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

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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 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".

This still leaves the matters of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32, what happens when we modify Perfect Fourths and Perfect Fifths by 1331/1296, and, what happens when we either lower Major intervals or raise Minor intervals by 1331/1296. However, we can cover these topics in the next section, as we need to delve even deeper into the 11-limit to cover these intervals on account of their complexity. Before we do that, however, we first need to both take stock of the rules for the current naming system, and compile a list of relatively simple 11-limit intervals which are all classified as Alpharabian intervals, as we have now covered most of the basics for 11-limit interval terminology in this system. Do note that when composite interval terms like "Greater Neutral" are qualified by tuning terms like "Alpharabian", at least in English, the tuning term is inserted between the elements of the interval term, thus, for instance, [[88/81]], the Greater Neutral Second in Alpharabian tuning, is labeled as the "Greater Alpharabian Neutral Second".

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~~After doing a little math~~, we can ~~see~~ that the ~~intervals in this table- and all~~ Alpharabian ~~intervals in general- all meet one or more of the following criteria~~:

From this particular sample, we can deduce that there are three fundamental premises of the Alpharabian tuning system:

* ~~They~~ are ~~directly~~ in the 2.11 subgroup

* Intervals that are in the 2.11 subgroup are all considered "Alpharabian".

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

* Intervals that result from the modification of a Pythagorean interval by [[1089/1024]] are labeled similarly to those modified in the equivalent fashion by [[2187/2048]], the only difference being that modification by 1089/1024 results in an Alpharabian interval rather than a Pythagorean interval.

* ~~They are formed through the modification of relatively simple~~ Pythagorean ~~intervals~~ by ~~intervals~~ in ~~the 2~~.~~11 subgroup, with octave equivalency assumed~~

* Since 1089/1024 is (33/32)^2, modifying a Pythagorean interval by [[33/32]] always results in an interval that is considered "Alpharabian".

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~~.

I should also point out that there's also at least one known secondary premise at play:

* As both the [[243/242|rastma]] and [[1331/1296]] are subchromas that form differences between members of the 2.11 subgroup and Pythagorean intervals, both of these subchromas belong to a set of intervals defining different interval sets within Alpharabian tuning, and subchromas within this particular interval set help define the differences between Pythagorean, Alpharabian and Betarabian intervals- the latter of which will be covered in the next section.

The following rules are directly derived from the above premises:

* Generally, intervals that result from the modification of a Pythagorean interval by 33/32 take either the 'parasuper' or 'parasub' prefixes, however, there are a number of special cases...

:* Augmentation of a Perfect Fourth or Perfect Fifth by 33/32 results in a Paramajor interval

:* Dimunition of a Perfect Fourth or Perfect Fifth by 33/32 results in a Paraminor interval

:* Augmentation of a Pythagorean Minor interval by 33/32 results in a Lesser Neutral interval

:* Dimunition of a Pythagorean Major interval by 33/32 results in a Greater Neutral interval

* Generally, intervals that result that result from the modification of a Pythagorean interval by 1331/1296 take either the 'super' or 'sub' prefixes, with these prefixes generally being stacked where multiple such modifications occur, however, there are some significant caveats...

:* Augmentation of a Pythagorean Minor interval by a single 1331/1296 results in a Supraminor interval, but a second such augmentation results in a Betarabian Major interval due to said interval differing from the nearby Alpharabian Major (covered under modifications by 1089/1024) by a rastma.

:* Dimunition of a Pythagorean Major interval by a single 1331/1296 results in a Submajor interval, but a second such dimunition results in a Betarabian Minor interval due to said interval differing from the nearby Alpharabian Minor (covered under modifications by 1089/1024) by a rastma.

