - vD♯ - vE - vF - ^G - ^A♭ - B - C	Kleeth	Aeolian / Ionian
3	sLsLsLs	C - D♭ - vE - vF - ^G - ^A♭ - B - C	Bish	Ionian / Phrygian
2	sLsLssL	C - D♭ - vE - vF - ^G - ^A♭ - ^B𝄫 - C	Fish	Phrygian / Mixolydian
1	sLssLsL	C - D♭ - vE - vF - vG♭ - ^A♭ - ^B𝄫 - C	Jwl	Mixolydian / Locrian
0	ssLsLsL	C - D♭ - E𝄫 - vF - vG♭ - ^A♭ - ^B𝄫 - C	Led	Locrian / Dorian

5L 1s

Just like the thirds, we can notice that the whole tones in 29edo do not close at the octave; instead, we see that five whole tones exceed the minor seventh by an edostep. However, the octave can still be closed by employing a diminished third to act as a "wolf" version of the whole tone; this leads to the scale having six distinct modes, rather than having an identical pattern on every degree as 12edo had.

The mode names for this scale are given by Lilly Flores.

Modes of 5L 1s

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

5L 4s

The 5L 4s scale is the first truly unusual scale in 29edo, being created via a stack of perfect chthonic intervals. This means that every second interval in the chain will represent an interval from the familiar circle of fifths, whereas each interval between them will be an entirely alien interordinal.

The mode names for this scale are given by Inthar.

Modes of 5L 4s

Gens Up	Step Pattern	Notation	Name (Inthar)
8	LLsLsLsLs	C - D - E - ^E - ^F♯ - G - A - vB♭ - ^B - C	Cristacan
7	LsLLsLsLs	C - D - ^D - ^E - ^F♯ - G - A - vB♭ - ^B - C	Pican
6	LsLsLLsLs	C - D - ^D - ^E - F - G - A - vB♭ - ^B - C	Stellerian
5	LsLsLsLLs	C - D - ^D - ^E - F - G - ^G - vB♭ - ^B - C	Podocian
4	LsLsLsLsL	C - D - ^D - ^E - F - G - ^G - vB♭ - B♭ - C	Nucifragan
3	sLLsLsLsL	C - ^C - ^D - ^E - F - G - ^G - vB♭ - B♭ - C	Coracian
2	sLsLLsLsL	C - ^C - ^D - E♭ - F - G - ^G - vB♭ - B♭ - C	Frugilegian
1	sLsLsLLsL	C - ^C - ^D - E♭ - F - ^F - ^G - vB♭ - B♭ - C	Temnurial
0	sLsLsLsLL	C - ^C - ^D - E♭ - F - ^F - ^G - A♭ - B♭ - C	Pyrrhian

Functional Harmony

Useful harmonic progressions may arise in a number of ways depending on the scale being used and depending on what chord the composer wishes to tonicize. Here, I will document some examples of how functional harmonic progressions may be created in the different scales of 29edo, with concepts that can be extended to apply to any scale.

Note that I will be constructing these chord progressions nonlinearly; more specifically, I will begin with a tonic, then find a dominant, and then a predominant, etc. with mediant chords added in between to supplement the harmony if need be.

Elements of Functional Harmony

Just like in common-practice music theory, chords in 29edo have a tendency to rotate about the Circle of Fifths. This means that in Diatonic music and other scales that contain 17\29, the chord built off of that note will be useful as a dominant; additionally, nearby intervals such as the upfifth and downfifth can create a weaker version of that pull, and as such are useful substitutes for the perfect fifth in scales such as 4L 3s.

29edo has three unique types of leading tones: from narrowest to widest, they are the diesis (1\29), the semitone (2\29), and the chroma (3\29). Of the three, the semitone has the strongest pull; it is narrow enough to create tension (whereas the wider chroma is often more recognizable as a regular melodic small step) while being wide enough to be recognized as a distinct interval (whereas the diesis acts more like an enharmonic alteration of the same note).

Finally, it is important to recognize certain tense intervals that resolve via contrary motion to certain perfect consonances. Notably, 14th century composer and theorist Marchetto de Padova used the interordinal intervals as counterpoint dissonances: two notes a semisixth apart (11\29) can resolve outwards by a chroma to create a perfect fifth, and two notes a semifourth apart (23\29) can resolve outwards by a chroma to reach a perfect fourth, or outwards to reach a unison. These paradigms can be reversed to account for the octave complements of those notes.

Example: Progression in C Vivecan

The Vivecan mode of 4L 3s does not contain a perfect fifth over the root, which may make it difficult to root the mode; however, it does contain an upfifth over the root, whereas five of the other six degrees have downfifths instead, so we may be able to create believable resolutions by using harmonic patterns to "convince" ourselves that the upfifth is more resolved than the downfifth.

Firstly, we can notice that vA and vB are separated by a downchthonic, which can resolve by contrary motion to ^G and C; this mimics the perfect chthonic's tendency to resolve to the perfect fifth in a similar fashion. Thus, we can use the vA vct ^4 chord (with degrees vA, vB, and D) as a useful lead into the C ^min ^5 tonic (with degrees C, ^E♭, and ^G).

By noticing that the vA and D also occur in the D ^min v5 supertonic, we can use that triad as a predominant that leads nicely into the vA chord. The movement by an upfourth from the dyad ^F-vA to vB-D creates a pseudo circle-of-fifths rotation, making this progression feel more coherent than it might look at first.

Finally, the ^F and vA of the D chord are shared by the chord ^F ^min v5 (with degrees ^F, vA, and C); additionally, the ^F-vA dyad is an upfourth above the C-^E♭ dyad in the tonic chord, which makes the ^F ^min v5 chord a logical mediant from the tonic to the predominant.

Thus, our final progression is C ^min ^5 - ^F ^min v5 - D ^min v5 - vA vct ^4. This progression uses a combination of voice leading and circle of fifths movement to create a sound that is both dynamic and functional.