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

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With the 11-limit established as perhaps the best p-limit for representing quartertones in terms of ratio simplicity, we need to look more closely at its properties- something which most microtonal systems at the time of this writing have yet to do. First off, one will notice that a stack of two 33/32 quartertones falls short of an apotome by a comma called the [[243/242|rastma]], which means that the rastma, when not tempered out, is a very important "parasubchroma"- a term derived from [[Diatonic, Chromatic, Enharmonic, Subchromatic|subchroma]]. One will also notice that because two parachromatic quartertones equals a chromatic semitone, and because two 33/32 quartertones fall short of an apotome by a rastma, we can deduce that other parachromatic quartertones- or, more generally "parachromas"- can be derived from 33/32 by adding or subtracting the rastma. When one adds the rastma to 33/32, one arrives at [[729/704]]- the parachromatic quartertone that adds up together with 33/32 to form an apotome. Since 33/32 is smaller than 729/704, and both can be described as "undecimal quartertones", we need to disambiguate them somehow, not to mention acknowledge the 11-limit's newfound status as a navigational prime. Thus, for purposes of continuing this discussion at the moment, we'll start referring to 33/32 as the "primary parachromatic quartertone", and, we'll refer to 729/704 as the "secondary parachromatic quartertone". However, the rastma doesn't only derive other parachromatic intervals from 33/32, but also derives other paradiatonic intervals from [[4096/3993]]- which we shall refer to for the moment as the "primary paradiatonic quartertone". For instance, if one subtracts the rastma from the primary paradiatonic quartertone, we get [[8192/8019]], which, when added to 33/32, yields [[256/243]]- the Pythagorean Diatonic Semitone. Because of this, and because 8192/8019 is derived from the primary paradiatonic quartertone, we shall refer to 8192/8019 as the "secondary paradiatonic quartertone".

However, the "primary" versus "secondary" distinction is temporary at best, as in truth there is more nuance to consider. Furthermore, we have to contend with the idea of a "Subchroma" from earlier, as well as the idea of a "Diesis", and define how these concept relate to the idea of "Parachromatic" and "Paradiatonic" intervals, and for this we should begin by looking at the distinction between a "Paradiatonic" interval and a "Diesis". In order to define a "Paradiatonic" interval as it contrasts with a "Diesis", we need to consider that "Paradiatonic" consists of the prefix "Para-" and the word "Diatonic", with "Para-" meaning "alongside" in this case, as paradiatonic intervals are those that are relatively easy to use as accidentals in otherwise diatonic keys. More importantly, we need to consider that diatonic intervals- as the term "diatonic" pertains to intervals other than semitones- are the intervals found in those heptatonic scales in which the ~~notes~~ are [https://en.wikipedia.org/wiki/Maximal_evenness spread out as much as possible], as per the more strict definition of "Diatonic" listed on [https://en.wikipedia.org/wiki/Diatonic_scale Wikipedias article on the Diatonic Scale]. In light of all this, it should follow that we can define "Paradiatonic" intervals as being those microtonal intervals which are as distant from the boundaries of the nearest semitone-based intervals as possible. Furthermore, since quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches, this means that Paradiatonic intervals are inevitably quartertone-based, as are their "Parachromatic" counterparts. In contrast, terms such as "Subchroma" and "Diesis" are a bit broader, as they are not restricted to quartertone-based intervals- in fact, they are most often used to refer to intervals ''smaller'' than a quartertone. That said, there are such things as "Subchromatic Quartertones", which contrast with their Paracrhomatic counterparts in that they have more complicated ratios whereas the Parachromatic quartertones have fairly simple ratios.

