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".

Doubling - The notes that are arrived at through stacking two instances of either 3/2 or 4/3 (or their tempered counterparts) are dubbed the "Double Dominant" and "Double Serviant" respectively. This 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".