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

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== Delving into the 11-Limit: Alpharabian Tuning ==

With the 11-limit established as a second navigational prime, 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".

With the 11-limit established as a second navigational prime, 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, that only partially covers the 11-limit's quartertones, and the "primary" versus "secondary" distinction is temporary at best. 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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== Delving into the 11-Limit: ~~Rastmic and~~ Betarabian Intervals ==

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

Now that we've covered the Alpharabian intervals, it's time to continue our journey into the 11-limit, and cover the matter of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32. When we subject a 3-limit Augmented interval to augmentation by 33/32, we can refer to the resulting interval as being "Superaugmented", and when we subject a 3-limit Diminished interval to dimunition by 33/32, we can refer to the resulting interval as being "Subdiminished". Furthermore, when the rastma is not tempered out, we can subject 3-limit Augmented intervals to dimunition by 33/32 without arriving at the same location as an Alpharabian interval. Likewise, we can subject a 3-limit Diminished interval to augmentation by 33/32 without arriving in the same location as an Alpharabian interval.

Now that we've covered the basic Alpharabian intervals, it's time to continue our journey into the 11-limit, and cover the matter of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32. When we subject a 3-limit Augmented interval to augmentation by 33/32, we can refer to the resulting interval as being "Superaugmented", and when we subject a 3-limit Diminished interval to dimunition by 33/32, we can refer to the resulting interval as being "Subdiminished". Furthermore, when the rastma is not tempered out, we can subject 3-limit Augmented intervals to dimunition by 33/32 without arriving at the same location as an Alpharabian interval. Likewise, we can subject a 3-limit Diminished interval to augmentation by 33/32 without arriving in the same location as an Alpharabian interval. However, because these alterations by a rastma result in intervals that 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.

Now, if one does the math, they will realize that an Alpharabian ~~Supermajor~~ Second, having a ratio of [[297/256]], is larger than an Alpharabian ~~Subminor~~ Third with its ratio of [[1024/891]], and that the difference between these two intervals is 264627/262144. Despite the fact that 264627/262144 is the sum of the rastma and the Alpharabian comma, its function can be contrasted with that of the Alpharabian comma in that 264627/262144 not only separates 297/256 and 1024/891, but also other similar enharmonic quartertone-based interval pairs, whereas the Alpharabian comma merely distinguishes enharmonic 11-limit semitones. Yet, the term "Alpharabian" contains the word "Alpha", which can be taken as signifying the Alpharabian comma's status a primary 11-limit comma. Therefore, if we take the "Alpha" off of "Alpharabian" and put the term "Beta" in its place, we can thus call 264627/262144 the "[[Betarabian comma|Betarabian Comma]]".

Now, if one does the math, they will realize that an Alpharabian Parasupermajor Second, having a ratio of 297/256, is larger than an Alpharabian Parasubminor Third with its ratio of 1024/891, and that the difference between these two intervals is 264627/262144. Despite the fact that 264627/262144 is the sum of the rastma and the Alpharabian comma, its function can be contrasted with that of the Alpharabian comma in that 264627/262144 not only separates 297/256 and 1024/891, but also other similar enharmonic quartertone-based interval pairs, whereas the Alpharabian comma merely distinguishes enharmonic 11-limit semitones. Yet, the term "Alpharabian" contains the word "Alpha", which can be taken as signifying the Alpharabian comma's status a primary 11-limit comma. Therefore, if we take the "Alpha" off of "Alpharabian" and put the term "Beta" in its place, we can thus call 264627/262144 the "[[Betarabian comma|Betarabian Comma]]". On another note, you may have noticed that I didn't include the 729/704 quartertone in the list of Alpharabian intervals. This was because I couldn't exactly find a place for 729/704 in the list of 11-limit intervals that can be considered "basic". However, I think it's fair to said that I have opened up another layer of 11-limit intervals- intervals that can't exactly be considered "basic" due to other more important intervals like 11/8 and 16/11 taking priority, yet can still be derived from the basic intervals by means of either adding or subtracting a rastma. With this in mind, you should recall that two 33/32 Parachromatic Quartertones fall short of the apotome by a rastma, and that if you add a rastma to 33/32, you get 729/704. However, there's more to the story here, as 729/704 differs from the 4096/3993 Paradiatonic Quartertone by the Betarabian comma. With both of these things in mind, it's safe to say that we can classify 729/704 as a Betarabian interval- specifically, we can call it the "Betarabian Parasuperprime" or the "Betarabian Parachromatic Quartertone".

