User:Unque/29edo Counterpoint Treatise: Difference between revisions

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~~'''Note: This page is currently under construction, and will be subject to major expansion in the near future. Come back soon!'''~~

On this page, I attempt to document in full my experience with writing counterpoint using [[29edo]] and the 29-form, with extrapolations that I've made into guidelines of a sort, which others may use as a jumping-off point for their own excursions into writing counterpoint, whether in 29edo itself or in other systems where these ideas or extrapolations therefrom may apply. These guidelines arise in large part from a combination of the writings of [[wikipedia:Johann_Joseph_Fux|Johann Fux]] and [[wikipedia:Marchetto_da_Padova|Marchetto da Padova]], analysis of the works of [[wikipedia:Johann_Sebastian_Bach|J.S. Bach]], and parameters that arose simply from trial and error in my own composition.

== Preface on Notation ==

A few notable features will be considered for this page. Intervals will be noted using extended diatonic notation (with ♯/♭ representing raising and lowering by a chromatic semitone, and ^/v representing raising or lowering by a diesis). Intervals included in diatonic modes will be noted with their diatonic names (where "major" and "minor" forms of an interval are respectively the wider and narrower interval that fall on a given degree), and intervals outside those modes will be denoted as "super-"/"supra-" and "sub-" forms of the closest diatonic interval. Where necessary, interval qualities may be abbreviated: m for minor, M for major, s for sub-, and S for super-/supra-.

A few notable notational features will be considered for this page. Intervals will be noted using extended diatonic notation (with ♯/♭ representing raising and lowering by a chromatic semitone, and ^/v representing raising or lowering by a diesis). Intervals included in diatonic modes will be noted with their diatonic names (where "major" and "minor" forms of an interval are respectively the wider and narrower interval that fall on a given degree), and intervals outside those modes will be denoted as "super-"/"supra-" and "sub-" forms of the closest diatonic interval. Where necessary, interval qualities may be abbreviated: m for minor, M for major, s for sub-, and S for super-/supra-. EDOstep notation (where n\29 represents n steps of 29edo, not to be confused with the JI interval n/29) may also be used for clarification when extended diatonic notation may provide difficulty to parse.

Sheet music examples, however, will employ the fewest number of ups and downs necessary; for instance, C♭ will be favored over ^B, though in the key of C this note will still be called the submajor seventh rather than the diminished octave. EDOstep notation (where n\29 represents n steps of 29edo, not to be confused with the JI interval n/29) may also be used for clarification when extended diatonic notation may provide difficulty to parse.

Interval equivalences created by 29edo will be considered equivalent here: the ~~interval at 6\29,~~ for instance, will be ~~noted~~ as a "subminor third" ~~even if spelled in sheet music as a~~ doubly augmented unison or a doubly diminished fourth, ~~because this name is more descriptive and easier to parse when considering an equal temperament sequence.~~

Interval equivalences created by 29edo will be considered equivalent here. Where multiple spellings are available in notation, sharps/flats will be prioritized over ups/downs, as it makes clear the movement by chromatic semitones; for instance, C♭ will be favored over vB. However, extended diatonic names will be favored over fifthwise names when possible, as these names are easier to parse when discussing interval sizes; for instance, "supermajor second" or "subminor third" will be favored over "doubly augmented unison" or "doubly diminished fourth," even if the interval using spellings such as C♯-F♭ or C♭-C♯

== General Principles ==

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=== Tonics and Resolution ===

A phrase should begin and end with all voices on the tonic, or on a perfect consonance with respect to it. At the end of a phrase, the tonic is to be approached from above or below by a conjunction (see Interval Categories). The chromatic semitone is primarily encouraged, but other resolutions are also possible.

A phrase should begin and end with all voices on the tonic, or on a perfect consonance with respect to it, unless employing a specific tonal resolution chord. At the end of a phrase, the tonic is to be approached from above or below by a conjunction (see Interval Categories). The chromatic semitone is primarily encouraged, but other resolutions are also possible.

=== Interval Categories ===

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#* Perfect Fourth (12\29)

# '''Imperfect consonances''' are permitted as melodic steps. Between voices, imperfect consonances may be approached through any means, and as such are encouraged as the main class of interval to use in the middle of phrases.

