29edo/Unque's compositional approach: Difference between revisions

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!Degree

!Cents

!~~Category~~

!Native Fifths

!Additional Categories

!Notation

!Notes

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|0.000

|P1

|

|C

|

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|1

|41.379

|[[Diesis (interval region)|Diesis]]~~/A7~~

|A7

|[[Diesis (interval region)|Diesis]]

|B♯

|Distinct from the octave. Three major thirds reach this augmented seventh.

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|82.759

|m2

|

|D♭

|

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|124.138

|[[Chroma]] (A1)

|Supraminor 2nd

|C♯

|Distinct from the minor second~~. This distinction nullifies~~ the familiar enharmonic equivalences.

|Distinct from the minor second, nullifying the familiar enharmonic equivalences.

|-

|4

|165.517

|d3

|Submajor 2nd

|B𝄪, E𝄫

|

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|206.897

|M2

|

|D

|

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|6

|248.276

|[[Chthonic]]/[[Extraclassical tonality|r3]]

|[[Chthonic]]

|[[Extraclassical tonality|Arto]] 3rd

|C𝄪, F𝄫

|New region in between M2 and m3; two of them make a perfect fourth.

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|289.655

|m3

|

|E♭

|

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|331.034

|A2

|Supraminor 3rd

|D♯

|Distinct from the minor third, as can be seen in the Harmonic Minor modes.

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|372.414

|d4

|Submajor 3rd

|F♭

|May be used as a neutral third, but does not bisect the perfect fifth.

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|413.793

|M3

|

|E

|

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|11

|455.172

|[[Naiadic]]/[[Extraclassical tonality|T3]]

|[[Naiadic]]

|[[Extraclassical tonality|Tendo]] 3rd

|D𝄪, G𝄫

|New region in between M3 and P4; two of them make a major sixth.

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|496.552

|P4

|

|F

|Just barely flatter than the fourth of 12edo, and closer to justly-tuned 4/3.

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|537.931

|A3

|Wolf 4th

|E♯

|

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|579.310

|d5

|Minor tritone

|G♭

|Distinct from the augmented fourth. Two minor thirds reach this diminished fifth.

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|620.690

|A4

|Major tritone

|F♯

|Distinct from the diminished fifth. Three whole tones reach this augmented fourth.

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|662.069

|d6

|Wolf 5th

|E𝄪, A𝄫

|

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|703.448

|P5

|

|G

|Just barely sharper than the fifth of 12edo, and closer to justly-tuned 3/2.

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|18

|744.828

|[[Cocytic]]~~/[[Extraclassical tonality~~|~~r6]]~~

|[[Cocytic]]

|Arto 6th

|F𝄪

|New region between P5 and m6; two of them reduce to a minor third.

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|786.207

|m6

|

|A♭

|

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|827.586

|A5

|Supraminor 6th

|G♯

|Distinct from the minor sixth. Two major thirds reach this augmented fifth.

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|868.966

|d7

|Submajor 6th

|B𝄫

|Distinct from the major sixth. Three minor thirds reach this diminished seventh.

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|910.345

|M6

|

|A

|

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|23

|951.724

|[[Ouranic]]~~/[[Extraclassical tonality~~|~~T6]]~~/~~[[Extraclassical tonality|r7]]~~

|[[Ouranic]]

|Tendo 6th/arto 7th

|G𝄪, C𝄫

|New region between M6 and m7; two of them reduce to a perfect fifth.

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|993.103

|m7

|

|B♭

|

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|1034.483

|A6

|Supraminor 7th

|A♯

|[[wikipedia:Augmented_sixth_chord|Augmented Sixth chords]] use this interval, not the typical minor seventh.

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|1075.862

|d8

|Submajor 7th

|C♭

|

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|1117.241

|M7

|

|B

|

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|28

|1158.621

|d2~~/[[Extraclassical tonality~~|~~T7]]~~

|d2

|Tendo 7th

|A𝄪, D𝄫

|Distinct from the octave. Four minor thirds reach this diminished ninth.

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|1200.000

|P8

|

|C

|

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===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.

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, with the notable exception of the tritone (which comes in two distinct forms depending on which mode it's found in).

{| class="wikitable sortable mw-collapsible"

|+ style="font-size: 105%;" |Modes of 5L 2s

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=== 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.

See [[User:Unque/Dietic Minor|Dietic Minor]] for a more in-depth discussion of how the Harmonic Minor structure can be treated in 29edo, and how this idea generalizes to other tuning systems.

