User:Aura/Aura's Ideas on Tonality: Difference between revisions
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== Introduction == | == Introduction == | ||
It seems that some people in the community want to know how my system relates to the more well-known approach of Aaron Hunt- a simple question with a complicated answer. Although this page is late in coming, I think it is time to really begin to let people see the underpinnings of my approach to music in general and microtonal music in particular, and hopefully begin to answer this question. | It seems that some people in the community want to know how my system relates to the more well-known approach of Aaron Hunt- a simple question with a complicated answer. Although this page is late in coming, I think it is time to really begin to let people see the underpinnings of my approach to music in general and microtonal music in particular, and hopefully begin to answer this question. | ||
== A Strange New World == | == A Strange New World == | ||
From looking things up, it seems that Hunt and I have both been influenced by music theory of [https://en.wikipedia.org/wiki/Harry_Partch Harry Partch]. However, there are significant differences, as while Hunt has been influenced by the work of [https://de.wikipedia.org/wiki/Martin_Vogel_(Musikwissenschaftler) Martin Vogel] and Dr. Patrick Ozzard-Low, I, for my part, have been influenced rather heavily by what little I know of the works of [https://en.wikipedia.org/wiki/Hugo_Riemann Hugo Riemann], and I've even picked up a few tricks concerning Locrian mode from [https://alexanderlafollett.com/site/the-locrian-mode-no-its-not-unusable/ Alexander LaFollett], as well as learning from my own experimentations with Locrian. While I can't remember which came first, I must say that the influence of Riemann's concept of Harmonic Duality on my work is strongly connected to my discovery that [https://music.stackexchange.com/questions/11274/what-are-the-greek-modes-and-how-do-they-differ-from-modern-modes Ancient Greek modes were built from the Treble downwards], and, because when the Ancient Romans borrowed the Greek terminology, they evidently made the mistake of assuming that the Greek note names were built from the Bass upwards, [[wikipedia:el:%CE%A4%CF%81%CF%8C%CF%80%CE%BF%CF%82_(%CE%BC%CE%BF%CF%85%CF%83%CE%B9%CE%BA%CE%AE)|resulting in a disconnect]] between the Ancient Greek musical system and Modern Western Music Theory. | |||
From looking things up, it seems that Hunt and I have both been influenced by music theory of [https://en.wikipedia.org/wiki/Harry_Partch Harry Partch]. However, there are significant differences, as while Hunt has been influenced by the work of [https://de.wikipedia.org/wiki/Martin_Vogel_(Musikwissenschaftler) Martin Vogel] and Dr. Patrick Ozzard-Low, I, for my part, have been influenced rather heavily by what little I know of the works of [https://en.wikipedia.org/wiki/Hugo_Riemann Hugo Riemann], and I've even picked up a few tricks concerning Locrian mode from [https://alexanderlafollett.com/site/the-locrian-mode-no-its-not-unusable/ Alexander LaFollett], as well as learning from my own experimentations with Locrian. While I can't remember which came first, I must say that the influence of Riemann's concept of Harmonic | |||
