User:Aura/Aura's Ideas on Functional Harmony (Part 1): Difference between revisions
Finally trying to put down my thoughts on some recurring questions |
Found a better follow-up to a sycophant chord that stays in key. |
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Instead, I contend that it's the root of a chord and the relationship between the chord root and the [[Tonic]] that dictates the bulk of the context for the function of the other notes in a given chord, with other bits of information being dictated by the relationship of other notes in the chord to both the Tonic and the actual chord root- do note that which note is considered to be the Tonic can in fact change based on additional context, such as the location of tritones- and, to a lesser extent, wolf fifths and wolf fourths- in a scale, as these, in combination with a tonality's direction of construction, can tonicize certain notes. From there, I think that only perfect fourths or perfect fifths that are either above or below a chord root can actually create stable frameworks for building chords, while dividing such intervals in two pieces without causing crowding creates the notes that impart character and color to chords. However, because perfect fifths are larger than perfect fourths, there's greater ease and a greater selection of options in dividing a perfect fifth without causing crowding than there is in doing the same with a perfect fourth. | Instead, I contend that it's the root of a chord and the relationship between the chord root and the [[Tonic]] that dictates the bulk of the context for the function of the other notes in a given chord, with other bits of information being dictated by the relationship of other notes in the chord to both the Tonic and the actual chord root- do note that which note is considered to be the Tonic can in fact change based on additional context, such as the location of tritones- and, to a lesser extent, wolf fifths and wolf fourths- in a scale, as these, in combination with a tonality's direction of construction, can tonicize certain notes. From there, I think that only perfect fourths or perfect fifths that are either above or below a chord root can actually create stable frameworks for building chords, while dividing such intervals in two pieces without causing crowding creates the notes that impart character and color to chords. However, because perfect fifths are larger than perfect fourths, there's greater ease and a greater selection of options in dividing a perfect fifth without causing crowding than there is in doing the same with a perfect fourth. | ||
I must also admit that I think additional harmonic information can be supplied by the likes of both otonal and utonal [[primodality]]. While | I must also admit that I think additional harmonic information can be supplied by the likes of both otonal and utonal [[primodality]], albeit my approach is a bit more unusual. While primes other than 2 can form the basis of tonality, it should be mentioned that the higher the prime involved as common numerator and or common denominator, the weaker the tonicization effect. Furthermore, I'm of the opinion that if you want to add intervals from segments with higher-prime denominators such as /3 or /5 to an otherwise /2^n segment to help flesh out what is essentially a Bass-Up tonality, it will usually work out best if that /3 or /5 interval is also a 2^n/ type of interval- in this case, 4/3 or 8/5. Similarly, I'm of the opinion that if you want to add intervals from segments with higher-prime denominators such as /5 or /7 to an otherwise /3*2^n segment to help flesh out what is essentially a Bass-Up tonality, it will usually work out best if that /5 or /7 interval is also a 3*2^n/ type of interval- in this case, 6/5 or 12/7. | ||
== Facets Derived from German Theory == | == Facets Derived from German Theory == | ||
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The three basic functions have their roots in [[LCJI]]. The functions themselves are labeled as follows: | The three basic functions have their roots in [[LCJI]]. The functions themselves are labeled as follows: | ||
'''Tonic''' - This functionality has its roots in the fundamental at the root of both the harmonic and subharmonic series, which for all intents and purposes, can be thought of as [[1/1]], and, in [[octave equivalence|octave equivalent]] systems, [[2/1]]. To use a character metaphor for how the Tonic acts in functional harmony, the Tonic is the king of the Kingdom of Tonality- a very good king who not only exercises the highest authority in matters of governing the kingdom and does not tolerate challenges to his leadership, but also knows how be a top-notch confidante to his subjects both wherever and whenever possible. For more specifics on the functions of the Tonic, see the linked article. | '''[[Tonic]]''' - This functionality has its roots in the fundamental at the root of both the harmonic and subharmonic series, which for all intents and purposes, can be thought of as [[1/1]], and, in [[octave equivalence|octave equivalent]] systems, [[2/1]]. To use a character metaphor for how the Tonic acts in functional harmony, the Tonic is the king of the Kingdom of Tonality- a very good king who not only exercises the highest authority in matters of governing the kingdom and does not tolerate challenges to his leadership, but also knows how be a top-notch confidante to his subjects both wherever and whenever possible. For more specifics on the functions of the Tonic, see the linked article. | ||
