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

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* They are formed through the modification of intervals in the 2.11 subgroup by relatively simple Pythagorean intervals, with octave equivalency assumed

* They are formed through the modification of relatively simple Pythagorean intervals by intervals in the 2.11 subgroup, with octave equivalency assumed

On one hand this seems to account for just about every interval in the 2.3.11 subgroup, so in a sense, every 2.3.11 interval is an Alpharabian interval. However it is quite evident that intervals like 729/704 are not on the list, so there must be additional criteria as to what counts as an Alpharabian interval and what doesn't.

== Delving into the 11-Limit: Betarabian Intervals and the Nexus Comma ==

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 that we've covered the basic Alpharabian intervals, it's time to continue our journey into the 11-limit. However, because the intervals in this section are not covered by the basic classification scheme for Alpharabian intervals, it would be better if we called these intervals by a different name. While the term "Rastmic" has historically been used as a descriptor for intervals like the 27/22 neutral third, this naming scheme fails to take the importance of the 11-limit into account, and also fails to consider the rastma's additional properties when not tempered out. Nevertheless, the term "Rastmic" as an interval descriptor retains its usefulness, even when all of the basic Alpharabian intervals are properly accounted for, as while there are infinitely many Alpharabian intervals, there are still many intervals that are ''not'' basic Alpharabian intervals yet only differ from these same intervals by a rastma- or two, or three, and so on. However, I'm under the impression that we need to save the "Rastmic" interval descriptor for when we move past a second layer of 11-limit intervals, and it is this second layer of 11-limit intervals that we shall cover in this section.

Now, if one does the math, they will realize that an Alpharabian Parasupermajor Second, having a ratio of 297/256, is larger than an Alpharabian Parasubminor Third with its ratio of 1024/891, and that the difference between these two intervals is 264627/262144. Despite the fact that 264627/262144 is the sum of the rastma and the Alpharabian comma, its function can be contrasted with that of the Alpharabian comma in that 264627/262144 not only separates 297/256 and 1024/891, but also other similar enharmonic quartertone-based interval pairs, whereas the Alpharabian comma merely distinguishes enharmonic 11-limit semitones. Yet, the term "Alpharabian" contains the word "Alpha", which can be taken as signifying the Alpharabian comma's status a primary 11-limit comma. Therefore, if we take the "Alpha" off of "Alpharabian" and put the term "Beta" in its place, we can thus call 264627/262144 the "[[Betarabian comma|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".

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 * They are formed through the modification of intervals in the 2.11 subgroup by relatively simple Pythagorean intervals, with octave equivalency assumed
 * They are formed through the modification of relatively simple Pythagorean intervals by intervals in the 2.11 subgroup, with octave equivalency assumed
+On one hand this seems to account for just about every interval in the 2.3.11 subgroup, so in a sense, every 2.3.11 interval is an Alpharabian interval.  However it is quite evident that intervals like 729/704 are not on the list,  so there must be additional criteria as to what counts as an Alpharabian interval and what doesn't.
 == Delving into the 11-Limit: Betarabian Intervals and the Nexus Comma ==
-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 that we've covered the basic Alpharabian intervals, it's time to continue our journey into the 11-limit.  However, because the intervals in this section are not covered by the basic classification scheme for Alpharabian intervals, it would be better if we called these intervals by a different name.  While the term "Rastmic" has historically been used as a descriptor for intervals like the 27/22 neutral third, this naming scheme fails to take the importance of the 11-limit into account, and also fails to consider the rastma's additional properties when not tempered out.  Nevertheless, the term "Rastmic" as an interval descriptor retains its usefulness, even when all of the basic Alpharabian intervals are properly accounted for, as while there are infinitely many Alpharabian intervals, there are still many intervals that are ''not'' basic Alpharabian intervals yet only differ from these same intervals by a rastma- or two, or three, and so on.  However, I'm under the impression that we need to save the "Rastmic" interval descriptor for when we move past a second layer of 11-limit intervals, and it is this second layer of 11-limit intervals that we shall cover in this section.
 Now, if one does the math, they will realize that an Alpharabian Parasupermajor Second, having a ratio of 297/256, is larger than an Alpharabian Parasubminor Third with its ratio of 1024/891, and that the difference between these two intervals is 264627/262144.  Despite the fact that 264627/262144 is the sum of the rastma and the Alpharabian comma, its function can be contrasted with that of the Alpharabian comma in that 264627/262144 not only separates 297/256 and 1024/891, but also other similar enharmonic quartertone-based interval pairs, whereas the Alpharabian comma merely distinguishes enharmonic 11-limit semitones.  Yet, the term "Alpharabian" contains the word "Alpha", which can be taken as signifying the Alpharabian comma's status a primary 11-limit comma.  Therefore, if we take the "Alpha" off of "Alpharabian" and put the term "Beta" in its place, we can thus call 264627/262144 the "[[Betarabian comma|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".