29edo: Difference between revisions

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Created page with "'''29 equal divisions of the octave''', or '''29edo''', is the tuning system which divides the 2/1 ratio into 29 equal parts of approximately 41.3 cents each. It is notable for its extremely accurate tuning of prime 3, and for unique melodic properties that proponents of the system consider particularly desirable. == Tuning Theory == === JI Approximation === While 29edo excels at prime 3, the rest of the primes up to 31 are relatively lacking. However, primes 5 t..."
 
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Note that double-sharp / double-flat intervals always differ from a natural ordinal by a single step up or down.
Note that double-sharp / double-flat intervals always differ from a natural ordinal by a single step up or down.


== [[MOS]] Scales ==
== Diatonic scales ==


=== Diatonic ===
=== Neomajor, neominor (MOS diatonic) ===
The [[Diatonic]] 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).
The MOS [[Diatonic]] 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"
{| class="wikitable sortable mw-collapsible"
|+Modes of Diatonic
|+Modes of MOS Diatonic
|-
|-
!Gens Up
!Gens Up
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|-
|-
|6
|6
|LLLsLLs
|5-5-5-2-5-5-2
|C - D - E - F♯ - G - A - B - C
|C - D - E - F♯ - G - A - B - C
|Lydian
|Lydian
|-
|-
|5
|5
|LLsLLLs
|5-5-2-5-5-5-2
|C - D - E - F - G - A - B - C
|C - D - E - F - G - A - B - C
|Ionian
|Ionian
|-
|-
|4
|4
|LLsLLsL
|5-5-2-5-5-2-5
|C - D - E - F - G - A - B♭ - C
|C - D - E - F - G - A - B♭ - C
|Mixolydian
|Mixolydian
|-
|-
|3
|3
|LsLLLsL
|5-2-5-5-5-2-5
|C - D - E♭ - F - G - A - B♭ - C
|C - D - E♭ - F - G - A - B♭ - C
|Dorian
|Dorian
|-
|-
|2
|2
|LsLLsLL
|5-2-5-5-2-5-5
|C - D - E♭ - F - G - A♭ - B♭ - C
|C - D - E♭ - F - G - A♭ - B♭ - C
|Aeolian
|Aeolian
|-
|-
|1
|1
|sLLLsLL
|2-5-5-5-2-5-5
|C - D♭ - E♭ - F - G - A♭ - B♭ - C
|C - D♭ - E♭ - F - G - A♭ - B♭ - C
|Phrygian
|Phrygian
|-
|-
|0
|0
|sLLsLLL
|2-5-5-2-5-5-5
|C - D♭ - E♭ - F - G♭ - A♭ - B♭ - C
|C - D♭ - E♭ - F - G♭ - A♭ - B♭ - C
|Locrian
|Locrian
|}
|}
=== Submajor, supraminor (Zarlino diatonic) ===
''TODO: complete section''
=== Ultramajor, inframinor (Omnidiatonic) ===
''TODO: complete section''
== MOS scales ==


=== Chromatic ===
=== Chromatic ===
The Chromatic scale is an extension of the Diatonic scale, which can be found by continuing the sequence along the circle of fifths.  Because the circle can be traversed in two possible ways, scales can be extended in an "acute" direction or a "grave" direction.
The Chromatic scale is an extension of the MOS Diatonic scale, which can be found by continuing the sequence along the circle of fifths.  Because the circle can be traversed in two possible ways, scales can be extended in an "acute" direction or a "grave" direction.
 
