User:Aura/Aura's Ideas on Functional Harmony (Part 1): Difference between revisions

[[File:Diatonic Function Map (Version 4).png|thumb|Diagram of diatonic and paradiatonic function locations as of an older edition of this article- note the use of the now obsolete terms "Interregnant" and "Intersubiant" for what are now the Imponent and Subient functions respectively.  Apart from these, and the new spelling standardization of the term "Servient", there have only been minor changes, and a new diagram should be forthcoming.]]

'''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 Servient results in the creation of the "Bidominant" or "Biservient" respectively.  This concept comes from the German language's way of referring to the chord built on the second scale degree of the Diatonic scale as the "Doppeldominante", which literally means "Double Dominant".

'''Parallelism''' - Notes located in the same primary [[tetrachord]] as either the Tonic, the Dominant, or the Servient but that don't crowd them tend to take on similar functions to said notes, with the caveat that functions derived from the Tonic in this fashion are still technically Nontonic functions.  This process is one of two that create what in traditional music theory are referred to as [[Wikipedia:Parallel and counter parallel|"''parallels''" and "''counter parallels''"]].  It should be noted that the ability of an interval to relate to the Tonic through Parallelism, as well as the surrounding of more dissonant intervals by consonant intervals in the same region displaying such relationships to the Tonic, results in a tendency towards harmonic stagnation.

'''Adjacency''' - Notes within a suitable voice leading distance from either the Dominant or Servient tend to have the opposite function relative to the Tonic- this process even extends to the relationship between the Dominant and Servient themselves.  On the other hand, notes within this same kind of distance from the Tonic often tend to have their functions colored more by their relationships to both the Dominant and Servient.  This process is one of two that create what in traditional music theory are referred to as "''parallels''" and "''counter parallels''", however, unlike Parallelism proper, this process can establish these kinds of relationships outside the primary tetrachord.

'''Preparation''' - Notes that "prepare the way" for either a Dominant or a Servient through any of the above operations, or through some other mechanism, relate to said notes by this process.  Functions which have this kind of role relative to some other function are denoted with a "pre-" prefix here.

'''Detempering''' - When the comma or subchroma that separates a note from one of the three primary functions is not tempered out, it results in the appearance of notes with either similar functions to the Dominant or Servient, or, in the case of this function being applied to the Tonic, a Nontonic function similar to one of the Chromatic functions.  Often, though not always, these notes fall within the uncanny valleys of the three primary functions.

'''Mediant''' - This is the note that maps to 2\7 from the Tonic in the scale's direction of construction and is named due to being roughly halfway between the Tonic and the Dominant.  This is the first of the two diatonic scale degrees that are located relatively far from the Tonic along the Circle of Third Harmonics, and, as a consequence, not only has the most possibilities for realization, but also a tendency towards stagnation, leading to the historical designation as a "weak harmony".  Aside from these, the properties that are central to the Mediant function are all most easily derived through the Tonic Parallel function and the Servient Adjacent function, and in addition, Mediants have both Preservient and Predominant functions.  Furthermore, since there's such a large range of Mediants, there are a multitude of subtypes, see the Mediant Subtypes 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 the note that maps to 5\7 from the Tonic in the scale's direction of construction, and is the second of two that are located relatively far from the Tonic along the Circle of Third Harmonics, and, as a consequence, is not only tied with the Mediant for having the most possibilities for realization, but also has a tendency towards stagnation, leading to the historical designation as a "weak harmony".  From a functional standpoint, the properties that are central to the Contramediant function are most easily derived through the Tonic Parallel function and the Dominant Adjacent function, and in addition, Contramediants have both Preservient and Predominant functions.  Furthermore, since there's such a large range of Contramediants, there are a multitude of subtypes, see the Contramediant Subtypes section for more discussion of this function.

