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.
-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.  While in earlier drafts of this page, I said we should use the "Super-" prefix for the augmentation of Major intervals by 33/32, and "Sub-" for the dimunition of Minor intervals by 33/32, I have since realized that such a system fails to account for the fact that 4096/3993, the primary limma-like interval of the 11-limit, differs from the Pythagorean diatonic semitone by 1331/1296- a type of parachromatic interval.  Since 4096/3993 is rightly deemed a type of "Subminor Second" while 33/32 is the difference between the Just Paramajor Fourth and the Just Perfect Fourth, I now propose that we refer to 1331/1296 as the "Alpharabian Superprime", and that we use the "Super-" and "Sub-" prefixes to refer to alterations of all intervals other than Fourths or Fifths by 1331/1296.  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.  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.
+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.  While in earlier drafts of this page, I said we should use the "Super-" prefix for the augmentation of Major intervals by 33/32, and "Sub-" for the dimunition of Minor intervals by 33/32, I have since realized that such a system fails to account for the fact that 4096/3993, the primary limma-like interval of the 11-limit, differs from the Pythagorean diatonic semitone by 1331/1296- a type of parachromatic interval.  Since 4096/3993 is rightly deemed a type of "Subminor Second" while 33/32 is the difference between the Just Paramajor Fourth and the Just Perfect Fourth, I now propose that we refer to 1331/1296 as the "Alpharabian Superprime", and that we use the "Super-" and "Sub-" prefixes to not only refer to the respective augmentation and dimunition of both Unisons and Octaves by 1331/1296, but also to the augmentation of Major intervals and dimunition of Minor intervals respectively by 1331/1296.  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.
-While this still leaves the matter of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32, as well as what happens when we modify Perfect Fourths and Fifths by 1331/1296, we can cover these topics 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".
+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 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".
 {| class="wikitable"
 |-
 !| Interval
+!| Cents
 !| Names
 |-
 | [[1331/1296]]
+|
 | Alpharabian Superprime
 |-
 | [[33/32]]
+|
 | Alpharabian Parasuperprime, Alpharabian Parachromatic Quartertone, al-Farabi Quartertone
 |-
 | [[1089/1024]]
+|
 | Alpharabian Augmented Unison, Alpharabian Chromatic Semitone
 |-
 | [[8192/8019]]
+|
 | Alpharabian Parasubminor Second
 |-
 | [[4096/3993]]
+|
 | Alpharabian Subminor Second, Alpharabian Paradiatonic Quartertone
 |-
 | [[128/121]]
+|
 | Alpharabian Minor Second, Alpharabian Diatonic Semitone
 |-
 | [[12/11]]
+|
 | Lesser Alpharabian Neutral Second
 |-
 | [[88/81]]
+|
 | Greater Alpharabian Neutral Second
 |-
 | [[297/256]]
+|
 | Alpharabian Supermajor Second
 |-
 | [[1024/891]]
+|
 | Alpharabian Subminor Third
 |-
 | [[144/121]]
+|
 | Alpharabian Minor Third
 |-
 | [[11/9]]
+|
 | Lesser Alpharabian Neutral Third
 |-
 | [[27/22]]
+|
 | Greater Alpharabian Neutral Third
 |-
 | [[121/96]]
+|
 | Alpharabian Major Third
 |-
 | [[2673/2048]]
+|
 | Alpharabian Supermajor Third
 |-
 | [[128/99]]
+|
 | Alpharabian Paraminor Fourth
 |-
 | [[11/8]]
+|
 | Alpharabian Paramajor Fourth, Just Paramajor Fourth
 |-
 | [[363/256]]
+|
 | Alpharabian Augmented Fourth
 |-
 | [[512/363]]
+|
 | Alpharabian Diminished Fifth
 |-
 | [[16/11]]
+|
 | Alpharabian Paraminor Fifth, Just Paraminor Fifth
 |-
 | [[99/64]]
+|
 | Alpharabian Paramajor Fifth
 |-
 | [[4096/2673]]
+|
 | Alpharabian Subminor Sixth
 |-
 | [[192/121]]
+|
 | Alpharabian Minor Sixth
 |-
 | [[44/27]]
+|
 | Lesser Alpharabian Neutral Sixth
 |-
 | [[18/11]]
+|
 | Greater Alpharabian Neutral Sixth
 |-
 | [[121/72]]
+|
 | Alpharabian Major Sixth
 |-
 | [[891/512]]
+|
 | Alpharabian Supermajor Sixth
 |-
 | [[512/297]]
+|
 | Alpharabian Subminor Seventh
 |-
 | [[81/44]]
+|
 | Lesser Alpharabian Neutral Seventh
 |-
 | [[11/6]]
+|
 | Greater Alpharabian Neutral Seventh
 |-
 | [[121/64]]
+|
 | Alpharabian Major Seventh
 |-
 | [[3993/2048]]
+|
 | Alpharabian Supermajor Seventh
 |-
 | [[2401/1089]]
+|
 | Alpharabian Diminished Octave
 |-
 | [[64/33]]
+|
 | Alpharabian Suboctave
 |-
 | [[2592/1331]]
+|
 | Alpharabian Parasuboctave
 |-