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

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Major rearrangement of interval table
 
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!Degree
!Degree
!Cents
!Cents
!Native Fifths
!Interval
!Additional Categories
!Notation
!Notation
!JI*
!Notes
!Notes
|-
|-
Line 16: Line 16:
|0.000
|0.000
|P1
|P1
|
|C
|C
|1/1
|
|
|-
|-
|1
|1
|41.379
|41.379
|A7
|A7 ([[Diesis (interval region)|Diesis]])
|[[Diesis (interval region)|Diesis]]
|B♯
|B♯
|[[45/44]], [[50/49]], [[37/36]]
|Distinct from the octave.  Three major thirds reach this augmented seventh.
|Distinct from the octave.  Three major thirds reach this augmented seventh.
|-
|-
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|82.759
|82.759
|m2
|m2
|
|D♭
|D♭
|[[256/243]], [[22/21]]
|
|
|-
|-
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|124.138
|124.138
|[[Chroma]] (A1)
|[[Chroma]] (A1)
|Supraminor 2nd
|C♯
|C♯
|[[15/14]], [[14/13]], [[29/27]]
|Distinct from the minor second, nullifying the familiar enharmonic equivalences.
|Distinct from the minor second, nullifying the familiar enharmonic equivalences.
|-
|-
|4
|4
|165.517
|165.517
|d3
|[[Porcupine|Quill]] (d3)
|Submajor 2nd
|B𝄪, E𝄫
|B𝄪, E𝄫
|[[11/10]], [[32/29]]
|
|
|-
|-
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|206.897
|206.897
|M2
|M2
|
|D
|D
|[[9/8]]
|
|
|-
|-
|6
|6
|248.276
|248.276
|[[Chthonic]]
|[[Chthonic]] ([[Extraclassical tonality|r3]])
|[[Extraclassical tonality|Arto]] 3rd
|C𝄪, F𝄫
|C𝄪, F𝄫
|[[37/32]]
|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.
|-
|-
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|289.655
|289.655
|m3
|m3
|
|E♭
|E♭
|[[32/27]], [[13/11]]
|
|
|-
|-
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|331.034
|331.034
|A2
|A2
|Supraminor 3rd
|D♯
|D♯
|[[29/24]]
|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.
|-
|-
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|372.414
|372.414
|d4
|d4
|Submajor 3rd
|F♭
|F♭
|[[36/29]]
|May be used as a neutral third, but does not bisect the perfect fifth.
|May be used as a neutral third, but does not bisect the perfect fifth.
|-
|-
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|413.793
|413.793
|M3
|M3
|
|E
|E
|[[81/64]], [[14/11]]
|
|
|-
|-
|11
|11
|455.172
|455.172
|[[Naiadic]]
|[[Naiadic]] ([[Extraclassical tonality|T3]])
|[[Extraclassical tonality|Tendo]] 3rd
|D𝄪, G𝄫
|D𝄪, G𝄫
|[[13/10]], [[48/37]]
|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.
|-
|-
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|496.552
|496.552
|P4
|P4
|
|F
|F
|[[4/3]]
|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.
|-
|-
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|537.931
|537.931
|A3
|A3
|Wolf 4th
|E♯
|E♯
|[[37/27]]
|
|
|-
|-
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|579.310
|579.310
|d5
|d5
|Minor tritone
|G♭
|G♭
|[[7/5]]
|Distinct from the augmented fourth.  Two minor thirds reach this diminished fifth.
|Distinct from the augmented fourth.  Two minor thirds reach this diminished fifth.
|-
|-
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|620.690
|620.690
|A4
|A4
|Major tritone
|F♯
|F♯
|[[10/7]]
|Distinct from the diminished fifth.  Three whole tones reach this augmented fourth.
|Distinct from the diminished fifth.  Three whole tones reach this augmented fourth.
|-
|-
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|662.069
|662.069
|d6
|d6
|Wolf 5th
|E𝄪, A𝄫
|E𝄪, A𝄫
|[[54/37]]
|
|
|-
|-
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|703.448
|703.448
|P5
|P5
|
|G
|G
|[[3/2]]
|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.
|-
|-
|18
|18
|744.828
|744.828
|[[Cocytic]]
|[[Cocytic]] ([[Extraclassical tonality|r6]])
|Arto 6th
|F𝄪
|F𝄪
|[[20/13]], [[37/24]]
|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.
|-
|-
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|786.207
|786.207
|m6
|m6
|
|A♭
|A♭
|[[128/81]], [[11/7]]
|
|
|-
|-
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|827.586
|827.586
|A5
|A5
|Supraminor 6th
|G♯
|G♯
|[[29/18]]
|Distinct from the minor sixth.  Two major thirds reach this augmented fifth.
|Distinct from the minor sixth.  Two major thirds reach this augmented fifth.
|-
|-
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|868.966
|868.966
|d7
|d7
|Submajor 6th
|B𝄫
|B𝄫
|[[48/29]]
|Distinct from the major sixth.  Three minor thirds reach this diminished seventh.
|Distinct from the major sixth.  Three minor thirds reach this diminished seventh.
|-
|-
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|910.345
|910.345
|M6
|M6
|
|A
|A
|[[27/16]], [[22/13]]
|
|
|-
|-
|23
|23
|951.724
|951.724
|[[Ouranic]]
|[[Ouranic]] ([[Extraclassical tonality|T6/r7]])
|Tendo 6th/arto 7th
|G𝄪, C𝄫
|G𝄪, C𝄫
|[[64/37]]
|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.
|-
|-
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|993.103
|993.103
|m7
|m7
|
|B♭
|B♭
|[[16/9]]
|
|
|-
|-
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|1034.483
|1034.483
|A6
|A6
|Supraminor 7th
|A♯
|A♯
|[[20/11]], [[29/16]]
|[[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.
|-
|-
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|1075.862
|1075.862
|d8
|d8
|Submajor 7th
|C♭
|C♭
|[[13/7]], [[54/29]]
|
|
|-
|-
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|1117.241
|1117.241
|M7
|M7
|
|B
|B
|[[243/128]], [[21/11]]
|
|
|-
|-
|28
|28
|1158.621
|1158.621
|d2
|d2 ([[Extraclassical tonality|T7]])
|Tendo 7th
|A𝄪, D𝄫
|A𝄪, D𝄫
|[[88/45]], [[72/37]]
|Distinct from the octave.  Four minor thirds reach this diminished ninth.
|Distinct from the octave.  Four minor thirds reach this diminished ninth.
|-
|-
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|1200.000
|1200.000
|P8
|P8
|
|C
|C
|[[2/1]]
|
|
|}
|}
<small>*Assuming the 2.3.7/5.11/5.13/5.29.37 subgroup; see Harmonic Approximations for details.</small>
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.
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.


