User:Aura/Aura's Ideas on Tonality
Introduction
It seems that some people in the community want to know how my system relates to the more well-known approach of Aaron Hunt- a simple question with a complicated answer. Although this page is late in coming, I think it is time to really begin to let people see the underpinnings of my approach to music in general and microtonal music in particular, and hopefully begin to answer this question.
A Strange New World
From looking things up, it seems that Hunt and I have both been influenced by music theory of Harry Partch. However, there are significant differences, as while Hunt has been influenced by the work of Martin Vogel and Dr. Patrick Ozzard-Low, 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, something which Hunt's system fails to do. 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. Nevertheless, 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 terms of what intervals can be regarded as "commas" as opposed to "chromas", as due to my prior experience with 24edo, I can only assume that chromas, in addition to the standard definitions, can also be intervals that are less than 50 cents, yet greater than 25 cents.
Hunt is right in pointing out the fundamental nature of the Octave in Bass-Up tonality, as well as his reasoning for why this is the case. However, it seems to be that he holds to the spurious Roman view that the Ancient Greeks built their music from the Bass Upwards, thus, he 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, which I cover in more detail here and here respectively. In the context of Mircotonality, 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, and how direction of construction affects chord progressions.
While 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 dependant 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 it's 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 analysed as dissonances that need to be resolved.
Of course, there are more examples of things that need to be reevaluated in light of the existence of both Bass-Up and Treble-Down tonality, however, I cannot begin to cover all of these things here on this page. 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 like those of Hunt.
Now, many if not most musicians who are not microtonalists are acquainted with standard music notation, with its clefs and staves, key signatures and time signatures. However, when you take all of this into the microtonal realm, it becomes readily apparent that- in all of the most intuitive systems- it is the 3-limit that defines both the standard location and structure of the various standard notes and key signatures that one finds in 12edo. This even extends to the fact that the standard sharp and flat accidentals modify the base note by an apotome, and how the double sharp and double flat accidentals modify the base note by two apotomes. Furthermore, it is the Pythagorean Diatonic Scales that arise as the standard variants for the various key signatures, as these are the simplest diatonic scales that can be formed with the 3-limit.
Because the 3-limit is a prime that has all of this foundational functionality, it is naturally very important in musical systems that have their roots in 12edo, and its pivotal role in laying the groundwork for key signatures means that it can be referred as a "navigational prime". Meanwhile, when one comes from a background in 24edo as I have, and has even used quartertone-based keys signatures as I have, a second p-limit seems to join together with the 3-limit in defining the standard location and structure of the various notes and quartertone-based key signatures that one would see in 24edo. Judging on my findings- which I shall cover in the next section- the 11-limit seems to be the best candidate for this second navigational prime despite the fact that the pure 11-limit is not capable of forming diatonic scales at all.
Parachromatic, Paradiatonic, and the 11-Limit's Significance
Now, some may question the musical grounds for using quartertones in light of their dissonance, as well as the idea that there is any merit to the idea of the 11-limit being considered a navigational prime. Well, we should start with the reasons for considering quartertones musically important in the first place- namely the fact that quartertones are the most readily accessible among microtones, and that current research seems to show that quartertones are the smallest musical intervals that can be regularly used in musical capacities without being considered a variation of one of the surrounding pitches. On this basis, we can proceed to look at the musical functions of semitones, and then go on to define the musical functions of the quartertones themselves.
Most 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. As mentioned to me by Kite Giedraitis in a conversation about this topic, there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and a chromatic semitone that is otherwise known as the apotome- which, when added together, add up to a 9/8 whole tone. Furthermore, 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, we can then 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, it should be noted that for quartertones, there are sometimes multiple correct options, and thus, things are more complicated. We shall begin to define the musical functions of quartertones by drawing a distinction between the terms "parachromatic" and "paradiatonic" for purposes of classifying quartertone intervals. For starters, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as seconds, albeit subminor seconds, while parachromatic quartertones are denoted as primes, albeit superprimes. 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, we can deduce 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, we have to choose 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. From here, we have to select simple parachromatic quartertones from the lowest p-limit that, when subtracted from 9/8, yield the paradiatonic interval with the lowest odd limit.
