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

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considered scales, chords, and intervals related to extraclassical tonality, plus a few additional scales
Line 20: Line 20:
|1
|1
|41.379
|41.379
|A7
|[[Diesis (interval region)|Diesis]]/A7
|B♯
|B♯
|Distinct from the octave.  Three major thirds reach this augmented seventh.
|Distinct from the octave.  Three major thirds reach this augmented seventh.
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|3
|3
|124.138
|124.138
|A1
|[[Chroma]] (A1)
|C♯
|C♯
|Distinct from the minor second.  This distinction nullifies the familiar enharmonic equivalences.
|Distinct from the minor second.  This distinction nullifies the familiar enharmonic equivalences.
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|6
|6
|248.276
|248.276
|[[Chthonic]]
|[[Chthonic]]/[[Extraclassical tonality|r3]]
|C𝄪, F𝄫
|C𝄪, F𝄫
|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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|d4
|d4
|F♭
|F♭
|
|May be used as a neutral third, but does not bisect the perfect fifth.
|-
|-
|10
|10
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|11
|11
|455.172
|455.172
|[[Naiadic]]
|[[Naiadic]]/[[Extraclassical tonality|T3]]
|D𝄪, G𝄫
|D𝄪, G𝄫
|New region in between M3 and P4; two of them make a major sixth.
|New region in between M3 and P4; two of them make a major sixth.
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|18
|18
|744.828
|744.828
|[[Cocytic]]
|[[Cocytic]]/[[Extraclassical tonality|r6]]
|F𝄪
|F𝄪
|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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|23
|23
|951.724
|951.724
|[[Ouranic]]
|[[Ouranic]]/[[Extraclassical tonality|T6]]/[[Extraclassical tonality|r7]]
|G𝄪, C𝄫
|G𝄪, C𝄫
|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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|28
|28
|1158.621
|1158.621
|d2
|d2/[[Extraclassical tonality|T7]]
|A𝄪, D𝄫
|A𝄪, D𝄫
|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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|}
|}


=== 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.
== 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===
Line 1,012: Line 1,072:
|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 downmajor 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 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, with 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 downmajor 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
|}
|}


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