In light of this information, and in light of the development of Western Music Theory since the time of the Romans, I think it would be a good idea to also build on the more historically accurate version of the Ancient Greek modes and Treble-Down tonality in general to the same extent as has been done for Bass-Up tonality. However, doing this involves discarding the commonly-held dogmatic assumption in Modern Western Music Theory that all music is built from the Bass Upwards. Furthermore, it involves renaming some of the diatonic functions encountered in Modern Western Music Theory to be better accommodating to Treble-Down Tonality, something which Hunt's system fails to do. So, in order to do this, what sort of foundation shall we use? Well, for one thing, I propose we take Riemann's concept of harmonic duality- as well as Partch's argument that the Overtone Series and the Undertone Series are equally fundamental- much more seriously. Nevertheless, Hunt has done a fantastic job in integrating the ancient idea of a comma and the modern idea of the Just-Noticeable Difference in pitch perception, and I have even taken from [http://musictheory.zentral.zone/huntsystem2.html#2 the research on his site in this area] to establish core aspects of my standards in terms of pitch representation quality. However, I differ significantly with him in terms of what intervals can be regarded as "commas" as opposed to "chromas", as due to my prior experience with [[24edo]], I can only assume that chromas, in addition to the standard definitions, can also be intervals that are less than 50 cents, yet greater than 25 cents. | In light of this information, and in light of the development of Western Music Theory since the time of the Romans, I think it would be a good idea to also build on the more historically accurate version of the Ancient Greek modes and Treble-Down tonality in general to the same extent as has been done for Bass-Up tonality. However, doing this involves discarding the commonly-held dogmatic assumption in Modern Western Music Theory that all music is built from the Bass Upwards. Furthermore, it involves renaming some of the diatonic functions encountered in Modern Western Music Theory to be better accommodating to Treble-Down Tonality, something which Hunt's system fails to do. So, in order to do this, what sort of foundation shall we use? Well, for one thing, I propose we take Riemann's concept of harmonic duality- as well as Partch's argument that the Overtone Series and the Undertone Series are equally fundamental- much more seriously. Nevertheless, Hunt has done a fantastic job in integrating the ancient idea of a comma and the modern idea of the Just-Noticeable Difference in pitch perception, and I have even taken from [http://musictheory.zentral.zone/huntsystem2.html#2 the research on his site in this area] to establish core aspects of my standards in terms of pitch representation quality. However, I differ significantly with him in terms of what intervals can be regarded as "commas" as opposed to "chromas", as due to my prior experience with [[24edo]], I can only assume that chromas, in addition to the standard definitions, can also be intervals that are less than 50 cents, yet greater than 25 cents. | ||
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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. | 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. | ||
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== Delving into the 11-Limit: Alpharabian Tuning and Additional Interval Classifications == | == Delving into the 11-Limit: Alpharabian Tuning and Additional Interval Classifications == | ||
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, | 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 | 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 Wikipedia's 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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Continuing along this same line of thought, I propose we refer to 33/32 as the "Alpharabian Parasuperprime", and, I have now returned to my [[User talk:Aura #Getting Started|initial idea]] of using the "Parasuper-" and "Parasub-" prefixes to refer to the augmentation of Major intervals and dimunition of Minor intervals respectively by 33/32, after foolishly thinking it untenable in light of the the 11-limit's status