However, the "primary" versus "secondary" distinction is temporary at best, as in truth there is more nuance to consider. Furthermore, we have to contend with the idea of a "Subchroma" from earlier, as well as the idea of a "Diesis", and define how these concept relate to the idea of "Parachromatic" and "Paradiatonic" intervals, and for this we should begin by looking at the distinction between a "Paradiatonic" interval and a "Diesis". In order to define a "Paradiatonic" interval as it contrasts with a "Diesis", we need to consider that "Paradiatonic" consists of the prefix "Para-" and the word "Diatonic", with "Para-" meaning "alongside" in this case, as paradiatonic intervals are those that are relatively easy to use as accidentals in otherwise diatonic keys. More importantly, we need to consider that diatonic intervals- as the term "diatonic" pertains to intervals other than semitones- are the intervals found in those heptatonic scales consisting of five whole tones and two semitones in each octave in which the semitones are [https://en.wikipedia.org/wiki/Maximal_evenness spread out as much as possible], as per the more strict definition of "Diatonic" listed on [https://en.wikipedia.org/wiki/Diatonic_scale Wikipedias article on the Diatonic Scale]. In light of all this, it should follow that we can define "Paradiatonic" intervals as being those microtonal intervals which are as distant from the boundaries of the nearest semitone-based intervals as possible. Furthermore, since quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches, this means that Paradiatonic intervals are inevitably quartertone-based, as are their "Parachromatic" counterparts. In contrast, terms such as "Subchroma" and "Diesis" are a bit broader, as they are not restricted to quartertone-based intervals- in fact, they are most often used to refer to intervals ''smaller'' than a quartertone. That said, there are such things as "Subchromatic Quartertones", which contrast with their Paracrhomatic counterparts in that they have more complicated ratios whereas the Parachromatic quartertones have fairly simple ratios.

However, all of this only partially 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 can immediately see that the "primary" versus "secondary" distinction is untenable here due to the apotome being a 3-limit interval. So, we should instead look to another source for more proper terminology for 11-limit intervals that are distinguished from each other by the rastma. Since 33/32 is also called the "al-Farabi Quartertone" and is the primary apotome-like interval of the of the 11-limit, 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 to no-fives no-sevens just 11-limit tuning in the same way that we can use the "Pythagorean" to refer to just 3-limit tuning. Therefore, we can use the term "Alpharabian" to refer to 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.

