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. Not only are the traditional key signatures all related to each other along an axis formed by the 3-limit, but the standard sharp and flat accidentals modify the base note by an apotome, and 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"- specifically, we can refer to it as the "Diatonic 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, one sees that 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, and 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. 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 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 superprimes of some sort. 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. 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.
In order to be thorough, I've since checked the 11-limit's representation of quartertones against that of the other rational intervals called "quarter tones" on Wikipedia's list of pitch intervals and again found the 11-limit's 33/32 to be better than any of them in terms of ratio simplicity. Therefore, the 11-limit is the most suitable p-limit for representing quartertones. While must confess that I didn't initially choose the 11-limit on these exact bases- 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. However, this still leaves the question of whether the 11-limit can serve as a navigational axis of any kind, as this is another question that must be answered in order to determine whether or not there is any merit to the idea of the 11-limit being considered a navigational prime. However, in order to even begin to consider this, we must familiarize ourselves with some of the 11-limit's inner workings.
Delving into the 11-Limit: Alpharabian Tuning and Additional Interval Classifications
With the 11-limit established as perhaps the best p-limit for representing quartertones in terms of ratio simplicity, we need to look more closely at its properties- something which most microtonal systems at the time of this writing have yet to do. First off, one will notice that a stack of two 33/32 quartertones falls short of an apotome by a comma called the rastma, which means that the rastma, when not tempered out, is a very important "parasubchroma"- a term derived from subchroma. One will also notice that because two parachromatic quartertones equals a chromatic semitone, and because two 33/32 quartertones fall short of an apotome by a rastma, we can deduce that other parachromatic quartertones- or, more generally "parachromas"- can be derived from 33/32 by adding or subtracting the rastma. When one adds the rastma to 33/32, one arrives at 729/704- the parachromatic quartertone that adds up together with 33/32 to form an apotome. Since 33/32 is smaller than 729/704, and both can be described as "undecimal quartertones", we need to disambiguate them somehow, not to mention acknowledge the 11-limit's newfound status as a navigational prime. Thus, for purposes of continuing this discussion at the moment, we'll start referring to 33/32 as the "primary parachromatic quartertone", and, we'll refer to 729/704 as the "secondary parachromatic quartertone". However, the rastma doesn't only derive other parachromatic intervals from 33/32, but also derives other paradiatonic intervals from 4096/3993- which we shall refer to for the moment as the "primary paradiatonic quartertone". For instance, if one subtracts the rastma from the primary paradiatonic quartertone, we get 8192/8019, which, when added to 33/32, yields 256/243- the Pythagorean Diatonic Semitone. Because of this, and because 8192/8019 is derived from the primary paradiatonic quartertone, we shall refer to 8192/8019 as the "secondary paradiatonic quartertone".
However, the "primary" versus "secondary" distinction is temporary at best, as in truth there is more nuance to consider. Furthermore, we have to contend with the idea of a "Subchroma" from earlier, as well as the idea of a "Diesis", and define how these concept relate to the idea of "Parachromatic" and "Paradiatonic" intervals, and for this we should begin by looking at the distinction between a "Paradiatonic" interval and a "Diesis". In order to define a "Paradiatonic" interval as it contrasts with a "Diesis", we need to consider that "Paradiatonic" consists of the prefix "Para-" and the word "Diatonic", with "Para-" meaning "alongside" in this case, as paradiatonic intervals are those that are relatively easy to use as accidentals in otherwise diatonic keys. More importantly, we need to consider that diatonic intervals- as the term "diatonic" pertains to intervals other than semitones- are the intervals found in those heptatonic scales consisting of five whole tones and two semitones in each octave in which the semitones are spread out as much as possible, as per the more strict definition of "Diatonic" listed on Wikipedia's article on the Diatonic Scale. In light of all this, it should follow that we can define "Paradiatonic" intervals as being those microtonal intervals which are as distant from the boundaries of the nearest semitone-based intervals as possible. Furthermore, since quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches, this means that Paradiatonic intervals are inevitably quartertone-based, as are their "Parachromatic" counterparts. In contrast, terms such as "Subchroma" and "Diesis" are a bit broader, as they are not restricted to quartertone-based intervals- in fact, they are most often used to refer to intervals smaller than a quartertone. That said, there are such things as "Subchromatic Quartertones", which contrast with their Paracrhomatic counterparts in that they have more complicated ratios whereas the Parachromatic quartertones have fairly simple ratios.
