Diatonic functional harmony: Difference between revisions

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== Paradiatonic Functions ==

The idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees is one that has at least some roots in [[Wikipedia:Ivan Wyschnegradsky|Wyschnegradsky]]'s concept of [[Wikipedia:Major fourth and minor fifth|"Major Fourth" and "Minor Fifth"]]. However, Aura wanted to use LCJI as a basis for defining these intervals and thus decided to take [[11/8]] as being the just version of Wyschnegradsky's "Major Fourth", and while he drew up sketches based loosely on [[24edo]] for early versions of this concept, he also realized that that two instances of [[33/32]] added up to an interval smaller than [[2187/2048]] but which had a similar function. Furthermore, since two instances of 11/8 resulted in an interval in the vicinity of a Major seventh, Aura decided to takes stacks of 11/8 to form a second navigational axis~~, the "'''Paradiatonic Axis'''",~~ which works together with the Diatonic Axis in order to define the microtonal functions positioned roughly halfway between the German and Viennese Diatonic functions, though there are a few other microtonal functions as well that are not immediately covered by this axis.

In addition to all the aforementioned Diatonic functions, there is an additional set of categories for dealing with the notes in between the various Diatonic scale degrees.

=== History ===

The idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees is one that has at least some roots in [[Wikipedia:Ivan Wyschnegradsky|Wyschnegradsky]]'s concept of [[Wikipedia:Major fourth and minor fifth|"Major Fourth" and "Minor Fifth"]]. However, Aura wanted to use LCJI as a basis for defining these intervals and thus decided to take [[11/8]] as being the just version of Wyschnegradsky's "Major Fourth", and while he drew up sketches based loosely on [[24edo]] for early versions of this concept, he also realized that that two instances of [[33/32]] added up to an interval smaller than [[2187/2048]] but which had a similar function. Furthermore, since two instances of 11/8 resulted in an interval in the vicinity of a Major seventh, Aura decided to takes stacks of 11/8 to form a second navigational axis which works together with the Diatonic Axis in order to define the microtonal functions positioned roughly halfway between the German and Viennese Diatonic functions, though there are a few other microtonal functions as well that are not immediately covered by this second axis.

[[File:Diatonic_Function_Map.png|thumb|Initial diagram of paradiatonic function locations by Aura made around the time of officially joining the Xenharmonic community. Note that a number of the functions listed on this page are missing, while the Contralead, the Semicontralead, the Intersubiant and Interregnant initially had different names.]]

~~Through interactions~~ with ~~others~~ in ~~the Xenharmonic community~~, ~~Aura was later influenced~~ by ~~others on Discord~~ to ~~take~~ [[~~MOS~~]]-~~based structural considerations into account~~. ~~This eventually resulted~~ in the ~~first formal definition~~ of things ~~like~~ a "'''~~parachroma~~'''", ~~an interval that can be easily tempered to equal half~~ of ~~a MOS~~-~~chroma~~.

Most traditional music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime. They also know that a diatonic semitone and a chromatic semitone add up to a whole tone. The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo. In [[Talk:159edo_notation#My_Second_Idea_for_a_Notation System|a conversation]] between Aura and [[Kite Giedraitis]] about this topic, Kite mentioned that there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and the aforementioned 2187/2048- a chromatic semitone that is otherwise known as the Apotome- which, when added together, add up to a 9/8 whole tone. Furthermore, Kite also mentioned how in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by [[81/80]]. On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields [[27/25]]- two additional diatonic semitones. On the other hand, subtracting 81/80 from the Apotome yields [[135/128]], and subtracting another 81/80 yields 25/24- two additional chromatic semitones. When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone. In light of all this, Kite argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.

Building on Kite's logic, Aura then decided to apply similar distinctions among quartertones, and thus make the argument that quartertones don't have to denote exactly one fourth of a whole tone in as of themselves, but rather, they only have to add up to a whole tone when paired up correctly. However, the catch was that that for quartertones, there are sometimes multiple correct options, making things more complicated. Aura decided to define the musical functions of quartertones initially on an informal basis by drawing a distinction between the terms "'''Parachromatic'''" (from the prefix ''para-'' in both the senses of ''alongside'' and ''resembling''<ref>[[Wiktionary: para- #Etymology 1]]</ref>, and the word ''chromatic'') and "'''Paradiatonic'''" (from the same two senses of the prefix ''para-'' and the word ''diatonic'') for purposes of classifying quartertone intervals.

