User:Aura/Aura's Ideas on Functional Harmony

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One construct from Western Classical music with potential implications for Microtonalists is harmonic function- especially as it pertains to the diatonic MOS scale and its various relatives. While in Mainstream Music Theory there are two prevailing schools of thought in regards to diatonic functional harmony- German Theory and Viennese Theory- ideas from both schools, as well as a number of other ideas, find a home in my microtonal theory and practice. If the reader will bear with me, I shall use narrative set-ups and character metaphors to describe how the various harmonic functions act in composition and the way they relate to one another.

Facets Derived from German Theory

Among the chief ideas that come from German Theory is that there are three basic, or primary functions, and that there are multiple operations that can be applied to these three basic functions in order to derive new functions.

Basic Diatonic Functions

The three basic functions have their roots in LCJI, and each has an uncanny valley of sorts around it. The functions themselves are labeled as follows:

Tonic - This is the note that serves as the tonal center, and thus, the main resolution tone, and is the note for which scales are named (e.g. the key of C major is so-named because in this scale, C serves as the Tonic). This functionality has its roots in the fundamental at the root of both the harmonic and subharmonic series, which for all intents and purposes, can be thought of as 1/1, and, in octave equivalent systems, 2/1. Beyond being simply one of the primary three functions in German Theory, it is the only function that is known to be universal when it comes to tonal music, with the various other functions being collectively defined as Nontonic, thus, it shouldn't come as a great surprise that the Tonic exerts a very powerful influence on the context of functional harmony regardless of the nature of the tonal music system in question, even helping to define aspects of the other primary functions on the chord level as opposed to the root level. In the realm of microtonality, the Tonic is not an interval that admits a lot of competition outside of modulation, and thus, deviations from a perfect 1/1 of up to 3.5 cents away from the Tonic are considered here to be found in the Tonic's "event horizon", in which they are either absorbed into the bandwidth of the Tonic itself, altered through fudging, or simply tempered out. Furthermore, the uncanny valley around the Tonic is noticeably larger than those around the other two functions, with wolf intervals around the Tonic only being tolerable in melody and ornamentation but not harmony. To use a character metaphor for how the Tonic acts in functional harmony, the Tonic is the king of the Kingdom of Tonality- a very good king who not only exercises the highest authority in matters of governing the kingdom, but also knows how be a top-notch confidante to his subjects both wherever and whenever possible.

Dominant - As per the name, and as noted on the Wikipedia article, the Dominant is the second most important after the Tonic, and, to use a character metaphor for how the Dominant acts in functional harmony, the Dominant is both the Head Steward of the Tonic's castle, and the one that executes the Tonic's directives as a Manager of Civil Service in the Kingdom of Tonality. However, in contrast to what is stated about the Dominant in the article, there are several caveats which must be addressed in the realm of microtonality. Firstly, one must take stock of the fact that, aside from the Unison and Octave, each octave-reduced harmonic and corresponding subharmonic interval come together to generate their own axis which has a preferred direction of travel[1] which is determined by a Tonality's direction of construction. Secondly, one must take stock of the fact that when you take the notes that occur before the Tonic on each of these axes when moving in the preferred direction of travel and place them in a sequence, one finds that a clear hierarchy of functional strength based on the closeness of harmonic and subharmonic connection to the Tonic becomes apparent, with the 3/2 Perfect 5th away from the Tonic in the direction of tonality construction naturally emerging as the note with the strongest connection to the Tonic, though it should be noted that the relationships in this hierarchy are quite sensitive to detuning. Thus, the term "Dominant"- in its most basic form as referred to in this article, and specifically at the root level- is restricted to where it only refers to such notes that occur roughly at a 3/2 interval away from the Tonic in the scale's direction of construction, with acceptable detuning levels being at around 3.5 cents from JI on either side. With all that said, it should be noted that the level of importance typically associated with the Dominant goes instead to a different note instead of a 3/2 Perfect 5th in scales where the 5th scale degree is too far away from 3/2, and that there are a variety of other interval constructions which have the tendency to create tension which requires the Tonic to resolve- these surrogate Dominants are called "Paradominants". On the chord level, not only is the root level definition of the Dominant function at play, but it should also be noted that no Dominant or Paradominant sees the Tonic occurring in the proximal structure of its chord- that is, as a third or fifth.

