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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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'''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.
[[File:29edo.png|thumb|29edo as the division of the fourth into 3 and 4 equal parts.]]
'''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 providing an accurate approximation of [[Pythagorean tuning]], and for tuning the [[perfect fourth]] to a highly divisible interval of 12 steps, allowing 29edo to support structures such as [[Porcupine]] and [[Negri]].


== Tuning Theory ==
== General theory ==
=== JI approximation ===
29edo excels at approximating prime 3, but it does not have good approximations to most of the following prime harmonics. However, while 5, 7, 11, and 13 are not tuned well individually, they are all flat by nearly the same amount, and therefore 29edo represents ratios between these primes, such as 13/11 or 7/5, very well. Looking further afield, we find primes 29 and 37 approximated rather well; it should be noted that [[32/29]] splits 4/3 in three, and [[37/32]] splits it in two, garnering these two primes important structural roles.


=== JI Approximation ===
If the patent approximations of 5, 7, 11, and 13 are used, since they are at least rather unambiguous, the accuracy of prime 3 and their shared flatness make 29edo, in fact, [[consistent]] to the [[15-odd-limit]]. We find that 19 and 23 share a flat tendency in common with them as well, and if we skip prime 17, the 23-odd-limit is tuned consistently.  
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.
{{Harmonics in ED|29|37|0}}
{| class="wikitable"
|+Approximations of prime harmonics in 29edo
! colspan="2" |Harmonic
!3
!5
!7
!11
!13
!17
!19
!23
!29
!31
|-
! rowspan="2" |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
|-
! colspan="2" |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 ==
== 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.
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; thus, the sharp and flat represent 3 steps 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.
To represent finer distinctions, up (^) and down (v) accidentals may be used to represent one step of 29edo. This system results in many equivalences in intervals: C𝄪, ^D, vE♭, and F𝄫 are all the same note corresponding to six steps above C.
{| class="wikitable"
|+
| colspan="5" |D
|-
|C𝄪
|^D
|vE♭
|F𝄫
|
|-
|^C𝄪
|vD♯
|E♭
|^F𝄫
|
|-
|
|D♯
|^E♭
|vF♭
|
|-
|
|^D♯
|vE
|F♭
|
|-
|
|vD𝄪
|E
|^F♭
|vG𝄫
|-
|
|D𝄪
|^E
|vF
|G𝄫
|-
| colspan="5" |F
|}
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.
{| 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
Line 182: Line 88:
|-
|-
|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
Line 188: Line 94:
|-
|-
|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 106:
|-
|-
|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
Line 206: Line 112:
|-
|-
|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
Line 212: Line 118:
|-
|-
|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 124:
|-
|-
|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 130:
|-
|-
|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
Line 230: Line 136:
|-
|-
|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
Line 236: Line 142:
|-
|-
|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
Line 242: Line 148:
|-
|-
|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
Line 249: Line 155:


=== 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]] (4L 3s) [[MOS]] scale can be found as a circle of supraminor thirds (augmented seconds), or via an "evened out" form of the Harmonic Minor scale. That is, one can derive a mode of smitonic through examining a harmonic minor mode (containing one augmented second, three major seconds, and three minor seconds), narrowing the augmented second by three steps to a major second, and distributing the three steps equally across the minor seconds. 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 173:
|-
|-
|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 179:
|-
|-
|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
Line 304: Line 210:


=== 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: add ups and downs''


{| class="wikitable sortable mw-collapsible"
{| class="wikitable sortable mw-collapsible"
Line 315: Line 223:
|-
|-
|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: add 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 323:


=== 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 augmented seventh, which exceeds the size of the octave by an edostep.  However, the octave can still be closed by employing one diminished third (equivalent to a downmajor second) 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.


{| class="wikitable sortable"
{| class="wikitable sortable"
Line 424: Line 334:
|-
|-
|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
|}
|}
== Multiples ==
As 29edo's chain of fifths is quite accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 29edo. The temperament they share is called Mystery, preserving 29edo's 2.3.7/5.11/5.13/5 subgroup.
=== 58edo ===
58edo splits the octave and perfect fifth in half, and provides a greatly improved 17th harmonic, along with improved mappings for 5, 11, 13, and especially 7, that all share a strong sharp tendency. It is a good tuning for temperaments such as [[Diaschismic]], based on the half-octave, and [[Hemififths]], based on the neutral third. It is also rather strong for the purposes of interval categorization schemes such as [[ADIN]] due to the fact that it has both [[neutral]] and [[interordinal]] intervals.
{{Harmonics in ED|58|37|0}}
=== 87edo ===
87edo splits the octave and perfect fifth into three. As 29edo's primes 5, 11, and 13 are close to 1/3 of a step off, 87edo tunes them near just, especially prime 5. Prime 7 (and especially the interval [[9/7]]) receives the most damage out of the 13-limit, but it is dented flatwards precisely enough to provide an essentially ideal tuning for [[Rodan]] while still maintaining 15-odd-limit consistency. Other important structures that 87edo supports include [[Kleismic]] and [[Didacus]].
{{Harmonics in ED|87|37|0}}
{{Navbox EDO}}
{{Cat|Edos}}

Latest revision as of 20:49, 8 March 2026

29edo as the division of the fourth into 3 and 4 equal parts.