'''Dominant''' - As per the name, and as noted on the [[Wikipedia:Dominant (music)|Wikipedia article]], the Dominant is the second most important after the Tonic, However, in contrast to what is stated about the Dominant in the article, there are several caveats which must be addressed in the realm of microtonality. Firstly, there's the matter of its origins- specifically, it is generated by the tonic as the first [[3-limit]] interval and indeed the first nontonic note in the "majoresque" direction. Secondly, there's the matter of just what it does, as it typically serves as one, or often more, of the following- a primary creator of instability in the "majoresque" direction that requires the Tonic for resolution, the second most important melodic and or harmonic anchor after the Tonic- a function that requires it to be tuned with a great deal of accuracy in order to blend well with the Tonic- as well as a generator of many of the "majoresque" notes in a [[5L 2s|diatonic MOS]], and or a discourager against the usage of other microtonally nearby pitches. Thirdly, one must take stock of the fact that, aside from the Unison and Octave, each octave-reduced harmonic and corresponding subharmonic interval come together to generate their own axis which has a preferred direction of travel<ref>[https://www.youtube.com/watch?v=HBdWxSxxe1M Quartertone Harmony - Beyond the Circle of Fifths SD 480p]</ref> which is determined by a Tonality's direction of construction. Finally, one must take stock of the fact that when you take the notes that occur before the Tonic on each of these axes when moving in the preferred direction of travel and place them in a sequence, one finds that a clear hierarchy of functional strength based on the closeness of harmonic and subharmonic connection to the Tonic becomes apparent, with the [[3/2]] Perfect 5th away from the Tonic in a tonality's direction of construction naturally emerging as the note with the strongest connection to the Tonic, though it should be noted that the relationships in this hierarchy are quite sensitive to detuning, and can even be scrambled by such detuning. Thus, the term "Dominant"- in its most basic form as referred to in this article, and specifically at the root level- is restricted to where it only refers to such notes that occur roughly at a 3/2 interval away from the Tonic in the scale's direction of construction, with acceptable detuning levels being at around 3.5 cents from JI on either side. On the chord level, not only is the root level definition of the Dominant function at play, but it should also be noted that the Dominant does not see the Tonic occurring in the proximal tertian structure of its chord- that is, as a third or fifth. To use a character metaphor for how the Dominant acts in functional harmony, the Dominant is both the Head Steward of the Tonic's castle, and the one that executes the Tonic's directives as a Manager of Civil Service in the Kingdom of Tonality. | '''Dominant''' - As per the name, and as noted on the [[Wikipedia:Dominant (music)|Wikipedia article]], the Dominant is the second most important after the Tonic, However, in contrast to what is stated about the Dominant in the article, there are several caveats which must be addressed in the realm of microtonality. Firstly, there's the matter of its origins- specifically, it is generated by the tonic as the first [[3-limit]] interval and indeed the first nontonic note in the "majoresque" direction. Secondly, there's the matter of just what it does, as it typically serves as one, or often more, of the following- a primary creator of instability in the "majoresque" direction that requires the Tonic for resolution, the second most important melodic and or harmonic anchor after the Tonic- a function that requires it to be tuned with a great deal of accuracy in order to blend well with the Tonic- as well as a generator of many of the "majoresque" notes in a [[5L 2s|diatonic MOS]], and or a discourager against the usage of other microtonally nearby pitches. Thirdly, one must take stock of the fact that, aside from the Unison and Octave, each octave-reduced harmonic and corresponding subharmonic interval come together to generate their own axis which has a preferred direction of travel<ref>[https://www.youtube.com/watch?v=HBdWxSxxe1M Quartertone Harmony - Beyond the Circle of Fifths SD 480p]</ref> which is determined by a Tonality's direction of construction. Finally, one must take stock of the fact that when you take the notes that occur before the Tonic on each of these axes when moving in the preferred direction of travel and place them in a sequence, one finds that a clear hierarchy of functional strength based on the closeness of harmonic and subharmonic connection to the Tonic becomes apparent, with the [[3/2]] Perfect 5th away from the Tonic in a tonality's direction of construction naturally emerging as the note with the strongest connection to the Tonic, though it should be noted that the relationships in this hierarchy are quite sensitive to detuning, and can even be scrambled by such detuning. Thus, the term "Dominant"- in its most basic form as referred to in this article, and specifically at the root level- is restricted to where it only refers to such notes that occur roughly at a 3/2 interval away from the Tonic in the scale's direction of construction, with acceptable detuning levels being at around 3.5 cents from JI on either side. On the chord level, not only is the root level definition of the Dominant function at play, but it should also be noted that the Dominant does not see the Tonic occurring in the proximal tertian structure of its chord- that is, as a third or fifth. To use a character metaphor for how the Dominant acts in functional harmony, the Dominant is both the Head Steward of the Tonic's castle, and the one that executes the Tonic's directives as a Manager of Civil Service in the Kingdom of Tonality. | ||