However, all of this only partially covers the 11-limit's quartertones. Since two parachromatic quartertones add up to a chromatic semitone, and since 33/32 is the "primary" parachromatic quartertone, we can safely assume that two 33/32 intervals adds up to a chromatic semitone of some sort, and indeed, 1089/1024 is a chromatic semitone under this definition, but as it differs from the apotome by the rastma, and we can immediately see that the "primary" versus "secondary" distinction is untenable here due to the apotome being a 3-limit interval. So, we should instead look to another source for more proper terminology for 11-limit intervals that are distinguished from each other by the rastma. Since 33/32 is also called the "al-Farabi Quartertone" and is the primary apotome-like interval of the of the 11-limit, and, since al-Farabi himself was also referred to as "Alpharabius" according to Wikipedia's article on him, we can use the term "Alpharabian" to refer to no-fives no-sevens just 11-limit tuning in the same way that we can use the "Pythagorean" to refer to just 3-limit tuning. Therefore, we can use the term "Alpharabian" to refer to the 11-limit semitones in the same way that we use "Pythagorean" to refer to the 3-limit semitones. Thus, just as the apotome can also be referred to as the "Pythagorean Chromatic Semitone", we can refer to 1089/1024 as the "Alpharabian Chromatic Semitone". Furthermore, just as there's an Alpharabian Chromatic Semitone, there is an "Alpharabian Diatonic Semitone" which adds up together with the Alpharabian Chromatic Semitone to create a 9/8 whole tone, and, when you take 9/8 and subtract 1089/1024, you arrive at 128/121 as the ratio for the Alpharabian Diatonic Semitone. Not only that, but just as there's the Pythagorean comma which forms the difference between a stack of two Pythagorean Diatonic Semitones and a 9/8 whole tone, so there is an "Alpharabian Comma" which forms the difference between a stack of two Alpharabian Diatonic Semitones and a 9/8 whole tone, and doing the math yields 131769/131072 as the ratio for the Alpharabian Comma.
Basic 11-Limit Interval Classifications
With all of the aforementioned stuff about Alpharabian tuning and the need for terminology that distinguishes 11-limit intervals that differ by the rastma, one can easily go on to ask what all this means in terms of the classification of more familiar 11-limit ratios like 11/8, seeing as the 11/8 can be derived from 4/3- the Just Perfect Fourth- through the addition of the "primary" parachromatic quartertone. Since the addition of the "primary" parachromatic quartertone to the Perfect Unison results in the "primary" parachromatic quartertone, one would assume that this means that 11/8 would be classified as the "Alpharabian parachromatic superfourth" or something like that. In actuality, however, while one would be correct in asserting 11/8 is both an Alpharabian interval and a parachromatic alteration of the perfect fourth, interpreting 11/8 as a derivative of 33/32 would in many respects be akin to interpreting 3/2- the Just Perfect Fifth- as a derivation of the apotome, when in fact, it is the other way around. Recall that the prime factorization of 33 is 3*11, so that means that 33/32 is not a pure 11-limit interval. Therefore, rather than assume 33/32 to be the basic 11-limit interval, we instead must recognize that that title properly belongs to 11/8. Furthermore we should take stock of the fact that two 11/8 intervals stacked on top of one another yields 121/64, the octave complement of the Alpharabian diatonic semitone. Since 121/64 is arguably a form of major seventh as a diatonic semitone always has a major seventh as its octave complement, and since a stack of two fourths equals a seventh, what does that mean for 11/8? Well, it means we need more terms, and we need to define those terms.
