Diatonic functional harmony: Difference between revisions

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Among the chief ideas that come from German Theory is that there are three basic, or primary functions, and that there are multiple operations that can be applied to these three basic functions in order to derive new functions.

=== Basic Functions ===

=== Basic Diatonic Functions ===

According to [[User:Aura|Aura]], the three basic functions have their roots in [[LCJI]], and are labeled as follows:

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== Facets Derived from Viennese Theory ==

Among the chief ideas that come from Viennese Theory is the idea that each degree has its own function relative to the Tonic. However, while in Viennese Theory proper, the degrees are strictly defined only relative to the cycle of fifths, Aura, for the realm of Microtonality, additionally takes things like Bass-Up Tonality (that is, music built from the Bass upwards) and Treble-Down Tonality (that is, music built from the Treble ~~downards~~) into consideration. On top of that, Aura also contends that many of the functions described by Viennese Theory find their roots in specific combinations of different operations on the different basic functions described in German Theory.

Among the chief ideas that come from Viennese Theory is the idea that each degree has its own function relative to the Tonic. However, while in Viennese Theory proper, the degrees are strictly defined only relative to the cycle of fifths, Aura, for the realm of Microtonality, not only takes stacks of 3/2 to form a key navigational axis called the "'''Diatonic Axis'''", but also additionally takes things like Bass-Up Tonality (that is, music built from the Bass upwards) and Treble-Down Tonality (that is, music built from the Treble downwards) into consideration. On top of that, Aura also contends that many of the functions described by Viennese Theory find their roots in specific combinations of different operations on the different basic functions described in German Theory.

=== Scale Degree Functions ===

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'''Lead''' - This is the note typically referred to when people say "the leading-tone". This is a note that occurs at intervals between 13/7 and 27/14 away from the Tonic, which serves as a leading tone in the scale's direction of construction. In Bass-Up Tonality, this functionality has its roots in the fifteenth harmonic. Although triads built on this scale degree are regarded by some as simply incomplete Dominant Seventh chords, Aura's own analysis, while acknowledging the functional similarities between the Lead and the Dominant, sees this interval as functionally distinct from the Dominant due to the the Lead also being related to the Mediant in the same way that the Dominant is related to the Tonic- a key functionality that is often exploited in [[Wikipedia:Vi–ii–V–I|circle progression]]s.

== ~~Additional Functions~~ ==

== Antitonic ==

~~In addition~~ to the ~~Functions derived from both German theory and Viennese theory~~, ~~there have been other functions described~~ by Aura and others.

Notes that occur around half an octave away from the Tonic- especially the region extending from 7/5 to 10/7- on account of harmonies built on notes in this area tend to oppose that of the Tonic, thus leading to the term '''Antitonic''' being used to describe the region as a whole by Aura and others. In addition, the term "Antitonic" acts as a generic term for any of a group of diatonic functions found in this region. While some microtonal theorists insist that the Antitonic functionality is more fundamental than perhaps even the Dominant or Serviant, others disagree.

=== Antitonic ~~Region~~ ===

=== Specific Types of Antitonic ===

Notes that occur around half an octave away from the Tonic- especially the region extending from 7/5 to 10/7- on account of harmonies built on notes in this area tend to oppose that of the Tonic, thus leading to the term '''Antitonic''' being used to describe the region as a whole. The exact outcome of ~~this opposition~~ depends on the ~~exact distance~~ of the ~~Antitonic from the Tonic~~:

The exact outcome and specific function of any given Antitonic depends on the interval between it and the Tonic in the scale's direction of construction, as well as by whether or not the interval in question is an augmented fourth or a diminished fifth:

*If the Antitonic is between 7/5 and 600 cents away from the Tonic and functions as an augmented fourth, it tends to cause the Dominant to become a new Tonic unless followed up by a different note- one that is usually a Major 7th away from the Tonic. Because of this tendency to "kiss up to" and tonicize the Dominant, this type of Antitonic can be dubbed a "'''Sycophant'''", with the prototypical Sycophant being 45/32.

*Conversely, if the Antitonic is between 600 cents and 10/7 away from the Tonic and functions as a diminished fifth, it tends to contrast with the Tonic in a manner somewhat akin to that of a Dominant, but by sheer brute force and contrary harmonic nature- e.g. if the Tonic harmony is Minor in nature, the Antitonic harmony will be Major- or more rarely, Supermajor- in nature. Furthermore, in scales such as the Locrian scale, any type of Serviant harmony tends to resolve towards either this type of Antitonic, or some other type of substitute for a Dominant. Because of these Dominant-esque tendencies, this type of Antitonic can be dubbed a "'''Tyrant'''", with the prototypical Tyrant being 64/45.