According to Aura, paradiatonic quartertones are be analogous to diatonic semitones in that they are denoted as seconds, albeit inframinor seconds by default, while parachromatic quartertones are analogous to chromatic semitones in that they are denoted as primes, albiet as ultraprimes by default. However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone. Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, it can be deduced that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone. Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, Aura ended up choosing the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone. As a result of multiple factors, Aura ended up choosing the combination of three 33/32 parachromatic quartertones and one [[4096/3993]] paradiatonic quartertone as the JI basis for this in regards to Diatonic theory, and, through interactions with others in the Xenharmonic community, Aura was later influenced by others on Discord to take [[MOS]]-based structural considerations into account. This eventually resulted in the first formal definition of a "'''parachroma'''" (an interval that can be easily tempered to equal half of a MOS-chroma), and, thorough applying the aforementioned thought process about classifying quartertones, the '''paralimma''' (the interval that remains after subtracting three parachromas from a MOS-step).

=== Basic Paradiatonic Functions ===

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'''Supercommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 20 cents above the Tonic. These intervals are little more than stepping stones in modulation, and extra intervals that can be used together with the Tonic for a sense of dissonance, or for a slightly less resolved version of a Unison or Octave.

'''Superdietic''' - This is a note that occurs at intervals between roughly 20 cents above the Tonic and 25/24 above the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, in Bass-Up Tonality, the Contralead. However, while some microtonalists would question the idea of there being a distinct Superdietic region, preferring to think of the quartertones in this region as being simply the junction between the Supercommatic and Contralead regions, there there are actually a few functional differences between a Superdietic and a Contralead in Bass-Up Tonality that are worth considering. For starters, Superdietics are often more likely to be passing tones than Contraleads, and, when they’re not merely passing non-chord tones, they are just as liable to resolve upward thought some sort of semitone-like motion to some form of Contralead, Semicontralead, or even a Supertonic, as they are to resolve downwards toward the Tonic, a property which intervals like [[33/32]] in particular are apt to demonstrate. Furthermore, whereas a Contralead can resolve to the Tonic in part through a strong harmonic connection, a Superdietic is much more likely to do so through sheer brute force when such a resolution is noticeable.

'''Superdietic''' - This is a note that occurs at intervals between roughly 20 cents above the Tonic and 25/24 above the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, in Bass-Up Tonality, the Contralead. However, while some microtonalists would question the idea of there being a distinct Superdietic region, preferring to think of the quartertones in this region as being simply the junction between the Supercommatic and Contralead regions, there there are actually a few functional differences between a Superdietic and a Contralead in Bass-Up Tonality that are worth considering. For starters, Superdietics are often more likely to be passing tones than Contraleads, and, when they’re not merely passing non-chord tones, they are just as liable to resolve upward thought some sort of semitone-like motion to some form of Contralead, Semicontralead, or even a Supertonic, as they are to resolve downwards toward the Tonic, a property which intervals like 33/32 in particular are apt to demonstrate. Furthermore, whereas a Contralead can resolve to the Tonic in part through a strong harmonic connection, a Superdietic is much more likely to do so through sheer brute force when such a resolution is noticeable.

'''Subdietic''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly 20 cents below the octave reduplication of the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, in Bass-Up Tonality, the Lead- however, there are functional differences between a Subdietic and a Lead in Bass-Up Tonality. In Bass-Up Tonality, Subdietics are often more likely to be passing tones than Leads, and, when they’re not merely passing non-chord tones, they are often harder to approach and or follow up without creating some kind of awkward tonal disconnect, with such a disconnect being especially noticeable for intervals like [[64/33]]. Furthermore, whereas a Lead can resolve to the Tonic in part through a strong harmonic connection, a Subdietic is much more likely to do so through sheer brute force.

