User:Aura/Aura's Ideas on Functional Harmony (Part 1): Difference between revisions
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'''Serviant''' - Compared to the term "''Subdominant''" from traditional music theory, the term "Serviant" is restricted to those notes that occur roughly at a [[4/3]] interval away from the Tonic in the scale's direction of construction, and act 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, and this is not the case for other intervals between 21/16 and 7/5. Secondly, in music built from the Treble downwards, the notes with these sorts of functions are actually located ''above'' the Dominant. Like with notes at other intervals between [[21/16]] and [[256/187]] away from the Tonic- the Serviant tends to resolve towards the Dominant, or else some other note that acts as a surrogate for the Dominant. As for the notes at other intervals between 21/16 and 256/187, they shall be divided into two classes depending on which side of the 4/3 Perfect 4th they fall on and discussed in another section. However, aside from [[27/20]], none of these other intervals occur in diatonic scales. | '''Serviant''' - Compared to the term "''Subdominant''" from traditional music theory, the term "Serviant" is restricted to those notes that occur roughly at a [[4/3]] interval away from the Tonic in the scale's direction of construction, and act 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, and this is not the case for other intervals between 21/16 and 7/5. Secondly, in music built from the Treble downwards, the notes with these sorts of functions are actually located ''above'' the Dominant. Like with notes at other intervals between [[21/16]] and [[256/187]] away from the Tonic- the Serviant tends to resolve towards the Dominant, or else some other note that acts as a surrogate for the Dominant. As for the notes at other intervals between 21/16 and 256/187, they shall be divided into two classes depending on which side of the 4/3 Perfect 4th they fall on and discussed in another section. However, aside from [[27/20]], none of these other intervals occur in diatonic scales. | ||
=== Basic Operations === | === Basic Diatonic Function-Deriving Operations === | ||
The way I see it, there are | The way I see it, there are six 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 "Double Dominant" or "Double Serviant" 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". | '''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 "Double Dominant" or "Double Serviant" 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". | ||
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'''Antipodism''' - Notes that are either opposite in tone color 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 tone color 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''' - 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. | |||
'''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 relative are denoted with a "mock-" prefix. | |||
== 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 combination of the Tonic Adjacent function, the | '''Contralead''' - This function, although not found in Viennese Theory proper, is easily derivable through a combination 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 combination of the Tonic Adjacent function, the Subdominant 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 combination of the Tonic Adjacent function, the Subdominant 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. From a functional standpoint, the properties that are central to the Mediant function are all most easily derived through a combination of the Tonic Parallel function and the Serviant Adjacent function. 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. | '''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. From a functional standpoint, the properties that are central to the Mediant function are all most easily derived through a combination 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. From a functional standpoint, the properties that are central to the Contramediant function are most easily derived through a combination of the Tonic Parallel function and the Dominant Adjacent function. 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. | '''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. From a functional standpoint, the properties that are central to the Contramediant function are most easily derived through a combination 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 combination of the Dominant Parallel function of [[9/5]] and Double Serviant function of [[16/9]], 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 combination 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 combination 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. | '''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 combination 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 600 cents and 10/7 away from the Tonic and functions as an augmented fourth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just above the range of a typical Sycophant in Bass-Up Tonality, it is called an '''Acuosycophant''' in this kind of tonal systems. A classic example of such an interval is 729/512. | *If the Antitonic is between 600 cents and 10/7 away from the Tonic and functions as an augmented fourth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just above the range of a typical Sycophant in Bass-Up Tonality, it is called an '''Acuosycophant''' in this kind of tonal systems. A classic example of such an interval is 729/512. | ||
== Paradiatonic Functions == | == Paradiatonic and Parachromatic Functions == | ||
In addition to all the aforementioned Diatonic functions, there is an additional set of categories for dealing with the notes in between the various Diatonic scale degrees. | In addition to all the aforementioned Diatonic functions, there is an additional set of categories for dealing with the notes in between the various Diatonic scale degrees. | ||