*If the Antitonic is between 7/5 and 600 cents away from the Tonic and functions as a diminished fifth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just below the range of a typical Tyrant, it is called a '''Gravotyrant''' in ~~Bass-Up tonality~~. A classic example of such an interval is 1024/729.

*If the Antitonic is between 7/5 and 600 cents away from the Tonic and functions as a diminished fifth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just below the range of a typical Tyrant in Bass-Up Tonality, it is called a '''Gravotyrant''' in this kind of tonal systems. A classic example of such an interval is 1024/729.

*If the Antitonic is between 600 cents and 10/7 away from the Tonic and functions as an augmented fourth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just above the range of a typical Sycophant, it is called an '''Acuosycophant''' in ~~Bass-Up Tonality~~. A classic example of such an interval is 729/512.

*If the Antitonic is between 600 cents and 10/7 away from the Tonic and functions as an augmented fourth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just above the range of a typical Sycophant in Bass-Up Tonality, it is called an '''Acuosycophant''' in this kind of tonal systems. A classic example of such an interval is 729/512.

== Paradiatonic Functions ==

In addition to all the aforementioned Diatonic functions, Aura not only takes [[11/8]] to be a more just version of [[Wikipedia:Ivan Wyschnegradsky|Wyschnegradsky]]'s [[Wikipedia:Major fourth and minor fifth|Major Fourth]] interval (under the name "paramajor fourth"), but also 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.

=== Basic Paradiatonic Functions ===

'''Intersubiant''' - This is a note that occurs roughly at intervals ranging from 256/187 to 25/18 away from the Tonic in the scale's direction of construction. Like both the Serviant and the Sycophant, intervals in this region tend to have a Predominant function, however, the way these intervals carry out this function is rather different from both as they neither act as a counterweight to the Dominant like a Serviant, nor do they completely tonicize the Dominant like a Sycophant. In Bass-Up tonality, this functionality has its roots in the eleventh harmonic.

'''Interregnant''' - This is a note that occurs roughly at intervals ranging from 36/25 to 187/128 away from the Tonic in the scale's direction of construction. Accordingly, intervals in this region behave as a cross between a Tyrant Antitonic on one hand and a Dominant on the other in that they often contrast with the Tonic through some combination of harmonic connection and brute force contrast. In Bass-Up tonality, this functionality has its roots in the eleventh subharmonic.

=== Interstep Functions ===

'''Varicant''' - Just as a Mediant lies roughly in the middle of the 3/2 interval separating the Tonic and the Dominant above it, a Varicant lies roughly in the middle of the 4/3 interval separating the Dominant and the Tonic above it. Intervals in this region often don’t consistently act as either sixths or sevenths, or even act as a cross between a sixth and a seventh- effectively straddling the border between these two diatonic categories, hence the name "Varicant", from Latin "vāricō"<ref>[[Wiktionary: varico #Latin]]</ref>. This is a note that occurs roughly at intervals between 128/75 and 225/128 away from the Tonic in the scale's direction of construction. In Bass-Up Tonality, this functionality is first encountered in the form of the [[7/4]] interval, though [[12/7]] is another notable interval included in this range. While many microtonalists think of 7/4 as being purely a type of seventh- and indeed, it most commonly acts as a sort of subminor seventh- Aura counterargues based on this same interval's relationships with 11/8 in particular that 7/4 is not merely a type of seventh, but rather, a type of a cross between a sixth and a seventh, with such a property explaining why [[14/11]] is generally considered to be a type of third.

'''Contravaricant''' - This is a note that occurs roughly at intervals between 256/225 and 75/64 away from the Tonic in the scale's direction of construction. This region is characterized by intervals that don’t consistently act as either seconds or thirds. In Bass-Up tonality, this functionality has its roots in the seventh subharmonic, though 7/6 is another notable interval included in this range.

== Additional Functions ==

In addition to the Functions derived from both German theory and Viennese theory, as well as the Antitonic and Paradiatonic functions, there have been other functions described, though the terms here are those used by Aura.

=== Circumdominant Regions ===

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'''Mocksycophant''' - In Bass-Up Tonality, this is an interval that ranges from 10/7 to 20/13 which acts as a type of fourth rather than either a type of fifth or a type of sixth. These intervals are generally not that common outside of Jazz and Blues.

~~== Paradiatonic Functions ==~~

In addition to all the aforementioned Diatonic functions, Aura takes a more just version of [[Wikipedia:Ivan Wyschnegradsky|Wyschnegradsky]]'s [[Wikipedia:Major fourth and minor fifth|Major Fourth]] interval (under a different name) to be a second axis for defining many of the various microtonal functions, which, in relation to the Diatonic functions, he dubs "'''paradiatonic functions'''".