Now, most music theorists know that Major and Minor intervals are chromatic alterations of one another. Furthermore, we have established that two parachromatic intervals equals a chromatic interval, and we have established that 11/8 is a parachromatic interval. So, what term shall we use to classify 11/8? Well, since "Major" and "Minor" intervals occur when there are two basic intervals of a given diatonic step size, and since we can also observe that Minor and Major relate directly to each other by chromatic alteration, we can thus argue that the term we need for classifying 11/8 that is comprised of the element "Para-" and either the word "Major" or the word "Minor", therefore, we can coin the terms "Paramajor" and "Paraminor". Since 11/8 is higher than the Just Perfect Fourth, that means that we must use the term "Paramajor" to describe 11/8- and since 11/8 is the primary 11-limit interval, we should refer to 11/8 as the "Alpharabian Paramajor Fourth" or "Just Paramajor Fourth". Furthermore, in the same way Major and Minor intervals are complements of each other, we can say that Paramajor and Paraminor intervals are complements of one another, so therefore, we can say that 16/11 is the "Alpharabian Paraminor Fifth" or "Just Paraminor Fifth". This arrangement seems works out very well, as 11/8 and 16/11 are basic intervals in their own right just as 3/2 and 4/3 are, with the name "Just Paramajor Fourth" for 11/8 reflecting how 11/8 is higher than the Just Perfect Fourth by a primary parachromatic quartertone, and the name "Just Paraminor Fifth" reflecting how 16/11 is lower than the Just Perfect Fifth by the same interval. However, it should be remembered that the Paramajor-Paraminor distinction can ultimately be thought of as referring to two different sizes of Fourth and two different sizes of Fifth in the same way that the Major-Minor distinction can be thought of as describing two different sizes of second, as well as two different sizes of third, two different sizes of sixth, and two different sizes of seventh- therefore the Paraminor Fourth and the Paramajor Fifth also exist, with these intervals being 128/99 and 99/64 respectively. However, we still have yet to cover the terminology for alterations of Major and Minor intervals by 33/32, and the introduction of "Paramajor" and "Paraminor" intervals leaves the question as to what terminology to use on this front for alterations of the Perfect Prime and the Octave.
In answering these questions one should note that the Prime and the Octave are the fundamental intervals in both my system and conventional music systems. Furthermore, it doesn't make sense to have dedicated names for intervals that go in the opposite direction from of a given tonality's direction of construction, such as "Paraminor Unison", and, since a term like "Paramajor Unison" would imply the existence of the nonsensical "Paraminor Unison" by definition, we can discard the idea of a "Paramajor Unison" also. While in earlier drafts of this page, I said we should use the "Super-" prefix for the augmentation of Major intervals by 33/32, and "Sub-" for the dimunition of Minor intervals by 33/32, I have since realized that such a system fails to account for the fact that 4096/3993, the primary limma-like interval of the 11-limit, differs from the Pythagorean diatonic semitone by 1331/1296- a type of parachromatic interval. Since 4096/3993 is rightly deemed a type of "Subminor Second" while 33/32 is the difference between the Just Paramajor Fourth and the Just Perfect Fourth, I now propose that we refer to 1331/1296 as the "Alpharabian Superprime", and that we use the "Super-" and "Sub-" prefixes to not only refer to the respective augmentation and dimunition of both Unisons and Octaves by 1331/1296, but also to the augmentation of Major intervals and dimunition of Minor intervals respectively by 1331/1296. While the "Super-" and "Sub-" prefixes are often associated with the 7-limit, it should be remembered that in this system, the 7-limit versions of these intervals are variations on the standard intervals as opposed to being the standard intervals themselves.
Continuing along this same line of thought, I propose we refer to 33/32 as the "Alpharabian Parasuperprime", and, I have now returned to my initial idea of using the "Parasuper-" and "Parasub-" prefixes to refer to the augmentation of Major intervals and dimunition of Minor intervals respectively by 33/32, after foolishly thinking it untenable in light of the the 11-limit's status as a navigational prime, a position which I now realize led to inconsistency in the naming scheme. Nevertheless, because the dimunition of a major interval by 33/32 does not result in the same interval as does the augmentation of a minor interval by 33/32, as these intervals differ by a rastma, and thus, as has been my idea since I first came onto this Wiki, I propose that we use the term "Greater Neutral" to refer to dimunition of a Major interval by 33/32, and the term "Lesser Neutral" to refer to the augmentation of a Minor interval by 33/32. As is to be expected, Supermajor and Subminor intervals are complements of one another; for example, when 243/128 is raised by 1331/1296, the result is 3993/2048- a supermajor seventh and the octave complement of 4096/3993, which has already been established as a subminor second. Similarly, Parasupermajor and Parasubminor intervals are also complements of one another, for instance 1024/891, the Alpharabian Parasubminor Third, is the octave complement of 891/512, the Alpharabian Parasuperjamor Sixth.