''TODO: switch to ups and downs''
{| class="wikitable sortable mw-collapsible"
{| class="wikitable sortable mw-collapsible"
|+Modes of Chromatic
|+Modes of Chromatic
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|-
|-
|11
|11
|LsLsLssLsLss
|3-2-3-2-3-2-2-3-2-3-2-2
|C - C♯ - D - D♯ - E - E♯ - F♯ - G - G♯ - A - A♯ - B - C
|C - C♯ - D - D♯ - E - E♯ - F♯ - G - G♯ - A - A♯ - B - C
|Grave Lydian
|Grave Lydian
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|-
|-
|10
|10
|LsLssLsLsLss
|3-2-3-2-2-3-2-3-2-3-2-2
|C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - A♯ - B - C
|C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - A♯ - B - C
|Grave Ionian
|Grave Ionian
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|-
|-
|9
|9
|LsLssLsLssLs
|3-2-3-2-2-3-2-3-2-2-3-2
|C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - B♭ - B - C
|C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - B♭ - B - C
|Grave Mixolydian
|Grave Mixolydian
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|-
|-
|8
|8
|LssLsLsLssLs
|3-2-2-3-2-3-2-3-2-2-3-2
|C - C♯ - D - E♭ - E - F - F♯ - G - G♯ - A - B♭ - B - C
|C - C♯ - D - E♭ - E - F - F♯ - G - G♯ - A - B♭ - B - C
|Grave Dorian
|Grave Dorian
Line 200: Line 210:
|-
|-
|7
|7
|LssLsLssLsLs
|3-2-2-3-2-3-2-2-3-2-3-2
|C - C♯ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C
|C - C♯ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C
|Grave Aeolian
|Grave Aeolian
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|-
|-
|6
|6
|sLsLsLssLsLs
|2-3-2-3-2-3-2-2-3-2-3-2
|C - D♭ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C
|C - D♭ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C
|Grave Phrygian
|Grave Phrygian
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|-
|-
|5
|5
|sLsLssLsLsLs
|2-3-2-3-2-2-3-2-3-2-3-2
|C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - B - C
|C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - B - C
|Acute Ionian
|Acute Ionian
Line 218: Line 228:
|-
|-
|4
|4
|sLsLssLsLssL
|2-3-2-3-2-2-3-2-3-2-2-3
|C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - C♭ - C
|C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - C♭ - C
|Acute Mixolydian
|Acute Mixolydian
Line 224: Line 234:
|-
|-
|3
|3
|sLssLsLsLssL
|2-3-2-2-3-2-3-2-3-2-2-3
|C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - A - B♭ - C♭ - C
|C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - A - B♭ - C♭ - C
|Acute Dorian
|Acute Dorian
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|-
|-
|2
|2
|sLssLsLssLsL
|2-3-2-2-3-2-3-2-2-3-2-3
|C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C
|C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C
|Acute Aeolian
|Acute Aeolian
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|-
|-
|1
|1
|ssLsLsLssLsL
|2-2-3-2-3-2-3-2-2-3-2-3
|C - D♭ - E𝄫 - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C
|C - D♭ - E𝄫 - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C
|Acute Phrygian
|Acute Phrygian
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|-
|-
|0
|0
|ssLsLssLsLsL
|2-2-3-2-3-2-2-3-2-3-2-3
|C - D♭ - E𝄫 - E♭ - F♭ - F - G♭ - A𝄫 - A♭ - B𝄫 - B♭ - C♭ - C
|C - D♭ - E𝄫 - E♭ - F♭ - F - G♭ - A𝄫 - A♭ - B𝄫 - B♭ - C♭ - C
|Acute Locrian
|Acute Locrian
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=== Smitonic ===
=== Smitonic ===
The [[Smitonic]] scale can be found as a circle of augmented seconds, or via an "evened out" form of the Harmonic Minor scale.  The mode names for this scale are given by Ayceman.
The [[Smitonic]] scale can be found as a circle of supraminor thirds (augmented seconds), or via an "evened out" form of the Harmonic Minor scale.  The mode names for this scale are given by Ayceman.


