User:Aura/Aura's Ideas on Functional Harmony (Part 1): Difference between revisions
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'''Tonic''' - This is the note that serves as the tonal center, and thus, the main resolution tone, and is the note for which scales are named (e.g. the key of C major is so-named because in this scale, C serves as the 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]]. Beyond being simply one of the primary three functions in German Theory, it is the only function that is known to be universal when it comes to tonal music, with the various other functions being collectively defined as ''Nontonic'', thus, it shouldn't come as a great surprise that the Tonic exerts a very powerful influence on the context of functional harmony regardless of the nature of the tonal music system in question, even helping to define aspects of the other primary functions on the chord level as opposed to the root level. In the realm of microtonality, the Tonic is not an interval that admits a lot of competition outside of modulation, and thus, deviations from a perfect 1/1 of up to 3.5 [[cent]]s away from the Tonic are considered here to be found in the Tonic's "event horizon", in which they are either absorbed into the bandwidth of the Tonic itself, altered through [[fudging]], or simply [[Tempering_out|tempered out]]. Furthermore, the uncanny valley around the Tonic is noticeably larger that those around the other two functions, with wolf intervals around the Tonic only being tolerable in melody and ornamentation but not harmony. | '''Tonic''' - This is the note that serves as the tonal center, and thus, the main resolution tone, and is the note for which scales are named (e.g. the key of C major is so-named because in this scale, C serves as the 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]]. Beyond being simply one of the primary three functions in German Theory, it is the only function that is known to be universal when it comes to tonal music, with the various other functions being collectively defined as ''Nontonic'', thus, it shouldn't come as a great surprise that the Tonic exerts a very powerful influence on the context of functional harmony regardless of the nature of the tonal music system in question, even helping to define aspects of the other primary functions on the chord level as opposed to the root level. In the realm of microtonality, the Tonic is not an interval that admits a lot of competition outside of modulation, and thus, deviations from a perfect 1/1 of up to 3.5 [[cent]]s away from the Tonic are considered here to be found in the Tonic's "event horizon", in which they are either absorbed into the bandwidth of the Tonic itself, altered through [[fudging]], or simply [[Tempering_out|tempered out]]. Furthermore, the uncanny valley around the Tonic is noticeably larger that those around the other two functions, with wolf intervals around the Tonic only being tolerable in melody and ornamentation but not harmony. | ||
'''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, though 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, 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. Secondly, 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 the direction of tonality 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. 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. With all that said, it should be noted that the level of importance typically associated with the Dominant goes instead to a different note instead of a 3/2 Perfect 5th in scales where the 5th scale degree is too far away from 3/2, and that there are a variety of other interval constructions which have the tendency to create tension which requires the Tonic to resolve- these surrogate Dominants are called "'''Paradominants'''" | '''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, though 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, 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. Secondly, 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 the direction of tonality 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. 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. With all that said, it should be noted that the level of importance typically associated with the Dominant goes instead to a different note instead of a 3/2 Perfect 5th in scales where the 5th scale degree is too far away from 3/2, and that there are a variety of other interval constructions which have the tendency to create tension which requires the Tonic to resolve- these surrogate Dominants are called "'''Paradominants'''". On the chord level, not only is the root level definition of the Dominant function at play, but it should also be noted that no Dominant or Paradominant sees the Tonic occurring in the proximal structure of its chord- that is, as a third or fifth. | ||
'''Serviant''' - Compared to the term "''Subdominant''" from traditional music theory, 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. 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 two-fold. 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 | '''Serviant''' - Compared to the term "''Subdominant''" from traditional music theory, 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. 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 two-fold. 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. The Serviant tends to resolve towards the Dominant, or else some other note that acts as a surrogate for the Dominant, though it can also create plagal cadences. With all that said, it should be noted that the level of importance typically associated with the Serviant goes instead to a different note instead of a 4/3 Perfect 4th in scales where the 4th scale degree is too far away from 4/3- these surrogate Serviants are called "'''Paraserviants'''". 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 or Paraserviants sees the Tonic occurring in the proximal structure of its chord- that is, as either a third or fifth- this 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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'''Preparation''' - These are notes that "prepare the way" for either a Dominant or a Serviant through any of the above operations, or through some other mechanism. Functions which have this kind of role relative are denoted with a "pre-" prefix here. | '''Preparation''' - These are notes that "prepare the way" for either a Dominant or a Serviant through any of the above operations, or through some other mechanism. Functions which have this kind of role relative are denoted with a "pre-" prefix here. | ||