Serviant - Compared to the term "Subdominant" from traditional music theory, the term "Serviant", specifically at the root level, is restricted to those notes that occur roughly at a 4/3 interval away from the Tonic in the scale's direction of construction since the Serviant function is essentially the inverse of the Dominant function, and acts as a sort of counterweight to the Dominant relative to the Tonic. To use a character metaphor for how the Serviant acts in functional harmony, the Serviant is a Servant who goes above and beyond the call of duty and acts as a confidante that witnesses things and reports back to the Dominant and Tonic about the way things are working both inside and outside the Tonic's castle due to its relationships to various Non-Tonic functions. 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. Secondly, in music built from the Treble downwards, the notes with these sorts of functions are actually located above the Dominant. The Serviant tends to resolve towards the Dominant, or else some other note that acts as a surrogate for the Dominant, though it can also create plagal cadences. With all that said, it should be noted that the level of importance typically associated with the Serviant goes instead to a different note instead of a 4/3 Perfect 4th in scales where the 4th scale degree is too far away from 4/3- these surrogate Serviants are called "Paraserviants". On the chord level, not only is the root level definition of the Serviant function at play, but it should also be noted that a Serviant or Paraserviants sees the Tonic occurring in the proximal structure of its chord- that is, as either a third or fifth- this explains why Serviant chords are weaker than their Dominant counterparts in both Bass-Up and Treble-Down Tonalities.

Basic Diatonic Function-Deriving Operations

The way I see it, there are seven 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 "Didominant" or "Diserviant" respectively. This concept comes from the German language's way of referring to the chord built on the second scale degree of the Diatonic scale as the "Doppeldominante", which literally means "Double Dominant".

Parallelism - Notes located in the same primary tetrachord as either the Tonic, the Dominant, or the Serviant take on similar functions to said notes, with the caveat that functions derived from the Tonic in this fashion are still technically Nontonic functions. This process is one of two that create what in traditional music theory are referred to as "parallels" and "counter parallels". It should be noted that the ability of an interval to relate to the Tonic through Parallelism, as well as the surrounding of more dissonant intervals by consonant intervals in the same region displaying such relationships to the Tonic, results in a tendency towards harmonic stagnation.

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

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 here.

Imitation - These are notes that can substitute for either the Dominant or Serviant functions through chromatic-type alteration. Functions which have this kind of role are denoted with a "mock-" prefix here.

Detempering - These are notes that appear when the comma or subchroma that separates them from the Tonic or from one of the primary Nontonic functions are not tempered out, and often, though not always, fall within the uncanny valley of the three primary functions.

Facets Derived from Viennese Theory

Among the chief ideas that come from Viennese Theory is the idea that each degree has its own function relative to the Tonic. However, while in Viennese Theory proper, the degrees are strictly defined only relative to the cycle of fifths, I, for the realm of Microtonality, not only take stacks of 3/2 to form a key navigational axis called the "Diatonic Axis", but also additionally take things like Bass-Up Tonality (that is, music built from the Bass upwards) and Treble-Down Tonality (that is, music built from the Treble downwards) into consideration. On top of that, I also contend that virtually all of the functions described by Viennese Theory find their roots in specific combinations of the different operations described above on the basic functions from German Theory.

Derivative Diatonic Functions

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 the Tonic Adjacent function, the Serviant Parallel function of 16/15, and the Antidominant function typified by the root of the 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 maps to both 1\7 and 2\24, 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 the Tonic Adjacent function, the Serviant Parallel function, and the Didominant function of 9/8, or, at least that's the case in Bass-Up tonality, where it is an interval that maps to both 1\7 and 4\24 and occurs above the Tonic as the second scale degree.

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 with both the most possibilities for realization and the ability to easily host chords built with wolf fifths. In terms of any sort of octave-reduced harmonic-subharmonic interval axis featuring harmonic motion by intervals in this region, one of the main features that characterizes this region is a tendency towards stagnation, leading to the historical characterization of these sorts of harmonies as "weak harmonies". Aside from this tendency towards stagnation, the properties that are central to the Mediant function are all most easily derived through 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.