{| class="wikitable sortable mw-collapsible"
{| class="wikitable sortable mw-collapsible"
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|-
|-
|6
|6
|LLsLsLs
|5-5-3-5-3-5-3
|C - D - E - ^F - ^G - vA - vB - C
|C - D - E - ^F - ^G - vA - vB - C
|Nerevarine
|Nerevarine
Line 267: Line 277:
|-
|-
|5
|5
|LsLLsLs
|5-3-5-5-3-5-3
|C - D - ^E♭ - ^F - ^G - vA - vB - C
|C - D - ^E♭ - ^F - ^G - vA - vB - C
|Vivecan
|Vivecan
Line 273: Line 283:
|-
|-
|4
|4
|LsLsLLs
|5-3-5-3-5-5-3
|C - D - ^E♭ - ^F - vG - vA - vB - C
|C - D - ^E♭ - ^F - vG - vA - vB - C
|Lorkhanic
|Lorkhanic
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|-
|-
|3
|3
|LsLsLsL
|5-3-5-3-5-3-5
|C - D - ^E♭ - ^F - vG - vA - B♭ - C
|C - D - ^E♭ - ^F - vG - vA - B♭ - C
|Sothic
|Sothic
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|-
|-
|2
|2
|sLLsLsL
|3-5-5-3-5-3-5
|C - ^D♭ - ^E♭ - ^F - vG - vA - B♭ - C
|C - ^D♭ - ^E♭ - ^F - vG - vA - B♭ - C
|Kagrenacan
|Kagrenacan
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|-
|-
|1
|1
|sLsLLsL
|3-5-3-5-5-3-5
|C - ^D♭ - ^E♭ - vF - vG - vA - B♭ - C
|C - ^D♭ - ^E♭ - vF - vG - vA - B♭ - C
|Almalexian
|Almalexian
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|-
|-
|0
|0
|sLsLsLL
|3-5-3-5-3-5-5
|C - ^D♭ - ^E♭ - vF - vG - A♭ - B♭ - C
|C - ^D♭ - ^E♭ - vF - vG - A♭ - B♭ - C
|Dagothic
|Dagothic
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=== Gramitonic ===
=== Gramitonic ===
The [[Gramitonic]] 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 nine-note scale.  Note how the four bright modes resemble the pattern of the familiar [[Tetrawood|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 [[Gramitonic]] scale takes the role of a diminished scale in 29edo: since four neominor thirds fall short of the octave, the chain of neominor thirds can be extended into this nine-note scale.  Note how the four bright modes resemble the pattern of the familiar [[Tetrawood|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.
 
''TODO: switch to ups and downs''


{| class="wikitable sortable mw-collapsible"
{| class="wikitable sortable mw-collapsible"
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|-
|-
|8
|8
|LsLsLsLss
|6-1-6-1-6-1-6-1-1
|C - F𝄫 - E♭ - A𝄫♭ - G♭ - C𝄫♭ - B𝄫 - E𝄫𝄫 - D𝄫 - C
|C - F𝄫 - E♭ - A𝄫♭ - G♭ - C𝄫♭ - B𝄫 - E𝄫𝄫 - D𝄫 - C
|Roi
|Roi
|-
|-
|7
|7
|LsLsLssLs
|6-1-6-1-6-1-1-6-1
|C - F𝄫 - E♭ - A𝄫♭ - G♭ - C𝄫♭ - B𝄫 - A - D𝄫 - C
|C - F𝄫 - E♭ - A𝄫♭ - G♭ - C𝄫♭ - B𝄫 - A - D𝄫 - C
|Steno
|Steno
|-
|-
|6
|6
|LsLssLsLs
|6-1-6-1-1-6-1-6-1
|C - F𝄫 - E♭ - A𝄫♭ - G♭ - F♯ - B𝄫 - A - D𝄫 - C
|C - F𝄫 - E♭ - A𝄫♭ - G♭ - F♯ - B𝄫 - A - D𝄫 - C
|Limni
|Limni
|-
|-
|5
|5
|LssLsLsLs
|6-1-1-6-1-6-1-6-1
|C - F𝄫 - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - D𝄫 - C
|C - F𝄫 - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - D𝄫 - C
|Telma
|Telma
|-
|-
|4
|4
|sLsLsLsLs
|1-6-1-6-1-6-1-6-1
|C - B♯ - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - D𝄫 - C
|C - B♯ - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - D𝄫 - C
|Krini
|Krini
|-
|-
|3
|3
|sLsLsLssL
|1-6-1-6-1-6-1-1-6
|C - B♯ - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - G𝄪 - C
|C - B♯ - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - G𝄪 - C
|Elos
|Elos
|-
|-
|2
|2
|sLsLssLsL
|1-6-1-6-1-1-6-1-6
|C - B♯ - E♭ - D♯ - G♭ - F♯ - E𝄪 - A - G𝄪 - C
|C - B♯ - E♭ - D♯ - G♭ - F♯ - E𝄪 - A - G𝄪 - C
|Mychos
|Mychos
|-
|-
|1
|1
|sLssLsLsL
|1-6-1-1-6-1-6-1-6
|C - B♯ - E♭ - D♯ - C𝄪♯ - F♯ - E𝄪 - A - G𝄪 - C
|C - B♯ - E♭ - D♯ - C𝄪♯ - F♯ - E𝄪 - A - G𝄪 - C
|Akti
|Akti
|-
|-
|0
|0
|ssLsLsLsL
|1-1-6-1-6-1-6-1-6
|C - B♯ - A𝄪♯ - D♯ - C𝄪♯ - F♯ - E𝄪 - A - G𝄪 - C
|C - B♯ - A𝄪♯ - D♯ - C𝄪♯ - F♯ - E𝄪 - A - G𝄪 - C
|Dini
|Dini
|}
|}