'''Supercollocant''' - This subtype of Supervicinant, although not found in Viennese Theory proper, is an interval that usually maps to both 1\7 and 2\24 in Bass-Up tonality.  Additionally, it has the Antidominant function as typified by the root of the [[Wikipedia:Neapolitan chord|Neapolitan chord]] in Bass-Up tonality and the Antiservient function in Treble-Down tonality.  In actuality, two subtypes of Supercollocant exist- the "'''Proximosupercollocant'''", which always maps to both 1\7 and 2\24 in Bass-Up tonality, and the "'''Distosupercollocant'''", which maps to 1\7 and either 2\24 or 3\24 depending on a variety of factors, and is distinct from both Proximosupercollocant and Superabrogant only in finer tuning systems where it is usually more consonant, and thus, weaker.  An example of a Proximosupercollocant is [[256/243]] while an example of a Distosupercollocant is [[16/15]].

'''Subtonic''' - This subtype of Subvicinant is an interval that maps to both 6\7 and 20\24 in Bass-Up tonality and occurs above the Tonic as the seventh scale degree.  However, it should be noted that the upper boundary of this function is situated at roughly 216/119 above the Tonic, since notes located at larger distances from the Tonic are liable to cause listeners to forget the ending pitch's relationship to the starting pitch- something which Subtonics don't do under any circumstances.  Conspicuously, one type of Supertonic is the "'''Biservient'''", which is the function specifically of [[16/9]].

'''Proximomediant''' - This type of Mediant is mapped to 2\7 and 6\24 and is so named due to being located on the end of the Mediant range closest to the Tonic.  Conspicuously, it is one of the possible the types of Mediant seen in Minor keys and is only distinct from the Mesoproximomediant in finer tuning systems where it is usually more dissonant.  One subtype of Proximomediant is the "'''Triservient'''", which is the function specifically of [[32/27]].

'''Mesoproximocontramediant''' - This type of Contramediant is mapped to 5\7 and either 17\24 or 18\24 depending on a variety of factors.  It is so named due to being located between the proximal end of the Contramediant range and the midline between the Tonic and the Servient.  Conspicuously, it is one of the possible the types of Contramediant seen in Major keys and is only distinct from the Proximocontramediant and Mesocontramediant in finer tuning systems where it's notable for containing the approximation range for [[5/3]] at its upper border, hence why it's not merely a simple compositing of the adjacent functions.

'''Mesocontramediant''' - This type of Contramediant is mapped to both 5\7 and 17\24 and is so named due to being located along the midline between the Tonic and the Servient.  This type of Contramediant is arguably the most dissonant, and is not a traditional diatonic function at all, and in fact, it doesn't serve well at phrase endings- rather, its Tonic Parallel function is only appropriate during the middle of musical phrases.  Intervals with this type of function include [[18/11]] and [[44/27]].

'''Mesodistocontramediant''' - This type of Contramediant is mapped to both 5\7 and either 16\24 or 17\24 depending on a variety of factors.  It is so named due to being located between the distal end of the Contramediant range and the midline between the Tonic and the Servient.  Conspicuously, it is one of the possible the types of Contramediant seen in Minor keys and is only distinct from the Mesocontramediant and Distocontramediant in finer tuning systems where it's notable for containing the approximation range for [[8/5]] at its lower border, hence why it's not merely a simple compositing of the adjacent functions.

'''Distocontramediant''' - This type of Contramediant is mapped to both 5\7 and 16\24 and is so named due to being located on the end of the Contramediant range furthest from the Tonic.  Conspicuously, it is one of the possible the types of Contramediant seen in Minor keys and is only distinct from the Mesodistocontramediant in finer tuning systems where it is usually more dissonant.  One subtype of Distomediant is the "'''Quadriservient'''", which is the function specifically of [[128/81]].  

Notes that occur around half an octave away from the Tonic, on account of harmonies built on notes in this area tending to oppose that of the Tonic, are referred to by the term "Antitonic" by myself and others.  It should be noted that the Antitonic is basically a first derivative function as it is derived from the Tonic through either perfect or imperfect Antipodism.  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 Servient, others, such as myself, disagree.

'''Sycophant''' - This type of Antitonic is mapped to both 3\7 and 12\24, and is named as such on account of it having a tendency to "kiss up to" and tonicize the Dominant- that is, to cause the Dominant to become a new Tonic- unless followed up by a different note such as some type of Mediant.  A prototypical example of this type of Antitonic is [[45/32]].

