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{{breadcrumb|29edo}}
{{breadcrumb|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.
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==
==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 introduced one interval to fulfill each of the four [[Interseptimal interval|interordinal]] functions, allowing for newcomers to explore the application of these unfamiliar intervals by using them in conjunction with the more familiar diatonic categories.
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 [[Interseptimal interval|interordinal]] functions, allowing for newcomers to explore the application of these unfamiliar intervals by using them in conjunction with the more familiar diatonic categories.
{| class="wikitable"
{| class="wikitable"
|+Intervals of 29edo
|+Intervals of 29edo
!Degree
!Degree
!Cents
!Cents
!Category
!Native Fifths
!Additional Categories
!Notation
!Notation
!Notes
!Notes
Line 17: Line 16:
|0.000
|0.000
|P1
|P1
|
|C
|C
|
|
Line 23: Line 23:
|41.379
|41.379
|A7
|A7
|[[Diesis (interval region)|Diesis]]
|B♯
|B♯
|Distinct from the octave.  Three major thirds reach this augmented seventh.
|Distinct from the octave.  Three major thirds reach this augmented seventh.
Line 29: Line 30:
|82.759
|82.759
|m2
|m2
|
|D♭
|D♭
|
|
Line 34: Line 36:
|3
|3
|124.138
|124.138
|A1
|[[Chroma]] (A1)
|Supraminor 2nd
|C♯
|C♯
|Distinct from the minor second.  This distinction nullifies the familiar enharmonic equivalences.
|Distinct from the minor second, nullifying the familiar enharmonic equivalences.
|-
|-
|4
|4
|165.517
|165.517
|d3
|d3
|Submajor 2nd
|B𝄪, E𝄫
|B𝄪, E𝄫
|
|
Line 47: Line 51:
|206.897
|206.897
|M2
|M2
|
|D
|D
|
|
Line 53: Line 58:
|248.276
|248.276
|[[Chthonic]]
|[[Chthonic]]
|[[Extraclassical tonality|Arto]] 3rd
|C𝄪, F𝄫
|C𝄪, F𝄫
|New region in between M2 and m3; two of them make a perfect fourth.
|New region in between M2 and m3; two of them make a perfect fourth.
Line 59: Line 65:
|289.655
|289.655
|m3
|m3
|
|E♭
|E♭
|
|
Line 65: Line 72:
|331.034
|331.034
|A2
|A2
|Supraminor 3rd
|D♯
|D♯
|Distinct from the minor third, as can be seen in the Harmonic Minor modes.
|Distinct from the minor third, as can be seen in the Harmonic Minor modes.
Line 71: Line 79:
|372.414
|372.414
|d4
|d4
|Submajor 3rd
|F♭
|F♭
|
|May be used as a neutral third, but does not bisect the perfect fifth.
|-
|-
|10
|10
|413.793
|413.793
|M3
|M3
|
|E
|E
|
|
Line 83: Line 93:
|455.172
|455.172
|[[Naiadic]]
|[[Naiadic]]
|[[Extraclassical tonality|Tendo]] 3rd
|D𝄪, G𝄫
|D𝄪, G𝄫
|New region in between M3 and P4; two of them make a major sixth.
|New region in between M3 and P4; two of them make a major sixth.
Line 89: Line 100:
|496.552
|496.552
|P4
|P4
|
|F
|F
|Just barely flatter than the fourth of 12edo, and closer to justly-tuned 4/3.
|Just barely flatter than the fourth of 12edo, and closer to justly-tuned 4/3.
Line 95: Line 107:
|537.931
|537.931
|A3
|A3
|Wolf 4th
|E♯
|E♯
|
|
Line 101: Line 114:
|579.310
|579.310
|d5
|d5
|Minor tritone
|G♭
|G♭
|Distinct from the augmented fourth.  Two minor thirds reach this diminished fifth.
|Distinct from the augmented fourth.  Two minor thirds reach this diminished fifth.