#* Major Second (5\29)†

#* Minor third (7\29)

#* Major third (10\29)

# '''Cadential consonances''' are permitted as melodic steps, but should be used with reserve. Between voices, cadential consonances may be approached through any means, but they tend to resolve by contrary motion to a perfect consonance.

#* Subminor Third (6\29)

#* Supermajor Second / Subminor Third (6\29)†

#* Supraminor Third (8\29)

#* Submajor Third (9\29)

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#* Supraminor Second / Chromatic Semitone (3\29)

#* Submajor Second (4\29)

#* Major Second (5\29)

#* Major Second (5\29)†

#* Supermajor Second / Subminor Third (6\29)†

#'''Dissonances''' are not permitted as melodic steps, and should generally be avoided between any two voices.

#*Augmented Third (13\29)

#*Diminished Fifth (14\29)

# The '''Diesis''' or '''Superunison''' (1\29) is permitted as a passing tone in melodies. Two voices should never be separated by the interval of a diesis except as a brief transitional position between a unison and another interval. The same guidelines apply to the Suboctave (28\29) or the Superoctave (30\29).

# The '''Diesis''' or '''Superunison''' (1\29) is permitted only as a passing tone in melodies; a voice should strongly avoid movement by a diesis on a strong beat, as the small size can destabilize the tonality. Two voices should never be separated by the interval of a diesis except as a brief transitional position between a unison and another interval. The same guidelines apply to the Suboctave (28\29) or the Superoctave (30\29).

†Note the overlap between certain categories: the Major Second (5\29) may function either as an imperfect consonance or a conjunction; the Supermajor Second (6\29), or Subminor Third, similarly may function as either a cadential consonance or a conjunction.

=== Contour ===

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The voices within a fugue should form a clear hierarchy of range; one voice should remain the lowest, another the highest, and so on for any additional voices. Two adjacent voices may play a unison, but a lower voice may not reach a pitch higher than the voice above it.

The interval between two adjacent voices generally should ~~not exceed~~ a Supermajor Tenth (40\29) unless absolutely necessary to avoid dissonances or provide melodic coherence within each individual voice.

The interval between two adjacent voices generally should avoid exceeding a Supermajor Tenth (40\29) unless absolutely necessary to avoid dissonances or provide melodic coherence within each individual voice.

=== Cadences ===

When resolving at the end of a phrase, cadences should generally involve two or more voices moving by conjunctions in opposite directions. One or more of the voices should land on the tonic pitch, and all others should land on perfect consonances with respect to one another.

When resolving at the end of a phrase, cadences should generally involve two or more voices moving by conjunctions in opposite directions. One or more of the voices should land on the tonic pitch, and all others should land on perfect consonances with respect to one another; where there are three or more voices, specific tonal chords may apply as well.

In 29edo, the intervals between two voices should generally begin on perfect consonances, move towards imperfect consonances, and finally utilize cadential consonances to create tension before resolving. The interordinal intervals (supermajor and subminor thirds and sixths) are extremely tense, and each one either exceeds or falls short of a perfect consonance by two chromatic semitones (a total of 6\29). This creates an extremely clear path to resolution via two types of movement:

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Note also that all four types of interordinal intervals have at least one type of resolution available. As such, the tension between two voices can be determined plainly by the type of cadence and the quality of the resolution, and conversely, the resolution can be determined plainly by the type of cadence and the quality of the tension.

{| class="wikitable"

|+Determining ~~tensions~~ for known resolutions

|+Determining cadential intervals for known resolutions

!Cadence Type

!Unison

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|}

{| class="wikitable"

|+Determining resolutions for known ~~tensions~~

|+Determining resolutions for known cadential intervals

!Cadence Type

!Subminor Third

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Due to the 29-form's inherent pull towards 3-limit structures, the diatonic scale is an integral part of its structure. The tonality of a fugue is most intuitively derived from the modes of the diatonic scale; specifically, the major and minor tonality respectively draw from the properties of the major and minor forms of the third degree of that scale.

No matter the tonality, perfect consonances with respect to the root take precedence over all other intervals, and should be the most emphasized intervals in a given melody. Subjects should be written primarily, if not entirely, in the diatonic scale, though the exact mode used depends on the tonality; ~~however~~, ~~countermelodies~~ and elaborations on the subject should access additional tones outside of the diatonic scale to create more diverse harmonies and eventually use interordinals as final tensions to create cadences.

