User:Aura/Aura's Music Theory: Introduction

Introduction

Aura has been influenced by some aspects of the music theory of Harry Partch, but would cite the little he knows of the works of Hugo Riemann as a more significant influence, as the influence of Riemann's concept of Harmonic Duality on his work with Treble-Down Tonality is strongly connected to his finding out that Ancient Greek modes were built from the Treble downwards, and how when the Ancient Romans borrowed the Greek terminology, they evidently made the mistake of assuming that the Greek note names were built from the Bass upwards, resulting in a disconnect between the Ancient Greek musical system and Modern Western Music Theory. In light of this information, and in light of the development of Western Music Theory since the time of the Romans, Aura proposes that Riemann's concept of harmonic duality- as well as Partch's argument that the Overtone Series and the Undertone Series are equally fundamental- should be taken much more seriously, and that there should be new innovations that build on the more historically accurate version of the Ancient Greek modes and Treble-Down Tonality in general to the same extent as has been done for Bass-Up Tonality. However, he is aware that doing this involves discarding the commonly-held dogmatic assumption in Modern Western Music Theory that all music is built from the Bass Upwards, and thus, has proposed terminology for renaming some of the diatonic functions encountered in Modern Western Music Theory to be better accommodating to the existence of Treble-Down Tonality.

I have been influenced by some aspects of the music theory of Harry Partch. However, I, for my part, have been influenced rather heavily by what little I know of the works of Hugo Riemann, and I've even picked up a few tricks concerning Locrian mode from Alexander LaFollett, as well as learning from my own experimentations with Locrian. While I can't remember which came first, I must say that the influence of Riemann's concept of Harmonic Duality on my work is strongly connected to my discovery that Ancient Greek modes were built from the Treble downwards, and, because when the Ancient Romans borrowed the Greek terminology, they evidently made the mistake of assuming that the Greek note names were built from the Bass upwards, resulting in a disconnect between the Ancient Greek musical system and Modern Western Music Theory.

In light of this information, and in light of the development of Western Music Theory since the time of the Romans, I think it would be a good idea to also build on the more historically accurate version of the Ancient Greek modes and Treble-Down tonality in general to the same extent as has been done for Bass-Up tonality. However, doing this involves discarding the commonly-held dogmatic assumption in Modern Western Music Theory that all music is built from the Bass Upwards. Furthermore, it involves renaming some of the diatonic functions encountered in Modern Western Music Theory to be better accommodating to Treble-Down Tonality. So, in order to do this, what sort of foundation shall we use? Well, for one thing, I propose we take Riemann's concept of harmonic duality- as well as Partch's argument that the Overtone Series and the Undertone Series are equally fundamental- much more seriously. In addition to all this I think that Aaron Andrew Hunt has done a fantastic job in integrating the ancient idea of a comma and the modern idea of the Just-Noticeable Difference in pitch perception, and I have even taken from the research on his site in this area to establish core aspects of my standards in terms of pitch representation quality. However, I differ significantly with him in that in addition to the categories of "comma" and "chroma", I also work with "subchromas" and "parachromas", as due to my prior experience with 24edo, I can only assume that intervals that are less than 50 cents yet greater than 25 cents can also act as their own musical intervals.

As far as I'm concerned, the Octave is a fundamental intervals in both Bass-Up Tonality amd Terble-Down Tonality, and Hunt is . However, it seems to be that Hunt only sees an incomplete picture when it comes to why the Octave is fundamental when it comes to the acoustic physics. As stated by the Wikipedia article on the Undertone Series, Henry Cowell has rightly pointed out that subharmonics are rather difficult to avoid in resonance, and this physical phenomenon can been demonstrated in systems such as this relatively simple one. Thus, the fact the Octave occupies the same positions relative to the fundamental in these systems as it does in harmonic systems lends itself to the reasonable assumption that physical properties of the subharmonic series act as an additional basis for the Octave being fundamental in acoustic physics. However, there's more to the physics of Treble-Down tonality than this, for as this video demonstrates, there are physical phenomenon in the analog world in which we live that produce notes that are not directly on the subharmonic spectrum, notes which our current understanding of physics fails to account for. Furthermore, as I was talking with Sam about our respective ideas of consonance, one concept that emerged from our discussion was the idea of "contra-linear consonance", which can be paired with Sam's own ideas on what I'll refer to here as "linear consonance", and my own observations on this comport with other observations on Treble-Down tonality.

Seeing Familiar Concepts in a Different Light

Since Treble-Down Tonality is a thing, however ancient or obscure it may be, it pays to revisit some of the fundamental areas of Modern Music Theory and not only reexamine them, but to also give them a facelift- for example, Musical Function, and the contrast between Consonance and Dissonance. In the context of Microtonality, it is perhaps all the more important that we do this even as we bring new concepts to the table in order to build scales and make good music with them. While it shouldn't be surprising that among the things that need to be reevaluated is the direction of chord construction as this aspect is literally the basis for the terms "Bass-Up" and "Treble-Down", among the things that also need to be reevaluated are the roles of the Bass and Treble, how direction of construction affects chord progressions, and how direction of construction affects the Diatonic Functions of notes other than the Tonic. While there are plenty of other things that need to be reevaluated that I can't cover here, suffice to say, however, that when one looks at the big picture, one will see that Treble-Down Tonality is the exact mirror image of the more conventional Bass-Up Tonality, a fact which lends to interesting and unexpected musical possibilities that are not present in more conventional systems.

