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

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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 ==

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

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, we'll start referring to 33/32 as the "primary parachromatic quartertone", and ~~from here on~~, 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 ~~here~~ 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".

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, we'll start referring to 33/32 as the "primary parachromatic quartertone", and for the moment, 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 just 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 ~~don't have the benefit of~~ the "primary" versus "secondary" distinction here due to the apotome being a 3-limit interval, we should instead look to another source for ~~a name~~. Since 33/32 is also called the "Al-Farabi Quartertone" and is the primary ~~limma~~ 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, that just covers the 11-limit's quartertones, and even then, 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.

== ~~Other~~ 11-Limit ~~Intervals~~ ==

== Basic 11-Limit Interval Classifications ==

With all of the aforementioned stuff about, 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 "parachromatic superfourth". In actuality, however, while one would be correct in asserting 11/8 is 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 ~~the primary parachromatic quartertone~~ 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". 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 "Just Paramajor Fourth". Furthermore, in the same way Major and Minor intervals are octave complements of each other, we can say that Paramajor and Paraminor intervals are octave complements of one another, so therefore, we can say that [[16/11]] is the "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.

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 octave complements of each other, we can say that Paramajor and Paraminor intervals are octave complements of one another, so therefore, we can say that [[16/11]] is the "Alpharabian Paraminor Fifth" or "Just Paraminor Fifth". This arrangement seems works out very well, as 11/8 and 16/11 are basic intervals in their own right just as 3/2 and 4/3 are, with the name "Just Paramajor Fourth" for 11/8 reflecting how 11/8 is higher than the Just Perfect Fourth by a primary parachromatic quartertone, and the name "Just Paraminor Fifth" reflecting how 16/11 is lower than the Just Perfect Fifth by the same interval. However, it should be remembered that the Paramajor-Paraminor distinction can ultimately be thought of as referring to two different sizes of Fourth and two different sizes of Fifth in the same way that the Major-Minor distinction can be thought of as describing two different sizes of second, as well as two different sizes of third, two different sizes of sixth, and two different sizes of seventh- therefore the Paraminor Fourth and the Paramajor Fifth also exist, with these intervals being [[128/99]] and [[99/64]] respectively. However, we still have yet to cover the terminology for alterations of Major and Minor intervals by 33/32, and the introduction of "Paramajor" and "Paraminor" intervals leaves the question as to what terminology to use on this front for alterations of the Perfect Prime and the Octave.

In answering these questions one should note that the Prime and the Octave are the fundamental intervals in both my system and conventional music systems. Furthermore, it doesn't make sense to have dedicated names for intervals that go in the opposite direction from of a given tonality's direction of construction, such as "Paraminor Unison", and, since a term like "Paramajor Unison" would imply the existence of the nonsensical "Paraminor Unison" by definition, we can discard the idea of a "Paramajor Unison" also. Therefore, we can reuse the term "Superprime", as well as the terms "Suboctave", and "Superoctave", as the "Super-" and "Sub-" prefixes imply widening and narrowing respectively. For the same reason, we should additionally use the "Super-" prefix for the augmentation of Major intervals by 33/32, and the dimunition of Minor intervals by 33/32. While the "Super-" and "Sub-" prefixes are often associated with the 7-limit, it should be remembered that in this system, the 7-limit versions of these intervals are variations on the standard intervals as opposed to being the standard intervals themselves. Furthermore, while I have [[User talk:Aura #Getting Started|previously]] advocated for the use of "Parasuper-" and "Parasub-" to refer to these 11-limit intervals, I now realize that such a distinction is largely untenable in light of the the 11-limit's status as a navigational prime, and the higher priority it thus carries over the 7-limit. 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 the rastma, I find it prudent to 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.

