Diatonic functional harmony: Difference between revisions

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'''Dominant''' - As per the name, and as noted on [[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, there are a variety of other intervals between [[187/128]] and [[32/21]] away from the Tonic in the scale's direction of construction which have the tendency to create tension which requires the Tonic to resolve. Thus, because of the extremely close harmonic connection between the Tonic and the [[3/2]] Perfect 5th, the term "Dominant"- in its most basic form as referred to in this article- is restricted to where it only refers to the 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. Secondly, 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. As for the notes at other intervals between 187/128 and 32/21, they 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.

'''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 ===

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In addition to the aforemented Basic Diatonic Functions, Aura contends that the following scale degree functions also exist:

'''Contralead''' - Although this function is not found in Viennese Theory proper and various schools of thought take different positions, Aura contends that due to a combination of this scale degree being native to Phrygian mode, often approaching the Tonic from above in a manner reminiscent of a [[Wikipedia:Leading-tone|leading tone]], and simultaneously serving as the root of the [[Wikipedia:Neapolitan chord|Neapolitan chord]], this lowered second scale degree in Bass-Up Tonality should be considered as more than just a simple chromatic alteration of the Supertonic. Furthermore, taking this idea into the realm of Microtonality, the Contralead is an interval that occurs at intervals between 28/27 and 14/13 away from the Tonic, and, which not only serves as a leading tone in the direction opposite that of the scale's direction of construction, but also contains all the functions of a Neapolitan chord, with both functionalities having their roots in [[16/15]].

'''Contralead''' - Although this function is not found in Viennese Theory proper and various schools of thought take different positions, Aura contends that due to a combination of this scale degree being native to Phrygian mode, often approaching the Tonic from above in a manner reminiscent of a [[Wikipedia:Leading-tone|leading tone]], and simultaneously serving as the root of the [[Wikipedia:Neapolitan chord|Neapolitan chord]], this lowered second scale degree in Bass-Up Tonality should be considered as more than just a simple chromatic alteration of the Supertonic. Furthermore, 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, which not only serves as a leading tone in the direction opposite that of the scale's direction of construction, but also contains all the functions of a Neapolitan chord, with both functionalities having their roots in [[16/15]].

'''Supertonic''' - This function is easily derivable through a combination of 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, Aura 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 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, Aura 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, though it is most often within 20 cents of either [[6/5]] or [[5/4]] due to these two LCJI intervals coming the closest to the halfway point between the Tonic and the Dominant. From a functional standpoint, the Mediant has multiple possible functions ranging from the Tonic Counter Parallel and Preserviant functions of both 5/4 and 6/5, which overlap with the Triple Serviant Function of [[32/27]] on one side, and the Quadruple Dominant function of [[81/64]] on the other side.

'''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, though it is most often within 20 cents of either [[6/5]] or [[5/4]] due to these two LCJI intervals coming the closest to the halfway point between the Tonic and the Dominant. From a functional standpoint, the Mediant has multiple possible functions ranging from the Tonic Counter Parallel and Preserviant functions of both 5/4 and 6/5, which overlap with the Triple Serviant Function of [[32/27]] on one side, and the Quadruple Dominant function of [[81/64]] on the other side.

'''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 Contramediant has multiple possible functions ranging from the Tonic Parallel and Predominant functions of both 8/5 and 5/3, which overlap with the Triple Dominant Function of [[27/16]] on one side, and the Quadruple Serviant function of [[128/81]] on the other side.

'''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 Contramediant has multiple possible functions ranging from the Tonic Parallel and Predominant functions of both [[8/5]] and [[5/3]], which overlap with the Triple Dominant Function of [[27/16]] on one side, and the Quadruple Serviant function of [[128/81]] on the other side.

'''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, Aura 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]], 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, Aura 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". This is a note that occurs at intervals between 13/7 and 27/14 away from the Tonic, which serves as a leading tone in the scale's direction of construction. In Bass-Up Tonality, this functionality has its roots in the fifteenth harmonic. Although triads built on this scale degree are regarded by some as simply incomplete Dominant Seventh chords, Aura's 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 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". 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. In Bass-Up Tonality, this functionality has its roots in the fifteenth harmonic. Although triads built on this scale degree are regarded by some as simply incomplete Dominant Seventh chords, Aura's 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 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.

== Antitonic ==

Notes that occur around half an octave away from the Tonic- especially the region extending from 7/5 to 10/7- on account of harmonies built on notes in this area tend to oppose that of the Tonic, thus leading to the term '''Antitonic''' being used to describe the region as a whole by Aura and others. In addition, the term "Antitonic" acts as a generic term for any of a group of diatonic functions found in this region. While some microtonal theorists insist that the Antitonic functionality is more fundamental than perhaps even the Dominant or Serviant, others disagree.

