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.

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-.

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.

== General Principles ==

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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.

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.

=== Cadences ===

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# In an '''expanding cadence''', two voices move apart from one another by chromatic semitones to form a perfect consonance.

# In a '''contracting cadence''', two voices move towards each other by chromatic semitones to form a perfect consonance.

# In a '''contracting cadence''', two voices move towards each other by chromatic semitones to form a perfect consonance.[[File:Expanding Cadence.png|thumb|An example of an expanding cadence that resolves from a subminor third to a perfect fourth.]]

Every expanding cadence has an inverse contracting cadence, and vice versa; for instance, the resolution from C♯ - G♭ to C - G is expanding, but from G♭ - C♯ to G - C is contracting.

Every expanding cadence has an inverse contracting cadence, and vice versa; for instance, the resolution from C♯ - G♭ to C - G (a perfect fifth in C) is expanding, but from G♭ - C♯ to G - C (a perfect fourth in G) is contracting.

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.

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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.]]

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.

~~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 except when this conflicts with melodic considerations of those voices~~.

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 d1 - A3 - A5 or d1 - d3 - A5, as 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.

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

=== Minor Tonality ===

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==== Minor Resolutions ====

In the minor tonality, contracting cadences are encouraged when resolving to fourths, and ~~expending~~ cadences are encouraged when resolving to fifths. This creates a darker tone, as it approaches the tonic via downwards motion from above.

In the minor tonality, contracting cadences are encouraged when resolving to fourths, and expanding cadences are encouraged when resolving to fifths. This creates a darker tone, as it approaches the tonic via downwards motion from above.

Just like in the major tonality, fuller chords are often desirable when using more voices. The following chords are considered for the minor tonality, in order of consonance potential:

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** 1 - M2 - P5 - M6

** 1 - P4 - P5 - m7

[[File:29edo Minor Cadence.png|thumb|An example of a minor key cadence. Notice the contracting motion between the upper two voices combined with the expanding motion between the lower two.]]

Just like in Major, inversions apply; that is, m3 - P5 - P8 has the same consonance potential as 1 - m3 - P5. Once again, the highest and lowest voice are the most prominent, and as such should tend to play the root or fifth of the key when possible.

Just like in Major, inversions apply; that is, m3 - P5 - P8 has the same consonance potential as 1 - m3 - P5. Once again, the highest and lowest voice are the most prominent, and as such should tend to play the root or fifth of the key except when this conflicts with melodic considerations.

Because the minor tonality is inherently darker than major, the more dissonant available resolutions are not discouraged as strongly as they are in major keys. However, the final resolution of a fugue often benefits from using a major resolution to create a bright turnaround, as this lends a greater sense of finality to the piece as a whole.

Because the minor tonality is inherently darker than major, the more dissonant available resolutions are not discouraged. However, the final resolution of a fugue often benefits from using a major resolution to create a bright turnaround, as this lends a greater sense of finality to the piece as a whole.

[[Category:29edo]]

[[Category:Approaches to tuning systems]]

@@ Line 5: / Line 5: @@
 '''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.
+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-.
+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.
 == General Principles ==
@@ Line 43: / Line 50: @@
 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.
+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.
 === Cadences ===
@@ Line 51: / Line 58: @@
 # In an '''expanding cadence''', two voices move apart from one another by chromatic semitones to form a perfect consonance.
-# In a '''contracting cadence''', two voices move towards each other by chromatic semitones to form a perfect consonance.
+# In a '''contracting cadence''', two voices move towards each other by chromatic semitones to form a perfect consonance.[[File:Expanding Cadence.png|thumb|An example of an expanding cadence that resolves from a subminor third to a perfect fourth.]]
-Every expanding cadence has an inverse contracting cadence, and vice versa; for instance, the resolution from C♯ - G♭ to C - G is expanding, but from G♭ - C♯ to G - C is contracting.
+Every expanding cadence has an inverse contracting cadence, and vice versa; for instance, the resolution from C♯ - G♭ to C - G (a perfect fifth in C) is expanding, but from G♭ - C♯ to G - C (a perfect fourth in G) is contracting.
 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.
@@ Line 136: / Line 143: @@
 ** 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.]]
+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.
-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 except when this conflicts with melodic considerations of those voices.
+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 d1 - A3 - A5 or d1 - d3 - A5, as 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.
-Finally, and perhaps most importantly, resolutions with lower consonance potential should be avoided as the final resolution of the piece, and instead be employed at the end of a phrase before a transition to another section.
 === Minor Tonality ===
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 ==== Minor Resolutions ====
-In the minor tonality, contracting cadences are encouraged when resolving to fourths, and expending cadences are encouraged when resolving to fifths.  This creates a darker tone, as it approaches the tonic via downwards motion from above.
+In the minor tonality, contracting cadences are encouraged when resolving to fourths, and expanding cadences are encouraged when resolving to fifths.  This creates a darker tone, as it approaches the tonic via downwards motion from above.
 Just like in the major tonality, fuller chords are often desirable when using more voices.  The following chords are considered for the minor tonality, in order of consonance potential:
@@ Line 172: / Line 179: @@
 ** 1 - M2 - P5 - M6
 ** 1 - P4 - P5 - m7
+[[File:29edo Minor Cadence.png|thumb|An example of a minor key cadence.  Notice the contracting motion between the upper two voices combined with the expanding motion between the lower two.]]
+Just like in Major, inversions apply; that is, m3 - P5 - P8 has the same consonance potential as 1 - m3 - P5.  Once again, the highest and lowest voice are the most prominent, and as such should tend to play the root or fifth of the key when possible.
-Just like in Major, inversions apply; that is, m3 - P5 - P8 has the same consonance potential as 1 - m3 - P5.  Once again, the highest and lowest voice are the most prominent, and as such should tend to play the root or fifth of the key except when this conflicts with melodic considerations.
+Because the minor tonality is inherently darker than major, the more dissonant available resolutions are not discouraged as strongly as they are in major keys.  However, the final resolution of a fugue often benefits from using a major resolution to create a bright turnaround, as this lends a greater sense of finality to the piece as a whole.
-Because the minor tonality is inherently darker than major, the more dissonant available resolutions are not discouraged.  However, the final resolution of a fugue often benefits from using a major resolution to create a bright turnaround, as this lends a greater sense of finality to the piece as a whole.
 [[Category:29edo]]
 [[Category:Approaches to tuning systems]]