as a navigational prime, a position which I now realize led to inconsistency in the naming scheme. Nevertheless, because the dimunition of a major interval by 33/32 does not result in the same interval as does the augmentation of a minor interval by 33/32, as these intervals differ by a rastma, and thus, as has been my idea since I first came onto this Wiki, I propose that we use the term "Greater Neutral" to refer to dimunition of a Major interval by 33/32, and the term "Lesser Neutral" to refer to the augmentation of a Minor interval by 33/32. As is to be expected, Supermajor and Subminor intervals are complements of one another; for example, when [[243/128]] is raised by 1331/1296, the result is 3993/2048- a supermajor seventh and the octave complement of 4096/3993, which has already been established as a subminor second. Similarly, Parasupermajor and Parasubminor intervals are also complements of one another, for instance 1024/891, the Alpharabian Parasubminor Third, is the octave complement of 891/512, the Alpharabian Parasuperjamor Sixth. | Continuing along this same line of thought, I propose we refer to 33/32 as the "Alpharabian Parasuperprime", and, I have now returned to my [[User talk:Aura #Getting Started|initial idea]] of using the "Parasuper-" and "Parasub-" prefixes to refer to the augmentation of Major intervals and dimunition of Minor intervals respectively by 33/32, after foolishly thinking it untenable in light of the the 11-limit's status as a navigational prime, a position which I now realize led to inconsistency in the naming scheme. Nevertheless, because the dimunition of a major interval by 33/32 does not result in the same interval as does the augmentation of a minor interval by 33/32, as these intervals differ by a rastma, and thus, as has been my idea since I first came onto this Wiki, I propose that we use the term "Greater Neutral" to refer to dimunition of a Major interval by 33/32, and the term "Lesser Neutral" to refer to the augmentation of a Minor interval by 33/32. As is to be expected, Supermajor and Subminor intervals are complements of one another; for example, when [[243/128]] is raised by 1331/1296, the result is 3993/2048- a supermajor seventh and the octave complement of 4096/3993, which has already been established as a subminor second. Similarly, Parasupermajor and Parasubminor intervals are also complements of one another, for instance 1024/891, the Alpharabian Parasubminor Third, is the octave complement of 891/512, the Alpharabian Parasuperjamor Sixth. | ||
This still leaves the matters of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32, what happens when we modify Perfect Fourths and Perfect Fifths by 1331/1296, and, what happens when we either lower Major intervals or raise Minor intervals by 1331/1296. However, we can cover these topics in the next section, as we need to delve even deeper into the 11-limit to cover these intervals on account of their complexity. Before we do that, however, we first need to compile a list of | This still leaves the matters of what happens when we modify 3-limit Augmented and Diminished intervals by 33/32, what happens when we modify Perfect Fourths and Perfect Fifths by 1331/1296, and, what happens when we either lower Major intervals or raise Minor intervals by 1331/1296. However, we can cover these topics in the next section, as we need to delve even deeper into the 11-limit to cover these intervals on account of their complexity. Before we do that, however, we first need to both take stock of the rules for the current naming system, and compile a list of relatively simple 11-limit intervals which are all classified as Alpharabian intervals, as we have now covered most of the basics for 11-limit interval terminology in this system. Do note that when composite interval terms like "Greater Neutral" are qualified by tuning terms like "Alpharabian", at least in English, the tuning term is inserted between the elements of the interval term, thus, for instance, [[88/81]], the Greater Neutral Second in Alpharabian tuning, is labeled as the "Greater Alpharabian Neutral Second". | ||