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The way I see it, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as seconds, albeit inframinor seconds by default, while parachromatic quartertones are analogous to chromatic semitones in that they are denoted as primes, albiet as ultraprimes by default. However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone. Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, it can be deduced that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone. Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, I ended up choosing the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone. As a result of multiple factors, I ended up choosing the combination of three 33/32 parachromatic quartertones and one [[4096/3993]] paradiatonic quartertone as the JI basis for this in regards to both Diatonic theory and [[Alpharabian tuning]], and, through interactions with others in the Xenharmonic community, I was later influenced by others on Discord to take [[MOS]]-based structural considerations into account. This eventually resulted in the first formal definition of a "'''parachroma'''" (an interval that can be easily tempered to equal half of a MOS-chroma), and later, the "'''parastep'''" (the interval that remains after subtracting as many parachromas from a Major MOS-step as possible without resulting in a negative interval). Finally, drawing from the concept of "parachromas" as applied to MOS-based contexts, I was able to finally give a formal definition of terms like "paramajor" (the result of adding a parachroma to either a MOS generator or its period-inverse) and "paraminor" (the result of subtracting a parachroma from a MOS generator or its period-inverse), which I had previously come up with on an informal basis. | The way I see it, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as seconds, albeit inframinor seconds by default, while parachromatic quartertones are analogous to chromatic semitones in that they are denoted as primes, albiet as ultraprimes by default. However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone. Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, it can be deduced that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone. Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, I ended up choosing the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone. As a result of multiple factors, I ended up choosing the combination of three 33/32 parachromatic quartertones and one [[4096/3993]] paradiatonic quartertone as the JI basis for this in regards to both Diatonic theory and [[Alpharabian tuning]], and, through interactions with others in the Xenharmonic community, I was later influenced by others on Discord to take [[MOS]]-based structural considerations into account. This eventually resulted in the first formal definition of a "'''parachroma'''" (an interval that can be easily tempered to equal half of a MOS-chroma), and later, the "'''parastep'''" (the interval that remains after subtracting as many parachromas from a Major MOS-step as possible without resulting in a negative interval). Finally, drawing from the concept of "parachromas" as applied to MOS-based contexts, I was able to finally give a formal definition of terms like "paramajor" (the result of adding a parachroma to either a MOS generator or its period-inverse) and "paraminor" (the result of subtracting a parachroma from a MOS generator or its period-inverse), which I had previously come up with on an informal basis. | ||
=== Basic Paradiatonic Functions === | Finally, the most recent development, which led to the separation of Paradiatonic and Parachromatic harmonic functions, was the experimentations of a YouTuber by the username "Quartertone Harmony", which resulted in a video in which he groups together a series of functions he refers to in the video as the "shadow scale"<ref>[https://www.youtube.com/watch?v=P6WJryxB_0Y Quartertone Harmony - Harmonic Functions of Quartertones SD 480p]</ref>, which I will refer to here as the '''paradiatonic scale'''. | ||
=== The Paradiatonic Scale and Basic Paradiatonic Functions === | |||
The Paradiatonic scale from a given tonic acts as a sort of "second shelf" of that tonality, and 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 | |||
Of these, 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 a note that occurs roughly at intervals between 256/225 and 75/64 away from the Tonic 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. | ||
'''Intersubiant''' - This is a note that occurs roughly at intervals ranging from 256/187 to [[25/18]] away from the Tonic in the scale's direction of construction. Like both the Serviant and the Sycophant, intervals in this region tend to have a Predominant function, however, the way these intervals carry out this function is rather different from both as they neither act as a counterweight to the Dominant like a Serviant, nor do they completely tonicize the Dominant like a Sycophant- at least to those who are more familiar with quartertones. In Bass-Up tonality, this functionality has its roots in the eleventh harmonic, and indeed 11/8 is perhaps one of the best examples of an interval within this range, since, as its name implies, it has a decent Predominant function without the risks of tonicizing the Dominant that arise with Sycophant Antitonics, while also having Preserviant functions. | |||
What's less expected, however, is that the Intersubiant also has Mocktyrant Functions. | |||
'''Interregnant''' - This is a note that occurs roughly at intervals ranging from [[36/25]] to 187/128 away from the Tonic in the scale's direction of construction. Accordingly, intervals in this region behave as a cross between a Tyrant Antitonic on one hand and a Dominant on the other in that they often contrast with the Tonic through some combination of harmonic connection and brute force contrast. In Bass-Up tonality, this functionality has its roots in the eleventh subharmonic, and indeed [[16/11]] is perhaps one of the best examples of an interval within this range, since, as its name implies, it has decent Preserviant and Predominant functions. However, it also has Mocksycophant Functions. | |||
