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

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When you lower a Perfect Fifth by 1331/1296, you get 1944/1331, which, like 352/243, differs from 16/11 by a rastma- albeit in the opposite direction- and there are two reasons that 1944/1331 can't be considered an Alpharabian interval despite its relative simplicity. The first reason is because there is only room for one Alpharabian Paraminor Fifth, and the most basic Paraminor Fifth is 16/11. The second reason is that since Paramajor and Paraminor intervals are basic interval categories for the 11-limit the way that Major and Minor are for the 3-limit, you can't exactly get away with calling 1944/1331 a "Subfifth" any more than you can get away with calling 16/11 a "Parasubfifth"- at least not in this system. The same reasoning applies when lowering a Perfect Fourth or raising a Perfect Fifth by 1331/1296. Therefore, when you modify a Perfect Fourth or Perfect Fifth by 1331/1296, the result must be a Betarabian interval that can be classified as either "Paramajor" or "Paraminor". When you lower a Major interval or raise a Minor interval by 1331/1296, you end up with similar issues, as Neutral intervals are basic interval categories for the 11-limit, and furthermore, while one may consider using terms like "Submajor" or "Supraminor" to describe these, at the end of the day, they will not be seen as distinct from Neutral intervals as quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches. At this point, someone might use this same argument to object to my distinction between the "Parasuper-" and "Parasub-" prefixes on one hand and "Super-" and "Sub-" prefixes on the other hand. However, I would say that while on one level, the would be right, the fact remains that on another level, the different types of quartertones add up differently, and those differences need to be respected when dealing with the 11-limit. Therefore, I would- for example- label 2662/2187 the "Lesser Betarabian Neutral Third", and label 6561/5324 the "Greater Betarabian Neutral Third".

While I can't cover all of the Betarabian intervals in this section as there are too many to cover, and the same is true of 11-limit intervals in general, I can perhaps by take note of a few more important 11-limit intervals. Firstly, there are a set of Betarabian Semitones- the "Betarabian Chromatic Semitone", 14641/13824, and the "Betarabian Diatonic Semitone", 15552/14641. The Betarabian Chromatic Semitone is smaller than the Alpharabian Chromatic Semitone by a rastma, while the Betarabian Diatonic Semitone is larger than the Alpharabian Diatonic Semitone by a rastma. When you subtract 33/32 from the Betarabian Chromatic Semitone, you get 1331/1296, however, when you subtract the 729/704 from the Betarabian Chromatic Semitone, you get the very complicated 161051/157464, the Betarabian Subchroma. On the other hand, when you subtract 33/32 from from the Betarabian Diatonic Semitone, you get the very complicated 165888/161051, the Betarabian Paradiatonic Quartertone, which differs from the Alpharabian Paradiatonic Quartertone by a rastma. Finally, I propose we bring our journey through the 11-limit to a fitting conclusion with a look at one final 11-limit comma. This comma has a ratio of 1771561/1769472, and forms the difference between the Rastma and the Alpharabian comma, furthermore, it also forms the difference between the Alpharabian Subminor Third and the larger Alpharabian Supermajor second. Furthermore, it also forms the difference between the the Alpharabian Parachromatic Quartertone and the Betarabian Paradiatonic Quartertone, as well as difference between innumerable pairs of other 11-limit intervals. However, what's most notable about this comma is that it is the amount by which a stack of three 128/121 Alpharabian ~~diatonic semitones~~ falls short of a 32/27 minor third. Considering that 128/121 is a pure 11-limit interval, while 32/27 is a pure 3-limit interval, this means that 1771561/1769472 is a very very important interval- especially in light of the fact that it is only slightly more than two cents in size and is thus not only an unnoticeable comma, but a prime target for tempering. Since tempering out 1771561/1769472 results in the formation of a nexus between the 3-limit and the 11-limit, the latter of which has so far already been established as perhaps the best p-limit for representing quartertones, 1771561/1769472 can thus be called the "[[Nexuma]]", or the "Nexus comma".

