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

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m Noted consideration for enharmonic equivalence in the ^♭II - I cadence
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Major rearrangement of interval table
 
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!Degree
!Degree
!Cents
!Cents
!Category
!Interval
!Notation
!Notation
!JI*
!Notes
!Notes
|-
|-
Line 16: Line 17:
|P1
|P1
|C
|C
|1/1
|
|
|-
|-
|1
|1
|41.379
|41.379
|A7
|A7 ([[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.
|-
|-
Line 28: Line 31:
|m2
|m2
|D♭
|D♭
|[[256/243]], [[22/21]]
|
|
|-
|-
|3
|3
|124.138
|124.138
|A1
|[[Chroma]] (A1)
|C♯
|C♯
|Distinct from the minor second.  This distinction nullifies the familiar enharmonic equivalences.
|[[15/14]], [[14/13]], [[29/27]]
|Distinct from the minor second, nullifying the familiar enharmonic equivalences.
|-
|-
|4
|4
|165.517
|165.517
|d3
|[[Porcupine|Quill]] (d3)
|B𝄪, E𝄫
|B𝄪, E𝄫
|[[11/10]], [[32/29]]
|
|
|-
|-
Line 46: Line 52:
|M2
|M2
|D
|D
|[[9/8]]
|
|
|-
|-
|6
|6
|248.276
|248.276
|[[Chthonic]]
|[[Chthonic]] ([[Extraclassical tonality|r3]])
|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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|m3
|m3
|E♭
|E♭
|[[32/27]], [[13/11]]
|
|
|-
|-
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|A2
|A2
|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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|d4
|d4
|F♭
|F♭
|
|[[36/29]]
|May be used as a neutral third, but does not bisect the perfect fifth.
|-
|-
|10
|10
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|M3
|M3
|E
|E
|[[81/64]], [[14/11]]
|
|
|-
|-
|11
|11
|455.172
|455.172
|[[Naiadic]]
|[[Naiadic]] ([[Extraclassical tonality|T3]])
|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.
|-
|-
Line 88: Line 101:
|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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|A3
|A3
|E♯
|E♯
|[[37/27]]
|
|
|-
|-
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|d5
|d5
|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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|A4
|A4
|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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|d6
|d6
|E𝄪, A𝄫
|E𝄪, A𝄫
|[[54/37]]
|
|
|-
|-
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|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]])
|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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|m6
|m6
|A♭
|A♭
|[[128/81]], [[11/7]]
|
|
|-
|-
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|A5
|A5
|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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|d7
|d7
|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.
|-
|-
Line 148: Line 171:
|M6
|M6
|A
|A
|[[27/16]], [[22/13]]
|
|
|-
|-
|23
|23
|951.724
|951.724
|[[Ouranic]]
|[[Ouranic]] ([[Extraclassical tonality|T6/r7]])
|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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|m7
|m7
|B♭
|B♭
|[[16/9]]
|
|
|-
|-
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|A6
|A6
|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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|d8
|d8
|C♭
|C♭
|[[13/7]], [[54/29]]
|
|
|-
|-
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|M7
|M7
|B
|B
|[[243/128]], [[21/11]]
|
|
|-
|-
|28
|28
|1158.621
|1158.621
|d2
|d2 ([[Extraclassical tonality|T7]])
|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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|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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|dd1 (C𝄫)
|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.
=== Harmonic Approximations ===
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.
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.
{| class="wikitable"
|+13-limit difference tones
! colspan="2" |Interval
!7/5
!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 ==
== Chords of 29edo ==
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|Primary consonance in 5L 2s
|Primary consonance in 5L 2s
|-
|-
|Upmajor
|Tendo
|C ^maj
|CT
|C - ^E - G
|C - ^E - G
|11\29 + 6\29
|11\29 + 6\29
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|
|
|-
|-
|Downminor
|Arto
|c vmin
|cr
|C - vE♭ - G
|C - vE♭ - G
|6\29 + 11\29
|6\29 + 11\29
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|
|
|-
|-
|Dietic Upmajor
|Dietic Tendo
|C ^maj ^5
|CT ^5
|C - ^E - ^G
|C - ^E - ^G
|11\29 + 7\29
|11\29 + 7\29
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|Primary consonance in 3L 4s
|Primary consonance in 3L 4s
|-
|-
|Dietic Downminor
|Dietic Arto
|c vmin ^5
|cr ^5
|C - vE♭ - ^G
|C - vE♭ - ^G
|6\29 + 12\29
|6\29 + 12\29
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|
|
|-
|-
|Wolf Upmajor
|Wolf Tendo
|C ^maj v5
|CT v5
|C - ^E - vG
|C - ^E - vG
|11\29 + 5\29
|11\29 + 5\29
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|
|
|-
|-
|Wolf Downminor
|Wolf Arto
|c vmin v5
|cr v5
|C - vE♭ - vG
|C - vE♭ - vG
|6\29 + 10\29
|6\29 + 10\29
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|
|
|-
|-
|Aug Upmajor
|Aug Tendo
|C ^maj ♯5
|CT ♯5
|C - ^E - G♯
|C - ^E - G♯
|11\29 + 9\29
|11\29 + 9\29
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|
|
|-
|-
|Aug Downminor
|Aug Arto
|c vmin ♯5
|cr ♯5
|C - vE♭ - G♯
|C - vE♭ - G♯
|6\29 + 14\29
|6\29 + 14\29
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|Primary consonance in 4L 5s
|Primary consonance in 4L 5s
|-
|-
|Dim Upmajor
|Dim Tendo
|C ^maj ♭5
|CT ♭5
|C - ^E - G♭
|C - ^E - G♭
|11\29 + 3\29
|11\29 + 3\29
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|
|
|-
|-
|Dim Downminor
|Dim Arto
|c vmin ♭5
|cr ♭5
|C - vE♭ - G♭
|C - vE♭ - G♭
|6\29 + 8\29
|6\29 + 8\29
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|-
|-
|Wolf Augchthonic
|Wolf Augchthonic
|C #ct ^4
|C ♯ct ^4
|C - D♯ - ^F
|C - D♯ - ^F
|8\29 + 5\29
|8\29 + 5\29
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|}
|}