This still leaves the matters of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32, what happens when we modify Perfect Fourths and Perfect Fifths by 1331/1296, and, what happens when we either lower Major intervals or raise Minor intervals by 1331/1296. However, we can cover these topics in the next section, as we need to delve even deeper into the 11-limit to cover these intervals on account of their complexity. Before we do that, however, we first need to both take stock of the rules for the current naming system, and compile a list of relatively simple 11-limit intervals which are all classified as Alpharabian intervals, as we have now covered most of the basics for 11-limit interval terminology in this system. Do note that when composite interval terms like "Greater Neutral" are qualified by tuning terms like "Alpharabian", at least in English, the tuning term is inserted between the elements of the interval term, thus, for instance, 88/81, the Greater Neutral Second in Alpharabian tuning, is labeled as the "Greater Alpharabian Neutral Second".
(Todo: Add more intervals and columns to the chart, with one extra column containing monzos)
| Interval | Cents | Shorthand Designations | Names |
|---|---|---|---|
| 1331/1296 | 46.133824 | rU1 | |
| 33/32 | 53.272943 | U1 | Alpharabian Ultraprime, Alpharabian Parachromatic Quartertone, al-Farabi Quartertone |
| 729/704 | 60.412063 | uA1 | Alpharabian Infraaugmented Prime, Alpharabian Infraapotomic Quartertone |
| 1089/1024 | 106.545886 | rA1 | Alpharabian Chromatic Semitone |
| 161051/147456 | 152.679710 | rrUA1 | Alpharabian Superaugmented Prime |
| 35937/32768 | 159.818830 | rUA1 | |
| 72171/65536 | 166.957949 | UA1 | Alpharabian Ultraaugmented Prime |
| 8192/8019 | 36.952052 | um2 | Alpharabian Inframinor Second |
| 4096/3993 | 44.091172 | Rum2 | Alpharabian Subminor Second, Alpharabian Paradiatonic Quartertone |
| 128/121 | 97.364115 | Alpharabian Minor Second, Alpharabian Diatonic Semitone | |
| 21296/19683 | 136.358819 | Alpharabian Supraminor Second | |
| 88/81 | 143.497938 | Lesser Alpharabian Neutral Second | |
| 12/11 | 150.637059 | Greater Alpharabian Neutral Second | |
| 1458/1331 | 157.776178 | Alpharabian Submajor Second | |
| 1331/1152 | 250.043825 | Alpharabian Supermajor Second | |
| 297/256 | 257.182945 | Alpharabian Parasupermajor Second | |
| 1024/891 | 240.862054 | Alpharabian Parasubminor Third | |
| 1536/1331 | 248.001174 | Alpharabian Subminor Third | |
| 144/121 | 301.274117 | Alpharabian Minor Third | |
| 2662/2187 | 340.268821 | Alpharabian Supraminor Third | |
| 11/9 | 347.407941 | Lesser Alpharabian Neutral Third | |
| 27/22 | 354.547060 | Greater Alpharabian Neutral Third | |
| 6561/5324 | 361.686180 | Alpharabian Submajor Third | |
| 121/96 | 400.680884 | Alpharabian Major Third | |
| 1331/1024 | 453.953827 | Alpharabian Supermajor Third | |
| 2673/2048 | 461.092947 | Alpharabian Parasupermajor Third | |
| 128/99 | 444.772056 | Alpharabian Paraminor Fourth | |
| 1728/1331 | 451.911176 | Alpharabian Subfourth | |
| 11/8 | 551.317942 | Alpharabian Paramajor Fourth, Just Paramajor Fourth | |
| 1331/972 | 544.178823 | Alpharabian Superfourth | |
| 363/256 | 604.590886 | Alpharabian Augmented Fourth | |
| 512/363 | 595.409114 | Alpharabian Diminished Fifth | |
| 16/11 | 648.682058 | Alpharabian Paraminor Fifth, Just Paraminor Fifth | |
| 99/64 | 755.227944 | Alpharabian Paramajor Fifth | |
| 4096/2673 | 738.907053 | Alpharabian Parasubminor Sixth | |
| 2048/1331 | 746.046173 | Alpharabian Subminor Sixth | |
