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 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, additionally considers Wyschnegradsky's Major Fourth interval to be a second axis for defining many of the various microtonal functions, and also 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 downards) 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 and the shared Pretonic functionalities of both scale degrees, 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.

Additional Diatonic Functions

In addition to the Functions derived from both German theory and Viennese theory, there have been other functions described by Aura and others.

Antitonic Region

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. The exact outcome of this opposition depends on the exact distance of the Antitonic from the Tonic:

• 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, it is called a Gravotyrant in Bass-Up tonality. 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, it is called an Acuosycophant in Bass-Up Tonality. A classic example of such an interval is 729/512.