'''Serviant''' - Compared to the term "''Subdominant''" from music theory of old, the term "Serviant", specifically at the root level, is restricted to those notes that occur roughly at a [[4/3]] interval away from the Tonic in the scale's direction of construction since the Serviant function is essentially the inverse of the Dominant function, and acts as a sort of counterweight to the Dominant relative to the Tonic. It typically serves as one, or often more, of the following— a primary creator of instability in the "minoresque" direction that can either be intensified with the dominant or resolved with the tonic, the third most important melodic and or harmonic anchor after the tonic and the dominant, a generator of many of the "minoresque" notes in a diatonic MOS, and or a discourager against the usage of other microtonally nearby pitches. To use a character metaphor for how the Serviant acts in functional harmony, the Serviant is a Servant who goes above and beyond the call of duty and acts as a confidante that observes things and reports back to the Dominant and Tonic about the way things are working both inside and outside the Tonic's castle due to its relationships to various Nontonic functions. Although one might think that the term "Subdominant" would be eligible for getting a similar treatment to the term "Dominant" here, the problems with such an option are threefold. Firstly, not all possible "Subdominant" harmonies have the same harmonic properties relative to the Tonic, as there is an extremely close connection between the Tonic and the 4/3 Perfect 4th. Secondly, in music built from the Treble downwards, the notes with these sorts of functions are actually located ''above'' the Dominant. Thirdly, in common parlance, "Subdominant" is often equated with "Predominant", however, while the Serviant does tend to resolve towards the Dominant, or else some other note that acts as a surrogate for the Dominant, the fact remains that it can also create plagal cadences, which break the modern Tonic-Dominant-Predominant paradigm and are better explained in part by the ideals of the two Germanic schools. On the chord level, not only is the root level definition of the Serviant function at play, but it should also be noted that a Serviant chord often sees the Tonic occurring in the proximal structure of its chord- that is, as either a third or fifth- which explains why Serviant chords are weaker than their Dominant counterparts in both Bass-Up and Treble-Down Tonalities. | '''Serviant''' - Compared to the term "''Subdominant''" from music theory of old, the term "Serviant" (or, alternatively "Servient"), specifically at the root level, is restricted to those notes that occur roughly at a [[4/3]] interval away from the Tonic in the scale's direction of construction since the Serviant function is essentially the inverse of the Dominant function, and acts as a sort of counterweight to the Dominant relative to the Tonic. It typically serves as one, or often more, of the following— a primary creator of instability in the "minoresque" direction that can either be intensified with the dominant or resolved with the tonic, the third most important melodic and or harmonic anchor after the tonic and the dominant, a generator of many of the "minoresque" notes in a diatonic MOS, and or a discourager against the usage of other microtonally nearby pitches. To use a character metaphor for how the Serviant acts in functional harmony, the Serviant is a Servant who goes above and beyond the call of duty and acts as a confidante that observes things and reports back to the Dominant and Tonic about the way things are working both inside and outside the Tonic's castle due to its relationships to various Nontonic functions. Although one might think that the term "Subdominant" would be eligible for getting a similar treatment to the term "Dominant" here, the problems with such an option are threefold. Firstly, not all possible "Subdominant" harmonies have the same harmonic properties relative to the Tonic, as there is an extremely close connection between the Tonic and the 4/3 Perfect 4th. Secondly, in music built from the Treble downwards, the notes with these sorts of functions are actually located ''above'' the Dominant. Thirdly, in common parlance, "Subdominant" is often equated with "Predominant", however, while the Serviant does tend to resolve towards the Dominant, or else some other note that acts as a surrogate for the Dominant, the fact remains that it can also create plagal cadences, which break the modern Tonic-Dominant-Predominant paradigm and are better explained in part by the ideals of the two Germanic schools. On the chord level, not only is the root level definition of the Serviant function at play, but it should also be noted that a Serviant chord often sees the Tonic occurring in the proximal structure of its chord- that is, as either a third or fifth- which explains why Serviant chords are weaker than their Dominant counterparts in both Bass-Up and Treble-Down Tonalities. | ||
=== Basic Diatonic Function-Deriving Operations === | === Basic Diatonic Function-Deriving Operations === | ||
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'''Antipodism''' - Notes that are either opposite in pitch hue or nearly so due to being approximately half an octave away from the starting point are harmonically opposed to the starting point. Non-tonic notes related through this process tend to have the opposite function relative to the Tonic. For the notes related to the Tonic by this process, see Antitonic below. | '''Antipodism''' - Notes that are either opposite in pitch hue or nearly so due to being approximately half an octave away from the starting point are harmonically opposed to the starting point. Non-tonic notes related through this process tend to have the opposite function relative to the Tonic. For the notes related to the Tonic by this process, see Antitonic below. | ||
'''Preparation''' - Notes that "prepare the way" for either a Dominant or a Serviant through any of the above operations, or through some other mechanism, relate to said notes by this process. Functions which have this kind of role relative are denoted with a "pre-" prefix here. | '''Preparation''' - Notes that "prepare the way" for either a Dominant or a Serviant through any of the above operations, or through some other mechanism, relate to said notes by this process. Functions which have this kind of role relative to some other function are denoted with a "pre-" prefix here. | ||