Line 107: Line 121:
|620.690
|620.690
|A4
|A4
|Major tritone
|F♯
|F♯
|Distinct from the diminished fifth.  Three whole tones reach this augmented fourth.
|Distinct from the diminished fifth.  Three whole tones reach this augmented fourth.
Line 113: Line 128:
|662.069
|662.069
|d6
|d6
|Wolf 5th
|E𝄪, A𝄫
|E𝄪, A𝄫
|
|
Line 119: Line 135:
|703.448
|703.448
|P5
|P5
|
|G
|G
|Just barely sharper than the fifth of 12edo, and closer to justly-tuned 3/2.
|Just barely sharper than the fifth of 12edo, and closer to justly-tuned 3/2.
Line 125: Line 142:
|744.828
|744.828
|[[Cocytic]]
|[[Cocytic]]
|Arto 6th
|F𝄪
|F𝄪
|New region between P5 and m6; two of them reduce to a minor third.
|New region between P5 and m6; two of them reduce to a minor third.
Line 131: Line 149:
|786.207
|786.207
|m6
|m6
|
|A♭
|A♭
|
|
Line 137: Line 156:
|827.586
|827.586
|A5
|A5
|Supraminor 6th
|G♯
|G♯
|Distinct from the minor sixth.  Two major thirds reach this augmented fifth.
|Distinct from the minor sixth.  Two major thirds reach this augmented fifth.
Line 143: Line 163:
|868.966
|868.966
|d7
|d7
|Submajor 6th
|B𝄫
|B𝄫
|Distinct from the major sixth.  Three minor thirds reach this diminished seventh.
|Distinct from the major sixth.  Three minor thirds reach this diminished seventh.
Line 149: Line 170:
|910.345
|910.345
|M6
|M6
|
|A
|A
|
|
Line 155: Line 177:
|951.724
|951.724
|[[Ouranic]]
|[[Ouranic]]
|Tendo 6th/arto 7th
|G𝄪, C𝄫
|G𝄪, C𝄫
|New region between M6 and m7; two of them reduce to a perfect fifth.
|New region between M6 and m7; two of them reduce to a perfect fifth.
Line 161: Line 184:
|993.103
|993.103
|m7
|m7
|
|B♭
|B♭
|
|
Line 167: Line 191:
|1034.483
|1034.483
|A6
|A6
|Supraminor 7th
|A♯
|A♯
|[[wikipedia:Augmented_sixth_chord|Augmented Sixth chords]] use this interval, not the typical minor seventh.
|[[wikipedia:Augmented_sixth_chord|Augmented Sixth chords]] use this interval, not the typical minor seventh.
Line 173: Line 198:
|1075.862
|1075.862
|d8
|d8
|Submajor 7th
|C♭
|C♭
|
|
Line 179: Line 205:
|1117.241
|1117.241
|M7
|M7
|
|B
|B
|
|
Line 185: Line 212:
|1158.621
|1158.621
|d2
|d2
|Tendo 7th
|A𝄪, D𝄫
|A𝄪, D𝄫
|Distinct from the octave.  Four minor thirds reach this diminished ninth.
|Distinct from the octave.  Four minor thirds reach this diminished ninth.
Line 191: Line 219:
|1200.000
|1200.000
|P8
|P8
|
|C
|C
|
|
Line 198: Line 227:
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.
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.


== Scales of 29edo ==
=== 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.
{| class="wikitable"
|+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𝄫)
|}
 
=== Extraclassical Tonality ===
{{main|Extraclassical tonality}}
 
In addition to the diatonic-based interval logic, 29edo contains excellent examples of extraclassical tonality; rather than the major and minor thirds seen in its diatonic scale, this entails the usage of arto and tendo thirds, which differ from the diatonic thirds by a diesis.  These thirds have a very distinct sound, and are an excellent option for introducing truly microtonal chords into an otherwise conventional progression.