The mode names for this scale are given by [[User:Ayceman|Ayceman]].

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|}

===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 [[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.

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|}

===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 [[3L 6s|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.

Similarly to the minor third, the major third of 29edo also does not close at the octave, allowing us to create an [[3L 5s|octatonic augmented scale]]. Just like the diminished scale, notice how the three brightest modes resemble the bright mode of the [[3L 6s|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 [[User:R-4981|R-4981]].

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=== 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.

29edo additionally offers scales that 12edo cannot remotely approximate; among these is its approximation of the [[3L 4s|neutral scale]], made via a stack of submajor sevenths. 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 a kind of augmented triad as its main 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.

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=== 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.

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 [[5L 1s|a whole tone scale that has 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.

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=== 3L 2M 2s ===

In terms of its harmonies and melodies, [[Nicetone|3L 2M 2s]] acts as a half-way point between 5L 2s and 3L 4s. The scale is generated using an alternating stack of wide and narrow neutral thirds; as such, the thirds are neutral like 3L 4s, but its fifths are perfect like 5L 2s, which makes it a very interesting extension of ideas from both MOS scales. Additionally, the wide neutral third is closer to 5/4 than the diatonic major third is (though admittedly not by a lot), which makes the ~~downmajor~~ triad a decent analog to the [[4:5:6]] triad that is so prevalent in 5-limit music.

In terms of its harmonies and melodies, [[Nicetone|3L 2M 2s]] acts as a half-way point between 5L 2s and 3L 4s. The scale is generated using an alternating stack of wide and narrow neutral thirds; as such, the thirds are neutral like 3L 4s, but its fifths are perfect like 5L 2s, which makes it a very interesting extension of ideas from both MOS scales. Additionally, the wide neutral third is closer to 5/4 than the diatonic major third is (though admittedly not by a lot), which makes the submajor triad a decent analog to the [[4:5:6]] triad that is so prevalent in 5-limit music.

There are two chiralities of the 3L 2M 2s scale based on which of the two neutral thirds you stack first; using the wider third first yields the right-hand version of the scale, while using the narrower third first yields the left-hand version.

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=== 5L 2M 3s ===

The [[Blackdye|5L 2M 3s]] scale is an extension of 3L 2M 2s that adds three additional tones, created by inserting a dietic step into each of the large steps, and unifies the two chiralities of the modes. It can also be constructed using two pentatonic scales of 2L 3s, where the roots of the scales differ by an interval of 4\29~~; this~~ construction allows us to separate the ten modes into five "acute" modes and five "grave" modes~~, with~~ the grave modes place the root somewhere in the flatter pentatonic, and the acute modes place the root on the sharper one.

The [[Blackdye|5L 2M 3s]] scale, often called Blackdye, is an extension of 3L 2M 2s that adds three additional tones, created by inserting a dietic step into each of the large steps, and unifies the two chiralities of the modes. It can also be constructed using two pentatonic scales of [[2L 3s]], where the roots of the scales differ by an interval of 4\29. This construction allows us to separate the ten modes into five "acute" modes and five "grave" modes: the grave modes place the root somewhere in the flatter pentatonic, and the acute modes place the root on the sharper one.

By noticing that the spacer 4\29 differs from the pentatonic large step by a chroma, we can most effectively notate the scale by treating that spacer as a diminished third; while the usage of double-flats in the grave modes and double-sharps in the acute modes makes this notation seem a bit unruly at first, it bypasses the additional ups and downs that would be necessitated if we were to treat the spacer as a ~~downmajor~~ second or an upchroma.

By noticing that the spacer 4\29 differs from the pentatonic large step by a chroma, we can most effectively notate the scale by treating that spacer as a diminished third; while the usage of double-flats in the grave modes and double-sharps in the acute modes makes this notation seem a bit unruly at first, it bypasses the additional ups and downs that would be necessitated if we were to treat the spacer as a submajor second or an upchroma.

{| class="wikitable"

|+Grave Modes of 5L 2M 3s

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29edo has three unique types of leading tones: from narrowest to widest, they are the [[Pythagorean comma|diesis]] (1\29), the [[256/243|semitone]] (2\29), and the [[2187/2048|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 [[wikipedia:Marchetto_da_Padova|Marchetto de Padova]] used the interordinal intervals as counterpoint dissonances: two notes a semisixth apart (11\29) can resolve outwards by a chroma (or more accurately, the enharmonically equivalent upminor second) 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~~.