'''Detempering''' - When the comma or subchroma that separates a note from one of the three primary functions is not tempered out, it results in the appearance of notes with either similar functions to the Dominant or Serviant, or, in the case of this function being applied to the Tonic, a Nontonic function similar to one of the Chromatic functions. Often, though not always, these notes fall within the uncanny valleys of the three primary functions. | '''Detempering''' - When the comma or subchroma that separates a note from one of the three primary functions is not tempered out, it results in the appearance of notes with either similar functions to the Dominant or Serviant, or, in the case of this function being applied to the Tonic, a Nontonic function similar to one of the Chromatic functions. Often, though not always, these notes fall within the uncanny valleys of the three primary functions. | ||
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'''Subaequient''' - This subtype of Subvicinant is an interval that maps to 6\7 and either 20\24 or 21\24 depending on a variety of factors. It is so named for the equable heptatonic region below the Tonic, which comprises the bulk of its range, and for its capacity to straddle the line between the Subtonic and Subabrogant functions in voice-leading. Conspicuously, it is distinct from Subtonic and Subabrogant functions only in finer tuning systems, with the approximation ranges for [[9/5]] and [[20/11]] being located at its borders. | '''Subaequient''' - This subtype of Subvicinant is an interval that maps to 6\7 and either 20\24 or 21\24 depending on a variety of factors. It is so named for the equable heptatonic region below the Tonic, which comprises the bulk of its range, and for its capacity to straddle the line between the Subtonic and Subabrogant functions in voice-leading. Conspicuously, it is distinct from Subtonic and Subabrogant functions only in finer tuning systems, with the approximation ranges for [[9/5]] and [[20/11]] being located at its borders. | ||
'''Subabrogant''' - This subtype of | '''Subabrogant''' - This subtype of Subvicinant is not a traditional diatonic function at all since it is an interval that maps to both 6\7 and 21\24 in Bass-Up tonality. As per this function's name, the intervals in this range are like Subcollocants in that they may cause listeners to forget the ending pitch's relationship to the starting pitch in voice-leading. However, they, like Superabrogants, feel as if they're rather disconnected from the Tonic, and furthermore, the occurrence of two successive instances of these sorts of intervals in the same melodic line in the same direction is liable to come across as jarring. | ||
'''Subcollocant''' - This subtype of Subvicinant is the note typically referred to when people say "the leading tone", and an interval that usually maps to both 6\7 and 22\24 in Bass-Up tonality. Additionally, it has the Antiserviant function in Bass-Up tonality and the Antidominant function in Treble-Down tonality. Although triads built on this scale degree are regarded by some as simply incomplete Dominant Seventh chords, my own analysis, while acknowledging the functional similarities between the Subcollocant and the Dominant in Bass-Up tonality, sees this interval as functionally distinct from the Dominant due to the Subcollocant also being potentially 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. In actuality, two subtypes of Subcollocant exist- the "'''Proximosubcollocant'''", which always maps to both 6\7 and 22\24 in Bass-Up tonality, and the "'''Distosubcollocant'''", which maps to 6\7 and either 21\24 or 22\24 depending on a variety of factors, and is distinct from both Proximosubcollocant and Subabrogant only in finer tuning systems where it is usually more consonant, and thus, weaker. An example of a Proximosubcollocant is [[243/128]] while an example of a Distosubcollocant is [[15/8]]. | '''Subcollocant''' - This subtype of Subvicinant is the note typically referred to when people say "the leading tone", and an interval that usually maps to both 6\7 and 22\24 in Bass-Up tonality. Additionally, it has the Antiserviant function in Bass-Up tonality and the Antidominant function in Treble-Down tonality. Although triads built on this scale degree are regarded by some as simply incomplete Dominant Seventh chords, my own analysis, while acknowledging the functional similarities between the Subcollocant and the Dominant in Bass-Up tonality, sees this interval as functionally distinct from the Dominant due to the Subcollocant also being potentially 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. In actuality, two subtypes of Subcollocant exist- the "'''Proximosubcollocant'''", which always maps to both 6\7 and 22\24 in Bass-Up tonality, and the "'''Distosubcollocant'''", which maps to 6\7 and either 21\24 or 22\24 depending on a variety of factors, and is distinct from both Proximosubcollocant and Subabrogant only in finer tuning systems where it is usually more consonant, and thus, weaker. An example of a Proximosubcollocant is [[243/128]] while an example of a Distosubcollocant is [[15/8]]. | ||
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The exact outcome and specific function of any given Antitonic depends on whether or not the interval in question is an augmented fourth or a diminished fifth. | The exact outcome and specific function of any given Antitonic depends on whether or not the interval in question is an augmented fourth or a diminished fifth. | ||
'''Sycophant''' - | '''Sycophant''' - This type of Antitonic is mapped to both 3\7 and 12\24, and is named as such on account of it having a tendency to "kiss up to" and tonicize the Dominant- that is, to cause the Dominant to become a new Tonic- unless followed up by a different note such as some type of Mediant. A prototypical example of this type of Antitonic is 45/32. | ||