While I can't cover all of the Betarabian intervals in this section as there are too many to cover, and the same is true of 11-limit intervals in general, I can perhaps by take note of a few more important 11-limit intervals. Firstly, there are a set of Betarabian Semitones- the "Betarabian Chromatic Semitone", 14641/13824, and the "Betarabian Diatonic Semitone", 15552/14641. The Betarabian Chromatic Semitone is smaller than the Alpharabian Chromatic Semitone by a rastma, while the Betarabian Diatonic Semitone is larger than the Alpharabian Diatonic Semitone by a rastma. When you subtract 33/32 from the Betarabian Chromatic Semitone, you get 1331/1296, however, when you subtract the 729/704 from the Betarabian Chromatic Semitone, you get the very complicated 161051/157464, the Betarabian Subchroma. On the other hand, when you subtract 33/32 from from the Betarabian Diatonic Semitone, you get the very complicated 165888/161051, the Betarabian Paradiatonic Quartertone, which differs from the Alpharabian Paradiatonic Quartertone by a rastma. Finally, I propose we bring our journey through the 11-limit to a fitting conclusion with a look at one final 11-limit comma. This comma has a ratio of 1771561/1769472, and forms the difference between the Rastma and the Alpharabian comma, furthermore, it also forms the difference between the Alpharabian Subminor Third and the larger Alpharabian Supermajor second. Furthermore, it also forms the difference between the the Alpharabian Parachromatic Quartertone and the Betarabian Paradiatonic Quartertone, as well as difference between innumerable pairs of other 11-limit intervals. However, what's most notable about this comma is that it is the amount by which a stack of three 128/121 Alpharabian Diatonic Semitones falls short of a 32/27 minor third. Considering that 128/121 is a pure 11-limit interval, while 32/27 is a pure 3-limit interval, this means that 1771561/1769472 is a very very important interval- especially in light of the fact that it is only slightly more than two cents in size and is thus not only an unnoticeable comma, but a prime target for tempering. Since tempering out 1771561/1769472 results in the formation of a nexus between the 3-limit and the 11-limit, the latter of which has so far already been established as perhaps the best p-limit for representing quartertones, 1771561/1769472 can thus be called the "[[Nexuma]]", or the "Nexus comma".