=== Checker ===
=== Checkertonic ===
Similarly to the minor third, the major third of 29edo also does not close at the octave, allowing us to create an [[Checkertonic|8-note augmented scale]].  Just like the previous "diminished" scale, notice how the three brightest modes resemble the bright mode of the [[Tcherepnin]] scale, with one of the nine steps omitted; the three darkest modes similarly resemble the dark mode of that scale; and the remaining two modes both resemble the symmetrical mode of Tcherepnin.  The mode names for this scale are given by R-4981.
Similarly to the neominor third, the neomajor third of 29edo also does not close at the octave, allowing us to create an [[Checkertonic|8-note augmented scale]].  Just like the previous "diminished" scale, notice how the three brightest modes resemble the bright mode of the [[Tcherepnin]] scale, with one of the nine steps omitted; the three darkest modes similarly resemble the dark mode of that scale; and the remaining two modes both resemble the symmetrical mode of Tcherepnin.  The mode names for this scale are given by R-4981.
 
''TODO: switch to ups and downs''


{| class="wikitable sortable mw-collapsible"
{| class="wikitable sortable mw-collapsible"
|+Modes of Checker
|+Modes of Checkertonic
|-
|-
!Gens Up
!Gens Up
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|-
|-
|7
|7
|LsLssLss
|8-1-8-1-1-8-1-1
|C - G𝄫♭ - F♭ - C𝄫♭ - B𝄫♭ - A♭ - E𝄫♭ - D𝄫 - C
|C - G𝄫♭ - F♭ - C𝄫♭ - B𝄫♭ - A♭ - E𝄫♭ - D𝄫 - C
|King
|King
|-
|-
|6
|6
|LssLsLss
|8-1-1-8-1-8-1-1
|C - G𝄫♭ - F♭ - E - B𝄫♭ - A♭ - E𝄫𝄫 - D𝄫 - C
|C - G𝄫♭ - F♭ - E - B𝄫♭ - A♭ - E𝄫𝄫 - D𝄫 - C
|Queen
|Queen
|-
|-
|5
|5
|LssLssLs
|8-1-1-8-1-1-8-1
|C - G𝄫♭ - F♭ - E - B𝄫♭ - A♭ - G♯ - D𝄫 - C
|C - G𝄫♭ - F♭ - E - B𝄫♭ - A♭ - G♯ - D𝄫 - C
|Marshall
|Marshall
|-
|-
|4
|4
|sLsLssLs
|1-8-1-8-1-1-8-1
|C - B♯ - F♭ - E - B𝄫♭ - A♭ - G♯ - D𝄫 - C
|C - B♯ - F♭ - E - B𝄫♭ - A♭ - G♯ - D𝄫 - C
|Cardinal
|Cardinal
|-
|-
|3
|3
|sLssLsLs
|1-8-1-1-8-1-8-1
|C - B♯ - F♭ - E - D𝄪 - A♭ - G♯ - D𝄫 - C
|C - B♯ - F♭ - E - D𝄪 - A♭ - G♯ - D𝄫 - C
|Rook
|Rook
|-
|-
|2
|2
|sLssLssL
|1-8-1-1-8-1-1-8
|C - B♯ - F♭ - E - D𝄪 - A♭ - G♯ - F𝄪♯ - C
|C - B♯ - F♭ - E - D𝄪 - A♭ - G♯ - F𝄪♯ - C
|Bishop
|Bishop
|-
|-
|1
|1
|ssLsLssL
|1-1-8-1-8-1-1-8
|C - B♯ - A𝄪♯ - E - D𝄪 - A♭ - G♯ - F𝄪♯ - C
|C - B♯ - A𝄪♯ - E - D𝄪 - A♭ - G♯ - F𝄪♯ - C
|Knight
|Knight
|-
|-
|0
|0
|ssLssLsL
|1-1-8-1-1-8-1-8
|C - B♯ - A𝄪♯ - E - D𝄪 - C𝄪𝄪 - G♯ - F𝄪♯ - C
|C - B♯ - A𝄪♯ - E - D𝄪 - C𝄪𝄪 - G♯ - F𝄪♯ - C
|Pawn
|Pawn
Line 413: Line 427:


=== Machinoid ===
=== Machinoid ===
Just like the thirds, we can notice that the whole tones in 29edo do not close at the octave; instead, we see that six whole tones reach an up-octave, not a perfect octave.  However, the octave can still be closed by employing one diminished third to act as a "wolf" version of the whole tone; this leads to [[Machinoid]], a whole tone scale that has six distinct modes.  The mode names for this scale are given by Lilly Flores.
Just like the thirds, we can notice that the whole tones in 29edo do not close at the octave; instead, we see that six whole tones reach an up-octave, not a perfect octave.  However, the octave can still be closed by employing one downmajor second (diminished third) to act as a "wolf" version of the whole tone; this leads to [[Machinoid]], a whole tone scale that has six distinct modes.  The mode names for this scale are given by Lilly Flores.
 
''TODO: switch to ups and downs''


{| class="wikitable sortable"
{| class="wikitable sortable"
Line 424: Line 440:
|-
|-
|5
|5
|LLLLLs
|5-5-5-5-5-4
|C - D - E - F♯ - G♯ - A♯ - C
|C - D - E - F♯ - G♯ - A♯ - C
|Erev
|Erev
|-
|-
|4
|4
|LLLLsL
|5-5-5-5-4-5
|C - D - E - F♯ - G♯ - B♭ -C
|C - D - E - F♯ - G♯ - B♭ -C
|Oplen
|Oplen
|-
|-
|3
|3
|LLLsLL
|5-5-5-4-5-5
|C - D - E - F♯ - A♭ - B♭ - C
|C - D - E - F♯ - A♭ - B♭ - C
|Layla
|Layla
|-
|-
|2
|2
|LLsLLL
|5-5-4-5-5-5
|C - D - E - G♭ - A♭ - B♭ - C
|C - D - E - G♭ - A♭ - B♭ - C
|Shemesh
|Shemesh
|-
|-
|1
|1
|LsLLLL
|5-4-5-5-5-5
|C - D - F♭ - G♭ - A♭ - B♭ - C
|C - D - F♭ - G♭ - A♭ - B♭ - C
|Boqer
|Boqer
|-
|-
|0
|0
|sLLLLL
|4-5-5-5-5-5
|C - E𝄫 - F♭ - G♭ - A♭ - B♭ - C
|C - E𝄫 - F♭ - G♭ - A♭ - B♭ - C
|Tsohorayim
|Tsohorayim
|}
|}

Revision as of 02:48, 13 January 2026

29 equal divisions of the octave, or 29edo, is the tuning system which divides the 2/1 ratio into 29 equal parts of approximately 41.3 cents each. It is notable for its extremely accurate tuning of prime 3, and for unique melodic properties that proponents of the system consider particularly desirable.

Tuning Theory

JI Approximation

While 29edo excels at prime 3, the rest of the primes up to 31 are relatively lacking. However, primes 5 through 13 have roughly the same amount of error, and in the same direction; difference tones such as 7/5 and 11/7 are thus rather accurate.

Approximations of prime harmonics in 29edo
Harmonic 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +1.5 -13.9 -17.1 -13.4 -12.9 +19.2 -7.9 -7.6 +4.9 +13.6
Relative (%) +3.6 -33.6 -41.3 -32.4 -31.3 +46.4 -19.0 -18.3 +11.9 +32.8
Steps

(reduced)

46

(17)

67

(9)

81

(23)

100

(13)

107

(20)

119

(3)

123

(7)

131

(15)

141

(25)

144

(28)

Intervals and Notation

Due to its accurate tuning of prime 3, 29edo can be notated quite cleanly with the familiar circle of fifths; the whole tone is five steps of 29edo, the diatonic semitone is two steps, and the chromatic semitone is three steps. Intervals with double-flats and double-sharps may alternatively be written using ups and downs, smaller accidentals which represent one step of 29edo.

To display how enharmonic equivalences behave, below is a table of the many ways to notate the six notes between D to F, which may be considered to roughly constitute the range of thirds above a tonic C.