'''Tyrant''' - This type of Antitonic is mapped to both 4\7 and 12\24, and it tends to contrast with the Tonic in a manner somewhat akin to that of a Dominant, but by sheer brute force and contrary harmonic nature, and indeed these brute force Dominant-esque tendencies are the source of the name "Tyrant".  For example, if the Tonic harmony is Minor in nature, the Antitonic harmony will be Major- or more rarely, Supermajor- in nature.  Furthermore, in scales such as the Locrian scale, any type of Serviant harmony tends to resolve towards some other type of substitute for a Dominant, often bypassing this type of Antitonic, though on rare occasions, a Tyrant will act as a leading tone to the Servient.  A prototypical example of this type of Antitonic is [[64/45]].

The way I see it, there is at least one known operation which can be used to derive additional Chromatic functions from various Diatonic functions.

'''Supercommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 35 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 35 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.

'''Geminodominant''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents beyond [[22/15]] 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, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Dominant through detempering, occurring in [[5-limit]] diatonic environments, and acting as a sort of "fraternal twin" to the Dominant, hence its name.  Specifically, as typified by intervals like [[40/27]], Geminodominants are dissonant intervals that simultaneously act as alternatives to the Dominant in both chord progressions and chord construction, and often require resolution, though they also have a Preservient function.  The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the traditional weak harmonies- the Mediant and the Contramediant- which also has the benefits of strengthening interrupted cadences and creating the sense of impending movement, but outside of these usages and well supported chords, this kind of thing is best avoided since things repel the similar but not identical.  Apart from diatonic contexts, Geminodominants only rise to prominence in systems where what might otherwise function as a Dominant is found just short of the sweet spot range near the standard issue 3/2.

=== Circumservient Regions ===

The circumservient regions are the two main regions on either side of the Servient proper, and since the uncanny valleys around the Servient are not as deep as those around the Tonic, there is more room for actual first derivative diatonic functionality, as well as other functionalities.  Note that together with the Servient itself, these functions are collectively called "'''Protoservients'''", as there are many systems where what would be the Servient fails to get distinguished from either one or the other of the functions listed here.

'''Geminoservient''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents beyond the 4/3 perfect fourth to about 3.5 cents short of [[15/11]] in the scale's direction of construction.  Although often overlooked or even outright shunned by traditional theorists, the Geminoservient is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Servient through detempering, occurring in 5-limit diatonic environments, and acting as a sort of "fraternal twin" to the Servient, hence its name.  As typified by intervals like [[27/20]], Geminoservients are dissonant intervals that often act as a sort of Predominant and or as the inverses of Geminodominants.  The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the Mediant and the Contramediant, but outside of these usages and well supported chords, this kind of thing is best avoided.  Apart from diatonic contexts, Geminoservients only rise to prominence in systems where what might otherwise function as a Servient is found just beyond the sweet spot range near the standard issue 4/3.

'''Pseudoservient''' - This is a note that occurs roughly at intervals ranging from about 25 cents to about 3.5 cents short of the 4/3 perfect fourth in the scale's direction of construction.  Because none of these intervals arise naturally in a [[5-limit]] diatonic scale, and because they fall within one of the Servient's uncanny valleys, it is rare to see this functionality outside of systems where what might otherwise function as a Servient is found just short the sweet spot range near the standard issue 4/3.  However, these intervals naturally arise in the Circle of Thirteenth Harmonics, in which case they have a Preservient function.

'''Primary Adpositive Purity''' - This rule is that for every chord root located one step away from either the Tonic, Dominant or Servient along the Circle of Fifths, there is a demand for the fifth of the chord in question to be within 3.5 cents of a just 3/2, though in the case of a chord that has a root located at two steps away from the Tonic along the along the Circle of Fifths, the fifth of the chord can be deliberately subjected to the process of Displacement as mentioned before and not run afoul of this restriction.  However, chord roots that are located three or more steps away from the Tonic along the circle of fifths are not subject to the aforementioned restriction due to their distance.  This means that the Tonic, Dominant, Servient, Supertonic and Subtonic chords all demand a perfect fifth as the fifth of the chord, whether you are building the Tonality upwards or downwards, but wolf fifths can occur easily as the fifth of chords which are located three or more steps away from the Tonic.

