User:Aura/Aura's Ideas on Tonality: Difference between revisions
Brought the tour of the 11-limit to a conclusion |
Additional section on 11-limit axis functionality |
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== Navigational Primes and the Parachromatic-Paradiatonic Contrast == | == Navigational Primes and the Parachromatic-Paradiatonic Contrast == | ||
Now, many if not most musicians who are not microtonalists are acquainted with standard music notation, with its clefs and staves, key signatures and time signatures. However, when you take all of this into the microtonal realm, it becomes readily apparent that- in all of the most intuitive systems- it is the 3-limit that defines both the standard location and structure of the various standard notes and key signatures that one finds in [[12edo]]. | Now, many if not most musicians who are not microtonalists are acquainted with standard music notation, with its clefs and staves, key signatures and time signatures. However, when you take all of this into the microtonal realm, it becomes readily apparent that- in all of the most intuitive systems- it is the 3-limit that defines both the standard location and structure of the various standard notes and key signatures that one finds in [[12edo]]. Not only are the traditional key signatures all related to each other along an axis formed by the 3-limit, but the standard sharp and flat accidentals modify the base note by an [[2187/2048|apotome]], and the double sharp and double flat accidentals modify the base note by two apotomes. Furthermore, it is the Pythagorean Diatonic Scales that arise as the standard variants for the various key signatures, as these are the simplest diatonic scales that can be formed with the 3-limit. Because the 3-limit is a prime that has all of this foundational functionality, it is naturally very important in musical systems that have their roots in 12edo, and its pivotal role in laying the groundwork for key signatures means that it can be referred as a "navigational prime". | ||
Meanwhile, when one comes from a background in 24edo as I have, and has even used quartertone-based keys signatures as I have, one sees that a second p-limit seems to join together with the 3-limit in defining the standard location and structure of the various notes and quartertone-based key signatures that one would see in 24edo, and the 11-limit seems to be the best candidate for this second navigational prime despite the fact that the pure 11-limit is not capable of forming diatonic scales at all. Now, some may question the musical grounds for using quartertones in light of their dissonance, as well as the idea that there is any merit to the idea of the 11-limit being considered a navigational prime. Well, we should start with the reasons for considering quartertones musically important in the first place- namely the fact that quartertones are the most readily accessible among microtones, and that current research seems to show that quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches. On this basis, we can proceed to look at the musical functions of semitones, and then go on to define the musical functions of the quartertones themselves. | Meanwhile, when one comes from a background in 24edo as I have, and has even used quartertone-based keys signatures as I have, one sees that a second p-limit seems to join together with the 3-limit in defining the standard location and structure of the various notes and quartertone-based key signatures that one would see in 24edo, and the 11-limit seems to be the best candidate for this second navigational prime despite the fact that the pure 11-limit is not capable of forming diatonic scales at all. Now, some may question the musical grounds for using quartertones in light of their dissonance, as well as the idea that there is any merit to the idea of the 11-limit being considered a navigational prime. Well, we should start with the reasons for considering quartertones musically important in the first place- namely the fact that quartertones are the most readily accessible among microtones, and that current research seems to show that quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches. On this basis, we can proceed to look at the musical functions of semitones, and then go on to define the musical functions of the quartertones themselves. | ||
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Building on Kite's logic, we can then apply similar distinctions among quartertones, and thus make the argument that quartertones don't have to denote exactly one fourth of a whole tone in as of themselves, but rather, they only have to add up to a whole tone when paired up correctly. However, it should be noted that for quartertones, there are sometimes multiple correct options, and thus, things are more complicated. We shall begin to define the musical functions of quartertones by drawing a distinction between the terms "Parachromatic" and "Paradiatonic" for purposes of classifying quartertone intervals. For starters, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as seconds, albeit subminor seconds, while parachromatic quartertones are denoted as superprimes of some sort. However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone. Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, we can deduce that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone. Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, we have to choose the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone. From here, we have to select simple parachromatic quartertones from the lowest p-limit that, when subtracted from 9/8, yield the paradiatonic interval with the lowest odd limit. | Building on Kite's logic, we can then apply similar distinctions among quartertones, and thus make the argument that quartertones don't have to denote exactly one fourth of a whole tone in as of themselves, but rather, they only have to add up to a whole tone when paired up correctly. However, it should be noted that for quartertones, there are sometimes multiple correct options, and thus, things are more complicated. We shall begin to define the musical functions of quartertones by drawing a distinction between the terms "Parachromatic" and "Paradiatonic" for purposes of classifying quartertone intervals. For starters, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as seconds, albeit subminor seconds, while parachromatic quartertones are denoted as superprimes of some sort. However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone. Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, we can deduce that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone. Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, we have to choose the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone. From here, we have to select simple parachromatic quartertones from the lowest p-limit that, when subtracted from 9/8, yield the paradiatonic interval with the lowest odd limit. | ||
Now that we have answered the questions as to both the musical significance and musical function of quartertones, we can take a look at the 7-limit, the 11-limit, the 13-limit, along with the 17-limit and the 19-limit, and compare the various quartertones of these limits, | Now that we have answered the questions as to both the musical significance and musical function of quartertones, we can take a look at the 7-limit, the 11-limit, the 13-limit, along with the 17-limit and the 19-limit, and compare the various quartertones of these limits, . For the 7-limit, you have 36/35, the septimal quartertone; when a stack of three septimal quartertones is subtracted from 9/8, we get a paradiatonic quartertone with a ratio of 42875/41472- not exactly a simple interval. Next we have the 11-limit, and for the 11-limit, you have 33/32, the undecimal quartertone; when a stack of three undecimal quartertones is subtracted from 9/8, we get a paradiatonic quartertone with a ratio of 4096/3993- this is better, but we still have to look at the remaining contenders. For the 13-limit, we have 40/39; when a stack of three 40/39 intervals is subtracted from 9/8, we get an interval with a ratio of 533871/512000- this is even worse than for the 7-limit. For the 17-limit, we have 34/33, the septendecimal quartertone; when a stack of three septendecimal quartertones is subtracted from 9/8, we get a quartertone with a ratio of 323433/314432- this is also worse than for the 7-limit. Finally, we have the 19-limit, and for the 19-limit, we have 39/38; when a stack of three 39/38 intervals is subtracted from 9/8, we get an interval with a ratio of 6859/6591- better than for the 7-limit, but still not as good as for the 11-limit. | ||
In order to be thorough, I've since checked the 11-limit's representation of quartertones against that of the other rational intervals called "quarter tones" [https://en.wikipedia.org/wiki/List_of_pitch_intervals on Wikipedia's list of pitch intervals] and again found the 11-limit's 33/32 to be better than any of them in terms of ratio simplicity. Therefore, the 11-limit is the most suitable p-limit for representing quartertones. While must confess that I didn't initially choose the 11-limit on these exact bases- rather, it was because of how well the 11-limit is represented in 24edo- the math indicates that I somehow managed to make the best choice in spite of myself. However, this still leaves the question of whether the 11-limit can serve as a navigational axis of any kind, as this is another question that must be answered in order to determine whether or not there is any merit to the idea of the 11-limit being considered a navigational prime. However, in order to even begin to consider this, we must familiarize ourselves with some of the 11-limit's inner workings. | |||
== Delving into the 11-Limit: Alpharabian Tuning and Additional Interval Classifications == | == Delving into the 11-Limit: Alpharabian Tuning and Additional Interval Classifications == | ||
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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, 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 "Paradiatonic" intervals | 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, 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. | 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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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". | 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 as there are too many to cover, and the same is true of 11-limit intervals in general, I can perhaps by take note of a few more important 11-limit intervals. Firstly, 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 bring our journey through the 11-limit to a fitting conclusion with a look at one final 11-limit comma. This comma has a ratio of 1771561/1769472, and 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 this comma 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 1771561/1769472 is a very very important interval- especially in light of the fact that it is only slightly more than two cents in size and is thus not only an unnoticeable comma, but a prime target for tempering. Since tempering out 1771561/1769472 results in the formation of a nexus between two navigational prime limits, 1771561/1769472 can thus be called the "[[Nexuma]]", or the "Nexus comma". | While I can't cover all of the Betarabian intervals in this section as there are too many to cover, and the same is true of 11-limit intervals in general, I can perhaps by take note of a few more important 11-limit intervals. Firstly, 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 bring our journey through the 11-limit to a fitting conclusion with a look at one final 11-limit comma. This comma has a ratio of 1771561/1769472, and 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 this comma 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 1771561/1769472 is a very very important interval- especially in light of the fact that it is only slightly more than two cents in size and is thus not only an unnoticeable comma, but a prime target for tempering. Since tempering out 1771561/1769472 results in the formation of a nexus between two navigational prime limits, 1771561/1769472 can thus be called the "[[Nexuma]]", or the "Nexus comma". | ||