 
One benefit of using extraclassical intervals is that the art and tendo thirds (unlike their diatonic counterparts) do not crowd one another when used in a chord together, being separated by a whole tone rather than a chroma.
== 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.
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+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
|-
|Tendo
|CT
|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
|
|-
|Arto
|cr
|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 Tendo
|CT ^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 Arto
|cr ^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 Tendo
|CT 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 Arto
|cr 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 Tendo
|CT ♯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 Arto
|cr ♯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 Tendo
|CT ♭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 Arto
|cr ♭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).
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+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.
{| class="wikitable"
|+Quartal Inversions
!Gens Up
!Symbol
!Notation
!Formula
!Resolves to
|-
|2
|C<sup>4</sup>
|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:
{| class="wikitable"
|+Quartal Alterations
!Symbol
!Formula
!Rooted
!First Inversion
!Second Inversion
|-
|C<sup>^4</sup>
|13\29 + 12\29
|C - ^F - ^B♭
|C - F - vG
|C - vD - G
|-
|C<sup>4</sup> ^7
|12\29 + 13\29
|C - F - ^B♭
|C - ^F - G
|C - vD - vG
|-
|C<sup>v4</sup>
|11\29 + 12\29
|C - vF - vB♭
|C - F - ^G
|C - ^D - G
|-
|C<sup>4</sup> v7
|12\29 + 11\29
|C - F - vB♭
|C - vF - G
|C - ^D - ^G
|}
 
=== Extraclassical Alterations ===
As mentioned in the section on intervals, extraclassical chords can form tetrads that are capped by the perfect fifth, rather than simple triads.  Alterations of this chord can be formed by moving any of the voices up or down by a diesis.
{| class="wikitable"
|+Extraclassical Alterations
!Alteration
!Symbol
!Notation
!Step Sizes
!Notes
|-
|None
|CTr
|C - vE♭ - ^E - G
|6 + 5 + 6
|
|-
|^Arto
|CTm
|C - E♭ - ^E - G
|7 + 5 + 6
|
|-
|vArto
|CTvr
|C - ^E𝄫 - ^E - G
|5 + 6 + 6
|Downarto third is enharmonically equivalent to the major second
|-
|^Tendo
|C^Tr
|C - vE♭ - vE♯ - G
|6 + 6 + 5
|Uptendo third is enharmonically equivalent to the perfect fourth
|-
|vTendo
|CMr
|C - vE♭ - E - G
|6 + 4 + 7
|
|-
|^Fifth
|CTr ^5
|C - vE♭ - ^E - ^G
|6 + 5 + 7
|Upfifth is the octave complement of the tendo third
|-
|vFifth
|CTr v5
|C - vE♭ - ^E - vG
|6 + 5 + 5
|
|}
 
== MOS Scales of 29edo ==
[[MOS scale|Moment of Symmetry scales]] can be constructed by repeatedly stacking a [[generator]] until the resultant scale has a [[maximum variety]] of 2.  Because 29 is a prime number, every possible interval in the system can generate a MOS scale; however, this also means that 29edo lacks any [[wikipedia:Mode_of_limited_transposition|modes of limited transposition]], which may be considered detrimental to approaches that value symmetrical chords, as well as systems akin to the [[wikipedia:Axis_system|Axis System]] of [[wikipedia:Béla_Bartók|Béla Bartók]].


===5L 2s===
===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"
{| class="wikitable sortable mw-collapsible"
|+ style="font-size: 105%;" |Modes of 5L&nbsp;2s
|+ style="font-size: 105%;" |Modes of 5L 2s
|-
|-
!Gens Up
!Gens Up
Line 246: Line 702:
|}
|}
===5L 7s===
===5L 7s===
The [[5L&nbsp;7s]] scale is an extension of 5L&nbsp;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.
The [[5L&nbsp;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.
{| class="wikitable sortable"
{| class="wikitable sortable mw-collapsible"
|+ style="font-size: 105%;" |Modes of 5L&nbsp;7s
|+ style="font-size: 105%;" |Modes of 5L 7s
|-
|-
!Gens Up
!Gens Up
Line 331: Line 787:
=== 4L 3s ===
=== 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 [[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]].
