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

Line 45:

With all of this in mind, one can easily then go on to ask what all this means in terms of the classification of more familiar 11-limit ratios like [[11/8]], seeing as the 11/8 can be derived from [[4/3]]- the Just Perfect Fourth- through the addition of the primary parachromatic quartertone. Since the addition of the primary parachromatic quartertone to the Perfect Unison results in the primary parachromatic quartertone, one would assume that this means that 11/8 would be classified as the "parachromatic superfourth". In actuality, however, while one would be correct in asserting 11/8 is a parachromatic alteration of the perfect fourth, interpreting 11/8 as a derivative of 33/32 would in many respects be akin to interpreting [[3/2]]- the Just Perfect Fifth- as a derivation of the apotome, when in fact, it is the other way around. Recall that the prime factorization of 33 is 3*11, so that means that, 33/32 is ''not'' a pure 11-limit interval. Therefore, rather than assume the primary parachromatic quartertone to be the basic 11-limit interval, we instead must recognize that that title properly belongs to 11/8. Furthermore we should take stock of the fact that two 11/8 intervals stacked on top of one another yields [[121/64]], the octave complement of the Alpharabian diatonic semitone. Since 121/64 is arguably a form of major seventh as a diatonic semitone always has a major seventh as its octave complement, and since a stack of two fourths equals a seventh, what does that mean for 11/8? Well, it means we need more terms, and we need to define those terms.

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 ~~a stack~~ of ~~two perfect fourths creates~~ a ~~minor seventh~~, and since ~~a Major interval is a chromatically raised variation on a~~ Minor ~~interval~~ and ~~vice versa~~, we can thus ~~say~~ 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 "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 "Just Paraminor Fifth".

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 "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 "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.

== Measuring EDO Approximation Quality ==

@@ Line 45: / Line 45: @@
 With all of this in mind, one can easily then go on to ask what all this means in terms of the classification of more familiar 11-limit ratios like [[11/8]], seeing as the 11/8 can be derived from [[4/3]]- the Just Perfect Fourth- through the addition of the primary parachromatic quartertone.  Since the addition of the primary parachromatic quartertone to the Perfect Unison results in the primary parachromatic quartertone, one would assume that this means that 11/8 would be classified as the "parachromatic superfourth".  In actuality, however, while one would be correct in asserting 11/8 is a parachromatic alteration of the perfect fourth, interpreting 11/8 as a derivative of 33/32 would in many respects be akin to interpreting [[3/2]]- the Just Perfect Fifth- as a derivation of the apotome, when in fact, it is the other way around.  Recall that the prime factorization of 33 is 3*11, so that means that, 33/32 is ''not'' a pure 11-limit interval.  Therefore, rather than assume the primary parachromatic quartertone to be the basic 11-limit interval, we instead must recognize that that title properly belongs to 11/8.  Furthermore we should take stock of the fact that two 11/8 intervals stacked on top of one another yields [[121/64]], the octave complement of the Alpharabian diatonic semitone.  Since 121/64 is arguably a form of major seventh as a diatonic semitone always has a major seventh as its octave complement, and since a stack of two fourths equals a seventh, what does that mean for 11/8?  Well, it means we need more terms, and we need to define those terms.
-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 a stack of two perfect fourths creates a minor seventh, and since a Major interval is a chromatically raised variation on a Minor interval and vice versa, we can thus say 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 "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 "Just Paraminor Fifth".
+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 "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 "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.
 == Measuring EDO Approximation Quality ==