== 11-limit Axis Functionality == | |||
Now that we have explored some of the 11-limit's inner workings, we can return to the question of whether the 11-limit can serve as a navigational axis of any kind, and thus whether or not there is any merit to the idea of the 11-limit being considered a navigational prime. When trying to answer this question, we would do well to remember not only the fact that in Just Intonation, the 3-limit and the 11-limit never meet, but also the fact that the 11-limit and the 3-limit operate on fundamentally different levels- the 3-limit operating on the diatonic level, and the 11-limit on the paradiatonic level. With this in mind, what we need to look for is similarities and parallels between how these two p-limits function in their purest form. | |||
The fundamental interval of the 3-limit is 3/2, the Just Perfect Fifth. When a stack of two Perfect Fifths is octave-reduced, we get 9/8- a diatonic whole tone. The diatonic nature of this whole tone is confirmed with further stacking and octave reduction of Perfect Fifths- when you stack three Perfect Fifths and octave-reduce them, you end up with 27/16, which is a whole tone away from the original Just Perfect Fifth. Continuing along this sequence, we should take stock of the fact that we only have seven base note names to work with- let's use "A", "B", "C", "D", "E", "F" and "G" for the sake of simplicity- as the term "diatonic" pertains to the intervals found in those heptatonic scales in which the notes are spread out as much as possible. Furthermore, since spreading out in both directions along this 3/2 chain ends up creating multiple variants of the same diatonic step, which these variations being best described as "chromatic", and thus, we need to introduce sharps and flats. Of course, since the 3-limit and the 2-limit never meet beyond the fundamental, continuing to expand along the 3-limit chain in both directions eventually creates what are essentially chromatic whole tones- frequently referred to as double sharps and double flats, but we have to draw the line somewhere as continuing beyond a certain point is impractical. | |||
When we look at the 11-limit, we again need to draw the line as to how far we go as continuing beyond a certain point is impractical. However, we need to take note of the fact that while the 3-limit works with whole tones and semitones, the 11-limit works with quartertones and semitones. Therefore, the most direct place of comparison between the 3-limit and the 11-limit is in their handling of semitones. However, since the fundamental interval of the 11-limit is 11/8, we should not go into our examination of the 11-limit expecting the same results as for the 3-limit. Since we have already established that is 11/8 a paramajor fourth, we should treat it as such here, and furthermore, we should recall that a stack of two 11/8 paramajor fourths is equal to a 121/64 major seventh, as per the way paradiatonic intervals and parachromatic intervals relate to diatonic and chromatic intervals. So, if we take things a step further and continue on stacking paramajor fourths, what happens? Well, stacking a third paramajor fourth onto 121/64 and octave-reducing the reulting interval yields 1331/1024, a supermajor third in terms of the classification system we have established earlier, and adding a fourth paramajor fourth yields 14641/8192- a type of augmented sixth. Adding a fifth paramajor fourth to the stack and octave reducing the resulting interval yields the complicated 161051/131072- a type of second that differs from 9/8 by 161051/147456, which, in turn, is smaller than a stack of three 33/32 Parachromatic Quartertones by a rastma. Thus, for the purposes of this discussion, we will refer to 161051/131072 as a type of "Sesquiaugmented Second". Finally, stacking six paramajor fourths and octave-reducing the resulting interval yields 1771561/1048576- a type of double augmented fifth- which differs from 27/16 by only a nexuma. | |||
Judging from this, it seems that there is indeed a clear sequence of intervals along the 11-limit, with every other member in this sequence being the octave complement of a stack of Alpharabian Diatonic Semitones. As the 11-limit handles stacks of Alpharabian Diatonic Semitones in much the same way that the 3-limit handles stacks of Pythagorean diatonic semitones, it can thus be argued that the 11-limit meets the standards set by the 3-limit in this respect. In fact, since the Alpharabian and Betarabian Semitones are actually closer to half of a whole tone that either one of the 3-limit semitones- and especially since the 11-limit's version of a double sharp fifth is only an unnoticeable comma's distance away from the 3-limit's major sixth- it can be argued that the 11-limit passes the semitone test with flying colors. With this information in hand, we can now safely assume that the 11-limit ''does'' is fact, carry the function of a navigational axis. Between this, and the previous mathematical confirmation of the 11-limit as being the best p-limit for representing quartertones, we can now safely say that there is indeed sufficient merit to the idea of the 11-limit being considered a navigational prime. This in turn means that although the 5-limit, 7-limit and 13-limit also play a role in defining key signatures, these primes define variations on the standard key signatures as opposed to the standard key signatures themselves. Furthermore, it can now be safely assumed that higher primes are ill-suited for serving as anything other than accidentals. | |||
== Measuring EDO Approximation Quality == | == Measuring EDO Approximation Quality == | ||