Finally, it is important to recognize certain tense intervals that resolve via contrary motion to certain perfect consonances. Notably, 14th century composer and theorist [[wikipedia:Marchetto_da_Padova|Marchetto de Padova]] used the interordinal intervals as counterpoint dissonances: two notes a semisixth apart (11\29) can resolve outwards by a chroma (or more accurately, the enharmonically equivalent upminor second) to create a perfect fifth, and two notes a semifourth apart (6\29) can resolve outwards by a chroma to reach a perfect fourth, or inwards to reach a unison. These paradigms can be reversed to account for the octave complements of those intervals.

=== 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).

Firstly, we can notice that the major second between vA and vB is enharmonically equivalent to 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.

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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.

== Mode, Key, and Scale Transitions ==

While voice leading and tension/release paradigms are useful ways to construct chord progressions given a scale, there are many other reasons why a chord progression may be employed. Another important one is as a method of moving roots within a scale, changing keys, or transitioning to a different scale.

=== Re-Rooting a Mode ===

For this example, presume that we are working with the Blackdye scale (5L 2M 3s), beginning on the Grave Ionian mode on C. Let's then presume that we want to transition into a section that is rooted on the Grave Mixolydian mode on G, a rotation of the original scale. Because the two modes contain the same fixed pitches, we need a way to solidify the new root without introducing new notes.

One of the easiest ways to do this is to begin with the tonic chord of the first mode, and gradually lead into the tonic of the second mode, while minimizing the amount of tension in order to make the progression as smooth as possible. In this case, we may notice that C Grave Ionian is rooted on the ''C vmaj'' triad (C - F♭ - G, or equivalently C - vE - G), and G Grave Mixolydian is rooted on the triad ''G vmaj'' triad (G - C♭ - D, or equivalently G - vB - D).

Because these two chords share the note G in common, we can smoothly move between the chords by changing one note at a time: the C of the ''C vmaj'' triad may become the C♭ or vB of the ''G vmaj'' triad, giving us the ''vE ^min'' triad (vE - G - vB) as our "mediator" chord. Then, the vE of that chord can become the D of ''G vmaj''.

These changes can also be performed in the opposite direction, giving us the progression ''C vmaj - C sus2 - G vmaj'', or equivalently ''C vmaj - G sus4 - G vmaj''; the former version of the progression, with all tertiary triads, creates a more blended sound due to the chords all being of similar quality, while the latter creates a more expressive sound with the ''G sus4'' chord creating a resolution to ''G vmaj''.

=== Moving Transpositions ===

For this example, presume that we are working with the 3L 4s "neutral" scale, beginning on the Kleeth mode on C. Let's then presume that we want to transition into a section that is rooted in the Kleeth move on B, a transposition of the original scale. Because the two scales contain the same relative pitches on different degrees, we need a way to make this change without sounding like a jarring leap.

Because the tonic chords are necessarily of the same quality (in this case, ''vmaj ^5'' chords), one of the easiest ways to do this is to find a relevant circle in which C and B are nearby, and move about that circle using chords that are all of that quality. Because 29edo is prime, all possible circles will contain every note in the system, so finding such a circle largely comes down to choosing an interval that is in the scale.

In this example, the circle of submajor thirds is extremely relevant to our scale, as this is the circle that defines the scale; if we were to visualize this circle, we would see that C - vE - ^G - B forms a continuous sequence, and thus we can move through ''vmaj ^5'' chords on each of these roots to smoothly move from C to B. If we treat the submajor third as an approximation of 5/4, this can also be seen as drifting by the comma [[128/125]].

[[Category:Approaches to tuning systems]]

[[Category:29edo]]