The mode names for this scale are given by [[User:Ayceman|Ayceman]].
{| class="wikitable sortable"
{| class="wikitable sortable mw-collapsible"
|+ style="font-size: 105%;" |Modes of 4L 3s
|+ style="font-size: 105%;" |Modes of 4L 3s
|-
|-
Line 385: Line 843:
|}
|}
===4L 5s===
===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.
The mode names for this scale are given by Lilly Flores.
{| class="wikitable sortable"
{| class="wikitable sortable mw-collapsible"
|+ style="font-size: 105%;" |Modes of 4L 5s
|+ style="font-size: 105%;" |Modes of 4L 5s
|-
|-
Line 442: Line 900:
|}
|}
===3L 5s===
===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]].
The mode names for this scale are given by [[User:R-4981|R-4981]].
{| class="wikitable sortable"
{| class="wikitable sortable mw-collapsible"
|+ style="font-size: 105%;" |Modes of 3L 5s
|+ style="font-size: 105%;" |Modes of 3L 5s
|-
|-
Line 492: Line 950:
|C - B♯ - A𝄪♯ - E - D𝄪 - C𝄪𝄪 - G♯ - F𝄪♯ - C
|C - B♯ - A𝄪♯ - E - D𝄪 - C𝄪𝄪 - G♯ - F𝄪♯ - C
|Pawn
|Pawn
|}
=== 3L 4s ===
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.
{| class="wikitable sortable mw-collapsible"
|+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 ===
=== 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.
The mode names for this scale are given by Lilly Flores.
Line 508: Line 1,021:
|5
|5
|LLLLLs
|LLLLLs
|C-D-E-F♯-G♯-A♯-C
|C - D - E - F♯ - G♯ - A♯ - C
|Erev
|Erev
|-
|-
|4
|4
|LLLLsL
|LLLLsL
|C-D-E-F♯-G♯-B♭-C
|C - D - E - F♯ - G♯ - B♭ -C
|Oplen
|Oplen
|-
|-
|3
|3
|LLLsLL
|LLLsLL
|C-D-E-F♯-A♭-B♭-C
|C - D - E - F♯ - A♭ - B♭ - C
|Layla
|Layla
|-
|-
|2
|2
|LLsLLL
|LLsLLL
|C-D-E-G♭-A♭-B♭-C
|C - D - E - G♭ - A♭ - B♭ - C
|Shemesh
|Shemesh
|-
|-
|1
|1
|LsLLLL
|LsLLLL
|C-D-F♭-G♭-A♭-B♭-C
|C - D - F♭ - G♭ - A♭ - B♭ - C
|Boqer
|Boqer
|-
|-
|0
|0
|sLLLLL
|sLLLLL
|C-E𝄫-F♭-G♭-A♭-B♭-C
|C - E𝄫 - F♭ - G♭ - A♭ - B♭ - C
|Tsohorayim
|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 [[User:Inthar|Inthar]].
{| class="wikitable sortable mw-collapsible"
|+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
|}
== Other Scales of 29edo ==
=== 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 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.
