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

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However, that just covers the 11-limit's quartertones. Since two parachromatic quartertones add up to a chromatic semitone, and since 33/32 is the primary parachromatic quartertone, we can safely assume that two 33/32 intervals adds up to a chromatic semitone of some sort, and indeed, [[1089/1024]] ''is'' a chromatic semitone under this definition, but as it differs from the apotome by the rastma, and we don't have the benefit of the "primary" versus "secondary" distinction here due to the apotome being a 3-limit interval, we should instead look to another source for a name. Since 33/32 is also called the "Al-Farabi Quartertone", and since Al-Farabi himself was also referred to as "Alpharabius" according to [https://en.wikipedia.org/wiki/Al-Farabi Wikipedia's article on him], we can use the term "Alpharabian" to refer the 11-limit semitones in the same way that we use "Pythagorean" to refer to the 3-limit semitones. Thus, just as the apotome can also be referred to as the "Pythagorean Chromatic Semitone", we can refer to 1089/1024 as the "Alpharabian Chromatic Semitone". Furthermore, just as there's an Alpharabian Chromatic Semitone, there is an "Alpharabian Diatonic Semitone" which adds up together with the Alpharabian Chromatic Semitone to create a 9/8 whole tone, and, when you take 9/8 and subtract 1089/1024, you arrive at [[128/121]] as the ratio for the Alpharabian Diatonic Semitone. Not only that, but just as there's the [[Pythagorean comma]] which forms the difference between a stack of two Pythagorean Diatonic Semitones and a 9/8 whole tone, so there is an "[[Alpharabian comma|Alpharabian Comma]]" which forms the difference between a stack of two Alpharabian Diatonic Semitones and a 9/8 whole tone, and doing the math yields 131769/131072 as the ratio for the Alpharabian Comma.

~~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.~~

== Paramajor, Paraminor, Supermajor, Subminor and Neutral Intervals ==

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.

With all of the aforementioned stuff about, one can easily 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 "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. 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, given that the Prime and the Octave are the fundamental intervals in both my system and conventional music systems, and that 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. Furthermore, while terms like these 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.

== Measuring EDO Approximation Quality ==

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 However, that just covers the 11-limit's quartertones.  Since two parachromatic quartertones add up to a chromatic semitone, and since 33/32 is the primary parachromatic quartertone, we can safely assume that two 33/32 intervals adds up to a chromatic semitone of some sort, and indeed, [[1089/1024]] ''is'' a chromatic semitone under this definition, but as it differs from the apotome by the rastma, and we don't have the benefit of the "primary" versus "secondary" distinction here due to the apotome being a 3-limit interval, we should instead look to another source for a name.  Since 33/32 is also called the "Al-Farabi Quartertone", and since Al-Farabi himself was also referred to as "Alpharabius" according to [https://en.wikipedia.org/wiki/Al-Farabi Wikipedia's article on him], we can use the term "Alpharabian" to refer the 11-limit semitones in the same way that we use "Pythagorean" to refer to the 3-limit semitones.  Thus, just as the apotome can also be referred to as the "Pythagorean Chromatic Semitone", we can refer to 1089/1024 as the "Alpharabian Chromatic Semitone". Furthermore, just as there's an Alpharabian Chromatic Semitone, there is an "Alpharabian Diatonic Semitone" which adds up together with the Alpharabian Chromatic Semitone to create a 9/8 whole tone, and, when you take 9/8 and subtract 1089/1024, you arrive at [[128/121]] as the ratio for the Alpharabian Diatonic Semitone.  Not only that, but just as there's the [[Pythagorean comma]] which forms the difference between a stack of two Pythagorean Diatonic Semitones and a 9/8 whole tone, so there is an "[[Alpharabian comma|Alpharabian Comma]]" which forms the difference between a stack of two Alpharabian Diatonic Semitones and a 9/8 whole tone, and doing the math yields 131769/131072 as the ratio for the Alpharabian Comma.
-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.
+== Paramajor, Paraminor, Supermajor, Subminor and Neutral Intervals ==
-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.
+With all of the aforementioned stuff about, one can easily 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 "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.  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, given that the Prime and the Octave are the fundamental intervals in both my system and conventional music systems, and that 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.  Furthermore, while terms like these 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.
 == Measuring EDO Approximation Quality ==