== Measuring EDO Approximation Quality ==

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 == Delving into the 11-Limit: Alpharabian Tuning ==
-With the 11-limit established as a second navigational prime, 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".
+With the 11-limit established as a second navigational prime, 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, that only partially covers the 11-limit's quartertones, and the "primary" versus "secondary" distinction is temporary at best.  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.
@@ Line 229: / Line 229: @@
 |}
-== Delving into the 11-Limit: Rastmic and Betarabian Intervals ==
+== Delving into the 11-Limit: Betarabian Intervals ==
-Now that we've covered the Alpharabian intervals, it's time to continue our journey into the 11-limit, and cover the matter of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32.  When we subject a 3-limit Augmented interval to augmentation by 33/32, we can refer to the resulting interval as being "Superaugmented", and when we subject a 3-limit Diminished interval to dimunition by 33/32, we can refer to the resulting interval as being "Subdiminished".  Furthermore, when the rastma is not tempered out, we can subject 3-limit Augmented intervals to dimunition by 33/32 without arriving at the same location as an Alpharabian interval.  Likewise, we can subject a 3-limit Diminished interval to augmentation by 33/32 without arriving in the same location as an Alpharabian interval.
+Now that we've covered the basic Alpharabian intervals, it's time to continue our journey into the 11-limit, and cover the matter of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32.  When we subject a 3-limit Augmented interval to augmentation by 33/32, we can refer to the resulting interval as being "Superaugmented", and when we subject a 3-limit Diminished interval to dimunition by 33/32, we can refer to the resulting interval as being "Subdiminished".  Furthermore, when the rastma is not tempered out, we can subject 3-limit Augmented intervals to dimunition by 33/32 without arriving at the same location as an Alpharabian interval.  Likewise, we can subject a 3-limit Diminished interval to augmentation by 33/32 without arriving in the same location as an Alpharabian interval.  However, because these alterations by a rastma result in intervals that 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.
-Now, if one does the math, they will realize that an Alpharabian Supermajor Second, having a ratio of [[297/256]], is larger than an Alpharabian Subminor Third with its ratio of [[1024/891]], and that the difference between these two intervals is 264627/262144.  Despite the fact that 264627/262144 is the sum of the rastma and the Alpharabian comma, its function can be contrasted with that of the Alpharabian comma in that 264627/262144 not only separates 297/256 and 1024/891, but also other similar enharmonic quartertone-based interval pairs, whereas the Alpharabian comma merely distinguishes enharmonic 11-limit semitones.  Yet, the term "Alpharabian" contains the word "Alpha", which can be taken as signifying the Alpharabian comma's status a primary 11-limit comma.  Therefore, if we take the "Alpha" off of "Alpharabian" and put the term "Beta" in its place, we can thus call 264627/262144 the "[[Betarabian comma|Betarabian Comma]]".
+Now, if one does the math, they will realize that an Alpharabian Parasupermajor Second, having a ratio of 297/256, is larger than an Alpharabian Parasubminor Third with its ratio of 1024/891, and that the difference between these two intervals is 264627/262144.  Despite the fact that 264627/262144 is the sum of the rastma and the Alpharabian comma, its function can be contrasted with that of the Alpharabian comma in that 264627/262144 not only separates 297/256 and 1024/891, but also other similar enharmonic quartertone-based interval pairs, whereas the Alpharabian comma merely distinguishes enharmonic 11-limit semitones.  Yet, the term "Alpharabian" contains the word "Alpha", which can be taken as signifying the Alpharabian comma's status a primary 11-limit comma.  Therefore, if we take the "Alpha" off of "Alpharabian" and put the term "Beta" in its place, we can thus call 264627/262144 the "[[Betarabian comma|Betarabian Comma]]".  On another note, you may have noticed that I didn't include the 729/704 quartertone in the list of Alpharabian intervals.  This was because I couldn't exactly find a place for 729/704 in the list of 11-limit intervals that can be considered "basic".  However, I think it's fair to said that I have opened up another layer of 11-limit intervals- intervals that can't exactly be considered "basic" due to other more important intervals like 11/8 and 16/11 taking priority, yet can still be derived from the basic intervals by means of either adding or subtracting a rastma.  With this in mind, you should recall that two 33/32 Parachromatic Quartertones fall short of the apotome by a rastma, and that if you add a rastma to 33/32, you get 729/704.  However, there's more to the story here, as 729/704 differs from the 4096/3993 Paradiatonic Quartertone by the Betarabian comma.  With both of these things in mind, it's safe to say that we can classify 729/704 as a Betarabian interval- specifically, we can call it the "Betarabian Parasuperprime" or the "Betarabian Parachromatic Quartertone".
 == Measuring EDO Approximation Quality ==