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 With the 11-limit established as perhaps the best p-limit for representing quartertones in terms of ratio simplicity, we need to look more closely at its properties- something which most microtonal systems at the time of this writing have yet to do.  First off, one will notice that a stack of two 33/32 quartertones falls short of an apotome by a comma called the [[243/242|rastma]], which means that the rastma, when not tempered out, is a very important "parasubchroma"- a term derived from [[Diatonic, Chromatic, Enharmonic, Subchromatic|subchroma]].  One will also notice that because two parachromatic quartertones equals a chromatic semitone, and because two 33/32 quartertones fall short of an apotome by a rastma, we can deduce that other parachromatic quartertones- or, more generally "parachromas"- can be derived from 33/32 by adding or subtracting the rastma.  When one adds the rastma to 33/32, one arrives at [[729/704]]- the parachromatic quartertone that adds up together with 33/32 to form an apotome.  Since 33/32 is smaller than 729/704, and both can be described as "undecimal quartertones", we need to disambiguate them somehow, not to mention acknowledge the 11-limit's newfound status as a navigational prime.  Thus, for purposes of continuing this discussion at the moment, we'll start referring to 33/32 as the "primary parachromatic quartertone", and, we'll refer to 729/704 as the "secondary parachromatic quartertone".  However, the rastma doesn't only derive other parachromatic intervals from 33/32, but also derives other paradiatonic intervals from [[4096/3993]]- which we shall refer to for the moment as the "primary paradiatonic quartertone".  For instance, if one subtracts the rastma from the primary paradiatonic quartertone, we get [[8192/8019]], which, when added to 33/32, yields [[256/243]]- the Pythagorean Diatonic Semitone.  Because of this, and because 8192/8019 is derived from the primary paradiatonic quartertone, we shall refer to 8192/8019 as the "secondary paradiatonic quartertone".
-However, the "primary" versus "secondary" distinction is temporary at best, as in truth there is more nuance to consider.  Furthermore, we have to contend with the idea of a "Subchroma" from earlier, as well as the idea of a "Diesis", and define how these concept relate to the idea of "Parachromatic" and "Paradiatonic" intervals, and for this we should begin by looking at the distinction between a "Paradiatonic" interval and a "Diesis".  In order to define a "Paradiatonic" interval as it contrasts with a "Diesis", we need to consider that "Paradiatonic" consists of the prefix "Para-" and the word "Diatonic", with "Para-" meaning "alongside" in this case, as paradiatonic intervals are those that are relatively easy to use as accidentals in otherwise diatonic keys.  More importantly, we need to consider that diatonic intervals- as the term "diatonic" pertains to intervals other than semitones- are the intervals found in those heptatonic scales in which the notes are [https://en.wikipedia.org/wiki/Maximal_evenness spread out as much as possible], as per the more strict definition of "Diatonic" listed on [https://en.wikipedia.org/wiki/Diatonic_scale Wikipedias article on the Diatonic Scale].   In light of all this, it should follow that we can define "Paradiatonic" intervals as being those microtonal intervals which are as distant from the boundaries of the nearest semitone-based intervals as possible.  Furthermore, since quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches, this means that Paradiatonic intervals are inevitably quartertone-based, as are their "Parachromatic" counterparts.  In contrast, terms such as "Subchroma" and "Diesis" are a bit broader, as they are not restricted to quartertone-based intervals- in fact, they are most often used to refer to intervals ''smaller'' than a quartertone.  That said, there are such things as "Subchromatic Quartertones", which contrast with their Paracrhomatic counterparts in that they have more complicated ratios whereas the Parachromatic quartertones have fairly simple ratios.
+However, the "primary" versus "secondary" distinction is temporary at best, as in truth there is more nuance to consider.  Furthermore, we have to contend with the idea of a "Subchroma" from earlier, as well as the idea of a "Diesis", and define how these concept relate to the idea of "Parachromatic" and "Paradiatonic" intervals, and for this we should begin by looking at the distinction between a "Paradiatonic" interval and a "Diesis".  In order to define a "Paradiatonic" interval as it contrasts with a "Diesis", we need to consider that "Paradiatonic" consists of the prefix "Para-" and the word "Diatonic", with "Para-" meaning "alongside" in this case, as paradiatonic intervals are those that are relatively easy to use as accidentals in otherwise diatonic keys.  More importantly, we need to consider that diatonic intervals- as the term "diatonic" pertains to intervals other than semitones- are the intervals found in those heptatonic scales consisting of five whole tones and two semitones in each octave in which the semitones are [https://en.wikipedia.org/wiki/Maximal_evenness spread out as much as possible], as per the more strict definition of "Diatonic" listed on [https://en.wikipedia.org/wiki/Diatonic_scale Wikipedias article on the Diatonic Scale].   In light of all this, it should follow that we can define "Paradiatonic" intervals as being those microtonal intervals which are as distant from the boundaries of the nearest semitone-based intervals as possible.  Furthermore, since quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches, this means that Paradiatonic intervals are inevitably quartertone-based, as are their "Parachromatic" counterparts.  In contrast, terms such as "Subchroma" and "Diesis" are a bit broader, as they are not restricted to quartertone-based intervals- in fact, they are most often used to refer to intervals ''smaller'' than a quartertone.  That said, there are such things as "Subchromatic Quartertones", which contrast with their Paracrhomatic counterparts in that they have more complicated ratios whereas the Parachromatic quartertones have fairly simple ratios.
 However, all of this only partially 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 can immediately see that the "primary" versus "secondary" distinction is untenable here due to the apotome being a 3-limit interval.  So, we should instead look to another source for more proper terminology for 11-limit intervals that are distinguished from each other by the rastma.  Since 33/32 is also called the "al-Farabi Quartertone" and is the primary apotome-like interval of the of the 11-limit, 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 to no-fives no-sevens just 11-limit tuning in the same way that we can use the "Pythagorean" to refer to just 3-limit tuning.  Therefore, we can use the term "Alpharabian" to refer to 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.