No matter the tonality, perfect consonances with respect to the root take precedence over all other intervals, and should be the most emphasized intervals in a given melody. Subjects should be written primarily, if not entirely, in the diatonic scale, though the exact mode used depends on the tonality; exceptions are made for the notes a chromatic semitone above or below the root, in order for the melody to form aforementioned cadences when a countermelody is added. Countermelodies and elaborations on the subject, however, should access additional tones outside of the diatonic scale to create more diverse harmonies and eventually use interordinals as final tensions to create cadences.

=== Major Tonality ===

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In the major tonality, expanding cadences are encouraged when resolving to fourths, and contracting cadences are encouraged when resolving to fifths. This motion ensures that the tonic pitch is approached from below rather than from above, which creates a brighter sound more indicative of a major key.

When using more than two distinct voices, it is often desirable to use fuller chords for resolutions at the end of a phrase, rather than simply doubling the root or fifth across octaves. The following chords are considered for major resolutions, listed in order of consonance potential:

When using more than two distinct voices, it is often desirable to use fuller chords for resolutions at the end of a phrase, rather than simply doubling the root or fifth across octaves. Ideally, these fuller resolutions should be constructed such that the interval between any two pitches is a perfect or imperfect consonance; . The following chords are considered for major resolutions, listed in order of consonance potential:

* Requiring three or more voices:

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** 1 - P4 - P5 - M6

** 1 - M2 - P5 - M6

[[File:3-voice Contracting Cadence.png|thumb|An example of a three-voice cadence. Notice how the lower two voices create ~~a contracting~~ cadence into the perfect fourth, while the highest voice moves in parallel with the middle voice.]]

[[File:3-voice Contracting Cadence.png|thumb|An example of a three-voice cadence. Notice how the lower two voices create an expanding cadence into the perfect fourth, while the highest voice moves in parallel with the middle voice.]]

Inversions of these chords may apply; for instance, M3 - P5 - P8 is considered to have the same consonance potential as 1 - M3 - P5. It should be noted, however, that the highest and lowest voice will be the most prominent in a given chord, and it is thus encouraged that those voices play the root or fifth of the key in resolutions when possible.

Tensions towards these resolutions can be made by combining types of expanding and contracting cadences; for instance, the chord 1 - M3 - P5 may be approached by a tension ~~SM7 - S4 - sm6 or~~ SM7 - m3 - sm6, as each voice in the tension differs from its resolution by one chroma.

Tensions towards these resolutions can be made by combining types of expanding and contracting cadences; for instance, the chord 1 - M3 - P5 may be approached by a tension such as SM7 - m3 - sm6, where each voice in the tension differs from its resolution by one chroma.

Finally, and perhaps most importantly, resolutions with lower consonance potential should be avoided as the final resolution of the piece, but are encouraged to be employed at the end of a phrase before a transition to another section.

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==== Minor Modes ====

Three diatonic modes contain minor triads over the root: Dorian, Aeolian, and Phrygian. Just like with the major modes, each of these modes has its own distinct sound that can provide different pros and cons for writing ~~fugues~~.

Three diatonic modes contain minor triads over the root: Dorian, Aeolian, and Phrygian. Just like with the major modes, each of these modes has its own distinct sound that can provide different pros and cons for writing fugue subjects.

The Dorian mode's major sixth degree is exceptionally useful as a lead into the diminished octave / submajor seventh, being four steps away rather than the awkward seven-step lurch that would occur in the other modes. The diminished octave is one of the most important types of leading intervals into the root, because it is included in four of the seven perfect types of cadences. This feature alone makes Dorian one of the most desirable minor modes for counterpoint writing, since the other modes must either employ a minor third step size, add more non-diatonic intervals to the subject to bypass this step size, or find another method of resolution.

