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

One construct from Western Classical music with potential implications for Microtonalists is '''[[Wikipedia:Function (music)|harmonic function]]'''- especially as it pertains to the [[5L 2s|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 the realm of microtonality.

According to Aura, there are at least four known operations which can be used to derive additional functions from the three basic functions listed above

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, Aura, for the realm of Microtonality, not only takes stacks of 3/2 to form a key navigational axis called the "'''Diatonic Axis'''", but also additionally takes 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, Aura also contends that many of the functions described by Viennese Theory find their roots in specific combinations of different operations on the different basic functions described in German Theory.

'''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.

'''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 [[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.

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.

The idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees is one that has at least some roots in [[Wikipedia:Ivan Wyschnegradsky|Wyschnegradsky]]'s concept of [[Wikipedia:Major fourth and minor fifth|"Major Fourth" and "Minor Fifth"]].  However, Aura 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 he drew up sketches based loosely on [[24edo]] for early versions of this concept, he 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, Aura decided to takes 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.

[[File:Diatonic_Function_Map.png|thumb|Initial diagram of paradiatonic function locations by Aura 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 and Interregnant 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 [[Talk:159edo_notation#My_Second_Idea_for_a_Notation System|a conversation]] between Aura 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, Aura 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 that for quartertones, there are sometimes multiple correct options, making things more complicated.  Aura 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''<ref>[[Wiktionary: para- #Etymology 1]]</ref>, 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.   

According to Aura, paradiatonic quartertones are be 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, Aura 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, Aura 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, Aura 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, thorough applying the aforementioned thought process about classifying quartertones, the '''paralimma''' (the interval that remains after subtracting three parachromas from a Major MOS-step).

'''Varicant''' - Just as a Mediant lies roughly in the middle of the 3/2 interval separating the Tonic and the Dominant above it, a Varicant lies roughly in the middle of the 4/3 interval separating the Dominant and the Tonic above it.  Intervals in this region often don’t consistently act as either sixths or sevenths, or even act as a cross between a sixth and a seventh- effectively straddling the border between these two diatonic categories, hence the name "Varicant", from Latin "vāricō"<ref>[[Wiktionary: varico #Latin]]</ref>.  This is a note that occurs roughly at intervals between 128/75 and 225/128 away from the Tonic in the scale's direction of construction.  In Bass-Up Tonality, this functionality is first encountered in the form of the [[7/4]] interval, though [[12/7]] is another notable interval included in this range.  While many microtonalists think of 7/4 as being purely a type of seventh- and indeed, it most commonly acts as a sort of subminor seventh- Aura counterargues 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.

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

In addition to the Functions derived from both German theory and Viennese theory, as well as the Antitonic and Paradiatonic functions, there have been other functions described, though the terms here are those used by Aura.

'''Gravodominant''' - This is a note that occurs roughly at intervals ranging from 187/128 away from the Tonic to about 3.5 cents short of the 3/2 perfect fifth in the scale's direction of construction.  Although often overlooked or even outright shunned by traditional theorists, Aura actually considers the Gravodominant to be a legitimate diatonic function- albeit one only existing in non-meantone environments.  Specifically, as typified by intervals like 40/27, Gravodominants are dissonant intervals that simultaneously act as alternatives to the Dominant in both chord progressions and chord construction, and often require resolution.  Apart from this, Gravodominants only rise to prominence in systems where what might otherwise function as a Dominant is found just short of the sweetspot range near the standard issue 3/2.