Diatonic functional harmony

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 find a home in the realm of microtonality.

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

According to Aura, the three basic functions have their roots in LCJI, and 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 Overtone and Undertone 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. 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.

Dominant - As per the name, and as noted on 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.

Basic Operations

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

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 creates what in traditional music theory are referred to as "parallels" and "counter parallels".

Stacking - The notes that are arrived at through stacking multiple instances of either 3/2 or 4/3 (or their tempered counterparts) are dubbed according to the number of instances stacked, and the nature of the notes separated by the interval being stacked. Thus, stacking two instances of the Dominant or the Serviant results in the creation of the "Double Dominant" or "Double Serviant" respectively. This idea comes from German'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".

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, 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.

Scale Degree Functions

In addition to the aforemented Basic 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 leading tone, and simultaneously serving as the root of the 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.

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.

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.

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 circle progressions.

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.

Specific Types of Antitonic

The exact outcome and specific function of any given Antitonic depends on the interval between it and the Tonic in the scale's direction of construction, as well as by whether or not the interval in question is an augmented fourth or a diminished fifth:

• If the Antitonic is between 7/5 and 600 cents away from the Tonic and functions as an augmented fourth, it tends to cause the Dominant to become a new Tonic unless followed up by a different note- one that is usually a Major 7th away from the Tonic. Because of this tendency to "kiss up to" and tonicize the Dominant, this type of Antitonic can be dubbed a "Sycophant", with the prototypical Sycophant being 45/32.
• Conversely, if the Antitonic is between 600 cents and 10/7 away from the Tonic and functions as a diminished fifth, 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- e.g. 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 either this type of Antitonic, or some other type of substitute for a Dominant. Because of these Dominant-esque tendencies, this type of Antitonic can be dubbed a "Tyrant", with the prototypical Tyrant being 64/45.
• 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 systems. 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 systems. A classic example of such an interval is 729/512.

Paradiatonic Functions

In addition to all the aforementioned Diatonic functions, Aura not only takes 11/8 to be a more just version of Wyschnegradsky's Major Fourth interval (under the name "paramajor fourth"), but also takes stacks of 11/8 to form a second navigational axis the "Paradiatonic 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.

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.

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.

Interstep Functions

Varicant - Just as a Mediant lies roughly in the middle of the 3/2 interval separating the Tonic and the Dominant above it, a Varicant lies roughly in the middle of the 4/3 interval separating the Dominant and the Tonic above it. Intervals in this region often don’t consistently act as either sixths or sevenths, or even act as a cross between a sixth and a seventh- effectively straddling the border between these two diatonic categories, hence the name "Varicant", from Latin "vāricō". 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.

Contravaricant - This is a note that occurs roughly at intervals between 256/225 and 75/64 away from the Tonic in the scale's direction of construction. This region is characterized by intervals that don’t consistently act as either seconds or thirds. In Bass-Up tonality, this functionality has its roots in the seventh subharmonic, though 7/6 is another notable interval included in this range.

Additional Functions

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.

Circumdominant Regions

The circumdomimant regions are the two main regions on either side of the Dominant proper, and there are several functions located in these regions, one of which is an actual diatonic function. Note that antitonic and paradiatonic functions falling in this region are not covered in this section.

Mocktyrant - In Bass-Up Tonality, this is an interval that ranges from 13/10 to 7/5 which acts as a type of fifth rather than either a type of fourth or a type of third. These intervals are generally not that common outside of Jazz and Blues.

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.

Acuodominant - This is a note that occurs roughly at intervals ranging from about 3.5 cents beyond the 3/2 perfect fifth to 32/21 away from the Tonic in the scale's direction of construction. These only rise to prominence in systems where what might otherwise function as a Dominant is found just beyond the sweetspot range near the standard issue 3/2.

Mockdominant - This is a note that occurs at intervals ranging from 32/21 to 64/39, specifically one which acts as a type of diminished fourth in contrast to a more typical function as a Contramediant, or else, acts as a surrogate for a note that would otherwise function as a Dominant. These only rise to prominence in systems where there’s no proper Dominant, and where what might otherwise function as a Dominant is found too far away from the standard issue 3/2. Beyond this, they tend mostly to be found in augmented chords, and in modes like Lydian Augmented in which the fifth is augmented.

Circumserviant Regions

The circumserviant regions are the two main regions on either side of the Serviant proper, and there are two functions located in these regions, one of which is an actual diatonic function. Note that antitonic and paradiatonic functions falling in this region are not covered in this section.

Mockserviant - This is a note that occurs at intervals ranging from 39/32 to 21/16, specifically one which acts as a type of diminished fourth in contrast to a more typical function as a Mediant, or else, acts as a surrogate for a note that would otherwise function as a Serviant. Thus, these intervals only rise to prominence either in modes like Superlocrian where the fourth is diminished or in systems where there’s no proper Serviant, and where what might otherwise function as a Serviant is found too far away from the standard issue 4/3.

Gravoserviant - This is a note that occurs roughly at intervals ranging from about 21/16 away from the Tonic to about 3.5 cents short of the 4/3 perfect fourth in the scale's direction of construction. These only rise to prominence in systems where what might otherwise function as a Serviant is found just short of the sweetspot range near the standard issue 4/3.

Acuoserviant - This is a note that occurs roughly at intervals ranging from 3.5 cents beyond the 4/3 perfect fourth to 256/187 away from the Tonic in the scale's direction of construction. As typified by intervals like 27/20, Acuoserviants are dissonant intervals that often act as a sort of predominant and or as the inverses of Gravodominants, making the Acuoserviant a legitimate diatonic function in terms of Aura's analysis. Apart from this, Acuoserviants only rise to prominence in systems where what might otherwise function as a Serviant is found just beyond the sweetspot range near the standard issue 4/3.

Mocksycophant - In Bass-Up Tonality, this is an interval that ranges from 10/7 to 20/13 which acts as a type of fourth rather than either a type of fifth or a type of sixth. These intervals are generally not that common outside of Jazz and Blues.