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 !Degree
 !Cents
-!Category
+!Native Fifths
+!Additional Categories
 !Notation
 !Notes
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 |0.000
 |P1
+|
 |C
 |
@@ Line 20: / Line 22: @@
 |1
 |41.379
-|[[Diesis (interval region)|Diesis]]/A7
+|A7
+|[[Diesis (interval region)|Diesis]]
 |B♯
 |Distinct from the octave.  Three major thirds reach this augmented seventh.
@@ Line 27: / Line 30: @@
 |82.759
 |m2
+|
 |D♭
 |
@@ Line 33: / Line 37: @@
 |124.138
 |[[Chroma]] (A1)
+|Supraminor 2nd
 |C♯
-|Distinct from the minor second.  This distinction nullifies the familiar enharmonic equivalences.
+|Distinct from the minor second, nullifying the familiar enharmonic equivalences.
 |-
 |4
 |165.517
 |d3
+|Submajor 2nd
 |B𝄪, E𝄫
 |
@@ Line 45: / Line 51: @@
 |206.897
 |M2
+|
 |D
 |
@@ Line 50: / Line 57: @@
 |6
 |248.276
-|[[Chthonic]]/[[Extraclassical tonality|r3]]
+|[[Chthonic]]
+|[[Extraclassical tonality|Arto]] 3rd
 |C𝄪, F𝄫
 |New region in between M2 and m3; two of them make a perfect fourth.
@@ Line 57: / Line 65: @@
 |289.655
 |m3
+|
 |E♭
 |
@@ Line 63: / Line 72: @@
 |331.034
 |A2
+|Supraminor 3rd
 |D♯
 |Distinct from the minor third, as can be seen in the Harmonic Minor modes.
@@ Line 69: / Line 79: @@
 |372.414
 |d4
+|Submajor 3rd
 |F♭
 |May be used as a neutral third, but does not bisect the perfect fifth.
@@ Line 75: / Line 86: @@
 |413.793
 |M3
+|
 |E
 |
@@ Line 80: / Line 92: @@
 |11
 |455.172
-|[[Naiadic]]/[[Extraclassical tonality|T3]]
+|[[Naiadic]]
+|[[Extraclassical tonality|Tendo]] 3rd
 |D𝄪, G𝄫
 |New region in between M3 and P4; two of them make a major sixth.
@@ Line 87: / Line 100: @@
 |496.552
 |P4
+|
 |F
 |Just barely flatter than the fourth of 12edo, and closer to justly-tuned 4/3.
@@ Line 93: / Line 107: @@
 |537.931
 |A3
+|Wolf 4th
 |E♯
 |
@@ Line 99: / Line 114: @@
 |579.310
 |d5
+|Minor tritone
 |G♭
 |Distinct from the augmented fourth.  Two minor thirds reach this diminished fifth.
@@ Line 105: / Line 121: @@
 |620.690
 |A4
+|Major tritone
 |F♯
 |Distinct from the diminished fifth.  Three whole tones reach this augmented fourth.
@@ Line 111: / Line 128: @@
 |662.069
 |d6
+|Wolf 5th
 |E𝄪, A𝄫
 |
@@ Line 117: / Line 135: @@
 |703.448
 |P5
+|
 |G
 |Just barely sharper than the fifth of 12edo, and closer to justly-tuned 3/2.
@@ Line 122: / Line 141: @@
 |18
 |744.828
-|[[Cocytic]]/[[Extraclassical tonality|r6]]
+|[[Cocytic]]
+|Arto 6th
 |F𝄪
 |New region between P5 and m6; two of them reduce to a minor third.
@@ Line 129: / Line 149: @@
 |786.207
 |m6
+|
 |A♭
 |
@@ Line 135: / Line 156: @@
 |827.586
 |A5
+|Supraminor 6th
 |G♯
 |Distinct from the minor sixth.  Two major thirds reach this augmented fifth.
@@ Line 141: / Line 163: @@
 |868.966
 |d7
+|Submajor 6th
 |B𝄫
 |Distinct from the major sixth.  Three minor thirds reach this diminished seventh.
@@ Line 147: / Line 170: @@
 |910.345
 |M6
+|
 |A
 |
@@ Line 152: / Line 176: @@
 |23
 |951.724
-|[[Ouranic]]/[[Extraclassical tonality|T6]]/[[Extraclassical tonality|r7]]
+|[[Ouranic]]
+|Tendo 6th/arto 7th
 |G𝄪, C𝄫
 |New region between M6 and m7; two of them reduce to a perfect fifth.
@@ Line 159: / Line 184: @@
 |993.103
 |m7
+|
 |B♭
 |
@@ Line 165: / Line 191: @@
 |1034.483
 |A6
+|Supraminor 7th
 |A♯
 |[[wikipedia:Augmented_sixth_chord|Augmented Sixth chords]] use this interval, not the typical minor seventh.
@@ Line 171: / Line 198: @@
 |1075.862
 |d8
+|Submajor 7th
 |C♭
 |
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 |1117.241
 |M7
+|
 |B
 |
@@ Line 182: / Line 211: @@
 |28
 |1158.621
-|d2/[[Extraclassical tonality|T7]]
+|d2
+|Tendo 7th
 |A𝄪, D𝄫
 |Distinct from the octave.  Four minor thirds reach this diminished ninth.
@@ Line 189: / Line 219: @@
 |1200.000
 |P8
+|
 |C
 |
@@ Line 626: / Line 657: @@
 ===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.
+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, with the notable exception of the tritone (which comes in two distinct forms depending on which mode it's found in).
 {| class="wikitable sortable mw-collapsible"
 |+ style="font-size: 105%;" |Modes of 5L 2s
@@ Line 756: / Line 787: @@
 === 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.
+See [[User:Unque/Dietic Minor|Dietic Minor]] for a more in-depth discussion of how the Harmonic Minor structure can be treated in 29edo, and how this idea generalizes to other tuning systems.
 The mode names for this scale are given by [[User:Ayceman|Ayceman]].
@@ Line 810: / Line 843: @@
 |}
 ===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 [[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.
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 |}
 ===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 [[3L 6s|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.