'''Contravaricant''' - Named in contrast to the Varicant function, this is an interval that maps to both 1\5 and 5\24 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 with slight potential for crowding in chords depending on the exact tuning.  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, Preservient, and Dominant Parallel functions, as well as an overlap between Tonic Adjacent and Tonic Parallel functions.  There are two different subtypes of this function that differ from each other based on how they are approached in interval stacking, the "'''Pseudobidominant'''", which arises from stacking two Pseudodominants, and the "'''Pseudotriservient'''", which arises from stacking three Pseudoservients.

'''Subient''' - This is an interval that maps to both 3\7 and 11\24 in the scale's direction of construction and comprises the intervals ranging from around 3.5 cents short of [[15/11]] to around 3.5 cents beyond [[11/8]].  Like both the Servient and the Sycophant, intervals in this region have a Predominant function, however, this predominant function is weaker than that of the surrounding interval regions since they neither act as a counterweight to the Dominant like a Servient nor do they completely tonicize the Dominant like a Sycophant- at least to those who are more familiar with quartertones.  What is even less expected is that these same intervals also have Preservient and Mocktyrant functions.  The name of this function comes from Latin "subeō"<ref>[[Wiktionary: subeo #Latin]]</ref>, specifically from the senses ranging from "approach" to "follow" and even "undergo", as all of these senses describe what the Subient actually does musically.  Since the Subient is not found in the Tonic's quartertone field, there is a tendency for Subient chords to be preceded and or followed by other chords with roots in the same quartertone field, though one could also reasonably approach it from the Tonic, the Servient, some kind of Contramediant, some kind of Supervicinant, or, in some tuning systems, certain kinds of Mediant.  In Bass-Up tonality, this functionality has its roots in the eleventh harmonic.  This function used to be called the "Intersubiant" on an ad hoc basis, though that term has since been rendered obsolete.

'''Imponent''' - This is an interval that maps to both 4\7 and 13\24 in the scale's direction of construction and comprises the intervals ranging from around 3.5 cents short of [[16/11]] to around 3.5 cents beyond [[22/15]].  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, though it unexpectedly has decent Predominant and Mocksycophant functions.  Like the Subient, the Imponent is not found in the Tonic's quartertone field and is either very distant from the Tonic along the circle of fifths or is completely missed by said circle of fifths, thus, there is a tendency for Imponent chords to be preceded and or followed by other chords with roots in the same quartertone field.  As if all that weren't enough, the Imponent can be used in the formation of chords, but because of [[delta-rational]] and other considerations, tertian chords framed with this type of interval usually have a markedly different structure imposed on them.  The name of this function comes from Latin "impōnō"<ref>[[Wiktionary: impono #Latin]]</ref>, specifically from the senses ranging from "impose upon" to "inflict" and even "establish", as all of these senses describe what the Imponent actually does musically.  Conspicuously, the distance between the Imponent and the Subient is only about a Minor Second- or, at the very most, a Supraminor Second- so as a result, the Imponent can be both a set-up and a follow-up to the Subient.  In Bass-Up tonality, the Imponent functionality has its roots in the eleventh subharmonic.  This function used to be called the "Interregnant" on an ad hoc basis, though that term has since been rendered obsolete.

'''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 with slight potential for crowding in chords depending on the exact tuning- effectively straddling the border between these two diatonic categories, hence the name "Varicant", from Latin "vāricō"<ref>[[Wiktionary: varico #Latin]]</ref>.  This is an interval that maps to both 4\5 and 19\24 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 act as Predominants, Preservients, and Serviant Parallels, as well as display an overlap between Tonic Adjacent and Tonic Parallel functions.  There are two different subtypes of this function that differ from each other based on how they are approached in interval stacking, the "'''Pseudotridominant'''", which arises from stacking three Pseudodominants, and the "'''Pseudobiservient'''", which arises from stacking two Pseudoservients.