== Scales of 29edo ==
=== 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 mw-collapsible"
{| class="wikitable sortable mw-collapsible"
|+ style="font-size: 105%;" |Modes of 5L 2s
|+ style="font-size: 105%;" |Modes of 5L 2s
Line 695: Line 843:


=== 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.


The mode names for this scale are given by [[User:Ayceman|Ayceman]].
The mode names for this scale are given by [[User:Ayceman|Ayceman]].
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|}
|}
===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.
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|}
|}
===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]].
Line 860: Line 1,010:


=== 3L 4s ===
=== 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.
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.
The modes names for this scale are given by [[Andrew Heathwaite]].  They can also be named by comparing two diatonic modes.
Line 915: Line 1,065:


=== 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 955: Line 1,105:
|C - E𝄫 - F♭ - G♭ - A♭ - B♭ - C
|C - E𝄫 - F♭ - G♭ - A♭ - B♭ - C
|Tsohorayim
|Tsohorayim
|}
=== 7L 1s ===
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 Willian Lynch.
{| class="wikitable sortable mw-collapsible"
|+Modes of 7L 1s
!Gens Up
!Step Pattern
!Notation
!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 ===
=== 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 [[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]].
The mode names for this scale are given by [[User:Inthar|Inthar]].
Line 970: Line 1,172:
|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 1,012: Line 1,214:
|C - ^C - ^D - E♭ - F - ^F - ^G - A♭ - B♭ - C
|C - ^C - ^D - E♭ - F - ^F - ^G - A♭ - B♭ - C
|Pyrrhian
|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
|}
|}


Line 1,024: Line 1,609:
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).
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 (23\29) can resolve outwards by a chroma to reach a perfect fourth, or outwards to reach a unison.  These paradigms can be reversed to account for the octave complements of those notes.
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 ===
=== 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.
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 vA and vB are separated by 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).
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.
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.
Line 1,036: 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:29edo]]
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