D
C𝄪 ^D vE♭ F𝄫
^C𝄪 vD♯ E♭ ^F𝄫
D♯ ^E♭ vF♭
^D♯ vE F♭
vD𝄪 E ^F♭ vG𝄫
D𝄪 ^E vF G𝄫
F

Note that double-sharp / double-flat intervals always differ from a natural ordinal by a single step up or down.

Diatonic scales

Neomajor, neominor (MOS diatonic)

The MOS Diatonic 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).

Modes of MOS Diatonic
Gens Up Step Pattern Notation Name
6 5-5-5-2-5-5-2 C - D - E - F♯ - G - A - B - C Lydian
5 5-5-2-5-5-5-2 C - D - E - F - G - A - B - C Ionian
4 5-5-2-5-5-2-5 C - D - E - F - G - A - B♭ - C Mixolydian
3 5-2-5-5-5-2-5 C - D - E♭ - F - G - A - B♭ - C Dorian
2 5-2-5-5-2-5-5 C - D - E♭ - F - G - A♭ - B♭ - C Aeolian
1 2-5-5-5-2-5-5 C - D♭ - E♭ - F - G - A♭ - B♭ - C Phrygian
0 2-5-5-2-5-5-5 C - D♭ - E♭ - F - G♭ - A♭ - B♭ - C Locrian

Submajor, supraminor (Zarlino diatonic)

TODO: complete section

Ultramajor, inframinor (Omnidiatonic)

TODO: complete section

MOS scales

Chromatic

The Chromatic scale is an extension of the MOS Diatonic scale, which can be found by continuing the sequence along the circle of fifths. Because the circle can be traversed in two possible ways, scales can be extended in an "acute" direction or a "grave" direction.

TODO: switch to ups and downs

Modes of Chromatic
Gens Up Step Pattern Notation Name Notes
11 3-2-3-2-3-2-2-3-2-3-2-2 C - C♯ - D - D♯ - E - E♯ - F♯ - G - G♯ - A - A♯ - B - C Grave Lydian Like the seven-note Lydian, lacks a Perfect Fourth over the root.
10 3-2-3-2-2-3-2-3-2-3-2-2 C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - A♯ - B - C Grave Ionian
9 3-2-3-2-2-3-2-3-2-2-3-2 C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - B♭ - B - C Grave Mixolydian
8 3-2-2-3-2-3-2-3-2-2-3-2 C - C♯ - D - E♭ - E - F - F♯ - G - G♯ - A - B♭ - B - C Grave Dorian
7 3-2-2-3-2-3-2-2-3-2-3-2 C - C♯ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C Grave Aeolian
6 2-3-2-3-2-3-2-2-3-2-3-2 C - D♭ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C Grave Phrygian Also accounts for Acute Lydian
5 2-3-2-3-2-2-3-2-3-2-3-2 C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - B - C Acute Ionian Also accounts for Grave Locrian
4 2-3-2-3-2-2-3-2-3-2-2-3 C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - C♭ - C Acute Mixolydian
3 2-3-2-2-3-2-3-2-3-2-2-3 C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - A - B♭ - C♭ - C Acute Dorian
2 2-3-2-2-3-2-3-2-2-3-2-3 C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C Acute Aeolian
1 2-2-3-2-3-2-3-2-2-3-2-3 C - D♭ - E𝄫 - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C Acute Phrygian
0 2-2-3-2-3-2-2-3-2-3-2-3 C - D♭ - E𝄫 - E♭ - F♭ - F - G♭ - A𝄫 - A♭ - B𝄫 - B♭ - C♭ - C Acute Locrian Like the seven-note Locrian, lacks a Perfect Fifth over the root.

Smitonic

The Smitonic scale can be found as a circle of supraminor thirds (augmented seconds), or via an "evened out" form of the Harmonic Minor scale. The mode names for this scale are given by Ayceman.

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

Gramitonic

The Gramitonic scale takes the role of a diminished scale in 29edo: since four neominor thirds fall short of the octave, the chain of neominor thirds can be extended into this nine-note scale. 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.