''' | '''Imitation''' - These are notes that can substitute for either the Dominant or Serviant functions through chromatic-type alteration. Functions which have this kind of role are denoted with a "mock-" prefix here. | ||
'''Detempering''' - These are notes that appear when the comma or subchroma that separates them from the Tonic or from one of the primary Nontonic functions are not tempered out, and often, though not always, fall within the uncanny valley of the three primary functions. | '''Detempering''' - These are notes that appear when the comma or subchroma that separates them from the Tonic or from one of the primary Nontonic functions are not tempered out, and often, though not always, fall within the uncanny valley of the three primary functions. | ||
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I should point out that all of the scale degree functions described in Viennese Theory, as well as a few additional functions listed on this page, can be classified as '''first derivative''' functions because only one instance of any given derivational process is needed to reach them. | I should point out that all of the scale degree functions described in Viennese Theory, as well as a few additional functions listed on this page, can be classified as '''first derivative''' functions because only one instance of any given derivational process is needed to reach them. | ||
'''Contralead''' - This function, although not found in Viennese Theory proper, is easily derivable through the Tonic Adjacent function, the Serviant Parallel function of [[16/15]], and the Antidominant function typified by the root of the [[Wikipedia:Neapolitan chord|Neapolitan chord]]. As such, this lowered second scale degree in Bass-Up | '''Contralead''' - This function, although not found in Viennese Theory proper, is easily derivable through the Tonic Adjacent function, the Serviant Parallel function of [[16/15]], and the Antidominant function typified by the root of the [[Wikipedia:Neapolitan chord|Neapolitan chord]]. As such, this lowered second scale degree in Bass-Up tonality should be considered as more than just a simple chromatic alteration of the Supertonic. Taking this idea into the realm of Microtonality, the Contralead is an interval that maps to both 1\7 and 2\24, and serves in part as a leading tone in the direction opposite that of the scale's direction of construction, but also has a Predominant function. | ||
'''Supertonic''' - This function is easily derivable through | '''Supertonic''' - This function is easily derivable through the Tonic Adjacent function, the Serviant Parallel function, and the Didominant function of [[9/8]], or, at least that's the case in Bass-Up tonality, where it is an interval that maps to both 1\7 and 4\24 and occurs above the Tonic as the second scale degree. | ||
'''Mediant''' - This is the note that | '''Mediant''' - This is the note that maps to 2\7 from the Tonic in the scale's direction of construction, and is named due to being roughly halfway between the Tonic and the Dominant. This is the first of the two diatonic scale degrees with both the most possibilities for realization and the ability to easily host chords built with wolf fifths. In terms of any sort of octave-reduced harmonic-subharmonic interval axis featuring harmonic motion by intervals in this region, one of the main features that characterizes this region is a tendency towards stagnation, leading to the historical characterization of these sorts of harmonies as "weak harmonies". Aside from this tendency towards stagnation, the properties that are central to the Mediant function are all most easily derived through a compositing of the Tonic Parallel function and the Serviant Adjacent function, and in addition, have both Preserviant and Predominant functions. However, there are other functional aspects of a Mediant that are determined by whether the interval in question is considered consonant or dissonant, with [[5/4]] and [[6/5]] being examples of consonant Mediants, and [[81/64]] and [[32/27]] being examples of dissonant Mediants. | ||
'''Antitonic''' - This is a special case, see the next section for more discussion of this function. | '''Antitonic''' - This is a special case, see the next section for more discussion of this function. | ||
'''Contramediant''' - Compared to the term "''Submediant''" from traditional music theory, the term "Contramediant" may have a slightly different frame of reference, as while a "Submediant" is halfway between the Tonic and a "Subdominant", the "Contramediant" is halfway between the Tonic and the Serviant. The Contramediant is | '''Contramediant''' - Compared to the term "''Submediant''" from traditional music theory, the term "Contramediant" may have a slightly different frame of reference, as while a "Submediant" is halfway between the Tonic and a "Subdominant", the "Contramediant" is halfway between the Tonic and the Serviant. The Contramediant is the note that maps to 5\7 from the Tonic in the scale's direction of construction, and is the second of two scale degrees both the most possibilities for realization and the ability to easily host chords built with wolf fifths. Furthermore, it also displays a tendency towards stagnation akin to that of the Mediant, leading to harmonies in this region being historically designated as "weak harmonies". From a functional standpoint, the properties that are central to the Contramediant function are most easily derived through a compositing of the Tonic Parallel function and the Dominant Adjacent function, and in addition, have both Preserviant and Predominant functions. However, there are other functional aspects of a Contramediant that are determined by whether the interval in question is considered consonant or dissonant, with [[5/3]] and [[8/5]] being examples of consonant Contramediants, and [[27/16]] and [[128/81]] being examples of dissonant Contramediants. | ||