@@ Line 1: / Line 1: @@
 {{Breadcrumb|29edo}}
-{{Todo|inline=1| add audio and sheet music examples}}
+{{Todo|inline=1|audio examples}}
-'''Note: This page is currently under construction, and will be subject to major expansion in the near future.  Come back soon!'''
 On this page, I attempt to document in full my experience with writing counterpoint using [[29edo]] and the 29-form, with extrapolations that I've made into guidelines of a sort, which others may use as a jumping-off point for their own excursions into writing counterpoint, whether in 29edo itself or in other systems where these ideas or extrapolations therefrom may apply.  These guidelines arise in large part from a combination of the writings of [[wikipedia:Johann_Joseph_Fux|Johann Fux]] and [[wikipedia:Marchetto_da_Padova|Marchetto da Padova]], analysis of the works of [[wikipedia:Johann_Sebastian_Bach|J.S. Bach]], and parameters that arose simply from trial and error in my own composition.
 == Preface on Notation ==
-A few notable features will be considered for this page.  Intervals will be noted using extended diatonic notation (with ♯/♭ representing raising and lowering by a chromatic semitone, and ^/v representing raising or lowering by a diesis).  Intervals included in diatonic modes will be noted with their diatonic names (where "major" and "minor" forms of an interval are respectively the wider and narrower interval that fall on a given degree), and intervals outside those modes will be denoted as "super-"/"supra-" and "sub-" forms of the closest diatonic interval.  Where necessary, interval qualities may be abbreviated: m for minor, M for major, s for sub-, and S for super-/supra-.
+A few notable notational features will be considered for this page.  Intervals will be noted using extended diatonic notation (with ♯/♭ representing raising and lowering by a chromatic semitone, and ^/v representing raising or lowering by a diesis).  Intervals included in diatonic modes will be noted with their diatonic names (where "major" and "minor" forms of an interval are respectively the wider and narrower interval that fall on a given degree), and intervals outside those modes will be denoted as "super-"/"supra-" and "sub-" forms of the closest diatonic interval.  Where necessary, interval qualities may be abbreviated: m for minor, M for major, s for sub-, and S for super-/supra-.  EDOstep notation (where n\29 represents n steps of 29edo, not to be confused with the JI interval n/29) may also be used for clarification when extended diatonic notation may provide difficulty to parse.
-Sheet music examples, however, will employ the fewest number of ups and downs necessary; for instance, C♭ will be favored over ^B, though in the key of C this note will still be called the submajor seventh rather than the diminished octave.  EDOstep notation (where n\29 represents n steps of 29edo, not to be confused with the JI interval n/29) may also be used for clarification when extended diatonic notation may provide difficulty to parse.
-Interval equivalences created by 29edo will be considered equivalent here: the interval at 6\29, for instance, will be noted as a "subminor third" even if spelled in sheet music as a doubly augmented unison or a doubly diminished fourth, because this name is more descriptive and easier to parse when considering an equal temperament sequence.
+Interval equivalences created by 29edo will be considered equivalent here.  Where multiple spellings are available in notation, sharps/flats will be prioritized over ups/downs, as it makes clear the movement by chromatic semitones; for instance, C♭ will be favored over vB.  However, extended diatonic names will be favored over fifthwise names when possible, as these names are easier to parse when discussing interval sizes; for instance, "supermajor second" or "subminor third" will be favored over "doubly augmented unison" or "doubly diminished fourth," even if the interval using spellings such as C♯-F♭ or C♭-C♯
 == General Principles ==
@@ Line 18: / Line 13: @@
 === Tonics and Resolution ===
-A phrase should begin and end with all voices on the tonic, or on a perfect consonance with respect to it.  At the end of a phrase, the tonic is to be approached from above or below by a conjunction (see Interval Categories).  The chromatic semitone is primarily encouraged, but other resolutions are also possible.
+A phrase should begin and end with all voices on the tonic, or on a perfect consonance with respect to it, unless employing a specific tonal resolution chord.  At the end of a phrase, the tonic is to be approached from above or below by a conjunction (see Interval Categories).  The chromatic semitone is primarily encouraged, but other resolutions are also possible.
 === Interval Categories ===
@@ Line 27: / Line 22: @@
 #* Perfect Fourth (12\29)
 # '''Imperfect consonances''' are permitted as melodic steps.  Between voices, imperfect consonances may be approached through any means, and as such are encouraged as the main class of interval to use in the middle of phrases.
+#* Major Second (5\29)<sup>†</sup>