@@ Line 57: / Line 57: @@
 == Paradiatonic Functions ==
-The idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees is one that has at least some roots in [[Wikipedia:Ivan Wyschnegradsky|Wyschnegradsky]]'s concept of [[Wikipedia:Major fourth and minor fifth|"Major Fourth" and "Minor Fifth"]].  However, Aura wanted to use LCJI as a basis for defining these intervals and thus decided to take [[11/8]] as being the just version of Wyschnegradsky's "Major Fourth", and while he drew up sketches based loosely on [[24edo]] for early versions of this concept, he also realized that that two instances of [[33/32]] added up to an interval smaller than [[2187/2048]] but which had a similar function.  Furthermore, since two instances of 11/8 resulted in an interval in the vicinity of a Major seventh, Aura decided to takes stacks of 11/8 to form a second navigational axis, the "'''Paradiatonic Axis'''", which works together with the Diatonic Axis in order to define the microtonal functions positioned roughly halfway between the German and Viennese Diatonic functions, though there are a few other microtonal functions as well that are not immediately covered by this axis.
+In addition to all the aforementioned Diatonic functions, there is an additional set of categories for dealing with the notes in between the various Diatonic scale degrees.
+=== History ===
+The idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees is one that has at least some roots in [[Wikipedia:Ivan Wyschnegradsky|Wyschnegradsky]]'s concept of [[Wikipedia:Major fourth and minor fifth|"Major Fourth" and "Minor Fifth"]].  However, Aura wanted to use LCJI as a basis for defining these intervals and thus decided to take [[11/8]] as being the just version of Wyschnegradsky's "Major Fourth", and while he drew up sketches based loosely on [[24edo]] for early versions of this concept, he also realized that that two instances of [[33/32]] added up to an interval smaller than [[2187/2048]] but which had a similar function.  Furthermore, since two instances of 11/8 resulted in an interval in the vicinity of a Major seventh, Aura decided to takes stacks of 11/8 to form a second navigational axis which works together with the Diatonic Axis in order to define the microtonal functions positioned roughly halfway between the German and Viennese Diatonic functions, though there are a few other microtonal functions as well that are not immediately covered by this second axis.
 [[File:Diatonic_Function_Map.png|thumb|Initial diagram of paradiatonic function locations by Aura made around the time of officially joining the Xenharmonic community.  Note that a number of the functions listed on this page are missing, while the Contralead, the Semicontralead, the Intersubiant and Interregnant initially had different names.]]
-Through interactions with others in the Xenharmonic community, Aura was later influenced by others on Discord to take [[MOS]]-based structural considerations into account.  This eventually resulted in the first formal definition of things like a "'''parachroma'''", an interval that can be easily tempered to equal half of a MOS-chroma.
+Most traditional music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime.  They also know that a diatonic semitone and a chromatic semitone add up to a whole tone.  The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo.  In [[Talk:159edo_notation#My_Second_Idea_for_a_Notation System|a conversation]] between Aura and [[Kite Giedraitis]] about this topic, Kite mentioned that there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and the aforementioned 2187/2048- a chromatic semitone that is otherwise known as the Apotome- which, when added together, add up to a 9/8 whole tone.  Furthermore, Kite also mentioned how in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by [[81/80]].  On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields [[27/25]]- two additional diatonic semitones.  On the other hand, subtracting 81/80 from the Apotome yields [[135/128]], and subtracting another 81/80 yields 25/24- two additional chromatic semitones.  When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone.  In light of all this, Kite argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.
+Building on Kite's logic, Aura then decided to apply similar distinctions among quartertones, and thus make the argument that quartertones don't have to denote exactly one fourth of a whole tone in as of themselves, but rather, they only have to add up to a whole tone when paired up correctly.  However, the catch was that that for quartertones, there are sometimes multiple correct options, making things more complicated.  Aura decided to define the musical functions of quartertones initially on an informal basis by drawing a distinction between the terms "'''Parachromatic'''" (from the prefix ''para-'' in both the senses of ''alongside'' and ''resembling''<ref>[[Wiktionary: para- #Etymology 1]]</ref>, and the word ''chromatic'') and "'''Paradiatonic'''" (from the same two senses of the prefix ''para-'' and the word ''diatonic'') for purposes of classifying quartertone intervals.
