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

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== Delving into the 11-Limit: Betarabian Intervals ==

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

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'' Alpharabian yet only differ from the Alphrarabian 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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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, I can at least take note of the fact that 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.

While I can't cover all of the Betarabian intervals in this section as there are too many to cover, I can at least take note of the fact that 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 look at one 11-limit comma in particular- the [[Nexuma]], or the Nexus comma. The Nexuma 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 it 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, and that both the 3-limit and the 11-limit are navigational primes, this means that the Nexuma is a very very important interval- especially in light of the fact that the Nexuma is only slightly more than two cents in size and is thus not only an unnoticeable comma, but a prime target for tempering.

== Measuring EDO Approximation Quality ==

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-== Delving into the 11-Limit: Betarabian Intervals ==
+== Delving into the 11-Limit: Betarabian Intervals and the Nexus Comma ==
 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'' Alpharabian yet only differ from the Alphrarabian 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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 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, I can at least take note of the fact that 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.
+While I can't cover all of the Betarabian intervals in this section as there are too many to cover, I can at least take note of the fact that 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 look at one 11-limit comma in particular- the [[Nexuma]], or the Nexus comma.  The Nexuma 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 it 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, and that both the 3-limit and the 11-limit are navigational primes, this means that the Nexuma is a very very important interval- especially in light of the fact that the Nexuma is only slightly more than two cents in size and is thus not only an unnoticeable comma, but a prime target for tempering.
 == Measuring EDO Approximation Quality ==