'''Subgradient''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly 25 cents below the octave reduplication of the Tonic.  These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Subcollocant- however, there are functional differences between a Subgradient and a Subcollocant in Bass-Up Tonality.  In Bass-Up Tonality, Subgradients are often more likely to be passing tones than Subcollocants, 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 Subcollocant can resolve to the Tonic in part through a strong harmonic connection, a Subgradient 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 Subgradient and the Tonic is likely to result in the Subgradient resolving back down to either the Subcollocant or the Subabrogant.  As if that weren't enough, the Subgradient also has the Antsubient function.  This function used to be called the "Subdietic", though that term has since been restricted to a related composite function (see below on Composite Functions).

'''Supergradient''' - This is a note that occurs at intervals between roughly 25 cents above the Tonic and 25/24 above the Tonic.  These intervals tend to act as parachromatic alterations of either the Tonic, or, more rarely in Bass-Up Tonality, the Supercollocant.  However, there are a few functional differences between a Supergradient and a Supercollocant in Bass-Up Tonality that are worth considering.  For starters, Supergradients are often more likely to be passing tones than Supercollocants, and, when they’re not merely passing non-chord tones, they are just as liable to resolve upward through some sort of semitone-like motion to some form of Supercollocant or Superabrogant, 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 Supercollocant can resolve to the Tonic in part through a strong harmonic connection, a Supergradient is much more likely to do so through sheer brute force when such a resolution is noticeable.  As if that weren't enough, the Subgradient also has the Antimponent function.  This function used to be called the "Superdietic", though that term has since been restricted to a related composite function (see below on Composite Functions).

'''Superobstant''' - This is a note which, in Bass-Up tonality, is mapped to 9\24.  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.  As per the name, which comes from "super-" and "obstant", the latter of which comes from Latin "obstō"<ref>[[Wiktionary: obsto #Latin]]</ref>, intervals in this region are also generally more dissonant and have a tendency to disrupt both chords and melodies, leading to their avoidance in chords outside of deliberate dissonances.  Chords of this type have Preservient, Presubient and Preimponent functions as well as Premediant functions.

'''Subobstant''' - This is a note which, in Bass-Up tonality, is mapped to 15\24.  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.  As per the name, which comes from "sub-" and "obstant", which, as mentioned before, comes from Latin "obstō", intervals in this region are also generally more dissonant, leading to their avoidance in chords outside of deliberate dissonances.  Chords of this type have Predominant, Presubient and Preimponent functions, as well as Precontramediant functions, and, perhaps very tellingly, tend to utilize Diminished Fourths instead of Major Thirds due to the functions of the Subgradient- which usually gets incorporated into these kinds of chords.

'''Acuotyrant''' - This function is a compositing of Prototyrant and Imponent in Bass-Up Tonality, and although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless operates more on the side of brute force when it contrasts with the Tonic.

'''Gravosycophant''' - This function is a compositing of Protosycophant and Subient in Bass-Up Tonality, and although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless often runs a high risk of tonicizing the Geminodominant.

'''Gravotyrant''' this function is a compositing of Prototyrant and Imponent in Treble-Down Tonality, and 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.

'''Acuosycophant''' - This function is a compositing of Protosycophant and Subient in Treble-Down Tonality, and although an interval like this generally fails to truly oppose the harmonies of the Tonic, it nevertheless often runs a high risk of tonicizing the Geminodominant.

'''Paraservient''' - This function is essentially the inverse of the Paradominant function, and acts as a sort of counterweight to the Paradominant relative to the Tonic.  Like the Paradominant function, it typically appears along prime axes other than that of the 3-limit in which they take Serviant-like function.  Thus, for example, in Bass-Up tonality, the note at 8/5 above the Tonic, being a direct prime subharmonic of the Tonic, gets to serve this function in addition to its Mesodistocontramediant function as defined by the 3-limit.  When considered alongside the actual Serviant function of 4/3, however, Paraservients take on the function of Preservient since the Serviant function of 4/3 is stronger.