{| class="wikitable"
|+Modes of Right-hand 3L 2M 2s
!Gens Up
!Pattern
!Notation
!5L 2s / 3L 4s modes
!Notes
|-
|6
|LMLsLMs
|C - D - vE - vF♯ - G - A - vB - C
|Lydian / Dril
|Fourth and sixth degrees are not neutral/perfect
|-
|5
|LsLMsLM
|C - D - ^E♭ - ^F - G - ^A♭ - ^B♭ - C
|Aeolian / Gil
|Fourth degree is imperfect
|-
|4
|LMsLMLs
|C - D - vE - F - G - vA - vB - C
|Ionian / Kleeth
|
|-
|3
|sLMLsLM
|C - ^D♭ - ^E♭ - F - G - ^A♭ - ^B♭ - C
|Phrygian / Bish
|
|-
|2
|MLsLMsL
|C - vD - vE - F - G - vA - B♭ - C
|Mixolydian / Fish
|
|-
|1
|sLMsLML
|C - ^D♭ - ^E♭ - F - ^G♭ - ^A♭ - B♭ - C
|Locrian / Jwl
|Fifth degree is imperfect
|-
|0
|MsLMLsL
|C - vD - E♭ - F - vG - vA - B♭ - C
|Dorian / Led
|Third and fifth degrees are not neutral/perfect
|}
{| class="wikitable"
|+Modes of Left-hand 3L 2M 2s
!Gens Up
!Pattern
!Notation
!5L 2s / 3L 4s modes
!Notes
|-
|6
|LsLMLsM
|C - D - ^E♭ - ^F - G - A - ^B♭ - C
|Dorian / Dril
|Fourth and sixth degrees are not neutral/perfect
|-
|5
|LMLsMLs
|C - D - vE - vF♯ - G - vA - vB - C
|Lydian / Gil
|Fourth degree is imperfect
|-
|4
|LsMLsLM
|C - D - ^E♭ - F - G - ^A♭ - ^B♭ - C
|Aeolian / Kleeth
|
|-
|3
|MLsLMLs
|C - vD - vE - F - G - vA - vB - C
|Ionian / Bish
|
|-
|2
|sLMLsML
|C - ^D♭ - ^E♭ - F - G - ^A♭ - B♭ - C
|Phrygian / Fish
|
|-
|1
|MLsMLsL
|C - vD - vE - F - vG - vA - B♭ - C
|Mixolydian / Jwl
|Fifth degree is imperfect
|-
|0
|sMLsLML
|C - ^D♭ - E♭ - F - ^G♭ - ^A♭ - B♭ - C
|Locrian / Led
|Third and fifth degrees are not neutral/perfect
|}
=== 5L 2M 3s ===
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 submajor second or an upchroma.
{| class="wikitable"
|+Grave Modes of 5L 2M 3s
!Gens Up
!Pattern
!Notation
!Name
|-
|4
|LmLsLmLsLs
|C - E𝄫 - E♭ - G𝄫 - F - A𝄫 - A♭ - C𝄫 - B♭ - D𝄫 - C
|Aeolian
|-
|3
|LmLsLsLmLs
|C - E𝄫 - E♭ - G𝄫 - F - A𝄫 - G - B𝄫 - B♭ - D𝄫 - C
|Dorian
|-
|2
|LsLmLsLmLs
|C - E𝄫 - D - F♭ - F - A𝄫 - G - B𝄫 - B♭ - D𝄫 - C
|Mixolydian
|-
|1
|LsLmLsLsLm
|C - E𝄫 - D - F♭ - F - A𝄫 - G - B𝄫 - A - C♭ - C
|Ionian
|-
|0
|LsLsLmLsLm
|C - E𝄫 - D - F♭ - E - G♭ - G - B𝄫 - A - C♭ - C
|Lydian
|}
{| class="wikitable"
|+Acute Modes of 5L 2M 3s
!Gens Up
!Pattern
!Notation
!Name
|-
|4
|mLsLmLsLsL
|C - C♯ - E♭ - D♯ - F - F♯ - A♭ - G♯ - B♭ - A♯ - C
|Locrian
|-
|3
|mLsLsLmLsL
|C - C♯ - E♭ - D♯ - F - E♯ - G - G♯ - B♭ - A♯ - C
|Phrygian
|-
|2
|sLmLsLmLsL
|C - B♯ - D - D♯ - F - E♯ - G - G♯ - B♭ - A♯ - C
|Aeolian
|-
|1
|sLmLsLsLmL
|C - B♯ - D - D♯ - F - E♯ - G - F𝄪 - A - A♯ - C
|Dorian
|-
|0
|sLsLmLsLmL
|C - B♯ - D - C𝄪 - E - E♯ - G - F𝄪 - A - A♯ - C
|Mixolydian
|}
=== 2L 3M 2s ===
The 2L 3M 2s scale is created by an alternating generator sequence, this time using the arto and tendo thirds.  It can be treated as an altered version of diatonic designed around extraclassical tonality.