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

(Todo: rewrite this section based on further modification of the rules for defining subsets of Alpharabian tuning)

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

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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 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".
+This still leaves the matters of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32, what happens when we modify Perfect Fourths and Perfect Fifths by 1331/1296, and, what happens when we either lower Major intervals or raise Minor intervals by 1331/1296.  However, we can cover these topics in the next section, as we need to delve even deeper into the 11-limit to cover these intervals on account of their complexity.  Before we do that, however, we first need to both take stock of the rules for the current naming system, and compile a list of relatively simple 11-limit intervals which are all classified as Alpharabian intervals, as we have now covered most of the basics for 11-limit interval terminology in this system.  Do note that when composite interval terms like "Greater Neutral" are qualified by tuning terms like "Alpharabian", at least in English, the tuning term is inserted between the elements of the interval term, thus, for instance, [[88/81]], the Greater Neutral Second in Alpharabian tuning, is labeled as the "Greater Alpharabian Neutral Second".
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 |}
-After doing a little math, we can see that the intervals in this table- and all Alpharabian intervals in general- all meet one or more of the following criteria:
+From this particular sample, we can deduce that there are three fundamental premises of the Alpharabian tuning system:
-* They are directly in the 2.11 subgroup
+* Intervals that are in the 2.11 subgroup are all considered "Alpharabian".
-* They are formed through the modification of intervals in the 2.11 subgroup by relatively simple Pythagorean intervals, with octave equivalency assumed
+* Intervals that result from the modification of a Pythagorean interval by [[1089/1024]] are labeled similarly to those modified in the equivalent fashion by [[2187/2048]], the only difference being that modification by 1089/1024 results in an Alpharabian interval rather than a Pythagorean interval.
-* They are formed through the modification of relatively simple Pythagorean intervals by intervals in the 2.11 subgroup, with octave equivalency assumed
+* Since 1089/1024 is (33/32)^2, modifying a Pythagorean interval by [[33/32]] always results in an interval that is considered "Alpharabian".
-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.
+I should also point out that there's also at least one known secondary premise at play:
+* As both the [[243/242|rastma]] and [[1331/1296]] are subchromas that form differences between members of the 2.11 subgroup and Pythagorean intervals, both of these subchromas belong to a set of intervals defining different interval sets within Alpharabian tuning, and subchromas within this particular interval set help define the differences between Pythagorean, Alpharabian and Betarabian intervals- the latter of which will be covered in the next section.
+The following rules are directly derived from the above premises:
+* Generally, intervals that result from the modification of a Pythagorean interval by 33/32 take either the 'parasuper' or 'parasub' prefixes, however, there are a number of special cases...
+:* Augmentation of a Perfect Fourth or Perfect Fifth by 33/32 results in a Paramajor interval
+:* Dimunition of a Perfect Fourth or Perfect Fifth by 33/32 results in a Paraminor interval
+:* Augmentation of a Pythagorean Minor interval by 33/32 results in a Lesser Neutral interval
+:* Dimunition of a Pythagorean Major interval by 33/32 results in a Greater Neutral interval
+* Generally, intervals that result that result from the modification of a Pythagorean interval by 1331/1296 take either the 'super' or 'sub' prefixes, with these prefixes generally being stacked where multiple such modifications occur, however, there are some significant caveats...
+:* Augmentation of a Pythagorean Minor interval by a single 1331/1296 results in a Supraminor interval, but a second such augmentation results in a Betarabian Major interval due to said interval differing from the nearby Alpharabian Major (covered under modifications by 1089/1024) by a rastma.
+:* Dimunition of a Pythagorean Major interval by a single 1331/1296 results in a Submajor interval, but a second such dimunition results in a Betarabian Minor interval due to said interval differing from the nearby Alpharabian Minor (covered under modifications by 1089/1024) by a rastma.
 == Delving into the 11-Limit: Betarabian Intervals and the Nexus Comma ==
+(Todo: rewrite this section based on further modification of the rules for defining subsets of Alpharabian tuning)
 Now that we've covered the basic Alpharabian intervals, it's time to continue our journey into the 11-limit.  However, because the intervals in this section are not covered by the basic classification scheme for Alpharabian intervals, it would be better if we called these intervals by a different name.  While the term "Rastmic" has historically been used as a descriptor for intervals like the 27/22 neutral third, this naming scheme fails to take the importance of the 11-limit into account, and also fails to consider the rastma's additional properties when not tempered out.  Nevertheless, the term "Rastmic" as an interval descriptor retains its usefulness, even when all of the basic Alpharabian intervals are properly accounted for, as while there are infinitely many Alpharabian intervals, there are still many intervals that are ''not'' basic Alpharabian intervals yet only differ from these same intervals by a rastma- or two, or three, and so on.  However, I'm under the impression that we need to save the "Rastmic" interval descriptor for when we move past a second layer of 11-limit intervals, and it is this second layer of 11-limit intervals that we shall cover in this section.