User:Aura/Aura's Ideas on Functional Harmony (Part 1): Difference between revisions
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The three basic functions have their roots in [[LCJI]], and each has an uncanny valley of sorts around it that extends to around at least 16 cents out from it. The functions themselves are labeled as follows: | The three basic functions have their roots in [[LCJI]], and each has an uncanny valley of sorts around it that extends to around at least 16 cents out from 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 '' | '''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- specifically at about 20 cents, with the syntonic comma being only just slightly outside this range. | ||
'''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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'''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. In some cases, whether or not such an interval falls in the uncanny valley of one of the primary functions is disputed, resulting in functional compositing. | |||
'''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. | |||
== Facets Derived from Viennese Theory == | == Facets Derived from Viennese Theory == | ||
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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 a | '''Contralead''' - This function, although not found in Viennese Theory proper, is easily derivable through a compositing of the Tonic Adjacent function, the Serviant Parallel functions 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 occurs at intervals between [[25/24]] and [[14/13]] away from the Tonic, 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 | '''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. | ||
'''Mediant''' - This is the note that occurs roughly at intervals between [[75/64]] and [[32/25]] away 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 the most possibilities for realization, and, 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 | '''Mediant''' - This is the note that occurs roughly at intervals between [[75/64]] and [[32/25]] away 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 the most possibilities for realization, and, 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 a note that occurs roughly at intervals between [[25/16]] and [[128/75]] away from the Tonic in the scale's direction of construction, and is the second of two scale degrees with the most possibilities for the realization. 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 | '''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 a note that occurs roughly at intervals between [[25/16]] and [[128/75]] away from the Tonic in the scale's direction of construction, and is the second of two scale degrees with the most possibilities for the realization. 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 | '''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. | ||
'''Lead''' - This is the note typically referred to when people say "the leading tone", and, from a harmonic standpoint it is easily derivable through a | '''Lead''' - This is the note typically referred to when people say "the leading tone", and, from a harmonic standpoint it is easily derivable through a compositing of the Tonic Adjacent function, the Dominant Parallel function of [[15/8]], and the Antiserviant function. This is a note that occurs at intervals between [[13/7]] and [[48/25]] away from the Tonic, which serves as a leading tone in the scale's direction of construction. 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 Lead and the Dominant, sees this interval as functionally distinct from the Dominant due to the the Lead 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- and, the possibility of being set-up through quartertone-like motion from above (see below on Paradiatonic and Parachromatic Functions). | ||
== Antitonic == | == Antitonic == | ||
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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. | ||
== 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. | |||
=== Circumtonic Regions === | |||
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 often avoided. | |||
'''Supercommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 20 cents above the Tonic. These intervals are little more than stepping stones in modulation, and extra intervals that can be used together with the Tonic for a sense of dissonance, or for a slightly less resolved version of a Unison or Octave. | |||
'''Subcommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 20 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 === | |||
Since the uncanny valley around the Dominant is smaller in size, there is more room for actual first derivative diatonic functionality, as well as other functionalities. | |||
The circumdomimant regions are the two main regions on either side of the Dominant proper, and there are several functions located in these regions, one of which is an actual diatonic function. Note that antitonic and paradiatonic functions falling in this region are not covered in this section. | |||
'''Acuodominant''' - This function is a compositing of Dominant detempering and Misodominant (see below on Basic Parachromatic Functions). | |||
'''Geminodominant''' - This is a note that occurs roughly at intervals ranging from 187/128 away from the Tonic to about 16 cents short of the 3/2 perfect fifth 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 Dominant through detempering, occurring in [[5-limit]] diatonic environments, and acting as a sort of "fraternal twin" to the Dominant, hence its name. Specifically, as typified by intervals like 40/27, Geminodominants are dissonant intervals that simultaneously act as alternatives to the Dominant in both chord progressions and chord construction, and often require resolution, though they also have a Preserviant function. Apart from this, Geminodominants only rise to prominence in systems where what might otherwise function as a Dominant is found just short of the sweetspot range near the standard issue 3/2. | |||
=== Circumserviant Regions === | |||
The circumserviant regions are the two main regions on either side of the Serviant proper, and there are two functions located in these regions, one of which is an actual diatonic function. Note that antitonic and paradiatonic functions falling in this region are not covered in this section. | |||
'''Geminoserviant''' - This is a note that occurs roughly at intervals ranging from 3.5 cents beyond the 4/3 perfect fourth to 256/187 away from the Tonic 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 Dominant, 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. Apart from this, Geminoserviants only rise to prominence in systems where what might otherwise function as a Serviant is found just beyond the sweetspot range near the standard issue 4/3. | |||
'''Gravoserviant''' - This function is a compositing of Serviant detempering and Misoserviant (see below on Basic Parachromatic Functions). | |||
== 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. | ||
== | == Other functionalities == | ||
'''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. | '''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 Gravodominant. | '''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 Gravodominant. | ||