'''Tyrant''' - | '''Tyrant''' - This type of Antitonic is mapped to both 4\7 and 12\24, and 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, and indeed these brute force Dominant-esque tendencies are the source of the name "Tyrant". For example, 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 some other type of substitute for a Dominant, often bypassing this type of Antitonic, though on rare occasions, a Tyrant will act as a leading tone to the Serviant. A prototypical example of this type of Antitonic is 64/45. | ||
== Chromatic Functions == | == Chromatic Functions == | ||
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'''Superdislocant''' - This is a note that is to the Supercollocant what a Tyrant is to a Sycophant. Specifically, it is the result of the Tonic being altered by some kind of chromatic semitone upwards and thus being displaced by a Nontonic function which leads away from the Tonic proper. | '''Superdislocant''' - This is a note that is to the Supercollocant what a Tyrant is to a Sycophant. Specifically, it is the result of the Tonic being altered by some kind of chromatic semitone upwards and thus being displaced by a Nontonic function which leads away from the Tonic proper. | ||
'''Subdislocant''' - This is a note that is to | '''Subdislocant''' - This is a note that is to the Subcollocant what a Tyrant is to a Sycophant. Specifically, it is the result of the Tonic being altered by some kind of chromatic semitone downwards and thus being displaced by a Nontonic function which leads away from the Tonic proper. | ||
'''Protosycophant''' - This is a note which can tonicize the Dominant, but because its harmonies fail to completely oppose those of the Tonic, it fails to count as a true Sycophant. A prototypical example of this type of interval is [[7/5]]. | '''Protosycophant''' - This is a note which can tonicize the Dominant, but because its harmonies fail to completely oppose those of the Tonic, it fails to count as a true Sycophant. A prototypical example of this type of interval is [[7/5]]. | ||
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'''Geminoserviant''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents beyond the 4/3 perfect fourth to about 3.5 cents short of [[15/11]] in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Serviant through detempering, occurring in 5-limit diatonic environments, and acting as a sort of "fraternal twin" to the Serviant, hence its name. As typified by intervals like [[27/20]], Geminoserviants are dissonant intervals that often act as a sort of Predominant and or as the inverses of Geminodominants. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the Mediant and the Contramediant, but outside of these usages and well supported chords, this kind of thing is best avoided. Apart from diatonic contexts, Geminoserviants only rise to prominence in systems where what might otherwise function as a Serviant is found just beyond the sweet spot range near the standard issue 4/3. | '''Geminoserviant''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents beyond the 4/3 perfect fourth to about 3.5 cents short of [[15/11]] in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Serviant through detempering, occurring in 5-limit diatonic environments, and acting as a sort of "fraternal twin" to the Serviant, hence its name. As typified by intervals like [[27/20]], Geminoserviants are dissonant intervals that often act as a sort of Predominant and or as the inverses of Geminodominants. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the Mediant and the Contramediant, but outside of these usages and well supported chords, this kind of thing is best avoided. Apart from diatonic contexts, Geminoserviants only rise to prominence in systems where what might otherwise function as a Serviant is found just beyond the sweet spot range near the standard issue 4/3. | ||
'''Pseudoserviant''' - This is a note that occurs roughly at intervals ranging from about 25 cents to about 3.5 cents short of the 4/3 perfect fourth in the scale's direction of construction. Because none of these intervals arise naturally in a [[5-limit]] diatonic scale, and because they fall within one of the | '''Pseudoserviant''' - This is a note that occurs roughly at intervals ranging from about 25 cents to about 3.5 cents short of the 4/3 perfect fourth in the scale's direction of construction. Because none of these intervals arise naturally in a [[5-limit]] diatonic scale, and because they fall within one of the Serviant's uncanny valleys, it is rare to see this functionality outside of systems where what might otherwise function as a Serviant is found just short the sweet spot range near the standard issue 4/3. However, these intervals naturally arise in the Circle of Thirteenth Harmonics, in which case they have a Preserviant function. | ||
=== Governing Rules === | === Governing Rules === | ||
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However, in non-meantone settings, it is possible to have a chord which shares its root with the Tertiary Dominant and shares both its third and fifth with the Relative Minor. Since such an arrangement results in a wolf fifth, and since the wolf fifth is a dissonance requiring resolution, one could argue that such a chord has a function distinct from the more traditional options- specifically one which is involved in interrupted cadences, as well as in starting cadences that "wind down" such as VImin-IVmaj-Imaj or even VImin-Vmaj-Imaj. This particular function is what I call the ''Major Contramediant Tensive''. | However, in non-meantone settings, it is possible to have a chord which shares its root with the Tertiary Dominant and shares both its third and fifth with the Relative Minor. Since such an arrangement results in a wolf fifth, and since the wolf fifth is a dissonance requiring resolution, one could argue that such a chord has a function distinct from the more traditional options- specifically one which is involved in interrupted cadences, as well as in starting cadences that "wind down" such as VImin-IVmaj-Imaj or even VImin-Vmaj-Imaj. This particular function is what I call the ''Major Contramediant Tensive''. | ||