== 11-limit Axis Functionality ==

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 When you lower a Perfect Fifth by 1331/1296, you get 1944/1331, which, like 352/243, differs from 16/11 by a rastma- albeit in the opposite direction- and there are two reasons that 1944/1331 can't be considered an Alpharabian interval despite its relative simplicity.  The first reason is because there is only room for one Alpharabian Paraminor Fifth, and the most basic Paraminor Fifth is 16/11.  The second reason is that since Paramajor and Paraminor intervals are basic interval categories for the 11-limit the way that Major and Minor are for the 3-limit, you can't exactly get away with calling 1944/1331 a "Subfifth" any more than you can get away with calling 16/11 a "Parasubfifth"- at least not in this system.  The same reasoning applies when lowering a Perfect Fourth or raising a Perfect Fifth by 1331/1296.  Therefore, when you modify a Perfect Fourth or Perfect Fifth by 1331/1296, the result must be a Betarabian interval that can be classified as either "Paramajor" or "Paraminor".  When you lower a Major interval or raise a Minor interval by 1331/1296, you end up with similar issues, as Neutral intervals are basic interval categories for the 11-limit, and furthermore, while one may consider using terms like "Submajor" or "Supraminor" to describe these, at the end of the day, they will not be seen as distinct from Neutral intervals as quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches.  At this point, someone might use this same argument to object to my distinction between the "Parasuper-" and "Parasub-" prefixes on one hand and "Super-" and "Sub-" prefixes on the other hand.  However, I would say that while on one level, the would be right, the fact remains that on another level, the different types of quartertones add up differently, and those differences need to be respected when dealing with the 11-limit.  Therefore, I would- for example- label 2662/2187 the "Lesser Betarabian Neutral Third", and label 6561/5324 the "Greater Betarabian Neutral Third".
-While I can't cover all of the Betarabian intervals in this section as there are too many to cover, and the same is true of 11-limit intervals in general, I can perhaps by take note of a few more important 11-limit intervals.  Firstly, there are a set of Betarabian Semitones- the "Betarabian Chromatic Semitone", 14641/13824, and the "Betarabian Diatonic Semitone", 15552/14641.  The Betarabian Chromatic Semitone is smaller than the Alpharabian Chromatic Semitone by a rastma, while the Betarabian Diatonic Semitone is larger than the Alpharabian Diatonic Semitone by a rastma.  When you subtract 33/32 from the Betarabian Chromatic Semitone, you get 1331/1296, however, when you subtract the 729/704 from the Betarabian Chromatic Semitone, you get the very complicated 161051/157464, the Betarabian Subchroma.  On the other hand, when you subtract 33/32 from from the Betarabian Diatonic Semitone, you get the very complicated 165888/161051, the Betarabian Paradiatonic Quartertone, which differs from the Alpharabian Paradiatonic Quartertone by a rastma.  Finally, I propose we bring our journey through the 11-limit to a fitting conclusion with a look at one final 11-limit comma.  This comma has a ratio of 1771561/1769472, and forms the difference between the Rastma and the Alpharabian comma, furthermore, it also forms the difference between the Alpharabian Subminor Third and the larger Alpharabian Supermajor second.  Furthermore, it also forms the difference between the the Alpharabian Parachromatic Quartertone and the Betarabian Paradiatonic Quartertone, as well as difference between innumerable pairs of other 11-limit intervals.  However, what's most notable about this comma is that it is the amount by which a stack of three 128/121 Alpharabian diatonic semitones falls short of a 32/27 minor third.  Considering that 128/121 is a pure 11-limit interval, while 32/27 is a pure 3-limit interval, this means that 1771561/1769472 is a very very important interval- especially in light of the fact that it is only slightly more than two cents in size and is thus not only an unnoticeable comma, but a prime target for tempering.  Since tempering out 1771561/1769472 results in the formation of a nexus between the 3-limit and the 11-limit, the latter of which has so far already been established as perhaps the best p-limit for representing quartertones, 1771561/1769472 can thus be called the "[[Nexuma]]", or the "Nexus comma".
+While I can't cover all of the Betarabian intervals in this section as there are too many to cover, and the same is true of 11-limit intervals in general, I can perhaps by take note of a few more important 11-limit intervals.  Firstly, there are a set of Betarabian Semitones- the "Betarabian Chromatic Semitone", 14641/13824, and the "Betarabian Diatonic Semitone", 15552/14641.  The Betarabian Chromatic Semitone is smaller than the Alpharabian Chromatic Semitone by a rastma, while the Betarabian Diatonic Semitone is larger than the Alpharabian Diatonic Semitone by a rastma.  When you subtract 33/32 from the Betarabian Chromatic Semitone, you get 1331/1296, however, when you subtract the 729/704 from the Betarabian Chromatic Semitone, you get the very complicated 161051/157464, the Betarabian Subchroma.  On the other hand, when you subtract 33/32 from from the Betarabian Diatonic Semitone, you get the very complicated 165888/161051, the Betarabian Paradiatonic Quartertone, which differs from the Alpharabian Paradiatonic Quartertone by a rastma.  Finally, I propose we bring our journey through the 11-limit to a fitting conclusion with a look at one final 11-limit comma.  This comma has a ratio of 1771561/1769472, and forms the difference between the Rastma and the Alpharabian comma, furthermore, it also forms the difference between the Alpharabian Subminor Third and the larger Alpharabian Supermajor second.  Furthermore, it also forms the difference between the the Alpharabian Parachromatic Quartertone and the Betarabian Paradiatonic Quartertone, as well as difference between innumerable pairs of other 11-limit intervals.  However, what's most notable about this comma is that it is the amount by which a stack of three 128/121 Alpharabian Diatonic Semitones falls short of a 32/27 minor third.  Considering that 128/121 is a pure 11-limit interval, while 32/27 is a pure 3-limit interval, this means that 1771561/1769472 is a very very important interval- especially in light of the fact that it is only slightly more than two cents in size and is thus not only an unnoticeable comma, but a prime target for tempering.  Since tempering out 1771561/1769472 results in the formation of a nexus between the 3-limit and the 11-limit, the latter of which has so far already been established as perhaps the best p-limit for representing quartertones, 1771561/1769472 can thus be called the "[[Nexuma]]", or the "Nexus comma".
 == 11-limit Axis Functionality ==