Now that we have answered the questions as to both the musical significance and musical function of quartertones, we can take a look at the 7-limit, the 11-limit, the 13-limit, along with the 17-limit and the 19-limit, and compare the various quartertones of these limits, and thus answer the question as to whether or not there is any merit to the idea of the 11-limit being considered a navigational prime. For the 7-limit, you have 36/35, the septimal quartertone; when a stack of three septimal quartertones is subtracted from 9/8, we get a paradiatonic quartertone with a ratio of 42875/41472- not exactly a simple interval. Next we have the 11-limit, and for the 11-limit, you have 33/32, the undecimal quartertone; when a stack of three undecimal quartertones is subtracted from 9/8, we get a paradiatonic quartertone with a ratio of 4096/3993- this is better, but we still have to look at the remaining contenders. For the 13-limit, we have 40/39; when a stack of three 40/39 intervals is subtracted from 9/8, we get an interval with a ratio of 533871/512000- this is even worse than for the 7-limit. For the 17-limit, we have 34/33, the septendecimal quartertone; when a stack of three septendecimal quartertones is subtracted from 9/8, we get a quartertone with a ratio of 323433/314432- this is also worse than for the 7-limit. Finally, we have the 19-limit, and for the 19-limit, we have 39/38; when a stack of three 39/38 intervals is subtracted from 9/8, we get an interval with a ratio of 6859/6591- better than for the 7-limit, but still not as good as for the 11-limit. Therefore, the 11-limit is the most suitable p-limit for representing quartertones, meaning that it is the best candidate after the 3-limit to be considered a navigational prime. While must confess that I didn't initially choose the 11-limit on this exact basis- rather, it was because of how well the 11-limit is represented in 24edo- the math indicates that I somehow managed to make the best choice in spite of myself.
With the 11-limit now reasonably well established as being the best p-limit for representing quartertones, we can safely assume that the 11-limit is therefore the second navigational prime. This in turn means that although the 5-limit, 7-limit and 13-limit also play a role in defining key signatures, these primes define variations on the standard key signatures as opposed to the standard key signatures themselves. Furthermore, it can now be safely assumed that higher primes are ill-suited for serving as anything other than accidentals.
Measuring EDO Approximation Quality
Hunt's system measures EDO approximation quality by taking the absolute errors between just intervals and their tempered counterparts, rounding them off, and assigning them ratings based on how many cents they differ once the error values have been rounded to the nearest whole number of cents, then averaging the values together in order to rate how a given EDO represents a random set of intervals. However, while there is some merit to most of Hunt's method, averaging the values together in order to rate how a given EDO represents a random set of intervals is not the best option, as not all possible intervals are good representations of any given p-limit, and simply averaging the errors together is not the best option for grading a given EDO's representation of any given p-limit as this system fails to take contortion into account. Rather, the standards for measuring EDO approximation quality need to be more strict.
For example, I measure representation quality not only by taking the absolute error between just intervals and their tempered counterparts as per Hunt's system, but also the absolute errors in cents as they accumulate when tempered p-limit intervals are stacked, and the number of such intervals I can stack without the absolute error exceeding an unnoticeable comma's distance of 3.5 cents determines the quality of representation, and thus the portions of the harmonic lattice that can be sufficiently represented by any given EDO. Furthermore, when the error accumulation between just intervals and their and tempered counterparts exceeds half an EDO step, contortion is considered to come into play, thus terminating the sequence of viable intervals for any given p-limit and limiting the portions of the harmonic lattice that can be considered viable in any given EDO. Because the sequence of intervals in any given p-limit extends to infinity, it would be wise to use the odd-limit as way of limiting the amount of intervals used in grading the approximation quality of various EDOs. Furthermore, as the octave is foundational in so many respects, we should set our odd-limit by way of selecting octaves of the Harmonic and Subharmonic systems as guides. As the 3-limit accounts for every pitch in 12edo using intervals with odd-limits less than 1024, we shall use 1024 as the cutoff for how high odd-limits can go- yes, 1024 is an even number, but it is also a power of 2, thus rendering suitable as a divider between different categories of odd-limits. This results in the following interval selections for representing p-limits up to 17- 3/2, 9/8, 27/16, 81/64, 243/128, and 729/512 for representing the 3-limit; 5/4, 25/16, 125/64, and 625/512 for representing the 5-limit; 7/4, 49/32 and 343/256 for representing the 7-limit; 11/8 and 121/64 for representing the 11-limit; 13/8 and 169/128 for representing the 13-limit; 17/16 and 289/256 for representing the 17-limit.
When two EDOs are both given a "P" rating in the Hunt system for representation quality, the tie between them is broken by which EDO-tempered version has the smaller absolute error. Furthermore, when the best representation of an interval in a given p-limit sequence cannot be reached by stacking tempered versions of the preceding intervals in that same p-limit sequence, the disconnected interval and any intervals following it in the same sequence are disqualified under my standards, no matter how good their representation rating in the Hunt system is.
Choice of EDO for Microtonal Systems
While Hunt's microtonal system is based on 205edo, my microtonal system is built on 159edo. Why this difference? Well, even though 205edo has better interval representation in a number of cases, the step size of 205edo is too small, as half of the distance between individual steps- that is, the distance between the center of a given step and the edge of that same step- is less than 3.5 cents, which is less than the average peak JND of human pitch perception. This results in individual steps blending into one another and thus being hard to tell apart- a problem which all EDOs higher than 171 have, and a significant deterrent for me. Secondly, while 171edo itself also has better interval representation in a number of cases, the comma created by one of 159edo's three circles of fifths is smaller than that created by one of 205edo's five circles of fifths, or even that created by the 171edo circle of fifths, leading to 159edo generally having better representations of the 3-limit and 11-limit in general- something which is especially important in light of the aforementioned highly important functions of these two prime limits in particular.