In answering these questions one should note that the Prime and the Octave are the fundamental intervals in both my system and conventional music systems. Furthermore, it doesn't make sense to have dedicated names for intervals that go in the opposite direction from of a given tonality's direction of construction, such as "Paraminor Unison", and, since a term like "Paramajor Unison" would imply the existence of the nonsensical "Paraminor Unison" by definition, we can discard the idea of a "Paramajor Unison" also. Therefore, we can reuse the term "Superprime", as well as the terms "Suboctave", and "Superoctave", as the "Super-" and "Sub-" prefixes imply widening and narrowing respectively. For the same reason, we should additionally use the "Super-" prefix for the augmentation of Major intervals by 33/32, and the dimunition of Minor intervals by 33/32. While the "Super-" and "Sub-" prefixes are often associated with the 7-limit, it should be remembered that in this system, the 7-limit versions of these intervals are variations on the standard intervals as opposed to being the standard intervals themselves. Furthermore, while I have [[User talk:Aura #Getting Started|previously]] advocated for the use of "Parasuper-" and "Parasub-" to refer to these 11-limit intervals, I now realize that such a distinction is largely untenable in light of the the 11-limit's status as a navigational prime, and the higher priority it thus carries over the 7-limit. 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 the rastma, I find it prudent to 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. However, we're not done just yet, for there's still the matter of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32. When we subject a 3-limit Augmented intervals 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 Paramajor interval. Likewise, we can subject a 3-limit Diminished interval to augmentation by 33/32 without arriving in the same location as an Alpharabian Paraminor interval. Therefore, we need to delve even deeper into the 11-limit.

Now, if one does the math, they will realize that an Alpharabian Supermajor Second, having a ratio of [[297/256]], is larger than an Alpharabian Subminor 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]]".

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

Now, if one does the math, they will realize that an Alpharabian Supermajor Second, having a ratio of [[297/256]], is larger than an Alpharabian Subminor 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]]". Furthermore, one will realize that just as the octave complement of a diatonic semitone is a major seventh, 3993/2048- the octave complement of a paradiatonic quartertone is a form of susupermajor seventh, with the same evaluation holding true when viewed in light of the fact that the difference between 121/64 and 3993/2048 is 33/32.