'''Subtonic''' - This function is easily derivable through a compositing of the Dominant Parallel function of [[9/5]] and Double Serviant function of [[16/9]], with both intervals additionally having Predominant functions, or at least that's the case in Bass-Up Tonality, and especially in meantone contexts where the two intervals are equated. As with the Supertonic, however, I would argue that due to both the close proximity of 9/5 and 16/9 even in non-meantone settings, these two functions often overlap to some degree or other. This is the note that occurs roughly at intervals between [[225/128]] and [[1024/567]] above the Tonic as the seventh scale degree. | '''Subtonic''' - This function is easily derivable through a compositing of the Dominant Parallel function of [[9/5]] and Double Serviant function of [[16/9]], with both intervals additionally having Predominant functions, or at least that's the case in Bass-Up Tonality, and especially in meantone contexts where the two intervals are equated. As with the Supertonic, however, I would argue that due to both the close proximity of 9/5 and 16/9 even in non-meantone settings, these two functions often overlap to some degree or other. This is the note that occurs roughly at intervals between [[225/128]] and [[1024/567]] above the Tonic as the seventh scale degree. | ||
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*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 system. 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 system. 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 in Bass-Up Tonality, it is called an '''Acuosycophant''' in this kind of tonal system. 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 system. A classic example of such an interval is 729/512. | ||
== Chromatic Functions == | |||
Although most of these functions are accounted for by applying the Imitation process to either the Dominant or Serviant, there are two for which this is not the case. | |||
'''Superdislocant''' - This is a note that is to either the Lead or the Contralead 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 either the Lead or the Contralead 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. | |||
== Additional Functions == | == Additional Functions == | ||
In addition to the functions derived from both German theory and Viennese theory, as well as the Antitonic functions, I describe other functions here. Do note that these tend to only show up in systems where intervals such as these are distinct, such as higher [[EDO]]s. | In addition to the functions derived from both German theory and Viennese theory, as well as the Antitonic functions, I describe other functions here. Do note that most of these these tend to only show up in systems where intervals such as these are distinct, such as higher [[EDO]]s. | ||
=== Circumtonic Regions === | === Circumtonic Regions === | ||
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The Circumtonic regions are the two main regions on either side of the Tonic, outside the Tonic's "event horizon". These intervals are invariably inside the Tonic's uncanny valleys, so they're usually avoided outside of modulation. | The Circumtonic regions are the two main regions on either side of the Tonic, outside the Tonic's "event horizon". These intervals are invariably inside the Tonic's uncanny valleys, so they're usually avoided outside of modulation. | ||
'''Supercommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly | '''Supercommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 32 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. | ||
'''Subcommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly | '''Subcommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 32 cents below the Tonic. As with Supercommatic intervals, 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. | ||
=== Circumdominant Regions === | === Circumdominant Regions === | ||
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'''Gravoserviant''' - This function is a compositing of Serviant detempering and Misoserviant (see below on Basic Parachromatic Functions). | '''Gravoserviant''' - This function is a compositing of Serviant detempering and Misoserviant (see below on Basic Parachromatic Functions). | ||
=== Circumantitonic Regions === | |||
'''Acuotyrant''' - This is a note that occurs at intervals ranging from 10/7 to 36/25 away from the Tonic. Although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless operates more on the side of brute force when it contrasts with the Tonic. | |||
'''Gravosycophant''' - This is a note that occurs at intervals ranging from 25/18 to 7/5 away from the Tonic. Although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless often runs a high risk of tonicizing either the Dominant or the Geminodominant. | |||
== Paradiatonic and Parachromatic Functions == | == Paradiatonic and Parachromatic Functions == | ||
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'''Semilead''' - This is a note that occurs roughly at intervals between 1024/567 and 13/7 above the Tonic as the seventh scale degree. Naturally, this interval functions as a sort of Dominant Parallel, though there are significant differences from both a Subtonic and a Lead. For starters, the Semilead also has the Misodominant Parallel and Antimisoserviant functions, and furthermore, in Bass-Up Tonality, a Semilead is also likely to either resolve downwards to a Subtonic, or, upwards to either the Lead. The [[11/6]] neutral seventh is a rather typical example of an interval with this function. | '''Semilead''' - This is a note that occurs roughly at intervals between 1024/567 and 13/7 above the Tonic as the seventh scale degree. Naturally, this interval functions as a sort of Dominant Parallel, though there are significant differences from both a Subtonic and a Lead. For starters, the Semilead also has the Misodominant Parallel and Antimisoserviant functions, and furthermore, in Bass-Up Tonality, a Semilead is also likely to either resolve downwards to a Subtonic, or, upwards to either the Lead. The [[11/6]] neutral seventh is a rather typical example of an interval with this function. | ||
== Differences from Traditional Neo-Riemannian Theory == | == Differences from Traditional Neo-Riemannian Theory == | ||