(Todo: Add more intervals and columns to the chart, with one extra column containing monzos) | |||
{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
| Line 59: | Line 59: | ||
!| Interval | !| Interval | ||
!| Cents | !| Cents | ||
!| Shorthand Designations | |||
!| Names | !| Names | ||
|- | |- | ||
| [[1331/1296]] | | [[1331/1296]] | ||
| 46.133824 | | 46.133824 | ||
| | | rU1 | ||
| | |||
|- | |- | ||
| [[33/32]] | | [[33/32]] | ||
| 53.272943 | | 53.272943 | ||
| Alpharabian | | U1 | ||
| Alpharabian Ultraprime, Alpharabian Parachromatic Quartertone, al-Farabi Quartertone | |||
|- | |||
| [[729/704]] | |||
| 60.412063 | |||
| uA1 | |||
| Alpharabian Infraaugmented Prime, Alpharabian Infraapotomic Quartertone | |||
|- | |- | ||
| [[1089/1024]] | | [[1089/1024]] | ||
| 106.545886 | | 106.545886 | ||
| | | rA1 | ||
| Alpharabian Chromatic Semitone | |||
|- | |||
| [[161051/147456]] | |||
| 152.679710 | |||
| rrUA1 | |||
| Alpharabian Superaugmented Prime | |||
|- | |||
| [[35937/32768]] | |||
| 159.818830 | |||
| rUA1 | |||
| | |||
|- | |||
| [[72171/65536]] | |||
| 166.957949 | |||
| UA1 | |||
| Alpharabian Ultraaugmented Prime | |||
|- | |- | ||
| [[8192/8019]] | | [[8192/8019]] | ||
| 36.952052 | | 36.952052 | ||
| Alpharabian | | um2 | ||
| Alpharabian Inframinor Second | |||
|- | |- | ||
| [[4096/3993]] | | [[4096/3993]] | ||
| 44.091172 | | 44.091172 | ||
| Rum2 | |||
| Alpharabian Subminor Second, Alpharabian Paradiatonic Quartertone | | Alpharabian Subminor Second, Alpharabian Paradiatonic Quartertone | ||
|- | |- | ||
| [[128/121]] | | [[128/121]] | ||
| 97.364115 | | 97.364115 | ||
| | |||
| Alpharabian Minor Second, Alpharabian Diatonic Semitone | | Alpharabian Minor Second, Alpharabian Diatonic Semitone | ||
|- | |||
| [[21296/19683]] | |||
| 136.358819 | |||
| | |||
| Alpharabian Supraminor Second | |||
|- | |- | ||
| [[88/81]] | | [[88/81]] | ||
| 143.497938 | | 143.497938 | ||
| | |||
| Lesser Alpharabian Neutral Second | | Lesser Alpharabian Neutral Second | ||
|- | |- | ||
| [[12/11]] | | [[12/11]] | ||
| 150.637059 | | 150.637059 | ||
| | |||
| Greater Alpharabian Neutral Second | | Greater Alpharabian Neutral Second | ||
|- | |||
| [[1458/1331]] | |||
| 157.776178 | |||
| | |||
| Alpharabian Submajor Second | |||
|- | |- | ||
| [[1331/1152]] | | [[1331/1152]] | ||
| 250.043825 | | 250.043825 | ||
| | |||
| Alpharabian Supermajor Second | | Alpharabian Supermajor Second | ||
|- | |- | ||
| [[297/256]] | | [[297/256]] | ||
| 257.182945 | | 257.182945 | ||
| | |||
| Alpharabian Parasupermajor Second | | Alpharabian Parasupermajor Second | ||
|- | |- | ||
| [[1024/891]] | | [[1024/891]] | ||
| 240.862054 | | 240.862054 | ||
| | |||
| Alpharabian Parasubminor Third | | Alpharabian Parasubminor Third | ||
|- | |- | ||
| [[1536/1331]] | | [[1536/1331]] | ||
| 248.001174 | | 248.001174 | ||
| | |||
| Alpharabian Subminor Third | | Alpharabian Subminor Third | ||
|- | |- | ||
| [[144/121]] | | [[144/121]] | ||
| 301.274117 | | 301.274117 | ||
| | |||
| Alpharabian Minor Third | | Alpharabian Minor Third | ||
|- | |||
| [[2662/2187]] | |||
| 340.268821 | |||
| | |||
| Alpharabian Supraminor Third | |||
|- | |- | ||
| [[11/9]] | | [[11/9]] | ||
| 347.407941 | | 347.407941 | ||
| | |||
| Lesser Alpharabian Neutral Third | | Lesser Alpharabian Neutral Third | ||
|- | |- | ||
| [[27/22]] | | [[27/22]] | ||
| 354.547060 | | 354.547060 | ||
| | |||
| Greater Alpharabian Neutral Third | | Greater Alpharabian Neutral Third | ||
|- | |||
| [[6561/5324]] | |||
| 361.686180 | |||
| | |||