 #* Minor third (7\29)
 #* Major third (10\29)
 # '''Cadential consonances''' are permitted as melodic steps, but should be used with reserve.  Between voices, cadential consonances may be approached through any means, but they tend to resolve by contrary motion to a perfect consonance.
-#* Subminor Third (6\29)
+#* Supermajor Second / Subminor Third (6\29)<sup>†</sup>
 #* Supraminor Third (8\29)
 #* Submajor Third (9\29)
@@ Line 38: / Line 34: @@
 #* Supraminor Second / Chromatic Semitone (3\29)
 #* Submajor Second (4\29)
-#* Major Second (5\29)
+#* Major Second (5\29)<sup>†</sup>
+#* Supermajor Second / Subminor Third (6\29)<sup>†</sup>
 #'''Dissonances''' are not permitted as melodic steps, and should generally be avoided between any two voices.
 #*Augmented Third (13\29)
 #*Diminished Fifth (14\29)
-# The '''Diesis''' or '''Superunison''' (1\29) is permitted as a passing tone in melodies.  Two voices should never be separated by the interval of a diesis except as a brief transitional position between a unison and another interval.  The same guidelines apply to the Suboctave (28\29) or the Superoctave (30\29).
+# The '''Diesis''' or '''Superunison''' (1\29) is permitted only as a passing tone in melodies; a voice should strongly avoid movement by a diesis on a strong beat, as the small size can destabilize the tonality.  Two voices should never be separated by the interval of a diesis except as a brief transitional position between a unison and another interval.  The same guidelines apply to the Suboctave (28\29) or the Superoctave (30\29).
+<small><sup>†</sup>Note the overlap between certain categories: the Major Second (5\29) may function either as an imperfect consonance or a conjunction; the Supermajor Second (6\29), or Subminor Third, similarly may function as either a cadential consonance or a conjunction.</small>
 === Contour ===
@@ Line 50: / Line 48: @@
 The voices within a fugue should form a clear hierarchy of range; one voice should remain the lowest, another the highest, and so on for any additional voices.  Two adjacent voices may play a unison, but a lower voice may not reach a pitch higher than the voice above it.
-The interval between two adjacent voices generally should not exceed a Supermajor Tenth (40\29) unless absolutely necessary to avoid dissonances or provide melodic coherence within each individual voice.
+The interval between two adjacent voices generally should avoid exceeding a Supermajor Tenth (40\29) unless absolutely necessary to avoid dissonances or provide melodic coherence within each individual voice.
 === Cadences ===
-When resolving at the end of a phrase, cadences should generally involve two or more voices moving by conjunctions in opposite directions.  One or more of the voices should land on the tonic pitch, and all others should land on perfect consonances with respect to one another.
+When resolving at the end of a phrase, cadences should generally involve two or more voices moving by conjunctions in opposite directions.  One or more of the voices should land on the tonic pitch, and all others should land on perfect consonances with respect to one another; where there are three or more voices, specific tonal chords may apply as well.
 In 29edo, the intervals between two voices should generally begin on perfect consonances, move towards imperfect consonances, and finally utilize cadential consonances to create tension before resolving.  The interordinal intervals (supermajor and subminor thirds and sixths) are extremely tense, and each one either exceeds or falls short of a perfect consonance by two chromatic semitones (a total of 6\29).  This creates an extremely clear path to resolution via two types of movement:
@@ Line 64: / Line 62: @@
 Note also that all four types of interordinal intervals have at least one type of resolution available.  As such, the tension between two voices can be determined plainly by the type of cadence and the quality of the resolution, and conversely, the resolution can be determined plainly by the type of cadence and the quality of the tension.
 {| class="wikitable"
-|+Determining tensions for known resolutions
+|+Determining cadential intervals for known resolutions
 !Cadence Type
 !Unison
@@ Line 84: / Line 82: @@
 |}
 {| class="wikitable"
-|+Determining resolutions for known tensions
+|+Determining resolutions for known cadential intervals
 !Cadence Type
 !Subminor Third
@@ Line 108: / Line 106: @@
 Due to the 29-form's inherent pull towards 3-limit structures, the diatonic scale is an integral part of its structure.  The tonality of a fugue is most intuitively derived from the modes of the diatonic scale; specifically, the major and minor tonality respectively draw from the properties of the major and minor forms of the third degree of that scale.