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 scale degrees both the most possibilities for realization and the ability to easily host chords built with wolf fifths. Furthermore, it also displays a tendency towards stagnation akin to that of the Mediant, leading to harmonies in this region being historically designated as "weak harmonies". From a functional standpoint, the properties that are central to the Contramediant function are most easily derived through 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 the Tonic Adjacent function, the Dominant Parallel function, and the Diserviant function of 16/9, with these intervals additionally having Predominant functions, or, at least that's the case in Bass-Up tonality, where it is an interval that maps to both 6\7 and 20\24 and occurs 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 the Tonic Adjacent function, the Dominant Parallel function of 15/8, and the Antiserviant function. This is an interval that maps to both 6\7 and 22\24, and 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 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 circle progressions- and, the possibility of being set-up through quartertone-like motion from above (see below on Paradiatonic and Parachromatic Functions).

Antitonic

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 Serviant, others, such as myself, disagree.

Specific Types of Antitonic

The exact outcome and specific function of any given Antitonic depends on whether or not the interval in question is an augmented fourth or a diminished fifth.

Sycophant - 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 a Lead, this type of Antitonic is mapped to both 3\7 and 12\24. A prototypical example of this type of Antitonic is 45/32.

Tyrant - If the Antitonic is mapped to both 4\7 and 12\24, 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 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 Serviant. A prototypical example of this type of Antitonic is 64/45.

Chromatic Functions

Although most of these functions are accounted for by applying the Imitation process to either the Dominant or Serviant, there are two that are not.

Primary Chromatic functions

There are two primary chromatic functions that arise from the Tonic through the Imitation process.

Superdislocant - This is a note that is to either the Lead or the Contralead what a Tyrant is to a Sycophant. Specifically, it is the result of the Tonic being altered by some kind of chromatic semitone upwards and thus being displaced by a Nontonic function which leads away from the Tonic proper.

Subdislocant - This is a note that is to either the Lead or the Contralead what a Tyrant is to a Sycophant. Specifically, it is the result of the Tonic being altered by some kind of chromatic semitone downwards and thus being displaced by a Nontonic function which leads away from the Tonic proper.

Additional Functions

In addition to the functions derived from both German theory and Viennese theory, as well as the Antitonic functions, I describe other functions here. Do note that most of these tend to only show up in systems where intervals such as these are distinct, such as higher EDOs.

Circumtonic Regions

The Circumtonic regions are the two main regions on either side of the Tonic, outside the Tonic's "event horizon". These intervals are invariably inside the Tonic's uncanny valleys, and as per Flora's later analysis[2], things repel the similar but not identical. Thus, these intervals cannot be directly approached, even melodically, and so they're usually avoided outside of modulation.

Supercommatic - This is a note that occurs at intervals from about 3.5 cents to roughly 30 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 30 cents below the Tonic. As with Supercommatic intervals, these intervals are little more than stepping stones in modulation, and extra intervals that can be used together with the Tonic for a sense of dissonance.

Circumdominant Regions

The circumdomimant regions are the two main regions on either side of the Dominant proper, and since the uncanny valley around the Dominant is around half the size of the one around the Tonic, there is more room for actual first derivative diatonic functionality, as well as other functionalities.

Geminodominant - This is a note that occurs roughly at intervals ranging from about 30 cents to about 15 cents short of the 3/2 perfect fifth in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Dominant through detempering, occurring in 5-limit diatonic environments, and acting as a sort of "fraternal twin" to the Dominant, hence its name. Specifically, as typified by intervals like 40/27, Geminodominants are dissonant intervals that simultaneously act as alternatives to the Dominant in both chord progressions and chord construction, and often require resolution, though they also have a Preserviant function. 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. 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.

Circumserviant Regions

The circumserviant regions are the two main regions on either side of the Serviant proper, and since the uncanny valley around the Serviant is around half the size of the one around the Tonic, there is more room for actual first derivative diatonic functionality, as well as other functionalities.