+According to Aura, paradiatonic quartertones are be analogous to diatonic semitones in that they are denoted as seconds, albeit inframinor seconds by default, while parachromatic quartertones are analogous to chromatic semitones in that they are denoted as primes, albiet as ultraprimes by default.  However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone.  Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, it can be deduced that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone.  Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, Aura ended up choosing the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone.  As a result of multiple factors, Aura ended up choosing the combination of three 33/32 parachromatic quartertones and one [[4096/3993]] paradiatonic quartertone as the JI basis for this in regards to Diatonic theory, and, through interactions with others in the Xenharmonic community, Aura was later influenced by others on Discord to take [[MOS]]-based structural considerations into account.  This eventually resulted in the first formal definition of a "'''parachroma'''" (an interval that can be easily tempered to equal half of a MOS-chroma), and, thorough applying the aforementioned thought process about classifying quartertones, the '''paralimma''' (the interval that remains after subtracting three parachromas from a MOS-step).
 === Basic Paradiatonic Functions ===
@@ Line 98: / Line 105: @@
 '''Supercommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 20 cents above the Tonic.  These intervals are little more than stepping stones in modulation, and extra intervals that can be used together with the Tonic for a sense of dissonance, or for a slightly less resolved version of a Unison or Octave.
-'''Superdietic''' - This is a note that occurs at intervals between roughly 20 cents above the Tonic and 25/24 above the Tonic.  These intervals tend to act as parachromatic alterations of either the Tonic, or, in Bass-Up Tonality, the Contralead.  However, while some microtonalists would question the idea of there being a distinct Superdietic region, preferring to think of the quartertones in this region as being simply the junction between the Supercommatic and Contralead regions, there there are actually a few functional differences between a Superdietic and a Contralead in Bass-Up Tonality that are worth considering.  For starters, Superdietics are often more likely to be passing tones than Contraleads, and, when they’re not merely passing non-chord tones, they are just as liable to resolve upward thought some sort of semitone-like motion to some form of Contralead, Semicontralead, or even a Supertonic, as they are to resolve downwards toward the Tonic, a property which intervals like [[33/32]] in particular are apt to demonstrate.  Furthermore, whereas a Contralead can resolve to the Tonic in part through a strong harmonic connection, a Superdietic is much more likely to do so through sheer brute force when such a resolution is noticeable.
+'''Superdietic''' - This is a note that occurs at intervals between roughly 20 cents above the Tonic and 25/24 above the Tonic.  These intervals tend to act as parachromatic alterations of either the Tonic, or, in Bass-Up Tonality, the Contralead.  However, while some microtonalists would question the idea of there being a distinct Superdietic region, preferring to think of the quartertones in this region as being simply the junction between the Supercommatic and Contralead regions, there there are actually a few functional differences between a Superdietic and a Contralead in Bass-Up Tonality that are worth considering.  For starters, Superdietics are often more likely to be passing tones than Contraleads, and, when they’re not merely passing non-chord tones, they are just as liable to resolve upward thought some sort of semitone-like motion to some form of Contralead, Semicontralead, or even a Supertonic, as they are to resolve downwards toward the Tonic, a property which intervals like 33/32 in particular are apt to demonstrate.  Furthermore, whereas a Contralead can resolve to the Tonic in part through a strong harmonic connection, a Superdietic is much more likely to do so through sheer brute force when such a resolution is noticeable.
 '''Subdietic''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly 20 cents below the octave reduplication of the Tonic.  These intervals tend to act as parachromatic alterations of either the Tonic, or, in Bass-Up Tonality, the Lead- however, there are functional differences between a Subdietic and a Lead in Bass-Up Tonality.   In Bass-Up Tonality, Subdietics are often more likely to be passing tones than Leads, and, when they’re not merely passing non-chord tones, they are often harder to approach and or follow up without creating some kind of awkward tonal disconnect, with such a disconnect being especially noticeable for intervals like [[64/33]].  Furthermore, whereas a Lead can resolve to the Tonic in part through a strong harmonic connection, a Subdietic is much more likely to do so through sheer brute force.