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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.
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 ===
=== Harmonic Approximations ===
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.
Critics of 29edo have often said that it fails to approximate most ratios beyond the 3-limit; primes 5, 7, 11, and 13 are all of relatively high error, and as such the tuning does not provide the necessary colors to suffice as an introductory system to tunings beyond 12edo.
{| class="wikitable sortable mw-collapsible mw-collapsed"
 
|+Tertian Triads
Assuming for the sake of the argument that approximating JI is somehow the only thing that one may want of a tuning, and further assuming that one even knows what that entails on their first venture into xenharmonic tunings, 29edo does not fall quite as flat as some would have you believe.  Notably, the aforementioned error on primes 5, 7, 11, and 13 are all in the same direction and of roughly the same amount, which leads their difference tones to be tuned quite well.
!Chord
{| class="wikitable"
!Symbol
|+13-limit difference tones
!Notation
! colspan="2" |Interval
!Formula
!7/5
!Notes
!11/5
!13/5
!11/7
!13/7
!13/11
|-
! rowspan="2" |Error
!Absolute (¢)
| -3.2
| +0.5
| +1.0
| +3.7
| +4.2
| +0.5
|-
!Relative (%)
| -7.7
| +1.2
| +2.3
| +8.9
| +10.0
| +1.1
|-
! colspan="2" |Steps
(Reduced)
|14
(14)
|33
(4)
|40
(11)
|19
(19)
|26
(26)
|7
(7)
|}
Because, for example, 11/7 can be found at (11/5)/(7/5), we only need to add three fractions to the subgroup; here, I'll use 7/5, 11/5, and 13/5.
 
Additionally, 4\29 can be interpreted as 32/29, adding prime 29 to the subgroup; this allows the chromatic semitone to be interpreted as 29/27, the supraminor third as 29/24, and the submajor third as 36/29.
 
Finally, the arto third / semifourth at 6\29 can be interpreted as 37/32, adding prime 37 to the subgroup; this allows the upfourth to be interpreted as 37/27, the tendo third as 48/37, and the diesis as 37/36.
 
So in total, our accurate subgroup for 29edo is '''2.3.7/5.11/5.13/5.29.37'''.  Not bad for a tuning that supposedly isn't useful beyond the 3-limit.
 
== 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
|Major
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=== 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 upminor thirds, 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.
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.
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|}
|}


=== 5L 4s ===
=== 7L 1s ===
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.
One interesting structural property of 29edo is that the perfect fourth, at twelve steps wide, is highly divisible.  This allows it to be split into various divisors that can be used to access harmonies that are all at once familiar and alien.  Specifically, splitting the fourth into three steps, we find a downmajor second and upminor third leading to it, and we can extend this pattern to create [[7L 1s|an 8-note scale]].