Notes that occur around half an octave away from the Tonic- especially the region extending from [[7/5]] to [[10/7]]- on account of harmonies built on notes in this area tend to oppose that of the Tonic, thus leading to the term '''Antitonic''' being used to describe the region as a whole by Aura and others. In addition, the term "Antitonic" acts as a generic term for any of a group of diatonic functions found in this region. While some microtonal theorists insist that the Antitonic functionality is more fundamental than perhaps even the Dominant or Serviant, others disagree.

=== Specific Types of Antitonic ===

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=== Basic Paradiatonic 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. In Bass-Up tonality, this functionality has its roots in the eleventh harmonic.

'''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.

'''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.

'''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.

=== Interstep Functions ===

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'''Varicoserviant''' - 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. Accordingly, 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. The name of this region comes from "Varicant" and "Serviant", with a linking "-o-" in place of the "-ant" of "Varicant".

=== Neutral Degree 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, set-ups and follow-ups for a Semicontralead- apart from root motion by a fifth- are generally best when moving in a given Tonality's direction of construction, as intervals like this moving stepwise in the opposite direction are likely to create a disjunct without proper context. 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, a Semilead is likely to either resolve downwards to a Subtonic, or, upwards to either the Lead, the Tonic, or something in between, though in the latter case, it's subject to the same sorts of restrictions as a Semicontralead. The [[11/6]] neutral seventh is a rather typical example of an interval with this function.

=== Circumtonic Regions ===

The Circumtonic regions are the two main regions on either side of the Tonic, beyond the Tonic's "event horizon". There are multiple paradiatonic functions hiding in these regions, though only two of them are really connected to the Tonic itself through stacking a combination 11/8 and 3/2 intervals.

'''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.

'''Superdietic''' - This is a note that occurs at intervals between roughly 20 cents above the Tonic and 28/27 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 to some form of Contralead or even 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.

'''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.

'''Subdietic''' - This is a note that occurs at intervals between roughly 27/14 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 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.

'''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.

