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

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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.
=== 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𝄫)
|}
== 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
|-
|Upmajor
|C ^maj
|C - ^E - G
|11\29 + 6\29
|
|-
|Upminor
|c ^min
|C - ^E♭ - G
|8\29 + 9\29
|
|-
|Downmajor
|C vmaj
|C - vE - G
|9\29 + 8\29
|
|-
|Downminor
|c vmin
|C - vE♭ - G
|6\29 + 11\29
|
|-
|Dietic Major
|C maj ^5
|C - E - ^G
|10\29 + 8\29
|
|-
|Dietic Minor
|c min ^5
|C - E♭ - ^G
|7\29 + 11\29
|
|-
|Dietic Upmajor
|C ^maj ^5
|C - ^E - ^G
|11\29 + 7\29
|Primary consonance in Nerevarine
|-
|Dietic Upminor
|c ^min ^5
|C - ^E♭ - ^G
|8\29 + 10\29
|Primary consonance in Vivecan
|-
|Dietic Downmajor
|C vmaj ^5
|C - vE - ^G
|9\29 + 9\29
|Primary consonance in 3L 4s
|-
|Dietic Downminor
|c vmin ^5
|C - vE♭ - ^G
|6\29 + 12\29
|
|-
|Wolf Major
|C maj v5
|C - E - vG
|10\29 + 6\29
|
|-
|Wolf Minor
|c min v5
|C - E♭ - vG
|7\29 + 9\29
|
|-
|Wolf Upmajor
|C ^maj v5
|C - ^E - vG
|11\29 + 5\29
|
|-
|Wolf Upminor
|c ^min v5
|C - ^E♭ - vG
|8\29 + 8\29
|Primary consonance in 4L 3s
|-
|Wolf Downmajor
|C vmaj v5
|C - vE - vG
|9\29 + 7\29
|
|-
|Wolf Downminor
|c vmin v5
|C - vE♭ - vG
|6\29 + 10\29
|
|-
|Aug Major
|C maj ♯5
|C - E - G♯
|10\29 + 10\29
|Primary consonance in 3L 5s and 5L 1s
|-
|Aug Minor
|c min ♯5
|C - E♭ - G♯
|7\29 + 13\29
|
|-
|Aug Upmajor
|C ^maj ♯5
|C - ^E - G♯
|11\29 + 9\29
|
|-
|Aug Upminor
|c ^min ♯5
|C - ^E♭ - G♯
|8\29 + 12\29
|
|-
|Aug Downmajor
|C vmaj ♯5
|C - vE - G♯
|9\29 + 11\29
|
|-
|Aug Downminor
|c vmin ♯5
|C - vE♭ - G♯
|6\29 + 14\29
|Upaug thirds are definitely pushing the definition of "thirds"
|-
|Dim Major
|C maj ♭5
|C - E - G♭
|10\29 + 4\29
|4\29 is a diminished third, so this is still tertian
|-
|Dim Minor
|c min ♭5
|C - E♭ - G♭
|7\29 + 7\29
|Primary consonance in 4L 5s
|-
|Dim Upmajor
|C ^maj ♭5
|C - ^E - G♭
|11\29 + 3\29
|Downdim thirds are definitely pushing the definition of "thirds"
|-
|Dim Upminor
|c ^min ♭5
|C - ^E♭ - G♭
|8\29 + 6\29
|
|-
|Dim Downmajor
|C vmaj ♭5
|C - vE - G♭
|9\29 + 5\29
|
|-
|Dim Downminor
|c vmin ♭5
|C - vE♭ - G♭
|6\29 + 8\29
|
|}
=== Chthonic Tetrads ===
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).
Whereas tertian harmony tends to be most practical in chords of three and four pitches, chthonic harmony tends to be most practical in chords of four or five.
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+Chthonic Tetrads
!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
|}


== Scales of 29edo ==
== Scales of 29edo ==
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===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.
{| class="wikitable sortable"
{| class="wikitable sortable mw-collapsible"
|+ style="font-size: 105%;" |Modes of 5L&nbsp;2s
|+ style="font-size: 105%;" |Modes of 5L&nbsp;2s
|-
|-
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===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&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.
{| class="wikitable sortable"
{| class="wikitable sortable mw-collapsible"
|+ style="font-size: 105%;" |Modes of 5L&nbsp;7s
|+ style="font-size: 105%;" |Modes of 5L&nbsp;7s
|-
|-
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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
|-
|-
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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
|-
|-
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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
|-
|-
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|C - B♯ - A𝄪♯ - E - D𝄪 - C𝄪𝄪 - G♯ - F𝄪♯ - C
|C - B♯ - A𝄪♯ - E - D𝄪 - C𝄪𝄪 - G♯ - F𝄪♯ - C
|Pawn
|Pawn
|}
=== 3L 4s ===
The first truly unheard-of scale that 29edo pulls off is its approximation of the neutral scale by stacking the downmajor third seven times.  Like 4L 3s, this scale uses harmony based on upfifths and downfifths rather than normal perfect fifths, which makes its harmony more distant from familiar structures.  Just like 5L 3s, it can be compared to the Tcherepnin scale, and as such it relies on augmented triads as its source of harmony; however, this scale pattern removes two of the nine Tcherepnin steps rather than three, reducing it to a more standard heptatonic form.
The modes names for this scale are given by [[Andrew Heathwaite]].  They can also be named by comparing two diatonic modes.
{| 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
|}
|}


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|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
|}
|}
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