Geminoserviant - This is a note that occurs roughly at intervals ranging from about 15 cents to about 30 cents beyond the 4/3 perfect fourth in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Serviant through detempering, occurring in 5-limit diatonic environments, and acting as a sort of "fraternal twin" to the Serviant, hence its name. As typified by intervals like 27/20, Geminoserviants are dissonant intervals that often act as a sort of predominant and or as the inverses of Geminodominants. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the Supertonic, Subtonic, 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, Geminoserviants only rise to prominence in systems where what might otherwise function as a Serviant is found just beyond the sweet spot range near the standard issue 4/3.

Paradiatonic and Parachromatic Functions

In addition to all the aforementioned Diatonic and Chromatic functions, as well as the detemperings of diatonic functions, there is an additional set of categories for dealing with the notes in between the various Diatonic scale degrees.

History

I've been in the process of developing this since well before I officially joined the Microtonal community, in fact, it all started for me with my discovery of the nature of the eleventh harmonic as a quartertone, however, while it is only thanks to a YouTuber who goes by "Quartertone Harmony" [3] that I've been able to fill in significant gaps in my theory, the reality is that the idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees, can trace at least some of its roots back to the work of Ivan Wyschnegradsky.

Specifically, idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees- an idea that has at least some roots in Wyschnegradsky's concept of "Major Fourth" and "Minor Fifth". However, I wanted to use LCJI as a basis for defining these intervals and thus decided to take 11/8 as being the just version of Wyschnegradsky's "Major Fourth", and while I drew up sketches based loosely on 24edo for early versions of this concept, I also realized that that two instances of 33/32 added up to an interval smaller than 2187/2048 but which had a similar function. Furthermore, since two instances of 11/8 resulted in an interval in the vicinity of a Major seventh, I decided to take stacks of 11/8 to form a second navigational axis which works together with the Diatonic Axis in order to define the microtonal functions positioned roughly halfway between the German and Viennese Diatonic functions, though there are a few other microtonal functions as well that are not immediately covered by this second axis.

Initial diagram of paradiatonic function locations I made around the time of officially joining the Xenharmonic community. Note that a number of the functions listed on this page are missing, while the Contralead, the Semicontralead, the Intersubiant, the Interregnant, the Misoserviant and the Misodominant initially had different names.

Most traditional music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime. They also know that a diatonic semitone and a chromatic semitone add up to a whole tone. The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo. In a conversation between myself and Kite Giedraitis about this topic, Kite mentioned that there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and the aforementioned 2187/2048- a chromatic semitone that is otherwise known as the Apotome- which, when added together, add up to a 9/8 whole tone. Furthermore, Kite also mentioned how in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by 81/80. On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields 27/25- two additional diatonic semitones. On the other hand, subtracting 81/80 from the Apotome yields 135/128, and subtracting another 81/80 yields 25/24- two additional chromatic semitones. When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone. In light of all this, Kite argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.

Building on Kite's logic, I then decided to apply similar distinctions among quartertones, and thus make the argument that quartertones don't have to denote exactly one fourth of a whole tone in as of themselves, but rather, they only have to add up to a whole tone when paired up correctly. However, the catch was that for quartertones, there are sometimes multiple correct options, making things more complicated. I decided to define the musical functions of quartertones initially on an informal basis by drawing a distinction between the terms "Parachromatic" (from the prefix para- in both the senses of alongside and resembling[4], and the word chromatic) and "Paradiatonic" (from the same two senses of the prefix para- and the word diatonic) for purposes of classifying quartertone intervals. This was easy, since I had found that two instances of 33/32 added up to 1089/1024 rather than 2187/2048, and, since I had informally added the "para-" prefix (in the same senses) to both "Major" and "Minor" to create the terms Paramajor and Paraminor to better describe how 11/8 and 16/11 related to 128/99 and 99/64 respectively in order to describe how, for instance, the notes at 99/64 and 16/11 above the Tonic relate to each other in much the same way as major and minor intervals do, except that this relationship occurs in a context where the note halfway between them is actually part of the base scale rather than the two notes in question, and there's a different interval between said two notes than the base scale's chroma.

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.

