User:Aura/Aura's Ideas on Functional Harmony (Part 1): Difference between revisions
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=== Basic Diatonic Functions === | === Basic Diatonic Functions === | ||
The three basic functions have their roots in [[LCJI]], and each has an uncanny valley of sorts around | The three basic functions have their roots in [[LCJI]], and each has an uncanny valley of sorts around it. The functions themselves are labeled as follows: | ||
'''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 | '''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'''". Accordingly, the intervals between [[187/128]] and [[32/21]] shall be divided into two classes depending on which side of the 3/2 Perfect Fifth they fall on and discussed in another section. However, aside from [[40/27]], none of these other intervals occur in diatonic scales. 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. | '''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'''". Accordingly, the intervals between [[187/128]] and [[32/21]] shall be divided into two classes depending on which side of the 3/2 Perfect Fifth they fall on and discussed in another section. However, aside from [[40/27]], none of these other intervals occur in diatonic scales. 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. | ||
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The way I see it, there are eight known operations which can be used to derive additional diatonic functions from the three basic functions listed above. | The way I see it, there are eight known operations which can be used to derive additional diatonic functions from the three basic functions listed above. | ||
'''Stacking''' - The notes that are arrived at through stacking multiple instances of either 3/2 or 4/3 (or their tempered counterparts) are dubbed according to the number of instances stacked, and the nature of the notes separated by the interval being stacked. Thus, stacking two instances of the Dominant or the Serviant results in the creation of the " | '''Stacking''' - The notes that are arrived at through stacking multiple instances of either 3/2 or 4/3 (or their tempered counterparts) are dubbed according to the number of instances stacked, and the nature of the notes separated by the interval being stacked. Thus, stacking two instances of the Dominant or the Serviant results in the creation of the "Didominant" or "Diserviant" respectively. This concept comes from the German language's way of referring to the chord built on the second scale degree of the Diatonic scale as the "Doppeldominante", which literally means "Double Dominant". | ||
'''Parallelism''' - Notes located in the same primary [[tetrachord]] as either the Tonic, the Dominant, or the Serviant take on similar functions to said notes, with the caveat that functions derived from the Tonic in this fashion are still technically | '''Parallelism''' - Notes located in the same primary [[tetrachord]] as either the Tonic, the Dominant, or the Serviant take on similar functions to said notes, with the caveat that functions derived from the Tonic in this fashion are still technically Nontonic functions. This process is one of two that create what in traditional music theory are referred to as [[Wikipedia:Parallel and counter parallel|"''parallels''" and "''counter parallels''"]]. It should be noted that the ability of an interval to relate to the Tonic through Parallelism, as well as the surrounding of more dissonant intervals by consonant intervals in the same region displaying such relationships to the Tonic, results in a tendency towards harmonic stagnation. | ||
'''Adjacency''' - Notes within a suitable voice leading distance from either the Dominant or Serviant tend to have the opposite function relative to the Tonic- this process even extends to the relationship between the Dominant and Serviant themselves. On the other hand, notes within this same kind of distance from the Tonic often tend to have their functions colored more by their relationships to both the Dominant and Serviant. This process is one of two that create what in traditional music theory are referred to as "''parallels''" and "''counter parallels''", however, unlike Parallelism proper, this process can establish these kinds of relationships outside the primary tetrachord. | '''Adjacency''' - Notes within a suitable voice leading distance from either the Dominant or Serviant tend to have the opposite function relative to the Tonic- this process even extends to the relationship between the Dominant and Serviant themselves. On the other hand, notes within this same kind of distance from the Tonic often tend to have their functions colored more by their relationships to both the Dominant and Serviant. This process is one of two that create what in traditional music theory are referred to as "''parallels''" and "''counter parallels''", however, unlike Parallelism proper, this process can establish these kinds of relationships outside the primary tetrachord. | ||
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'''Immitation''' - 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. | '''Immitation''' - 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. | ||
'''Compositing''' - These notes have multiple functions due to occurring at the boundary between different functions or due to occurring in close proximity to other functions. | '''Compositing''' - These notes have multiple functions due to occurring at the boundary between different functions or due to occurring in close proximity to other 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 | '''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 , 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 a compositing of the Tonic Adjacent function, the Serviant Parallel function of [[10/9]] and Double Dominant function of [[9/8]], or at least that's the case in Bass-Up Tonality, and especially in [[meantone]] contexts where the two intervals are equated. However, I would argue that due to both the close proximity of 10/9 and 9/8 even in non-meantone settings and the shared Predominant functionalities of both scale degrees, these two functions often overlap to some degree or other. Accordingly, this is the note that occurs roughly at intervals between [[567/512]] and [[256/225]] above the Tonic as the second scale degree. | '''Supertonic''' - This function is easily derivable through a compositing of the Tonic Adjacent function, the Serviant Parallel function of [[10/9]] and Double Dominant function of [[9/8]], or at least that's the case in Bass-Up Tonality, and especially in [[meantone]] contexts where the two intervals are equated. However, I would argue that due to both the close proximity of 10/9 and 9/8 even in non-meantone settings and the shared Predominant functionalities of both scale degrees, these two functions often overlap to some degree or other. Accordingly, this is the note that occurs roughly at intervals between [[567/512]] and [[256/225]] above the Tonic as the second scale degree. | ||