| Alpharabian Submajor Third | |||
|- | |- | ||
| [[121/96]] | | [[121/96]] | ||
| 400.680884 | | 400.680884 | ||
| | |||
| Alpharabian Major Third | | Alpharabian Major Third | ||
|- | |- | ||
| [[1331/1024]] | | [[1331/1024]] | ||
| 453.953827 | | 453.953827 | ||
| | |||
| Alpharabian Supermajor Third | | Alpharabian Supermajor Third | ||
|- | |- | ||
| [[2673/2048]] | | [[2673/2048]] | ||
| 461.092947 | | 461.092947 | ||
| | |||
| Alpharabian Parasupermajor Third | | Alpharabian Parasupermajor Third | ||
|- | |- | ||
| [[128/99]] | | [[128/99]] | ||
| 444.772056 | | 444.772056 | ||
| | |||
| Alpharabian Paraminor Fourth | | Alpharabian Paraminor Fourth | ||
|- | |||
| [[1728/1331]] | |||
| 451.911176 | |||
| | |||
| Alpharabian Subfourth | |||
|- | |- | ||
| [[11/8]] | | [[11/8]] | ||
| 551.317942 | | 551.317942 | ||
| | |||
| Alpharabian Paramajor Fourth, Just Paramajor Fourth | | Alpharabian Paramajor Fourth, Just Paramajor Fourth | ||
|- | |||
| [[1331/972]] | |||
| 544.178823 | |||
| | |||
| Alpharabian Superfourth | |||
|- | |- | ||
| [[363/256]] | | [[363/256]] | ||
| 604.590886 | | 604.590886 | ||
| | |||
| Alpharabian Augmented Fourth | | Alpharabian Augmented Fourth | ||
|- | |- | ||
| [[512/363]] | | [[512/363]] | ||
| 595.409114 | | 595.409114 | ||
| | |||
| Alpharabian Diminished Fifth | | Alpharabian Diminished Fifth | ||
|- | |- | ||
| [[16/11]] | | [[16/11]] | ||
| 648.682058 | | 648.682058 | ||
| | |||
| Alpharabian Paraminor Fifth, Just Paraminor Fifth | | Alpharabian Paraminor Fifth, Just Paraminor Fifth | ||
|- | |- | ||
| [[99/64]] | | [[99/64]] | ||
| 755.227944 | | 755.227944 | ||
| | |||
| Alpharabian Paramajor Fifth | | Alpharabian Paramajor Fifth | ||
|- | |- | ||
| [[4096/2673]] | | [[4096/2673]] | ||
| 738.907053 | | 738.907053 | ||
| | |||
| Alpharabian Parasubminor Sixth | | Alpharabian Parasubminor Sixth | ||
|- | |- | ||
| [[2048/1331]] | | [[2048/1331]] | ||
| 746.046173 | | 746.046173 | ||
| | |||
| Alpharabian Subminor Sixth | | Alpharabian Subminor Sixth | ||
|- | |- | ||
| [[192/121]] | | [[192/121]] | ||
| 799.319116 | | 799.319116 | ||
| | |||
| Alpharabian Minor Sixth | | Alpharabian Minor Sixth | ||
|- | |- | ||
| [[44/27]] | | [[44/27]] | ||
| 845.452940 | | 845.452940 | ||
| | |||
| Lesser Alpharabian Neutral Sixth | | Lesser Alpharabian Neutral Sixth | ||
|- | |- | ||
| [[18/11]] | | [[18/11]] | ||
| 852.592059 | | 852.592059 | ||
| | |||
| Greater Alpharabian Neutral Sixth | | Greater Alpharabian Neutral Sixth | ||
|- | |- | ||
| [[121/72]] | | [[121/72]] | ||
| 898.725883 | | 898.725883 | ||
| | |||
| Alpharabian Major Sixth | | Alpharabian Major Sixth | ||
|- | |- | ||
| [[1331/768]] | | [[1331/768]] | ||
| 951.998826 | | 951.998826 | ||
| | |||
| Alpharabian Supermajor Sixth | | Alpharabian Supermajor Sixth | ||
|- | |- | ||
| [[891/512]] | | [[891/512]] | ||
| 959.137946 | | 959.137946 | ||
| | |||
| Alpharabian Parasupermajor Sixth | | Alpharabian Parasupermajor Sixth | ||
|- | |- | ||
| [[512/297]] | | [[512/297]] | ||
| 942.817055 | | 942.817055 | ||
| | |||
| Alpharabian Parasubminor Seventh | | Alpharabian Parasubminor Seventh | ||
|- | |- | ||
| [[2304/1331]] | | [[2304/1331]] | ||
| 949.956175 | | 949.956175 | ||
| | |||
| Alpharabian Subminor Seventh | | Alpharabian Subminor Seventh | ||
|- | |- | ||
| [[11/6]] | | [[11/6]] | ||
| 1,049.362941 | | 1,049.362941 | ||
| | |||
| Lesser Alpharabian Neutral Seventh | | Lesser Alpharabian Neutral Seventh | ||
|- | |- | ||
| [[81/44]] | | [[81/44]] | ||
| 1,056.502061 | | 1,056.502061 | ||
| | |||
| Greater Alpharabian Neutral Seventh | | Greater Alpharabian Neutral Seventh | ||
|- | |- | ||
| [[121/64]] | | [[121/64]] | ||
| 1,102.635885 | | 1,102.635885 | ||
| | |||
| Alpharabian Major Seventh | | Alpharabian Major Seventh | ||
|- | |- | ||
| [[3993/2048]] | | [[3993/2048]] | ||