2L 3M 2s has two different chiralities; the right-hand form can be found when the tendo third is stacked first, and the left-hand form when the arto third is stacked first.
{| class="wikitable"
|+Modes of Right-hand 3L 2M 2s
!Gens Up
!Pattern
!Notation
!Altered 5L 2s
|-
|6
|MLMsMLs
|C - D - ^E - ^F♯ - G - A - ^B - C
|Tendo Lydian
|-
|5
|MsMLsML
|C - D - vE♭ - vF - G - vA♭ - vB♭ - C
|Arto Aeolian
|-
|4
|MLsMLMs
|C - D - ^E - F - G - ^A - ^B - C
|Tendo Ionian
|-
|3
|sMLMsML
|C - ^D - vE♭ - F - G - vA♭ - vB♭ - C
|Arto Phrygian
|-
|2
|LMsMLsM
|C - vD♭ - ^E - F - G - ^A - B♭ - C
|Tendo Mixolydian
|-
|1
|sMLsMLM
|C - ^D - vE♭ - F - vG♭ - vA♭ - B♭ - C
|Arto Locrian
|-
|0
|LsMLMsM
|C - vD♭ - E♭ - F - ^G - ^A - B♭ - C
|Wolf Dorian
|}
{| class="wikitable"
|+Modes of Left-hand 2L 3M 2s
!Gens Up
!Pattern
!Notation
!Altered 5L 2s
|-
|6
|MsMLMsL
|C - D - vE♭ - vF - G - A - vB♭ - C
|Arto Dorian
|-
|5
|MLMsLMs
|C - D - ^E - ^F# - G - ^A - ^B - C
|Tendo Lydian
|-
|4
|MsLMsML
|C - D - vE♭ - F - G - vA♭ - vB♭ - C
|Arto Aeolian
|-
|3
|LMsMLMs
|C - ^D - ^E - F - G - ^A - ^B - C
|Tendo Ionian
|-
|2
|sMLMsLM
|C - vD♭ - vE♭ - F - G - vA♭ - B♭ - C
|Arto Phrygian
|-
|1
|LMsLMsM
|C - ^D - ^E - F - ^G - ^A - B♭ - C
|Tendo Mixolydian
|-
|0
|sLMsMLM
|C - vD♭ - E♭ - F - vG♭ -vA♭ - B♭ - C
|Wolf Locrian
|}
=== Chromatic Genus ===
The Chromatic Genus is a scale used by Ancient Greek musicians, and is built by [[Tetrachord|splitting the perfect fourth]] into a minor third and two semitones; these semitones could be equal in some tunings, but were typically not so.  29edo can approximate this scale in three main ways: the neutral variant where the semitones are both minor seconds, the arto variant where the semitones are both chromata, and the Pythagorean variant where one of each type of semitone is used.  The Pythagorean variant of the scale can be further divided into left-hand and right-hand variants, depending on the order in which the semitones are placed.
The arto tuning of the chromatic genus is particularly notable since the arto third bisects the fourth, and the chroma bisects the arto third; because of these relationships, the number of minor thirds that occur in the modes of that scale are increased, as the third can be reached by one large step or by two small steps.
Here, I will notate the step pattern using s for semitones, L for minor thirds, and | for the separation between the two disjunct tetrachords.