'''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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 '''Dominant''' - As per the name, and as noted on [[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, there are a variety of other intervals between [[187/128]] and [[32/21]] away from the Tonic in the scale's direction of construction which have the tendency to create tension which requires the Tonic to resolve.  Thus, because of the extremely close harmonic connection between the Tonic and the [[3/2]] Perfect 5th, the term "Dominant"- in its most basic form as referred to in this article- is restricted to where it only refers to the 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.  Secondly, 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. As for the notes at other intervals between 187/128 and 32/21, they 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.
-'''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 ===
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 In addition to the aforemented Basic Diatonic Functions, Aura contends that the following scale degree functions also exist:
-'''Contralead''' - Although this function is not found in Viennese Theory proper and various schools of thought take different positions, Aura contends that due to a combination of this scale degree being native to Phrygian mode, often approaching the Tonic from above in a manner reminiscent of a [[Wikipedia:Leading-tone|leading tone]], and simultaneously serving as the root of the [[Wikipedia:Neapolitan chord|Neapolitan chord]], this lowered second scale degree in Bass-Up Tonality should be considered as more than just a simple chromatic alteration of the Supertonic.  Furthermore, taking this idea into the realm of Microtonality, the Contralead is an interval that occurs at intervals between 28/27 and 14/13 away from the Tonic, and, which not only serves as a leading tone in the direction opposite that of the scale's direction of construction, but also contains all the functions of a Neapolitan chord, with both functionalities having their roots in [[16/15]].
+'''Contralead''' - Although this function is not found in Viennese Theory proper and various schools of thought take different positions, Aura contends that due to a combination of this scale degree being native to Phrygian mode, often approaching the Tonic from above in a manner reminiscent of a [[Wikipedia:Leading-tone|leading tone]], and simultaneously serving as the root of the [[Wikipedia:Neapolitan chord|Neapolitan chord]], this lowered second scale degree in Bass-Up Tonality should be considered as more than just a simple chromatic alteration of the Supertonic.  Furthermore, 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, which not only serves as a leading tone in the direction opposite that of the scale's direction of construction, but also contains all the functions of a Neapolitan chord, with both functionalities having their roots in [[16/15]].
-'''Supertonic''' - This function is easily derivable through a combination of 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, Aura 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 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, Aura 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, though it is most often within 20 cents of either [[6/5]] or [[5/4]] due to these two LCJI intervals coming the closest to the halfway point between the Tonic and the Dominant.  From a functional standpoint, the Mediant has multiple possible functions ranging from the Tonic Counter Parallel and Preserviant functions of both 5/4 and 6/5, which overlap with the Triple Serviant Function of [[32/27]] on one side, and the Quadruple Dominant function of [[81/64]] on the other side.
+'''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, though it is most often within 20 cents of either [[6/5]] or [[5/4]] due to these two LCJI intervals coming the closest to the halfway point between the Tonic and the Dominant.  From a functional standpoint, the Mediant has multiple possible functions ranging from the Tonic Counter Parallel and Preserviant functions of both 5/4 and 6/5, which overlap with the Triple Serviant Function of [[32/27]] on one side, and the Quadruple Dominant function of [[81/64]] on the other side.
-'''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 Contramediant has multiple possible functions ranging from the Tonic Parallel and Predominant functions of both 8/5 and 5/3, which overlap with the Triple Dominant Function of [[27/16]] on one side, and the Quadruple Serviant function of [[128/81]] on the other side.
+'''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 Contramediant has multiple possible functions ranging from the Tonic Parallel and Predominant functions of both [[8/5]] and [[5/3]], which overlap with the Triple Dominant Function of [[27/16]] on one side, and the Quadruple Serviant function of [[128/81]] on the other side.
-'''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, Aura 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]], 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, Aura 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".  This is a note that occurs at intervals between 13/7 and 27/14 away from the Tonic, which serves as a leading tone in the scale's direction of construction.  In Bass-Up Tonality, this functionality has its roots in the fifteenth harmonic.  Although triads built on this scale degree are regarded by some as simply incomplete Dominant Seventh chords, Aura's 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 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".  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.  In Bass-Up Tonality, this functionality has its roots in the fifteenth harmonic.  Although triads built on this scale degree are regarded by some as simply incomplete Dominant Seventh chords, Aura's 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 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.
 == Antitonic ==
-Notes that occur around half an octave away from the Tonic- especially the region extending from 7/5 to 10/7- on account of harmonies built on notes in this area tend to oppose that of the Tonic, thus leading to the term '''Antitonic''' being used to describe the region as a whole by Aura and others.  In addition, the term "Antitonic" acts as a generic term for any of a group of diatonic functions found in this region.  While some microtonal theorists insist that the Antitonic functionality is more fundamental than perhaps even the Dominant or Serviant, others disagree.
+Notes that occur around half an octave away from the Tonic- especially the region extending from [[7/5]] to [[10/7]]- on account of harmonies built on notes in this area tend to oppose that of the Tonic, thus leading to the term '''Antitonic''' being used to describe the region as a whole by Aura and others.  In addition, the term "Antitonic" acts as a generic term for any of a group of diatonic functions found in this region.  While some microtonal theorists insist that the Antitonic functionality is more fundamental than perhaps even the Dominant or Serviant, others disagree.
 === Specific Types of Antitonic ===
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 === Basic Paradiatonic 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.  In Bass-Up tonality, this functionality has its roots in the eleventh harmonic.
+'''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.
-'''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.
+'''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.
 === Interstep Functions ===
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 '''Varicoserviant''' - 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.  Accordingly, 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.  The name of this region comes from "Varicant" and "Serviant", with a linking "-o-" in place of the "-ant" of "Varicant".
+=== Neutral Degree 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, set-ups and follow-ups for a Semicontralead- apart from root motion by a fifth- are generally best when moving in a given Tonality's direction of construction, as intervals like this moving stepwise in the opposite direction are likely to create a disjunct without proper context.  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, a Semilead is likely to either resolve downwards to a Subtonic, or, upwards to either the Lead, the Tonic, or something in between, though in the latter case, it's subject to the same sorts of restrictions as a Semicontralead.  The [[11/6]] neutral seventh is a rather typical example of an interval with this function.
 === Circumtonic Regions ===
+The Circumtonic regions are the two main regions on either side of the Tonic, beyond the Tonic's "event horizon".  There are multiple paradiatonic functions hiding in these regions, though only two of them are really connected to the Tonic itself through stacking a combination 11/8 and 3/2 intervals.
 '''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.
-'''Superdietic''' - This is a note that occurs at intervals between roughly 20 cents above the Tonic and 28/27 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 to some form of Contralead or even 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.
+'''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.
-'''Subdietic''' - This is a note that occurs at intervals between roughly 27/14 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 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.
+'''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.
 '''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.