+Similarly to the minor third, the major third of 29edo also does not close at the octave, allowing us to create an [[3L 5s|octatonic augmented scale]].  Just like the diminished scale, notice how the three brightest modes resemble the bright mode of the [[3L 6s|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 [[User:R-4981|R-4981]].
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 === 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.
+edo additionally offers scales that 12edo cannot remotely approximate; among these is its approximation of the [[3L 4s|neutral scale]], made via a stack of submajor sevenths.  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 a kind of augmented triad as its main 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.
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 === 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.
+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 [[5L 1s|a whole tone scale that has 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.
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 === 3L 2M 2s ===
-In terms of its harmonies and melodies, [[Nicetone|3L 2M 2s]] acts as a half-way point between 5L 2s and 3L 4s.  The scale is generated using an alternating stack of wide and narrow neutral thirds; as such, the thirds are neutral like 3L 4s, but its fifths are perfect like 5L 2s, which makes it a very interesting extension of ideas from both MOS scales.  Additionally, the wide neutral third is closer to 5/4 than the diatonic major third is (though admittedly not by a lot), which makes the downmajor triad a decent analog to the [[4:5:6]] triad that is so prevalent in 5-limit music.
+In terms of its harmonies and melodies, [[Nicetone|3L 2M 2s]] acts as a half-way point between 5L 2s and 3L 4s.  The scale is generated using an alternating stack of wide and narrow neutral thirds; as such, the thirds are neutral like 3L 4s, but its fifths are perfect like 5L 2s, which makes it a very interesting extension of ideas from both MOS scales.  Additionally, the wide neutral third is closer to 5/4 than the diatonic major third is (though admittedly not by a lot), which makes the submajor triad a decent analog to the [[4:5:6]] triad that is so prevalent in 5-limit music.
 There are two chiralities of the 3L 2M 2s scale based on which of the two neutral thirds you stack first; using the wider third first yields the right-hand version of the scale, while using the narrower third first yields the left-hand version.
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 === 5L 2M 3s ===
-The [[Blackdye|5L 2M 3s]] scale is an extension of 3L 2M 2s that adds three additional tones, created by inserting a dietic step into each of the large steps, and unifies the two chiralities of the modes.  It can also be constructed using two pentatonic scales of 2L 3s, where the roots of the scales differ by an interval of 4\29; this construction allows us to separate the ten modes into five "acute" modes and five "grave" modes, with the grave modes place the root somewhere in the flatter pentatonic, and the acute modes place the root on the sharper one.
+The [[Blackdye|5L 2M 3s]] scale, often called Blackdye, is an extension of 3L 2M 2s that adds three additional tones, created by inserting a dietic step into each of the large steps, and unifies the two chiralities of the modes.  It can also be constructed using two pentatonic scales of [[2L 3s]], where the roots of the scales differ by an interval of 4\29.  This construction allows us to separate the ten modes into five "acute" modes and five "grave" modes: the grave modes place the root somewhere in the flatter pentatonic, and the acute modes place the root on the sharper one.
-By noticing that the spacer 4\29 differs from the pentatonic large step by a chroma, we can most effectively notate the scale by treating that spacer as a diminished third; while the usage of double-flats in the grave modes and double-sharps in the acute modes makes this notation seem a bit unruly at first, it bypasses the additional ups and downs that would be necessitated if we were to treat the spacer as a downmajor second or an upchroma.
+By noticing that the spacer 4\29 differs from the pentatonic large step by a chroma, we can most effectively notate the scale by treating that spacer as a diminished third; while the usage of double-flats in the grave modes and double-sharps in the acute modes makes this notation seem a bit unruly at first, it bypasses the additional ups and downs that would be necessitated if we were to treat the spacer as a submajor second or an upchroma.
 {| class="wikitable"
 |+Grave Modes of 5L 2M 3s