== Measuring EDO Approximation Quality ==

@@ 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 ==
+== Delving into the 11-Limit: Alpharabian Tuning ==
-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, we'll start referring to 33/32 as the "primary parachromatic quartertone", and from here on, 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 here 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".
+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, we'll start referring to 33/32 as the "primary parachromatic quartertone", and for the moment, 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 just 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 don't have the benefit of the "primary" versus "secondary" distinction here due to the apotome being a 3-limit interval, we should instead look to another source for a name.  Since 33/32 is also called the "Al-Farabi Quartertone" and is the primary limma 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, that just covers the 11-limit's quartertones, and even then, 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.
-== Other 11-Limit Intervals ==
+== Basic 11-Limit Interval Classifications ==
-With all of the aforementioned stuff about, 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 "parachromatic superfourth".  In actuality, however, while one would be correct in asserting 11/8 is 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 the primary parachromatic quartertone 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".  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 "Just Paramajor Fourth".  Furthermore, in the same way Major and Minor intervals are octave complements of each other, we can say that Paramajor and Paraminor intervals are octave complements of one another,  so therefore, we can say that [[16/11]] is the "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.
+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 octave complements of each other, we can say that Paramajor and Paraminor intervals are octave complements of one another,  so therefore, we can say that [[16/11]] is the "Alpharabian Paraminor Fifth" or "Just Paraminor Fifth".  This arrangement seems works out very well, as 11/8 and 16/11 are basic intervals in their own right just as 3/2 and 4/3 are, with the name "Just Paramajor Fourth" for 11/8 reflecting how 11/8 is higher than the Just Perfect Fourth by a primary parachromatic quartertone, and the name "Just Paraminor Fifth" reflecting how 16/11 is lower than the Just Perfect Fifth by the same interval.  However, it should be remembered that the Paramajor-Paraminor distinction can ultimately be thought of as referring to two different sizes of Fourth and two different sizes of Fifth in the same way that the Major-Minor distinction can be thought of as describing two different sizes of second, as well as two different sizes of third, two different sizes of sixth, and two different sizes of seventh- therefore the Paraminor Fourth and the Paramajor Fifth also exist, with these intervals being [[128/99]] and [[99/64]] respectively.  However, we still have yet to cover the terminology for alterations of Major and Minor intervals by 33/32, and the introduction of "Paramajor" and "Paraminor" intervals leaves the question as to what terminology to use on this front for alterations of the Perfect Prime and the Octave.
-In answering these questions one should note that the Prime and the Octave are the fundamental intervals in both my system and conventional music systems.  Furthermore, it doesn't make sense to have dedicated names for intervals that go in the opposite direction from of a given tonality's direction of construction, such as "Paraminor Unison", and, since a term like "Paramajor Unison" would imply the existence of the nonsensical "Paraminor Unison" by definition, we can discard the idea of a "Paramajor Unison" also.  Therefore, we can reuse the term "Superprime", as well as the terms "Suboctave", and "Superoctave", as the "Super-" and "Sub-" prefixes imply widening and narrowing respectively.  For the same reason, we should additionally use the "Super-" prefix for the augmentation of Major intervals by 33/32, and the dimunition of Minor intervals by 33/32.  While the "Super-" and "Sub-" prefixes are often associated with the 7-limit, it should be remembered that in this system, the 7-limit versions of these intervals are variations on the standard intervals as opposed to being the standard intervals themselves.  Furthermore, while I have [[User talk:Aura #Getting Started|previously]] advocated for the use of "Parasuper-" and "Parasub-" to refer to these 11-limit intervals, I now realize that such a distinction is largely untenable in light of the the 11-limit's status as a navigational prime, and the higher priority it thus carries over the 7-limit.  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 the rastma, I find it prudent to 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.
+In answering these questions one should note that the Prime and the Octave are the fundamental intervals in both my system and conventional music systems.  Furthermore, it doesn't make sense to have dedicated names for intervals that go in the opposite direction from of a given tonality's direction of construction, such as "Paraminor Unison", and, since a term like "Paramajor Unison" would imply the existence of the nonsensical "Paraminor Unison" by definition, we can discard the idea of a "Paramajor Unison" also.  Therefore, we can reuse the term "Superprime", as well as the terms "Suboctave", and "Superoctave", as the "Super-" and "Sub-" prefixes imply widening and narrowing respectively.  For the same reason, we should additionally use the "Super-" prefix for the augmentation of Major intervals by 33/32, and the dimunition of Minor intervals by 33/32.  While the "Super-" and "Sub-" prefixes are often associated with the 7-limit, it should be remembered that in this system, the 7-limit versions of these intervals are variations on the standard intervals as opposed to being the standard intervals themselves.  Furthermore, while I have [[User talk:Aura #Getting Started|previously]] advocated for the use of "Parasuper-" and "Parasub-" to refer to these 11-limit intervals, I now realize that such a distinction is largely untenable in light of the the 11-limit's status as a navigational prime, and the higher priority it thus carries over the 7-limit.  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 the rastma, I find it prudent to 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.  However, we're not done just yet, for there's still the matter of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32.  When we subject a 3-limit Augmented intervals 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 Paramajor interval.  Likewise, we can subject a 3-limit Diminished interval to augmentation by 33/32 without arriving in the same location as an Alpharabian Paraminor interval.  Therefore, we need to delve even deeper into the 11-limit.
-Now, if one does the math, they will realize that an Alpharabian Supermajor Second, having a ratio of [[297/256]], is larger than an Alpharabian Subminor 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]]".
+== Delving into the 11-Limit: Betarabian Intervals ==
+Now, if one does the math, they will realize that an Alpharabian Supermajor Second, having a ratio of [[297/256]], is larger than an Alpharabian Subminor 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]]".  Furthermore, one will realize that just as the octave complement of a diatonic semitone is a major seventh, 3993/2048- the octave complement of a paradiatonic quartertone is a form of susupermajor seventh, with the same evaluation holding true when viewed in light of the fact that the difference between 121/64 and 3993/2048 is 33/32.
 == Measuring EDO Approximation Quality ==