{| class="wikitable"
|+Modes of Neutral-Chromatic
!Pattern
!Notation
!Greek Name
|-
|<nowiki>ssL | ssL</nowiki>
|C - D♭ - E𝄫 - F - G - A♭ - B𝄫 - C
|Dorian
|-
|<nowiki>sL | ssLs</nowiki>
|C - D♭ - E - F♯ - G - A♭ - B - C
|Hypolydian
|-
|<nowiki>L | ssLss</nowiki>
|C - D♯ - E♯ - F - G♭ - A♯ - B - C
|Hypophrygian
|-
|<nowiki>| ssLssL</nowiki>
|C - D - E♭ - F♭ - G - A♭ - B𝄫 - C
|Hypodorian
|-
|<nowiki>ssLssL |</nowiki>
|C - D♭ - E𝄫 - F - G♭ - A𝄫 - B♭ - C
|Mixolydian
|-
|<nowiki>sLssL | s</nowiki>
|C - D♭ - E - F - G♭ - A - B - C
|Lydian
|-
|<nowiki>LssL | ss</nowiki>
|C - D♯ - E - F - G♯ - A♯ - B - C
|Phrygian
|}
{| class="wikitable"
|+Modes of Arto-Chromatic
!Pattern
!Notation
!Greek Name
|-
|<nowiki>ssL | ssL</nowiki>
|C - ^D♭ - vE♭ - F - G - ^A♭ - vB♭ - C
|Dorian
|-
|<nowiki>sL | ssLs</nowiki>
|C - ^D♭ - vE - vF♯ - G - ^A♭ - vB - C
|Hypolydian
|-
|<nowiki>L | ssLss</nowiki>
|C - ^D - ^E - vF♯ - G - ^A - vB - C
|Hypophrygian
|-
|<nowiki>| ssLssL</nowiki>
|C - D - ^E♭ - vF♯ - G - ^A♭ - vB♭ - C
|Hypodorian
|-
|<nowiki>ssLssL |</nowiki>
|C - ^D♭ - vE♭ - F - ^G♭ - vA♭ - B♭ - C
|Mixolydian
|-
|<nowiki>sLssL | s</nowiki>
|C - ^D♭ - vE - F - ^G♭ - vA - vB - C
|Lydian
|-
|<nowiki>LssL | ss</nowiki>
|C - ^D - vE - F - ^G - ^A - vB - C
|Phrygian
|}
{| class="wikitable"
|+Modes of Pyth-Chromatic
!Pattern
!Notation (LH)
!Notation (RH)
!Greek Name
|-
|<nowiki>ssL | ssL</nowiki>
|C - D♭ - ^E𝄫 - F - G - A♭ - ^B𝄫 - C
|C - ^D♭ - ^E𝄫 - F - G - ^A♭ - ^B𝄫 - C
|Dorian
|-
|<nowiki>sL | ssLs</nowiki>
|C - ^D♭ - E - F♯ - G - ^A♭ - B - C
|C - D♭ - vE - vF♯ - G - A♭ - vB - C
|Hypolydian
|-
|<nowiki>L | ssLss</nowiki>
|C - vD♯ - vE♯ - vF - G♭ - vA - vB - C
|C - vD♯ - vE♯ - F - G♭ - vA - B - C
|Hypophrygian
|-
|<nowiki>| ssLssL</nowiki>
|C - D - E♭ - ^F♭ - G - A♭ - ^B𝄫 - C
|C - D - ^E♭ - ^F♭ - G - ^A♭ - ^B𝄫 - C
|Hypodorian
|-
|<nowiki>ssLssL |</nowiki>
|C - D♭ - ^E𝄫 - F - G♭ - ^A𝄫 - B♭ - C
|C - ^D♭ - ^E𝄫 - F - ^G♭ - ^A𝄫 - B♭ - C
|Mixolydian
|-
|<nowiki>sLssL | s</nowiki>
|C - ^D♭ - E - F - ^G♭ - A - B - C
|C - D♭ - vE - F - G♭ - vA - vB - C
|Lydian
|-
|<nowiki>LssL | ss</nowiki>
|C - vD♯ - vE - F - vG♯ - vA♯ - vB - C
|C - vD♯ - E - F - vG♯ - vA♯ - B - C
|Phrygian
|}
== 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 [[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 (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 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.
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.
== 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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