[[Category:Terms]]

[[Category:Theory]]

[[Category:Method]]

@@ Line 5: / Line 5: @@
 Among the chief ideas that come from German Theory is that there are three basic, or primary functions, and that there are multiple operations that can be applied to these three basic functions in order to derive new functions.
-=== Basic Functions ===
+=== Basic Diatonic Functions ===
 According to [[User:Aura|Aura]], the three basic functions have their roots in [[LCJI]], and are labeled as follows:
@@ Line 25: / Line 25: @@
 == Facets Derived from Viennese Theory ==
-Among the chief ideas that come from Viennese Theory is the idea that each degree has its own function relative to the Tonic.  However, while in Viennese Theory proper, the degrees are strictly defined only relative to the cycle of fifths, Aura, for the realm of Microtonality, additionally takes things like Bass-Up Tonality (that is, music built from the Bass upwards) and Treble-Down Tonality (that is, music built from the Treble downards) into consideration.  On top of that, Aura also contends that many of the functions described by Viennese Theory find their roots in specific combinations of different operations on the different basic functions described in German Theory.
+Among the chief ideas that come from Viennese Theory is the idea that each degree has its own function relative to the Tonic.  However, while in Viennese Theory proper, the degrees are strictly defined only relative to the cycle of fifths, Aura, for the realm of Microtonality, not only takes stacks of 3/2 to form a key navigational axis called the "'''Diatonic Axis'''", but also additionally takes things like Bass-Up Tonality (that is, music built from the Bass upwards) and Treble-Down Tonality (that is, music built from the Treble downwards) into consideration.  On top of that, Aura also contends that many of the functions described by Viennese Theory find their roots in specific combinations of different operations on the different basic functions described in German Theory.
 === Scale Degree Functions ===
@@ Line 43: / Line 43: @@
 '''Lead''' - This is the note typically referred to when people say "the leading-tone".  This is a note that occurs at intervals between 13/7 and 27/14 away from the Tonic, which serves as a leading tone in the scale's direction of construction.  In Bass-Up Tonality, this functionality has its roots in the fifteenth harmonic.  Although triads built on this scale degree are regarded by some as simply incomplete Dominant Seventh chords, Aura's own analysis, while acknowledging the functional similarities between the Lead and the Dominant, sees this interval as functionally distinct from the Dominant due to the the Lead also being related to the Mediant in the same way that the Dominant is related to the Tonic- a key functionality that is often exploited in [[Wikipedia:Vi–ii–V–I|circle progression]]s.
-== Additional Functions ==
+== Antitonic ==
-In addition to the Functions derived from both German theory and Viennese theory, there have been other functions described by Aura and others.
+Notes that occur around half an octave away from the Tonic- especially the region extending from 7/5 to 10/7- on account of harmonies built on notes in this area tend to oppose that of the Tonic, thus leading to the term '''Antitonic''' being used to describe the region as a whole by Aura and others.  In addition, the term "Antitonic" acts as a generic term for any of a group of diatonic functions found in this region.  While some microtonal theorists insist that the Antitonic functionality is more fundamental than perhaps even the Dominant or Serviant, others disagree.
-=== Antitonic Region ===
+=== Specific Types of Antitonic ===
-Notes that occur around half an octave away from the Tonic- especially the region extending from 7/5 to 10/7- on account of harmonies built on notes in this area tend to oppose that of the Tonic, thus leading to the term '''Antitonic''' being used to describe the region as a whole.  The exact outcome of this opposition depends on the exact distance of the Antitonic from the Tonic:
+The exact outcome and specific function of any given Antitonic depends on the interval between it and the Tonic in the scale's direction of construction, as well as by whether or not the interval in question is an augmented fourth or a diminished fifth:
 *If the Antitonic is between 7/5 and 600 cents away from the Tonic and functions as an augmented fourth, it tends to cause the Dominant to become a new Tonic unless followed up by a different note- one that is usually a Major 7th away from the Tonic.  Because of this tendency to "kiss up to" and tonicize the Dominant, this type of Antitonic can be dubbed a "'''Sycophant'''", with the prototypical Sycophant being 45/32.
 *Conversely, if the Antitonic is between 600 cents and 10/7 away from the Tonic and functions as a diminished fifth, it tends to contrast with the Tonic in a manner somewhat akin to that of a Dominant, but by sheer brute force and contrary harmonic nature- e.g. if the Tonic harmony is Minor in nature, the Antitonic harmony will be Major- or more rarely, Supermajor- in nature. Furthermore, in scales such as the Locrian scale, any type of Serviant harmony tends to resolve towards either this type of Antitonic, or some other type of substitute for a Dominant. Because of these Dominant-esque tendencies, this type of Antitonic can be dubbed a "'''Tyrant'''", with the prototypical Tyrant being 64/45.