-No matter the tonality, perfect consonances with respect to the root take precedence over all other intervals, and should be the most emphasized intervals in a given melody.  Subjects should be written primarily, if not entirely, in the diatonic scale, though the exact mode used depends on the tonality; however, countermelodies and elaborations on the subject should access additional tones outside of the diatonic scale to create more diverse harmonies and eventually use interordinals as final tensions to create cadences.
+No matter the tonality, perfect consonances with respect to the root take precedence over all other intervals, and should be the most emphasized intervals in a given melody.  Subjects should be written primarily, if not entirely, in the diatonic scale, though the exact mode used depends on the tonality; exceptions are made for the notes a chromatic semitone above or below the root, in order for the melody to form aforementioned cadences when a countermelody is added.  Countermelodies and elaborations on the subject, however, should access additional tones outside of the diatonic scale to create more diverse harmonies and eventually use interordinals as final tensions to create cadences.
 === Major Tonality ===
@@ Line 131: / Line 129: @@
 In the major tonality, expanding cadences are encouraged when resolving to fourths, and contracting cadences are encouraged when resolving to fifths.  This motion ensures that the tonic pitch is approached from below rather than from above, which creates a brighter sound more indicative of a major key.
-When using more than two distinct voices, it is often desirable to use fuller chords for resolutions at the end of a phrase, rather than simply doubling the root or fifth across octaves.  The following chords are considered for major resolutions, listed in order of consonance potential:
+When using more than two distinct voices, it is often desirable to use fuller chords for resolutions at the end of a phrase, rather than simply doubling the root or fifth across octaves.  Ideally, these fuller resolutions should be constructed such that the interval between any two pitches is a perfect or imperfect consonance; .  The following chords are considered for major resolutions, listed in order of consonance potential:
 * Requiring three or more voices:
@@ Line 143: / Line 141: @@
 ** 1 - P4 - P5 - M6
 ** 1 - M2 - P5 - M6
-[[File:3-voice Contracting Cadence.png|thumb|An example of a three-voice cadence.  Notice how the lower two voices create a contracting cadence into the perfect fourth, while the highest voice moves in parallel with the middle voice.]]
+[[File:3-voice Contracting Cadence.png|thumb|An example of a three-voice cadence.  Notice how the lower two voices create an expanding cadence into the perfect fourth, while the highest voice moves in parallel with the middle voice.]]
 Inversions of these chords may apply; for instance, M3 - P5 - P8 is considered to have the same consonance potential as 1 - M3 - P5.  It should be noted, however, that the highest and lowest voice will be the most prominent in a given chord, and it is thus encouraged that those voices play the root or fifth of the key in resolutions when possible.
-Tensions towards these resolutions can be made by combining types of expanding and contracting cadences; for instance, the chord 1 - M3 - P5 may be approached by a tension SM7 - S4 - sm6 or SM7 - m3 - sm6, as each voice in the tension differs from its resolution by one chroma.
+Tensions towards these resolutions can be made by combining types of expanding and contracting cadences; for instance, the chord 1 - M3 - P5 may be approached by a tension such as SM7 - m3 - sm6, where each voice in the tension differs from its resolution by one chroma.
 Finally, and perhaps most importantly, resolutions with lower consonance potential should be avoided as the final resolution of the piece, but are encouraged to be employed at the end of a phrase before a transition to another section.
@@ Line 153: / Line 151: @@
 ==== Minor Modes ====
-Three diatonic modes contain minor triads over the root: Dorian, Aeolian, and Phrygian.  Just like with the major modes, each of these modes has its own distinct sound that can provide different pros and cons for writing fugues.
+Three diatonic modes contain minor triads over the root: Dorian, Aeolian, and Phrygian.  Just like with the major modes, each of these modes has its own distinct sound that can provide different pros and cons for writing fugue subjects.
 The Dorian mode's major sixth degree is exceptionally useful as a lead into the diminished octave / submajor seventh, being four steps away rather than the awkward seven-step lurch that would occur in the other modes.  The diminished octave is one of the most important types of leading intervals into the root, because it is included in four of the seven perfect types of cadences.  This feature alone makes Dorian one of the most desirable minor modes for counterpoint writing, since the other modes must either employ a minor third step size, add more non-diatonic intervals to the subject to bypass this step size, or find another method of resolution.