The mode names for this scale are given by [[User:Inthar|Inthar]].
The mode names for this scale are given by Willian Lynch.
{| class="wikitable sortable mw-collapsible"
{| class="wikitable sortable mw-collapsible"
|+Modes of 5L 4s
|+Modes of 7L 1s
!Gens Up
!Gens Up
!Step Pattern
!Step Pattern
!Notation
!Notation
!Name (Inthar)
!Name (Lynch)
|-
|7
|LLLLLLLs
|C - vD - ^E♭ - F - vG - ^A♭ - B♭ - vC - C
|Octopus
|-
|6
|LLLLLLsL
|C - vD - ^E♭ - F - vG - ^A♭ - B♭ - ^B♭ - C
|Mantis
|-
|5
|LLLLLsLL
|C - vD - ^E♭ - F - vG - ^A♭ - vA - ^B♭ - C
|Dolphin
|-
|4
|LLLLsLLL
|C - vD - ^E♭ - F - vG - G - vA - ^B♭ - C
|Crab
|-
|3
|LLLsLLLL
|C - vD - ^E♭ - F - ^F - G - vA - ^B♭ - C
|Tuna
|-
|2
|LLsLLLLL
|C - vD - ^E♭ - vE - ^F - G - vA - ^B♭ - C
|Salmon
|-
|1
|LsLLLLLL
|C - vD - D - vE - ^F - G - vA - ^B♭ - C
|Starfish
|-
|0
|sLLLLLLL
|C - ^C - D - vE - ^F - G - vA - ^B♭ - C
|Whale
|}
 
=== 5L 4s ===
The [[5L 4s]] scale follows from similar principles to 7L 1s, this time splitting the perfect fourth in two parts rather than three.  Continuing this sequence provides a 9-note scale that is significantly more alien than the previous, due to its repeated inclusion of true interordinal intervals.
 
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
|8
|LLsLsLsLs
|LLsLsLsLs
|C - D - E - ^E - ^F♯ - G - A - vB♭ - ^B - C
|C - D - E - ^E - ^F♯ - G - A - ^A - ^B - C
|Cristacan
|Cristacan
|-
|-
|7
|7
|LsLLsLsLs
|LsLLsLsLs
|C - D - ^D - ^E - ^F♯ - G - A - vB♭ - ^B - C
|C - D - ^D - ^E - ^F♯ - G - A - ^A - ^B - C
|Pican
|Pican
|-
|-
|6
|6
|LsLsLLsLs
|LsLsLLsLs
|C - D - ^D - ^E - F - G - A - vB♭ - ^B - C
|C - D - ^D - ^E - F - G - A - ^A - ^B - C
|Stellerian
|Stellerian
|-
|-
|5
|5
|LsLsLsLLs
|LsLsLsLLs
|C - D - ^D - ^E - F - G - ^G - vB♭ - ^B - C
|C - D - ^D - ^E - F - G - ^G - ^A - ^B - C
|Podocian
|Podocian
|-
|-
|4
|4
|LsLsLsLsL
|LsLsLsLsL
|C - D - ^D - ^E - F - G - ^G - vB♭ - B♭ - C
|C - D - ^D - ^E - F - G - ^G - ^A - B♭ - C
|Nucifragan
|Nucifragan
|-
|-
|3
|3
|sLLsLsLsL
|sLLsLsLsL
|C - ^C - ^D - ^E - F - G - ^G - vB♭ - B♭ - C
|C - ^C - ^D - ^E - F - G - ^G - ^A - B♭ - C
|Coracian
|Coracian
|-
|-
|2
|2
|sLsLLsLsL
|sLsLLsLsL
|C - ^C - ^D - E♭ - F - G - ^G - vB♭ - B♭ - C
|C - ^C - ^D - E♭ - F - G - ^G - ^A - B♭ - C
|Frugilegian
|Frugilegian
|-
|-
|1
|1
|sLsLsLLsL
|sLsLsLLsL
|C - ^C - ^D - E♭ - F - ^F - ^G - vB♭ - B♭ - C
|C - ^C - ^D - E♭ - F - ^F - ^G - ^A - B♭ - C
|Temnurial
|Temnurial
|-
|-
Line 1,215: Line 1,324:


=== 5L 2M 3s ===
=== 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: 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 submajor 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.
Line 1,512: Line 1,621:


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.
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]].
== See Also ==
* For my treatise on counterpoint in particular: [[User:Unque/29edo Counterpoint Treatise|29edo Counterpoint Treatise]]
* For my discussion of divisions of the fourth and other intervals: [[User:Unque/On Voice Leading|On Voice Leading]]


[[Category:Approaches to tuning systems]]
[[Category:Approaches to tuning systems]]
[[Category:29edo]]
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