-*If the Antitonic is between 7/5 and 600 cents away from the Tonic and functions as a diminished fifth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just below the range of a typical Tyrant, it is called a '''Gravotyrant''' in Bass-Up tonality.  A classic example of such an interval is 1024/729.
+*If the Antitonic is between 7/5 and 600 cents away from the Tonic and functions as a diminished fifth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just below the range of a typical Tyrant in Bass-Up Tonality, it is called a '''Gravotyrant''' in this kind of tonal systems.  A classic example of such an interval is 1024/729.
-*If the Antitonic is between 600 cents and 10/7 away from the Tonic and functions as an augmented fourth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just above the range of a typical Sycophant, it is called an '''Acuosycophant''' in Bass-Up Tonality.  A classic example of such an interval is 729/512.
+*If the Antitonic is between 600 cents and 10/7 away from the Tonic and functions as an augmented fourth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just above the range of a typical Sycophant in Bass-Up Tonality, it is called an '''Acuosycophant''' in this kind of tonal systems.  A classic example of such an interval is 729/512.
+== Paradiatonic Functions ==
+In addition to all the aforementioned Diatonic functions, Aura not only takes [[11/8]] to be a more just version of [[Wikipedia:Ivan Wyschnegradsky|Wyschnegradsky]]'s [[Wikipedia:Major fourth and minor fifth|Major Fourth]] interval (under the name "paramajor fourth"), but also 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.
+=== Basic Paradiatonic Functions ===
+'''Intersubiant''' - This is a note that occurs roughly at intervals ranging from 256/187 to 25/18 away from the Tonic in the scale's direction of construction.  Like both the Serviant and the Sycophant, intervals in this region tend to have a Predominant function, however, the way these intervals carry out this function is rather different from both as they neither act as a counterweight to the Dominant like a Serviant, nor do they completely tonicize the Dominant like a Sycophant.  In Bass-Up tonality, this functionality has its roots in the eleventh harmonic.
+'''Interregnant''' - This is a note that occurs roughly at intervals ranging from 36/25 to 187/128 away from the Tonic in the scale's direction of construction.  Accordingly, intervals in this region behave as a cross between a Tyrant Antitonic on one hand and a Dominant on the other in that they often contrast with the Tonic through some combination of harmonic connection and brute force contrast.  In Bass-Up tonality, this functionality has its roots in the eleventh subharmonic.
+=== Interstep Functions ===
+'''Varicant''' - Just as a Mediant lies roughly in the middle of the 3/2 interval separating the Tonic and the Dominant above it, a Varicant lies roughly in the middle of the 4/3 interval separating the Dominant and the Tonic above it.  Intervals in this region often don’t consistently act as either sixths or sevenths, or even act as a cross between a sixth and a seventh- effectively straddling the border between these two diatonic categories, hence the name "Varicant", from Latin "vāricō"<ref>[[Wiktionary: varico #Latin]]</ref>.  This is a note that occurs roughly at intervals between 128/75 and 225/128 away from the Tonic in the scale's direction of construction.  In Bass-Up Tonality, this functionality is first encountered in the form of the [[7/4]] interval, though [[12/7]] is another notable interval included in this range.  While many microtonalists think of 7/4 as being purely a type of seventh- and indeed, it most commonly acts as a sort of subminor seventh- Aura counterargues based on this same interval's relationships with 11/8 in particular that 7/4 is not merely a type of seventh, but rather, a type of a cross between a sixth and a seventh, with such a property explaining why [[14/11]] is generally considered to be a type of third.
+'''Contravaricant''' - This is a note that occurs roughly at intervals between 256/225 and 75/64 away from the Tonic in the scale's direction of construction.  This region is characterized by intervals that don’t consistently act as either seconds or thirds.  In Bass-Up tonality, this functionality has its roots in the seventh subharmonic, though 7/6 is another notable interval included in this range.
+== Additional Functions ==
+In addition to the Functions derived from both German theory and Viennese theory, as well as the Antitonic and Paradiatonic functions, there have been other functions described, though the terms here are those used by Aura.
 === Circumdominant Regions ===
@@ Line 78: / Line 100: @@
 '''Mocksycophant''' - In Bass-Up Tonality, this is an interval that ranges from 10/7 to 20/13 which acts as a type of fourth rather than either a type of fifth or a type of sixth.  These intervals are generally not that common outside of Jazz and Blues.
-== Paradiatonic Functions ==
-In addition to all the aforementioned Diatonic functions, Aura takes a more just version of [[Wikipedia:Ivan Wyschnegradsky|Wyschnegradsky]]'s [[Wikipedia:Major fourth and minor fifth|Major Fourth]] interval (under a different name) to be a second axis for defining many of the various microtonal functions, which, in relation to the Diatonic functions, he dubs "'''paradiatonic functions'''".
 [[Category:Terms]]
 [[Category:Theory]]
 [[Category:Method]]