In January of 2022, Quartertone Harmony posted a video in which he grouped together a series of functions he referd to in the video as the "shadow scale"[5], which I will refer to here as the paradiatonic scale, and this in turn led to the separation of Paradiatonic and Parachromatic harmonic functions for me. This whole concept of a "shadow scale", in addition to everything else discuss in this section, paves the way for the my idea of MOS-Shadow theory, but, aside from how it applies to Diatonic-scale based functional harmony, MOS-Shadow theory is another whole discussion for another time.

The Paradiatonic Scales

The Paradiatonic scales from a given tonic acts as a sort of "second shelf" of that tonality.

The Bright Paradiatonic Scale 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

The Dark Paradiatonic Scale consists of the following scale degrees as analyzed relative to Viennese Theory's scale steps:

I, dbII tII/dbIII, dIII, tIV, dV, dVI, tVI/dbVII

While the Diatonic scale itself has seven notes, the two Paradiatonic scales each have eight notes, furthermore, the tunings of each note in each Paradiatonic scale not only depend upon the exact tuning of the Diatonic scale used as a basis, but also vary considerably when it comes to the notes of the Paradiatonic that occur between the main Diatonic interval category ranges.

Basic Paradiatonic Functions

Out of the various functions found in the Paradiatonic scale, 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 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 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 an interval that maps to both 3\7 and 11\24 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. What makes the Intersubiant different from both the Serviant and the Sycophant is that it tends to be preceded or followed by another chord with a root in the same quartertone field.

Interregnant - This is an interval that maps to both 4\7 and 13\24 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. What makes the Interregnant different from both the Dominant and the Tyrant is that it tends to be preceded or followed by another chord with a root in the same quartertone field.

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"[6]. 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 acts as Predominants and Preserviants, but not as Dominant Parallels.

Derivative Paradiatonic Functions

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.

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.

Subgradient - 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 Subgradient and a Lead in Bass-Up Tonality. In Bass-Up Tonality, Subgradients 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 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 Subdietic and the Tonic is likely to result in the Subgradient resolving back down to either the Lead or the Subtonic. As if that weren't enough, the Subgradient also has the Antintersubiant function. This function used to be called the "Subdietic", though that term has since been restricted a related composite function (see below on Composite Functions).

Supergradient - 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, there there are a few functional differences between a Supergradient and a Contralead in Bass-Up Tonality that are worth considering. For starters, Supergradients 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 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 Antinterregnant function. This function used to be called the "Superdietic", though that term has since been restricted a related composite function (see below on Composite Functions).

Basic 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, and, perhaps very tellingly, tend to utilize Diminished Fourths instead of Major Thirds due to the functions of the Subdietic- which usually gets incorporated into these kinds of chords.

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 "Varicoserviant" 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 the Lead. The 11/6 neutral seventh is a rather typical example of an interval with this function.

Composite Functions

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Notes on the boundaries of functional regions have multiple functions due to occurring at the boundary between different functions, the process by which this happens is known as Compositing. As for the actual composite functions themselves, there are quite a few of them.

Subdietic - This function is a compositing of Subcommatic and Subgradient, and as Subcommatic is part of its nature, it is effectively repelled harmonically.

Superdietic - This function is a compositing of Subcommatic and Subgradient, and as Supercommatic is part of its nature, it is effectively repelled harmonically.

Acuodominant - This function is a compositing of Dominant Detempering and Misodominant.

Gravoserviant - This function is a compositing of Serviant Detempering and Misoserviant.

These are composite functions immediately surrounding the Antitonic region

Acuotyrant - 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 - 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 Geminodominant.

  • If the Antitonic is between 7/5 and 600 cents away from the Tonic and functions as a diminished fifth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just below the range of a typical Tyrant in Bass-Up Tonality, it is called a Gravotyrant in this kind of tonal system. A classic example of such an interval is 1024/729.
  • If the Antitonic is between 600 cents and 10/7 away from the Tonic and functions as an augmented fourth, it demonstrates a mixture of both Sycophant-like and Tyrant-like properties, but because it is found just above the range of a typical Sycophant in Bass-Up Tonality, it is called an Acuosycophant in this kind of tonal system. A classic example of such an interval is 729/512.

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 Neo-Riemannian transformations are available, there are additional, derived types of transformation are needed.

References