In Modern Western Music Theory and in Bass-Up Tonality in general, the Bass largely plays the role of accompaniment, playing host to chords and the occasional countermelody as the Treble plays host to the melody, these roles are actually reversed in Treble-Down Tonality. This has the effect of switching the roles of numerous instruments, including various percussion instruments, therefore, the roles of Bass and Treble need to be seen as dependent on the tonality's direction of construction. Furthermore, one needs to be mindful of the fact that the way individual pitches are stacked together to make chords is also affected dramatically by the difference between Treble-Down Tonality and Bass-Up Tonality- specifically of the fact that while in Modern Western Music theory starts with the lower pitches and adds progressively higher pitches on top to make chords- hence the term "Bass-Up Tonality", Treble-Down Tonality, as per the name, sees one start with the higher pitches and add progressively lower pitches underneath. I should point out that the same types of intervals that are stacked in Bass-Up Tonality are the same types of intervals that are stacked in Treble-Down Tonality, and they are even stacked in the same order- however, due to the direction of chord construction being different between Bass-Up Tonality and Treble-Down Tonality, this results in the chords having different shapes, and even where Treble-Down chords sound identical to Bass-Up chords, the Treble-Down and Bass-Up chords have different names due to being constructed differently, and having different follow-ups in chord progression.

Take for example a chord consisting of the notes F-Natural, A-Flat and C-Natural. This would be immediately recognizable as an F-Minor triad in Bass-Up Tonality, and octave reduplication of the root would thus mean a second F-Natural is placed above the C-Natural. However, in Treble-Down Tonality, this same triad would actually be a C-Antimajor triad, as the interval pattern starting from the top note, C-Natural, is the same as that of the corresponding C-Major, with a major third interval between the first and fifth of the chord, and a minor third between the third and the fifth, and furthermore, when one wants to reduplicate the root for a C-Antimajor chord, one would add a second C-Natrual below the F-Natural. Just in reduplicating the root of the chord, the otherwise identical F-Minor and C-Antimajor triads can be differentiated. When one wants to add say a major seventh to these two triads, the results differ again due to the direction of construction. In this case, the F-Minor triad would see an E-Natural added above the C-Natural to create a F-Minor Major Seventh chord, while adding a major seventh to the C-Antimajor triad would result in adding a D-Flat below the F-Natural, with the resulting chord- a C-Antimajor-Seventh chord- sounding identical to a D-Flat Major Seventh chord when octave reduplication of the root is not present. When octave reduplication of the root is present for a C-Antimajor-Seventh chord, one will immediately think of this chord as dissonant because of the clash between the Seventh and the octave reduplicated root, however, the Antimajor-Seventh chord actually functions as the Treble-Down counterpart to the Major-Seventh chord, and thus, the Antimajor-Seventh chord is more properly considered a consonance of the same caliber as its Bass-Up counterpart despite the dissonance in the bass. I can already anticipate someone asking why these two chords have similar follow-ups when they sound so different to the ear, and the answer to that is that in both Bass-Up Tonality and Treble-Down Tonality, dissonant intervals close to the main iteration of the chord root are dispreferred, and are analyzed as dissonances that need to be resolved.

Definitions of Scale Degree Names

As mentioned before, Diatonic Functions of various notes need to be renamed and reevaluated. Therefore, In order to foster the understanding of some of the terms used for the scale degrees which I will use- to say nothing of their associated diatonic functions- it is necessary for me to define them, though these definitions will likely be amended in the future.

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). For more information about the general functionalities and properties of the Tonic, see Wikipedia:Tonic (music).

Contralead - This is my term for a note that occurs at intervals between 27/26 and 14/13 away from the Tonic, and, which serves as a leading tone in the direction opposite that of the scale's direction of construction- which, in most modern music, is from the Bass upwards. This means that in most music, the Contralead occurs as a lowered second scale degree.

Supertonic - This is the note that occurs roughly at intervals between 567/512 and 256/225 above the Tonic as the second scale degree, with 9/8 proving to be the most harmonically stable ratio between the Supertonic and the Tonic, and 10/9 often occurring as a common but less stable alternative, thus, in diatonic scales, the Supertonic generally occurs within 5 cents of either 10/9 or 9/8. For more information about the general functionalities and properties of the Supertonic, see Wikipedia:Supertonic, but do note that this article does not distinguish between a Supertonic and a Contralead.

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. This is the first of the two scale degrees with the most possibilities for realization, though in true diatonic scales, it is generally within 20 cents of either 6/5 or 5/4. As only notes at intervals with powers of 2 in either the numerator or the denominator are harmonically or subharmonically connected with the Tonic and 6/5 fails to meet this critera, I often replace the traditional 6/5 Minor 3rd with the 77/64 Minor Third. For more information about the general functionalities and properties of the Mediant, see Wikipedia:Mediant.