'''Primary Adpositive Purity''' - This rule is that for every chord root located one step away from either the Tonic, Dominant or Serviant along the Circle of Fifths, there is a demand for the fifth of the chord in question to be within 3.5 cents of a just 3/2, | '''Primary Adpositive Purity''' - This rule is that for every chord root located one step away from either the Tonic, Dominant or Serviant along the Circle of Fifths, there is a demand for the fifth of the chord in question to be within 3.5 cents of a just 3/2, though in the case of a chord that has a root located at two steps away from the Tonic along the along the Circle of Fifths, the fifth of the chord can be deliberately subjected to the process of Displacement as mentioned before and not run afoul of this restriction. However, chord roots that are located three or more steps away from the Tonic along the circle of fifths are not subject to the aforementioned restriction due to their distance. This means that the Tonic, Dominant, Serviant, Supertonic and Subtonic chords all demand a perfect fifth as the fifth of the chord, whether you are building the Tonality upwards or downwards, but wolf fifths can occur easily as the fifth of chords which are located three or more steps away from the Tonic. | ||
== Beyond Diatonic and Chromatic Functional Harmony == | == Beyond Diatonic and Chromatic Functional Harmony == | ||
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=== The Paradiatonic Scales === | === The Paradiatonic Scales === | ||
The Paradiatonic scales from a given tonic acts as a sort of "second shelf" of that tonality. | The Paradiatonic scales from a given tonic acts as a sort of "second shelf" of that tonality. Note that scale degrees in parentheses are optional. | ||
The '''Bright Paradiatonic Scale''' consists of the following scale degrees as analyzed relative to Viennese Theory's scale steps: | The '''Bright Paradiatonic Scale''' consists of the following scale degrees as analyzed relative to Viennese Theory's scale steps: | ||
I, tII/dbIII, dIII, tIV, dV, dVI, tVI/dbVII, tVII | I, tII/dbIII, (dIII), tIV, dV, (dVI), tVI/dbVII, tVII | ||
The '''Dark Paradiatonic Scale''' consists of the following scale degrees as analyzed relative to Viennese Theory's scale steps: | The '''Dark Paradiatonic Scale''' consists of the following scale degrees as analyzed relative to Viennese Theory's scale steps: | ||
I, dbII tII/dbIII, dIII, tIV, dV, dVI, tVI/dbVII | I, dbII tII/dbIII, (dIII), tIV, dV, (dVI), tVI/dbVII | ||
While the Diatonic scale itself has seven notes, the two Paradiatonic scales each have eight notes, furthermore, the tunings of each note in each Paradiatonic scale not only depend upon the exact tuning of the Diatonic scale used as a basis, but also vary considerably when it comes to the notes of the Paradiatonic that occur between the main Diatonic interval category ranges. | While the Diatonic scale itself has seven notes, the two Paradiatonic scales each have eight notes, furthermore, the tunings of each note in each Paradiatonic scale not only depend upon the exact tuning of the Diatonic scale used as a basis, but also vary considerably when it comes to the notes of the Paradiatonic that occur between the main Diatonic interval category ranges. | ||
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=== Basic Paradiatonic Functions === | === Basic Paradiatonic Functions === | ||
Out of the various functions found in the Paradiatonic | Out of the various functions found in the Paradiatonic scales, four of them- specifically, the tII/dbIII, tIV, dV and tVI/dbVII- can be considered basic, while the other three are first derivatives. As with the three basic diatonic functions, the four most basic paradiatonic functions have their roots in LCJI. In the order listed, the tII/dbIII, tIV, dV and tVI/dbVII functions are the following... | ||
'''Contravaricant''' - Named in contrast to the Varicant function, this is an interval that maps to both 1\5 and 5\24 in the scale's direction of construction, lying roughly in the middle of the 4/3 interval separating the Tonic and the Serviant above it. Intervals in the Contravaricant region often don't consistently act as either seconds or thirds, or even act as a cross between a second and a third, only without potential for crowding in chords. In Bass-Up tonality, this functionality is first encountered in the form of [[8/7]], though [[7/6]] is another notable interval included in this range, with intervals in this range having Predominant, Preserviant, and Dominant Parallel functions, as well as an overlap between Tonic Adjacent and Tonic Parallel functions. There are two different subtypes of this function that differ from each other based on how they are approached in interval stacking, the "'''Pseudobidominant'''", which arises from stacking two Pseudodominants, and the "'''Pseudotriserviant'''", which arises from stacking three Pseudoserviants. | '''Contravaricant''' - Named in contrast to the Varicant function, this is an interval that maps to both 1\5 and 5\24 in the scale's direction of construction, lying roughly in the middle of the 4/3 interval separating the Tonic and the Serviant above it. Intervals in the Contravaricant region often don't consistently act as either seconds or thirds, or even act as a cross between a second and a third, only without potential for crowding in chords. In Bass-Up tonality, this functionality is first encountered in the form of [[8/7]], though [[7/6]] is another notable interval included in this range, with intervals in this range having Predominant, Preserviant, and Dominant Parallel functions, as well as an overlap between Tonic Adjacent and Tonic Parallel functions. There are two different subtypes of this function that differ from each other based on how they are approached in interval stacking, the "'''Pseudobidominant'''", which arises from stacking two Pseudodominants, and the "'''Pseudotriserviant'''", which arises from stacking three Pseudoserviants. | ||
