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 ~~primes, albeit~~ superprimes. However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone. Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, we can deduce that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone. Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, we have to choose the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone. From here, we have to select simple parachromatic quartertones from the lowest p-limit that, when subtracted from 9/8, yield the paradiatonic interval with the lowest odd limit.

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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== 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". 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.

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.

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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 [[User talk:Aura #Getting Started|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 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".

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"

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== Delving into the 11-Limit: Betarabian Intervals ==

Now that we've covered the basic Alpharabian intervals, it's time to continue our journey into the 11-limit~~, and cover the matter of what happens 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 "Superaugmented", and when we subject a 3-limit Diminished interval to dimunition by 33/32, we can refer to the resulting interval as being "Subdiminished". 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. Likewise, we can subject a 3-limit Diminished interval to augmentation by 33/32 without arriving in the same location as an Alpharabian interval. However, because these alterations by a rastma result in ~~intervals that~~ 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'' Alpharabian yet only differ from the Alphrarabian 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 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'' Alpharabian yet only differ from the Alphrarabian 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|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.

== Measuring EDO Approximation Quality ==

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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 primes, albeit superprimes.  However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone.  Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, we can deduce that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone.  Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, we have to choose the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone.  From here, we have to select simple parachromatic quartertones from the lowest p-limit that, when subtracted from 9/8, yield the paradiatonic interval with the lowest odd limit.
+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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 == 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".  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.
+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.
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 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 [[User talk:Aura #Getting Started|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 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".
+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".
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 == Delving into the 11-Limit: Betarabian Intervals ==
-Now that we've covered the basic Alpharabian intervals, it's time to continue our journey into the 11-limit, and cover the matter of what happens 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 "Superaugmented", and when we subject a 3-limit Diminished interval to dimunition by 33/32, we can refer to the resulting interval as being "Subdiminished".  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.  Likewise, we can subject a 3-limit Diminished interval to augmentation by 33/32 without arriving in the same location as an Alpharabian interval.  However, because these alterations by a rastma result in intervals that 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'' Alpharabian yet only differ from the Alphrarabian 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 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'' Alpharabian yet only differ from the Alphrarabian 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|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.
 == Measuring EDO Approximation Quality ==