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 edo has three unique types of leading tones: from narrowest to widest, they are the [[Pythagorean comma|diesis]] (1\29), the [[256/243|semitone]] (2\29), and the [[2187/2048|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 [[wikipedia:Marchetto_da_Padova|Marchetto de Padova]] used the interordinal intervals as counterpoint dissonances: two notes a semisixth apart (11\29) can resolve outwards by a chroma (or more accurately, the enharmonically equivalent upminor second) 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.
+Finally, it is important to recognize certain tense intervals that resolve via contrary motion to certain perfect consonances.  Notably, 14th century composer and theorist [[wikipedia:Marchetto_da_Padova|Marchetto de Padova]] used the interordinal intervals as counterpoint dissonances: two notes a semisixth apart (11\29) can resolve outwards by a chroma (or more accurately, the enharmonically equivalent upminor second) to create a perfect fifth, and two notes a semifourth apart (6\29) can resolve outwards by a chroma to reach a perfect fourth, or inwards to reach a unison.  These paradigms can be reversed to account for the octave complements of those intervals.
 === 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).
+Firstly, we can notice that the major second between vA and vB is enharmonically equivalent to 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.
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 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.
+== Mode, Key, and Scale Transitions ==
+While voice leading and tension/release paradigms are useful ways to construct chord progressions given a scale, there are many other reasons why a chord progression may be employed.  Another important one is as a method of moving roots within a scale, changing keys, or transitioning to a different scale.
+=== Re-Rooting a Mode ===
+For this example, presume that we are working with the Blackdye scale (5L 2M 3s), beginning on the Grave Ionian mode on C.  Let's then presume that we want to transition into a section that is rooted on the Grave Mixolydian mode on G, a rotation of the original scale.  Because the two modes contain the same fixed pitches, we need a way to solidify the new root without introducing new notes.
+One of the easiest ways to do this is to begin with the tonic chord of the first mode, and gradually lead into the tonic of the second mode, while minimizing the amount of tension in order to make the progression as smooth as possible.  In this case, we may notice that C Grave Ionian is rooted on the ''C vmaj'' triad (C - F♭ - G, or equivalently C - vE - G), and G Grave Mixolydian is rooted on the triad ''G vmaj'' triad (G - C♭ - D, or equivalently G - vB - D).
+Because these two chords share the note G in common, we can smoothly move between the chords by changing one note at a time: the C of the ''C vmaj'' triad may become the C♭ or vB of the ''G vmaj'' triad, giving us the ''vE ^min'' triad (vE - G - vB) as our "mediator" chord.  Then, the vE of that chord can become the D of ''G vmaj''.
+These changes can also be performed in the opposite direction, giving us the progression ''C vmaj - C sus2 - G vmaj'', or equivalently ''C vmaj - G sus4 - G vmaj''; the former version of the progression, with all tertiary triads, creates a more blended sound due to the chords all being of similar quality, while the latter creates a more expressive sound with the ''G sus4'' chord creating a resolution to ''G vmaj''.
+=== Moving Transpositions ===
+For this example, presume that we are working with the 3L 4s "neutral" scale, beginning on the Kleeth mode on C.  Let's then presume that we want to transition into a section that is rooted in the Kleeth move on B, a transposition of the original scale.  Because the two scales contain the same relative pitches on different degrees, we need a way to make this change without sounding like a jarring leap.
+Because the tonic chords are necessarily of the same quality (in this case, ''vmaj ^5'' chords), one of the easiest ways to do this is to find a relevant circle in which C and B are nearby, and move about that circle using chords that are all of that quality.  Because 29edo is prime, all possible circles will contain every note in the system, so finding such a circle largely comes down to choosing an interval that is in the scale.
+In this example, the circle of submajor thirds is extremely relevant to our scale, as this is the circle that defines the scale; if we were to visualize this circle, we would see that C - vE - ^G - B forms a continuous sequence, and thus we can move through ''vmaj ^5'' chords on each of these roots to smoothly move from C to B.  If we treat the submajor third as an approximation of 5/4, this can also be seen as drifting by the comma [[128/125]].
 [[Category:Approaches to tuning systems]]
+[[Category:29edo]]