User:Aura/Aura's Ideas on Tonality: Difference between revisions
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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. | 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. | ||
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 "Sub-" for the dimunition of Minor intervals by 33/32. As is to be expected, Supermajor and Subminor intervals are octave compliments of one another; for example, when 121/64 is raised by 33/32, the result 3993/2048- the octave complement of 4096/3993, which is the primary | 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 "Sub-" for the dimunition of Minor intervals by 33/32. As is to be expected, Supermajor and Subminor intervals are octave compliments of one another; for example, when 121/64 is raised by 33/32, the result 3993/2048- the octave complement of 4096/3993, which is the primary limma-like interval of the 11-limit. 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. 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. | ||
Upon doing the math, one realizes that when you take a stack of three 11/8 Just Paramajor Fourths, and octave-reduce it, the resulting interval is [[1331/1024]], meanwhile, when you raise [[81/64]] by a 33/32 quartertone, you get [[2673/2048]] which differs from [[1331/1024]] by a rastma. While it can easily be established that 2673/2048 is an Alpharabian interval, the fact is that 1331/1024 is a pure 11-limit interval as it's prime factorization is 11*3/2*10, therefore, 1331/1024 is also an Alpharabian interval. So, what term shall we use to classify [[1331/1024]]? While I have [[User talk:Aura #Getting Started|previously]] advocated for the use of the "Parasuper-" and "Parasub-" prefixes to refer to the augmentation of Major intervals and dimunition of Minor intervals respectively by 33/32, I now realize that this type of labeling scheme is untenable in light of the the 11-limit's status as a navigational prime, and the 33/32 quartertone's status as the primary apotome-like interval of the of the 11-limit. Nevertheless, the "Parasuper-" and "Parasub-" prefixes are still useful coinages, are still useful coinages, so I think I'll bring back the terms and repurpose them, with the term "Parasupermajor" being used to describe the type of third that 1331/1024 is, and "Parasubminor" being used to describe the nature of its octave complement. This arrangement is fitting due to the simple connection between 1331/1024 and 3993/2048- the latter being the octave complement the primary paradiatonic quartertone. With this new layer of interval names in mind, we can proceed to subtract 81/64 from 1331/1024 to create [[1331/1296]], the Alpharabian Parasuperprime, which is smaller than 33/32 by the rastma. From there, we can then go on to derive a set of Parasupermajor and Parasubminor intervals, and even Greater Paraneutral and Lesser Paraneutral intervals, by modifying Pythagorean Major and Minor intervals by 1331/1296. We can also go on to derive such intervals as "Parasuperprimes" and "Parasuboctaves" by modifying the simple Unison and Octave by 1331/1296, though it should be noted that modifying Fourths and Fifths by 1331/1296 is a different story. | |||
While this still leaves the matter of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32, we can cover that in the next section, as we need to delve even deeper into the 11-limit to cover these intervals. 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". | While this still leaves the matter of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32, we can cover that in the next section, as we need to delve even deeper into the 11-limit to cover these intervals. 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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| [[11/8]] | | [[11/8]] | ||
| Alpharabian Paramajor Fourth | | Alpharabian Paramajor Fourth, Just Paramajor Fourth | ||
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| [[363/256]] | | [[363/256]] | ||
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| [[16/11]] | | [[16/11]] | ||
| Alpharabian Paraminor Fifth | | Alpharabian Paraminor Fifth, Just Paraminor Fifth | ||
|- | |- | ||
| [[99/64]] | | [[99/64]] | ||