TODO: switch to ups and downs

Modes of Gramitonic
Gens Up Step Pattern Notation Name (Flores)
8 6-1-6-1-6-1-6-1-1 C - F𝄫 - E♭ - A𝄫♭ - G♭ - C𝄫♭ - B𝄫 - E𝄫𝄫 - D𝄫 - C Roi
7 6-1-6-1-6-1-1-6-1 C - F𝄫 - E♭ - A𝄫♭ - G♭ - C𝄫♭ - B𝄫 - A - D𝄫 - C Steno
6 6-1-6-1-1-6-1-6-1 C - F𝄫 - E♭ - A𝄫♭ - G♭ - F♯ - B𝄫 - A - D𝄫 - C Limni
5 6-1-1-6-1-6-1-6-1 C - F𝄫 - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - D𝄫 - C Telma
4 1-6-1-6-1-6-1-6-1 C - B♯ - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - D𝄫 - C Krini
3 1-6-1-6-1-6-1-1-6 C - B♯ - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - G𝄪 - C Elos
2 1-6-1-6-1-1-6-1-6 C - B♯ - E♭ - D♯ - G♭ - F♯ - E𝄪 - A - G𝄪 - C Mychos
1 1-6-1-1-6-1-6-1-6 C - B♯ - E♭ - D♯ - C𝄪♯ - F♯ - E𝄪 - A - G𝄪 - C Akti
0 1-1-6-1-6-1-6-1-6 C - B♯ - A𝄪♯ - D♯ - C𝄪♯ - F♯ - E𝄪 - A - G𝄪 - C Dini

Checkertonic

Similarly to the neominor third, the neomajor third of 29edo also does not close at the octave, allowing us to create an 8-note augmented scale. Just like the previous "diminished" scale, notice how the three brightest modes resemble the bright mode of the Tcherepnin scale, with one of the nine steps omitted; the three darkest modes similarly resemble the dark mode of that scale; and the remaining two modes both resemble the symmetrical mode of Tcherepnin. The mode names for this scale are given by R-4981.

TODO: switch to ups and downs

Modes of Checkertonic
Gens Up Step Pattern Notation Name (R-4981)
7 8-1-8-1-1-8-1-1 C - G𝄫♭ - F♭ - C𝄫♭ - B𝄫♭ - A♭ - E𝄫♭ - D𝄫 - C King
6 8-1-1-8-1-8-1-1 C - G𝄫♭ - F♭ - E - B𝄫♭ - A♭ - E𝄫𝄫 - D𝄫 - C Queen
5 8-1-1-8-1-1-8-1 C - G𝄫♭ - F♭ - E - B𝄫♭ - A♭ - G♯ - D𝄫 - C Marshall
4 1-8-1-8-1-1-8-1 C - B♯ - F♭ - E - B𝄫♭ - A♭ - G♯ - D𝄫 - C Cardinal
3 1-8-1-1-8-1-8-1 C - B♯ - F♭ - E - D𝄪 - A♭ - G♯ - D𝄫 - C Rook
2 1-8-1-1-8-1-1-8 C - B♯ - F♭ - E - D𝄪 - A♭ - G♯ - F𝄪♯ - C Bishop
1 1-1-8-1-8-1-1-8 C - B♯ - A𝄪♯ - E - D𝄪 - A♭ - G♯ - F𝄪♯ - C Knight
0 1-1-8-1-1-8-1-8 C - B♯ - A𝄪♯ - E - D𝄪 - C𝄪𝄪 - G♯ - F𝄪♯ - C Pawn

Machinoid

Just like the thirds, we can notice that the whole tones in 29edo do not close at the octave; instead, we see that six whole tones reach an up-octave, not a perfect octave. However, the octave can still be closed by employing one downmajor second (diminished third) to act as a "wolf" version of the whole tone; this leads to Machinoid, a whole tone scale that has six distinct modes. The mode names for this scale are given by Lilly Flores.

TODO: switch to ups and downs

Modes of Machinoid
Gens Up Step Pattern Notation Name (Flores)
5 5-5-5-5-5-4 C - D - E - F♯ - G♯ - A♯ - C Erev
4 5-5-5-5-4-5 C - D - E - F♯ - G♯ - B♭ -C Oplen
3 5-5-5-4-5-5 C - D - E - F♯ - A♭ - B♭ - C Layla
2 5-5-4-5-5-5 C - D - E - G♭ - A♭ - B♭ - C Shemesh
1 5-4-5-5-5-5 C - D - F♭ - G♭ - A♭ - B♭ - C Boqer
0 4-5-5-5-5-5 C - E𝄫 - F♭ - G♭ - A♭ - B♭ - C Tsohorayim
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