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

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.  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.

== 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.