Serviant - This is my term for the note that occurs roughly at an interval of 4/3 away from the Tonic in the scale's direction of construction. Although this is commonly called the "Subdominant" in traditional music theory, the problems with that term 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 the 4/3 Perfect 4th, and this is not the case for other intervals between 22/17 and 7/5, which might otherwise be called "Subdominants". 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 32/25 and 7/5 away from the Tonic- the Serviant tends to resolve towards the Dominant, or, as is the case in Locrian, a certain type of Antitonic. As for the notes at other intervals between 22/17 and 7/5, I would divide them into two classes depending on what side of the 4/3 Perfect 4th they fall on, however, aside from 27/20, none of these other intervals occur in diatonic scales.

Antitonic - This is my general term for notes that occur around half an octave away from the Tonic- specifically the region extending from 7/5 to 10/7- on account of harmonies built on notes in this area tending to oppose that of the Tonic. The exact outcome of this opposition depends on the exact distance of the Antitonic from the Tonic. If the Antitonic is less than half an octave away from the Tonic, 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, I call any type Antitonic less than half an octave away from the Tonic a "Sycophant". Conversely, if the Antitonic is more than half an octave away from the Tonic, 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, I call any type of Antitonic more than half an octave away from the Tonic a "Tyrant".

Dominant - As per the name, and as noted in Wikipedia:Dominant (music), this note is the second most important after the Tonic, though in contrast to what is stated about the Dominant in the article, I would add several caveats. Firstly, I would prefer to restrict the term "Dominant" to where it only refers to the note that occurs roughly at an interval of 3/2 away from the Tonic in the the scale's direction of construction, not only because other intervals between 10/7 and 17/11 away from the Tonic in the scale's direction of construction have the tendency to create tension which requires the Tonic to resolve, but also because the 3/2 Fifth is by far the best choice for this sort of functionality on account of the the extremely close harmonic connection between the Tonic and the 3/2 Perfect 5th. Secondly, I would also add the caveat that the level of importance typically associated with the Dominant goes instead to the Tyrant Antitonic in those cases where one occurs on the 5th scale degree instead of a 3/2 Perfect 5th. As for the notes at other intervals between 10/7 and 25/16, I would divide them into two classes depending on which side of the 3/2 Perfect Fifth they fall on, however, aside from 40/27, none of these other intervals occur in diatonic scales.

Contramediant - This is my term for the note that occurs roughly at intervals between 25/16 and 128/75 away from the Tonic in the scale's direction of construction. The Contramediant is the second of two scale degrees with the most possibilities for the realization, however, in true diatonic scales of the variety I'm defining here, it is generally within 20 cents of either 8/5 or 27/16- the latter interval being used in place of 5/3 for reasong which are detailed below. If the Contramediant is set at the Tonic's 27th harmonic, a grave fifth is then positioned between the Contramediant and the Mediant, and the slightly-off sound of the resulting minor triad provides a more clear indication that one should expect a follow up- this is a deceptive cadence at its finest. Furthermore, setting the Contramediant at the Tonic's 27th harmonic alters the character of the Serviant chord to be less consonant, and thus allows the Serviant to unambiguously perform its harmonic functions relative to both the Tonic and the Dominant.

Subtonic - This is the note that occurs roughly at intervals between 225/128 and 1024/567 above the Tonic as the seventh scale degree, with 16/9 proving to be the most harmonically stable ratio between the Supertonic and the Tonic, and 9/5 often occurring as a common but less stable alternative, thus, in diatonic scales, the Subtonic generally occurs within 5 cents of either 16/9 or 9/5. For more information about the general functionalities and properties of the Subtonic, see Wikipedia's article Wikipedia:Subtonic.

Lead - This is a note that occurs at intervals between 13/7 and 52/27 away from the Tonic, which serves as a leading tone in the scale's direction of construction. For more information about the general functionalities and properties of the Lead, see Wikipedia:Leading-tone, but do note that this article does refers to what I call a "Contralead" by the term "Upper Leading-Tone".

In addition to the diatonic functions and scale degree names that I've listed, there are also others that are unique to Microtonality, so I'll have to define those as well.

Supercommatic - This is the region above the Tonic spanning from 3.5 cents away the Tonic out to about 20 cents away. Intervals in this region tend to not be very effective as leading-tones or as the direct targets of subchromatic motions on account of their size. That said, in EDOs that support step sizes that fall into this region, these small steps make for useful secondary targets, allowing the possibility of seamless modulation to keys that are not in the same series of fifths, though one should be wary of having many steps as to have individual steps blend completely into one another on account of their small size.

Superdietic - Extending from about 20 cents above the Tonic out to a distance represented by the ratio of 27/26, This is one of the regions where one is most likely to find subchromas, and dieses, the latter being the source of "dietic" part of the term "Superdietic". In Bass-Up Tonality, this is the region where you are most likely to find "uninds"- intervals that act as a cross between a unison and a second.

Semilead

Varicant

Contravaricant