User:Aura/Aura's Ideas on Tonality: Difference between revisions

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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[[KiteGiedraitis | Kite Giedraitis]] in [[Talk:159edo_notation#My_Second_Idea_for_a_Notation System|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.

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, 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. Nevertheless, my decision to consider the 11-limit a navigational prime in my system- with 33/32 being the primary parachroma for this limit- not only sets my system apart from the Hunt System, but also both the Helmholtz-Ellis Notation System and the Functional Just System.

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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.

== Delving into the 11-Limit: Alpharabian Tuning ==

== Delving into the 11-Limit: Alpharabian Tuning and Additional Interval Classifications ==

With the 11-limit established as a second navigational prime, 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 [[243/242|rastma]], which means that the rastma, when not tempered out, is a very important "parasubchroma"- a term derived from [[Diatonic, Chromatic, Enharmonic, Subchromatic|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, ~~that only partially covers the 11-limit's quartertones, and~~ the "primary" versus "secondary" distinction is temporary at best. 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 [https://en.wikipedia.org/wiki/Al-Farabi 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|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.

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 in which the notes are [[https://en.wikipedia.org/wiki/Maximal_evenness spread out as much as possible]], as per the more strict definition of "Diatonic" listed on [https://en.wikipedia.org/wiki/Diatonic_scale Wikipedias article on the Diatonic Scale]. In light of all this, it should follow that "Paradiatonic" intervals are simply those intervals smaller than a semitone which are as distant from the boundaries of a semitone 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.

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 [https://en.wikipedia.org/wiki/Al-Farabi 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|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 ==

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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 compile a list of all the 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".

{| class="wikitable"

{| class="wikitable center-1 right-2"

|-

!| Interval

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| [[1331/1296]]

| 46.133824

| Alpharabian Superprime

| Alpharabian Superprime, Alpharabian Subchromatic Quartertone

|-

| [[33/32]]

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| [[8192/8019]]

| 36.952052

| Alpharabian Parasubminor Second

| Alpharabian Parasubminor Second, Alpharabian Diesis

|-

| [[4096/3993]]

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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|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".

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".

~~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~~.

While I can't cover all of the Betarabian intervals in this section, I can at least take note of the fact that 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.

== Measuring EDO Approximation Quality ==

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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.

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 [http://musictheory.zentral.zone/huntsystem2.html#2 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.

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 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[[KiteGiedraitis | Kite Giedraitis]] in [[Talk:159edo_notation#My_Second_Idea_for_a_Notation System|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.
+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, 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.  Nevertheless, my decision to consider the 11-limit a navigational prime in my system- with 33/32 being the primary parachroma for this limit- not only sets my system apart from the Hunt System, but also both the Helmholtz-Ellis Notation System and the Functional Just System.
@@ Line 35: / Line 35: @@
 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.
-== Delving into the 11-Limit: Alpharabian Tuning ==
+== Delving into the 11-Limit: Alpharabian Tuning and Additional Interval Classifications ==
 With the 11-limit established as a second navigational prime, 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 [[243/242|rastma]], which means that the rastma, when not tempered out, is a very important "parasubchroma"- a term derived from [[Diatonic, Chromatic, Enharmonic, Subchromatic|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, that only partially covers the 11-limit's quartertones, and the "primary" versus "secondary" distinction is temporary at best.  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 [https://en.wikipedia.org/wiki/Al-Farabi 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|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.
+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 in which the notes are [[https://en.wikipedia.org/wiki/Maximal_evenness spread out as much as possible]], as per the more strict definition of "Diatonic" listed on [https://en.wikipedia.org/wiki/Diatonic_scale Wikipedias article on the Diatonic Scale].   In light of all this, it should follow that "Paradiatonic" intervals are simply those intervals smaller than a semitone which are as distant from the boundaries of a semitone 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.
+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 [https://en.wikipedia.org/wiki/Al-Farabi 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|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 ==
@@ Line 53: / Line 55: @@
 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 compile a list of all the 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".
-{| class="wikitable"
+{| class="wikitable center-1 right-2"
 |-
 !| Interval
@@ Line 61: / Line 63: @@
 | [[1331/1296]]
 | 46.133824
-| Alpharabian Superprime
+| Alpharabian Superprime, Alpharabian Subchromatic Quartertone
 |-
 | [[33/32]]
@@ Line 73: / Line 75: @@
 | [[8192/8019]]
 | 36.952052
-| Alpharabian Parasubminor Second
+| Alpharabian Parasubminor Second, Alpharabian Diesis
 |-
 | [[4096/3993]]
@@ Line 235: / Line 237: @@
 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|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".
+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".
-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.
+While I can't cover all of the Betarabian intervals in this section, I can at least take note of the fact that 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.
 == Measuring EDO Approximation Quality ==
@@ Line 245: / Line 249: @@
 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.
+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 [http://musictheory.zentral.zone/huntsystem2.html#2 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.