| 192/121 | 799.319116 | Alpharabian Minor Sixth | |
| 44/27 | 845.452940 | Lesser Alpharabian Neutral Sixth | |
| 18/11 | 852.592059 | Greater Alpharabian Neutral Sixth | |
| 121/72 | 898.725883 | Alpharabian Major Sixth | |
| 1331/768 | 951.998826 | Alpharabian Supermajor Sixth | |
| 891/512 | 959.137946 | Alpharabian Parasupermajor Sixth | |
| 512/297 | 942.817055 | Alpharabian Parasubminor Seventh | |
| 2304/1331 | 949.956175 | Alpharabian Subminor Seventh | |
| 11/6 | 1,049.362941 | Lesser Alpharabian Neutral Seventh | |
| 81/44 | 1,056.502061 | Greater Alpharabian Neutral Seventh | |
| 121/64 | 1,102.635885 | Alpharabian Major Seventh | |
| 3993/2048 | 1,155.908828 | Alpharabian Supermajor Seventh | |
| 8019/4096 | 1,163.047948 | Alpharabian Ultramajor Seventh | |
| 2048/1089 | 1,093.454114 | Alpharabian Diminished Octave | |
| 64/33 | 1,146.727057 | Alpharabian Infraoctave | |
| 2592/1331 | 1,153.866176 | Alpharabian Suboctave |
From this particular sample, we can deduce that there are several fundamental premises of the Alpharabian tuning system:
- Intervals that are in the 2.11 subgroup are all considered Alpharabian intervals.
- The intervals 3/2, 4/3, 9/8, 16/9, and so forth, have the same functions as in Pythagorean tuning.
- Intervals that result from the modification of a Pythagorean interval by 1089/1024 are labeled similarly to those modified in the equivalent fashion by 2187/2048, the only difference being that modification by 1089/1024 results in an Alpharabian interval rather than a Pythagorean interval.
- Since 1089/1024 is (33/32)^2, modifying a Pythagorean interval by 33/32 always results in an interval that is considered "Alpharabian".
- As both the rastma and 1331/1296 are subchromas that form differences between members of the 2.11 subgroup and Pythagorean intervals, both of these subchromas belong to a set of intervals defining different interval sets within Alpharabian tuning, and subchromas within this particular interval set help define the differences between Pythagorean, Alpharabian and Betarabian intervals.
The following rules are directly derived from the above premises:
- Generally, intervals that result from the modification of a Pythagorean interval by 33/32 take either the 'parasuper' or 'parasub' prefixes, however, there are a number of special cases...
- Augmentation of a Perfect Fourth or Perfect Fifth by 33/32 results in a Paramajor interval
- Dimunition of a Perfect Fourth or Perfect Fifth by 33/32 results in a Paraminor interval
- Augmentation of a Pythagorean Minor interval by 33/32 results in a Lesser Neutral interval
- Dimunition of a Pythagorean Major interval by 33/32 results in a Greater Neutral interval
- Generally, intervals that result that result from the modification of a Pythagorean interval by 1331/1296 take either the 'super' or 'sub' prefixes, with these prefixes generally being stacked where multiple such modifications occur, however, there are some significant caveats...
- Augmentation of a Pythagorean Minor interval by a single 1331/1296 results in a Supraminor interval, but a second such augmentation results in a Betarabian Major interval due to said interval differing from the nearby Alpharabian Major (covered under modifications by 1089/1024) by a rastma.
- Dimunition of a Pythagorean Major interval by a single 1331/1296 results in a Submajor interval, but a second such dimunition results in a Betarabian Minor interval due to said interval differing from the nearby Alpharabian Minor (covered under modifications by 1089/1024) by a rastma.