'''Varicant''' - Just as a Mediant lies roughly in the middle of the 3/2 interval separating the Tonic and the Dominant above it, a Varicant lies roughly in the middle of the 4/3 interval separating the Dominant and the Tonic above it. Intervals in this region often don’t consistently act as either sixths or sevenths, or even act as a cross between a sixth and a seventh- effectively straddling the border between these two diatonic categories, hence the name "Varicant", from Latin "vāricō"<ref>[[Wiktionary: varico #Latin]]</ref>. This is a note that occurs roughly at intervals between 128/75 and 225/128 away from the Tonic in the scale's direction of construction. In Bass-Up Tonality, this functionality is first encountered in the form of the [[7/4]] interval, though [[12/7]] is another notable interval included in this range. While many microtonalists think of 7/4 as being purely a type of seventh- and indeed, it most commonly acts as a sort of subminor seventh- 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. | '''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 a note that occurs roughly at intervals between 128/75 and 225/128 away from the Tonic in the scale's direction of construction. In Bass-Up Tonality, this functionality is first encountered in the form of the [[7/4]] interval, though [[12/7]] is another notable interval included in this range. While many microtonalists think of 7/4 as being purely a type of seventh- and indeed, it most commonly acts as a sort of subminor seventh- 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 acts as Predominants and Preserviants, but not as Dominant Parallels. | ||
=== Neutralization === | |||
In addition to the six known Diatonic Function-Deriving Operations listed above, there's also one Paradiatonic Function-Deriving Operation known as Neutralization, which, as the name suggests, creates paradiatonic functions from the neutralization and hybridization of Major and Minor Diatonic scale degrees. | |||
=== Derivative Paradiatonic Functions === | |||
'''Neutral Mediant''' - As per the name, this is nothing other than a neutralized Mediant, and thus, it has the Tonic Parallel, Serviant Adjacent, Preserviant and Predominant functions that you expect from a Mediant, only, it doesn't serve well at phrase endings, rather, its Tonic Parallel function is only appropriate during the middle of musical phrases. | |||
'''Neutral Contramediant''' - As per the name, this is nothing other than a neutralized Contramediant, and thus, it has the Tonic Parallel, Dominant Adjacent, Preserviant and Predominant functions that you expect from a Contramediant, but, like with the Neutral Mediant, it doesn't serve well at phrase endings, rather, its Tonic Parallel function is only appropriate during the middle of musical phrases. | |||
''' | '''Subdietic''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly 20 cents below the octave reduplication of the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, in Bass-Up Tonality, the Lead- however, there are functional differences between a Subdietic and a Lead in Bass-Up Tonality. In Bass-Up Tonality, Subdietics are often more likely to be passing tones than Leads, 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 Subdietic 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 Subdietic and the Tonic is likely to result in the Subdietic resolving back down to either the Lead or the Subtonic. As if that weren't enough, the Subdietic also has the Antintersubiant 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, a Semilead is likely to either resolve downwards to a Subtonic, or, upwards to either the Lead | === Parachromatic Functions === | ||
These are quartertone functions that are not on the Paradiatonic Scale. Of these, there are only two basic functions... | |||
'''Misodominant''' - This is a note that occurs roughly at intervals between 32/21 and 25/16 away from the Tonic in the scale's direction of construction. This region is characterized by intervals that don’t consistently act as either fifths or sixths, or even act as a cross between a fifth and a sixth, as well as by intervals that act as parachromatic alterations of either the Dominant or the Contramediant. Although originally named the "Varicodominant" region- the name coming from "Varicant" and "Dominant", with a linking "-o-" in place of the "-ant" of "Varicant"- the fact that intervals in this region are also generally more dissonant, leading to their avoidance in chords outside of deliberate dissonances, has lead to a name change for the region as a whole. The new name of this region comes from "miso-" and "Dominant". Chords of this type have Predominant, Preintersubiant and Preinterregnant functions, as well as Mockdominant and Precontramediant functions. | |||
'''Misoserviant''' - This is a note that occurs roughly at intervals between 32/25 and 21/16 away from the Tonic in the scale's direction of construction. This region is characterized by intervals that don’t consistently act as either thirds or fourths, or even act as a cross between a third and a fourth, as well as by intervals that act as parachromatic alterations of either the Mediant or the Serviant. Although originally named the "Varicodominant" region- the name coming from "Varicant" and "Serviant", with a linking "-o-" in place of the "-ant" of "Varicant"- the fact that intervals in this region are also generally more dissonant, leading to their avoidance in chords outside of deliberate dissonances, has lead to a name change for the region as a whole. The new name of this region comes from "miso-" and "Serviant". Chords of this type have Preserviant, Preintersubiant and Preinterregnant functions as well as Mockserviant and Premediant functions. | |||