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'''Intersubiant''' - This is an interval that maps to both 3\7 and 11\24 in the scale's direction of construction and comprises the intervals ranging from around 3.5 cents short of [[15/11]] to around 3.5 cents beyond [[11/8]]. Like both the Serviant and the Sycophant, intervals in this region have a Predominant function, however, this predominant function is weaker than that of the surrounding interval regions since they neither act as a counterweight to the Dominant like a Serviant nor do they completely tonicize the Dominant like a Sycophant- at least to those who are more familiar with quartertones. What is even less expected is that these same intervals also have Preserviant and Mocktyrant functions. Since the Intersubiant is not found in the Tonic's quartertone field, there is a tendency for Intersubiant chords to be preceded and or followed by other chords with roots in the same quartertone field, though one could also reasonably approach it from the Tonic, the Serviant, some kind of Contramediant, some kind of Supervicinant, or, in some tuning systems, certain kinds of Mediant. In Bass-Up tonality, this functionality has its roots in the eleventh harmonic. | '''Intersubiant''' - This is an interval that maps to both 3\7 and 11\24 in the scale's direction of construction and comprises the intervals ranging from around 3.5 cents short of [[15/11]] to around 3.5 cents beyond [[11/8]]. Like both the Serviant and the Sycophant, intervals in this region have a Predominant function, however, this predominant function is weaker than that of the surrounding interval regions since they neither act as a counterweight to the Dominant like a Serviant nor do they completely tonicize the Dominant like a Sycophant- at least to those who are more familiar with quartertones. What is even less expected is that these same intervals also have Preserviant and Mocktyrant functions. Since the Intersubiant is not found in the Tonic's quartertone field, there is a tendency for Intersubiant chords to be preceded and or followed by other chords with roots in the same quartertone field, though one could also reasonably approach it from the Tonic, the Serviant, some kind of Contramediant, some kind of Supervicinant, or, in some tuning systems, certain kinds of Mediant. In Bass-Up tonality, this functionality has its roots in the eleventh harmonic. | ||
'''Interregnant''' - This is an interval that maps to both 4\7 and 13\24 in the scale's direction of construction and comprises the intervals ranging from around 3.5 cents short of [[16/11]] to around 3.5 cents beyond [[22/15]]. 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, though it unexpectedly has decent Predominant and Mocksycophant functions | '''Interregnant''' - This is an interval that maps to both 4\7 and 13\24 in the scale's direction of construction and comprises the intervals ranging from around 3.5 cents short of [[16/11]] to around 3.5 cents beyond [[22/15]]. 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, though it unexpectedly has decent Predominant and Mocksycophant functions. Like the Intersubiant, the Interregnant is not found in the Tonic's quartertone field and is either very distant from the Tonic along the circle of fifths or is completely missed by said circle of fifths, thus, there is a tendency for Interregnant chords to be preceded and or followed by other chords with roots in the same quartertone field. Conspicuously, the distance between the Interregnant is only about a Minor Second- or, at the very most, a Supraminor Second- away from the Intersubiant, so as a result, the Interregnant can be both a set-up and a follow-up to the Intersubiant. In Bass-Up tonality, the Interregnant functionality has its roots in the eleventh subharmonic. | ||
'''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, only without potential for crowding in chords- effectively straddling the border between these two diatonic categories, hence the name "Varicant", from Latin "vāricō"<ref>[[Wiktionary: varico #Latin]]</ref>. This is an interval that maps to both 4\5 and 19\24 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- I counterargue 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. Furthermore, in contrast to the Subtonics of Bass-Up Tonality, Varicants are liable to | '''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, only without potential for crowding in chords- effectively straddling the border between these two diatonic categories, hence the name "Varicant", from Latin "vāricō"<ref>[[Wiktionary: varico #Latin]]</ref>. This is an interval that maps to both 4\5 and 19\24 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- I counterargue 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. Furthermore, in contrast to the Subtonics of Bass-Up Tonality, Varicants are liable to act as Predominants, Preserviants, and Serviant Parallels, as well as display an overlap between Tonic Adjacent and Tonic Parallel functions. There are two different subtypes of this function that differ from each other based on how they are approached in interval stacking, the "'''Pseudotridominant'''", which arises from stacking three Pseudodominants, and the "'''Pseudobiserviant'''", which arises from stacking two Pseudoserviants. | ||
=== Derivative Paradiatonic Functions === | === Derivative Paradiatonic Functions === | ||