Delving into the 11-Limit: Betarabian Intervals and the Nexus Comma
(Todo: rewrite this section based on further modification of the rules for defining subsets of Alpharabian tuning)
Now that we've covered the basic Alpharabian intervals, it's time to continue our journey into the 11-limit. However, because the intervals in this section are not covered by the basic classification scheme for Alpharabian intervals, it would be better if we called these intervals by a different name. While the term "Rastmic" has historically been used as a descriptor for intervals like the 27/22 neutral third, this naming scheme fails to take the importance of the 11-limit into account, and also fails to consider the rastma's additional properties when not tempered out. Nevertheless, the term "Rastmic" as an interval descriptor retains its usefulness, even when all of the basic Alpharabian intervals are properly accounted for, as while there are infinitely many Alpharabian intervals, there are still many intervals that are not basic Alpharabian intervals yet only differ from these same intervals by a rastma- or two, or three, and so on. However, I'm under the impression that we need to save the "Rastmic" interval descriptor for when we move past a second layer of 11-limit intervals, and it is this second layer of 11-limit intervals that we shall cover in this section.
Now, if one does the math, they will realize that an Alpharabian Parasupermajor Second, having a ratio of 297/256, is larger than an Alpharabian Parasubminor Third with its ratio of 1024/891, and that the difference between these two intervals is 264627/262144. Despite the fact that 264627/262144 is the sum of the rastma and the Alpharabian comma, its function can be contrasted with that of the Alpharabian comma in that 264627/262144 not only separates 297/256 and 1024/891, but also other similar enharmonic quartertone-based interval pairs, whereas the Alpharabian comma merely distinguishes enharmonic 11-limit semitones. Yet, the term "Alpharabian" contains the word "Alpha", which can be taken as signifying the Alpharabian comma's status a primary 11-limit comma. Therefore, if we take the "Alpha" off of "Alpharabian" and put the term "Beta" in its place, we can thus call 264627/262144 the "Betarabian Comma". On another note, you may have noticed that I didn't include the 729/704 quartertone in the list of Alpharabian intervals. This was because I couldn't exactly find a place for 729/704 in the list of 11-limit intervals that can be considered "basic". However, I think it's fair to said that I have opened up another layer of 11-limit intervals- intervals that can't exactly be considered "basic" due to other more important intervals like 11/8 and 16/11 taking priority, yet can still be derived from the basic intervals by means of either adding or subtracting a rastma. With this in mind, you should recall that two 33/32 Parachromatic Quartertones fall short of the apotome by a rastma, and that if you add a rastma to 33/32, you get 729/704. However, there's more to the story here, as 729/704 differs from the 4096/3993 Paradiatonic Quartertone by the Betarabian comma. With both of these things in mind, it's safe to say that we can classify 729/704 as a Betarabian interval- specifically, we can call it the "Betarabian Parasuperprime" or the "Betarabian Parachromatic Quartertone".
Of course, it stands to reason that there are more Betarabian intverals than just the Betarabian Parachromatic Quartertone and the Betarabian Comma- in fact, Betarabian intervals result when we modify 3-limit Augmented and Diminished intervals by 33/32. When we subject a 3-limit Augmented interval to augmentation by 33/32, we can refer to the resulting interval as being "Parasuperaugmented", and when we subject a 3-limit Diminished interval to dimunition by 33/32, we can refer to the resulting interval as being "Parasubdiminished". A good example of a parasuperaugmented interval is 24057/16384, the Betarabian Parasuperaugmented Fourth, which is larger than 16/11 by the Betarabian comma, while a good example of a parasubdiminished interval is 32768/24057, the Betarabian Parasubdiminished Fifth. Furthermore, when the rastma is not tempered out, we can subject 3-limit Augmented intervals to dimunition by 33/32 without arriving at the same location as an Alpharabian interval; for example, reducing the apotome or Augmented Unison by 33/32 yields 729/704. Likewise, we can subject a 3-limit Diminished interval to augmentation by 33/32 without arriving in the same location as an Alpharabian interval; for example, augmenting 1024/729 by 33/32 yields 352/243, which we shall call the "Lesser Betarabian Paraminor Fifth". One may inquire as to my reasoning for calling 352/243 the "Lesser Betarabian Paraminor Fifth" instead of simply calling it the "Betarabian Paraminor Fifth", and the answer is actually quite simple- there are two Betarabian intervals that can be considered "Paraminor Fifths".