=== Derivative Parachromatic Functions === | |||
While some neutralized scale degrees- such as the Neutral third and Neutral sixth- have many of the same diatonic functions as the adjacent Major and Minor scale degrees, this is not the case for neutral seconds and neutral sevenths due to the Major and Minor versions of these scale degrees having noticeably different functions. | |||
'''Semicontralead''' - This is a note that occurs roughly at intervals between 14/13 and 567/512 above the Tonic as the second scale degree. Naturally, this interval functions as a sort of cross between a Contralead and a Supertonic, and indeed chords built on this can function as some sort of cross between a Neapolitan chord and a Supertonic chord. However, there are ways in which the Semicontralead is distinct from both- notably, the Semicontralead also has the Misoserviant Parallel and Antimisodominant functions. The [[12/11]] neutral second 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. | |||
'''Superdietic''' - This is a note that occurs at intervals between roughly 20 cents above the Tonic and 25/24 above the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, in Bass-Up Tonality, the Contralead. However, while some microtonalists would question the idea of there being a distinct Superdietic region, preferring to think of the quartertones in this region as being simply the junction between the Supercommatic and Contralead regions, there there are actually a few functional differences between a Superdietic and a Contralead in Bass-Up Tonality that are worth considering. For starters, Superdietics are often more likely to be passing tones than Contraleads, and, when they’re not merely passing non-chord tones, they are just as liable to resolve upward thought some sort of semitone-like motion to some form of Contralead, Semicontralead, or even a Supertonic, 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 Contralead can resolve to the Tonic in part through a strong harmonic connection, a Superdietic is much more likely to do so through sheer brute force when such a resolution is noticeable. As if that weren't enough, the Subdietic also has the Antinterregnant function. | |||
=== Circumtonic Regions === | === Circumtonic Regions === | ||
The Circumtonic regions are the two main regions on either side of the Tonic, outside the Tonic's "event horizon" | The Circumtonic regions are the two main regions on either side of the Tonic, outside the Tonic's "event horizon". | ||
'''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. | '''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. | '''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. | ||
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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. | 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. | ||
'''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. | ||
'''Gravodominant''' - This is a note that occurs roughly at intervals ranging from 187/128 away from the Tonic to about 3.5 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, I actually consider the Gravodominant to be a legitimate diatonic function- albeit one only existing in non-meantone environments. Specifically, as typified by intervals like 40/27, Gravodominants are dissonant intervals that simultaneously act as alternatives to the Dominant in both chord progressions and chord construction, and often require resolution. Apart from this, Gravodominants 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. | '''Gravodominant''' - This is a note that occurs roughly at intervals ranging from 187/128 away from the Tonic to about 3.5 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, I actually consider the Gravodominant to be a legitimate diatonic function- albeit one only existing in non-meantone environments. Specifically, as typified by intervals like 40/27, Gravodominants are dissonant intervals that simultaneously act as alternatives to the Dominant in both chord progressions and chord construction, and often require resolution. Apart from this, Gravodominants 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 === | === 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. | 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. | ||
'''Acuoserviant''' - 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. As typified by intervals like 27/20, Acuoserviants are dissonant intervals that often act as a sort of predominant and or as the inverses of Gravodominants, making the Acuoserviant a legitimate diatonic function in terms of Aura's analysis. Apart from this, Acuoserviants 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. | '''Acuoserviant''' - 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. As typified by intervals like 27/20, Acuoserviants are dissonant intervals that often act as a sort of predominant and or as the inverses of Gravodominants, making the Acuoserviant a legitimate diatonic function in terms of Aura's analysis. Apart from this, Acuoserviants 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. | ||
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'''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. | ||
== Differences from Traditional Neo-Riemannian Theory == | |||
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The lack of [[Meantone]] temperament has some pretty significant implications for triadic transformations voiceleading in my theory- namely in that while all three basic [[Wikipedia: Neo-Riemannian theory|Neo-Riemannian]] transformations are available, there are additional, derived types of transformation are needed. | The lack of [[Meantone]] temperament has some pretty significant implications for triadic transformations voiceleading in my theory- namely in that while all three basic [[Wikipedia: Neo-Riemannian theory|Neo-Riemannian]] transformations are available, there are additional, derived types of transformation are needed. | ||