'''Subgradient''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly 25 cents below the octave reduplication of the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Subcollocant- however, there are functional differences between a Subgradient and a | '''Subgradient''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly 25 cents below the octave reduplication of the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Subcollocant- however, there are functional differences between a Subgradient and a Subcollocant in Bass-Up Tonality. In Bass-Up Tonality, Subgradients are often more likely to be passing tones than Subcollocants, 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 Subgradient is much more likely to do so through sheer brute force, and even these cases require a proper set-up, as otherwise, the awkward tonal disconnect between the Subgradient and the Tonic is likely to result in the Subgradient resolving back down to either the Subcollocant or the Subabrogant. As if that weren't enough, the Subgradient also has the Antintersubiant function. This function used to be called the "Subdietic", though that term has since been restricted to a related composite function (see below on Composite Functions). | ||
'''Supergradient''' - This is a note that occurs at intervals between roughly 25 cents above the Tonic and 25/24 above the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Supercollocant. However, there are a few functional differences between a Supergradient and a Supercollocant in Bass-Up Tonality that are worth considering. For starters, Supergradients are often more likely to be passing tones than | '''Supergradient''' - This is a note that occurs at intervals between roughly 25 cents above the Tonic and 25/24 above the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Supercollocant. However, there are a few functional differences between a Supergradient and a Supercollocant in Bass-Up Tonality that are worth considering. For starters, Supergradients are often more likely to be passing tones than Supercollocants, and, when they’re not merely passing non-chord tones, they are just as liable to resolve upward through some sort of semitone-like motion to some form of Supercollocant or Superabrogant, 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 Supercollocant can resolve to the Tonic in part through a strong harmonic connection, a Supergradient is much more likely to do so through sheer brute force when such a resolution is noticeable. As if that weren't enough, the Subgradient also has the Antinterregnant function. This function used to be called the "Superdietic", though that term has since been restricted to a related composite function (see below on Composite Functions). | ||
=== Parachromatic Functions === | === Parachromatic Functions === | ||
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'''Subsanguinant''' - This function is one of two "bleeding tone" functions- hence its Latin-derived name- and is a compositing of Subgradient and Proximosubcollocant functions, and thus, has a more tense feel than a Proximosubcollocant alone. It should be noted that this function is formed by the overlap between different functional regions rather than merely being located at a boundary. | '''Subsanguinant''' - This function is one of two "bleeding tone" functions- hence its Latin-derived name- and is a compositing of Subgradient and Proximosubcollocant functions, and thus, has a more tense feel than a Proximosubcollocant alone. It should be noted that this function is formed by the overlap between different functional regions rather than merely being located at a boundary. | ||
'''Supersanguinant''' - This function is one of two "bleeding tone" functions- hence its Latin-derived name- and is a compositing of Supergradient and Proximosupercollocant functions, and thus, has a more tense feel than a | '''Supersanguinant''' - This function is one of two "bleeding tone" functions- hence its Latin-derived name- and is a compositing of Supergradient and Proximosupercollocant functions, and thus, has a more tense feel than a Proximosupercollocant alone. It should be noted that this function is formed by the overlap between different functional regions rather than merely being located at a boundary. | ||
'''Submaculant''' - This function is a compositing of Subgradient and Subdislocant, and thus is like the chromatic version of the Subsanguinant. | '''Submaculant''' - This function is a compositing of Subgradient and Subdislocant, and thus is like the chromatic version of the Subsanguinant. | ||
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'''Paraserviant''' - This function is essentially the inverse of the Paradominant function, and acts as a sort of counterweight to the Paradominant relative to the Tonic. Like the Paradominant function, it typically appears along prime axes other than that of the 3-limit in which they take Serviant-like function. Thus, for example, in Bass-Up tonality, the note at 8/5 above the Tonic, being a direct prime subharmonic of the Tonic, gets to serve this function in addition to its Mesodistocontramediant function as defined by the 3-limit. When considered alongside the actual Serviant function of 4/3, however, Paraserviants take on the function of Preserviant since the Serviant function of 4/3 is stronger. | '''Paraserviant''' - This function is essentially the inverse of the Paradominant function, and acts as a sort of counterweight to the Paradominant relative to the Tonic. Like the Paradominant function, it typically appears along prime axes other than that of the 3-limit in which they take Serviant-like function. Thus, for example, in Bass-Up tonality, the note at 8/5 above the Tonic, being a direct prime subharmonic of the Tonic, gets to serve this function in addition to its Mesodistocontramediant function as defined by the 3-limit. When considered alongside the actual Serviant function of 4/3, however, Paraserviants take on the function of Preserviant since the Serviant function of 4/3 is stronger. | ||
== References == | == References == | ||