When you lower a Perfect Fifth by 1331/1296, you get 1944/1331, which, like 352/243, differs from 16/11 by a rastma- albeit in the opposite direction- and there are two reasons that 1944/1331 can't be considered an Alpharabian interval despite its relative simplicity. The first reason is because there is only room for one Alpharabian Paraminor Fifth, and the most basic Paraminor Fifth is 16/11. The second reason is that since Paramajor and Paraminor intervals are basic interval categories for the 11-limit the way that Major and Minor are for the 3-limit, you can't exactly get away with calling 1944/1331 a "Subfifth" any more than you can get away with calling 16/11 a "Parasubfifth"- at least not in this system. The same reasoning applies when lowering a Perfect Fourth or raising a Perfect Fifth by 1331/1296. Therefore, when you modify a Perfect Fourth or Perfect Fifth by 1331/1296, the result must be a Betarabian interval that can be classified as either "Paramajor" or "Paraminor". When you lower a Major interval or raise a Minor interval by 1331/1296, you end up with similar issues, as Neutral intervals are basic interval categories for the 11-limit, and furthermore, while one may consider using terms like "Submajor" or "Supraminor" to describe these, at the end of the day, they will not be seen as distinct from Neutral intervals as quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches. At this point, someone might use this same argument to object to my distinction between the "Parasuper-" and "Parasub-" prefixes on one hand and "Super-" and "Sub-" prefixes on the other hand. However, I would say that while on one level, the would be right, the fact remains that on another level, the different types of quartertones add up differently, and those differences need to be respected when dealing with the 11-limit. Therefore, I would- for example- label 2662/2187 the "Lesser Betarabian Neutral Third", and label 6561/5324 the "Greater Betarabian Neutral Third".
While I can't cover all of the Betarabian intervals in this section as there are too many to cover, and the same is true of 11-limit intervals in general, I can perhaps by take note of a few more important 11-limit intervals. Firstly, there are a set of Betarabian Semitones- the "Betarabian Chromatic Semitone", 14641/13824, and the "Betarabian Diatonic Semitone", 15552/14641. The Betarabian Chromatic Semitone is smaller than the Alpharabian Chromatic Semitone by a rastma, while the Betarabian Diatonic Semitone is larger than the Alpharabian Diatonic Semitone by a rastma. When you subtract 33/32 from the Betarabian Chromatic Semitone, you get 1331/1296, however, when you subtract the 729/704 from the Betarabian Chromatic Semitone, you get the very complicated 161051/157464, the Betarabian Subchroma. On the other hand, when you subtract 33/32 from from the Betarabian Diatonic Semitone, you get the very complicated 165888/161051, the Betarabian Paradiatonic Quartertone, which differs from the Alpharabian Paradiatonic Quartertone by a rastma. Finally, I propose we bring our journey through the 11-limit to a fitting conclusion with a look at one final 11-limit comma. This comma has a ratio of 1771561/1769472, and forms the difference between the Rastma and the Alpharabian comma, furthermore, it also forms the difference between the Alpharabian Subminor Third and the larger Alpharabian Supermajor second. Furthermore, it also forms the difference between the the Alpharabian Parachromatic Quartertone and the Betarabian Paradiatonic Quartertone, as well as difference between innumerable pairs of other 11-limit intervals. However, what's most notable about this comma is that it is the amount by which a stack of three 128/121 Alpharabian Diatonic Semitones falls short of a 32/27 minor third. Considering that 128/121 is a pure 11-limit interval, while 32/27 is a pure 3-limit interval, this means that 1771561/1769472 is a very very important interval- especially in light of the fact that it is only slightly more than two cents in size and is thus not only an unnoticeable comma, but a prime target for tempering. Since tempering out 1771561/1769472 results in the formation of a nexus between the 3-limit and the 11-limit, the latter of which has so far already been established as perhaps the best p-limit for representing quartertones, 1771561/1769472 can thus be called the "Nexuma", or the "Nexus comma".
11-limit Axis Functionality
Now that we have explored some of the 11-limit's inner workings, we can return to the question of whether the 11-limit can serve as a navigational axis of any kind, and thus whether or not there is any merit to the idea of the 11-limit being considered a navigational prime. When trying to answer this question, we would do well to remember not only the fact that in Just Intonation, the 3-limit and the 11-limit never meet, but also the fact that the 11-limit and the 3-limit operate on fundamentally different levels- the 3-limit operating on the diatonic level, and the 11-limit on the paradiatonic level. With this in mind, what we need to look for is similarities and parallels between how these two p-limits function in their purest form.