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 providing an accurate approximation of Pythagorean tuning, and for tuning the perfect fourth to a highly divisible interval of 12 steps, allowing 29edo to support structures such as Porcupine and Negri.

General theory

JI approximation

29edo excels at approximating prime 3, but it does not have good approximations to most of the following prime harmonics. However, while 5, 7, 11, and 13 are not tuned well individually, they are all flat by nearly the same amount, and therefore 29edo represents ratios between these primes, such as 13/11 or 7/5, very well. Looking further afield, we find primes 29 and 37 approximated rather well; it should be noted that 32/29 splits 4/3 in three, and 37/32 splits it in two, garnering these two primes important structural roles.

If the patent approximations of 5, 7, 11, and 13 are used, since they are at least rather unambiguous, the accuracy of prime 3 and their shared flatness make 29edo, in fact, consistent to the 15-odd-limit. We find that 19 and 23 share a flat tendency in common with them as well, and if we skip prime 17, the 23-odd-limit is tuned consistently.

Approximation of prime harmonics in 29edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) 0.0 +1.5 -13.9 -17.1 -13.4 -12.9 +19.2 -7.9 -7.6 +4.9 +13.6 -3.1
Relative (%) 0.0 +3.6 -33.6 -41.3 -32.4 -31.3 +46.4 -19.0 -18.3 +11.9 +32.8 -7.4
Steps

(reduced)

29

(0)

46

(17)

67

(9)

81

(23)

100

(13)

107

(20)

119

(3)

123

(7)

131

(15)

141

(25)

144

(28)

151

(6)

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; thus, the sharp and flat represent 3 steps of 29edo.

To represent finer distinctions, up (^) and down (v) accidentals may be used to represent one step of 29edo. This system results in many equivalences in intervals: C𝄪, ^D, vE♭, and F𝄫 are all the same note corresponding to six steps above C.

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.

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 (4L 3s) MOS scale can be found as a circle of supraminor thirds (augmented seconds), or via an "evened out" form of the Harmonic Minor scale. That is, one can derive a mode of smitonic through examining a harmonic minor mode (containing one augmented second, three major seconds, and three minor seconds), narrowing the augmented second by three steps to a major second, and distributing the three steps equally across the minor seconds. 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: add 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: add 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 augmented seventh, which exceeds the size of the octave by an edostep. However, the octave can still be closed by employing one diminished third (equivalent to a downmajor second) 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.

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

Multiples

As 29edo's chain of fifths is quite accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 29edo. The temperament they share is called Mystery, preserving 29edo's 2.3.7/5.11/5.13/5 subgroup.

58edo

58edo splits the octave and perfect fifth in half, and provides a greatly improved 17th harmonic, along with improved mappings for 5, 11, 13, and especially 7, that all share a strong sharp tendency. It is a good tuning for temperaments such as Diaschismic, based on the half-octave, and Hemififths, based on the neutral third. It is also rather strong for the purposes of interval categorization schemes such as ADIN due to the fact that it has both neutral and interordinal intervals.

Approximation of prime harmonics in 58edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) 0.0 +1.5 +6.8 +3.6 +7.3 +7.7 -1.5 -7.9 -7.6 +4.9 -7.1 -3.1
Relative (%) 0.0 +7.2 +32.8 +17.3 +35.3 +37.4 -7.3 -38.0 -36.7 +23.7 -34.3 -14.8
Steps

(reduced)

58

(0)

92

(34)

135

(19)

163

(47)

201

(27)

215

(41)

237

(5)

246

(14)

262

(30)

282

(50)

287

(55)

302

(12)

87edo

87edo splits the octave and perfect fifth into three. As 29edo's primes 5, 11, and 13 are close to 1/3 of a step off, 87edo tunes them near just, especially prime 5. Prime 7 (and especially the interval 9/7) receives the most damage out of the 13-limit, but it is dented flatwards precisely enough to provide an essentially ideal tuning for Rodan while still maintaining 15-odd-limit consistency. Other important structures that 87edo supports include Kleismic and Didacus.

Approximation of prime harmonics in 87edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) 0.0 +1.5 -0.1 -3.3 +0.4 +0.9 +5.4 +5.9 +6.2 +4.9 -0.2 -3.1
Relative (%) 0.0 +10.8 -0.8 -24.0 +2.9 +6.2 +39.1 +43.0 +45.0 +35.6 -1.5 -22.2
Steps

(reduced)

87

(0)

138

(51)

202

(28)

244

(70)

301

(40)

322

(61)

356

(8)

370

(22)

394

(46)

423

(75)

431

(83)

453

(18)


View • Talk • EditEqual temperaments
EDOs
Macrotonal 5 • 6 • 7 • 8 • 9 • 10 • 11
12-23 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23
24-35 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35
36-47 36 • 37 • 39 • 40 • 41 • 43 • 44 • 45 • 46 • 47
48-59 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58
60-71 60 • 63 • 64 • 65 • 67 • 68 • 70
72-83 72 • 77 • 80 • 81
84-95 84 • 87 • 89 • 90 • 93 • 94
Large EDOs 99 • 104 • 106 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 217 • 224 • 239 • 270 • 306 • 311 • 612 • 665
Nonoctave equal temperaments
Tritave 4 • 9 • 13 • 17 • 26 • 39
Fifth 8 • 9 • 11 • 20
Other
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