The fundamental interval of the 3-limit is 3/2, the Just Perfect Fifth. When a stack of two Perfect Fifths is octave-reduced, we get 9/8- a diatonic whole tone. The diatonic nature of this whole tone is confirmed with further stacking and octave reduction of Perfect Fifths- when you stack three Perfect Fifths and octave-reduce them, you end up with 27/16, which is a whole tone away from the original Just Perfect Fifth. Continuing along this sequence, we should take stock of the fact that we only have seven base note names to work with- let's use "A", "B", "C", "D", "E", "F" and "G" for the sake of simplicity- as the term "diatonic" pertains to the intervals found in those heptatonic scales in which the notes are spread out as much as possible. Furthermore, since spreading out in both directions along this 3/2 chain ends up creating multiple variants of the same diatonic step, which these variations being best described as "chromatic", and thus, we need to introduce sharps and flats. Of course, since the 3-limit and the 2-limit never meet beyond the fundamental, continuing to expand along the 3-limit chain in both directions eventually creates what are essentially chromatic whole tones- frequently referred to as double sharps and double flats, but we have to draw the line somewhere as continuing beyond a certain point is impractical.
When we look at the 11-limit, we again need to draw the line as to how far we go as continuing beyond a certain point is impractical. However, we need to take note of the fact that while the 3-limit works with whole tones and semitones, the 11-limit works with quartertones and semitones. Therefore, the most direct place of comparison between the 3-limit and the 11-limit is in their handling of semitones. However, since the fundamental interval of the 11-limit is 11/8, we should not go into our examination of the 11-limit expecting the same results as for the 3-limit. Since we have already established that is 11/8 a paramajor fourth, we should treat it as such here, and furthermore, we should recall that a stack of two 11/8 paramajor fourths is equal to a 121/64 major seventh, as per the way paradiatonic intervals and parachromatic intervals relate to diatonic and chromatic intervals. So, if we take things a step further and continue on stacking paramajor fourths, what happens? Well, stacking a third paramajor fourth onto 121/64 and octave-reducing the resulting interval yields 1331/1024, a supermajor third in terms of the classification system we have established earlier, and adding a fourth paramajor fourth yields 14641/8192- a type of augmented sixth. Adding a fifth paramajor fourth to the stack and octave reducing the resulting interval yields the complicated 161051/131072- a type of second that differs from 9/8 by 161051/147456, which, in turn, is larger than the Alpharabian Chromatic Semitone by 1331/1296. Thus, for the purposes of this discussion, we will refer to 161051/131072 as the "Alpharabian Superaugmented Second". Finally, stacking six paramajor fourths and octave-reducing the resulting interval yields 1771561/1048576- a type of double augmented fifth- which differs from 27/16 by only a nexuma.
Judging from this, it seems that there is indeed a clear sequence of intervals along the 11-limit, with every other member in this sequence being the octave complement of a stack of Alpharabian Diatonic Semitones. As the 11-limit handles stacks of Alpharabian Diatonic Semitones in much the same way that the 3-limit handles stacks of Pythagorean diatonic semitones, it can thus be argued that the 11-limit meets the standards set by the 3-limit in this respect. In fact, since the Alpharabian and Betarabian Semitones are actually closer to half of a whole tone that either one of the 3-limit semitones- and especially since the 11-limit's version of a double sharp fifth is only an unnoticeable comma's distance away from the 3-limit's major sixth- it can be argued that the 11-limit passes the semitone test with flying colors. With this information in hand, we can now safely assume that the 11-limit does is fact, carry the function of a navigational axis. Between this, and the previous mathematical confirmation of the 11-limit as being the best p-limit for representing quartertones, we can now safely say that there is indeed sufficient merit to the idea of the 11-limit being considered a navigational prime- in fact, we can now refer to it as the "Paradiatonic 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. With this in mind, it is now time to talk about my choice of which EDO to use as a basis for my microtonal system.
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 of major significance in light of the aforementioned highly important functions of these two prime limits in particular.