| 1,155.908828 | | 1,155.908828 | ||
| | |||
| Alpharabian Supermajor Seventh | | Alpharabian Supermajor Seventh | ||
|- | |- | ||
| [[8019/4096]] | | [[8019/4096]] | ||
| 1,163.047948 | | 1,163.047948 | ||
| Alpharabian | | | ||
| Alpharabian Ultramajor Seventh | |||
|- | |- | ||
| [[2048/1089]] | | [[2048/1089]] | ||
| 1,093.454114 | | 1,093.454114 | ||
| | |||
| Alpharabian Diminished Octave | | Alpharabian Diminished Octave | ||
|- | |- | ||
| [[64/33]] | | [[64/33]] | ||
| 1,146.727057 | | 1,146.727057 | ||
| Alpharabian | | | ||
| Alpharabian Infraoctave | |||
|- | |- | ||
| [[2592/1331]] | | [[2592/1331]] | ||
| 1,153.866176 | | 1,153.866176 | ||
| | |||
| Alpharabian Suboctave | | Alpharabian Suboctave | ||
|- | |- | ||
|} | |} | ||
From this particular sample, we can deduce that there are several fundamental premises of the Alpharabian tuning system: | |||
* Intervals that are in the 2.11 subgroup are all considered Alpharabian intervals. | |||
* The intervals [[3/2]], [[4/3]], [[9/8]], [[16/9]], and so forth, have the same functions as in [[Pythagorean tuning]]. | |||
* Intervals that result from the modification of a Pythagorean interval by [[1089/1024]] are labeled similarly to those modified in the equivalent fashion by [[2187/2048]], the only difference being that modification by 1089/1024 results in an Alpharabian interval rather than a Pythagorean interval. | |||
* Since 1089/1024 is (33/32)^2, modifying a Pythagorean interval by [[33/32]] always results in an interval that is considered "Alpharabian". | |||
* As both the [[243/242|rastma]] and [[1331/1296]] are [[diatonic, chromatic, enharmonic, subchromatic|subchromas]] that form differences between members of the 2.11 subgroup and Pythagorean intervals, both of these subchromas belong to a set of intervals defining different interval sets within Alpharabian tuning, and subchromas within this particular interval set help define the differences between Pythagorean, Alpharabian and Betarabian intervals. | |||
The following rules are directly derived from the above premises: | |||
* Generally, intervals that result from the modification of a Pythagorean interval by 33/32 take either the 'parasuper' or 'parasub' prefixes, however, there are a number of special cases... | |||
:* Augmentation of a Perfect Fourth or Perfect Fifth by 33/32 results in a Paramajor interval | |||
:* Dimunition of a Perfect Fourth or Perfect Fifth by 33/32 results in a Paraminor interval | |||
:* Augmentation of a Pythagorean Minor interval by 33/32 results in a Lesser Neutral interval | |||
:* Dimunition of a Pythagorean Major interval by 33/32 results in a Greater Neutral interval | |||
* Generally, intervals that result that result from the modification of a Pythagorean interval by 1331/1296 take either the 'super' or 'sub' prefixes, with these prefixes generally being stacked where multiple such modifications occur, however, there are some significant caveats... | |||
:* Augmentation of a Pythagorean Minor interval by a single 1331/1296 results in a Supraminor interval, but a second such augmentation results in a Betarabian Major interval due to said interval differing from the nearby Alpharabian Major (covered under modifications by 1089/1024) by a rastma. | |||
:* Dimunition of a Pythagorean Major interval by a single 1331/1296 results in a Submajor interval, but a second such dimunition results in a Betarabian Minor interval due to said interval differing from the nearby Alpharabian Minor (covered under modifications by 1089/1024) by a rastma. | |||
== Delving into the 11-Limit: Betarabian Intervals and the Nexus Comma == | == Delving into the 11-Limit: Betarabian Intervals and the Nexus Comma == | ||
(Todo: rewrite this section based on further modification of the rules for defining subsets of Alpharabian tuning) | |||
Now that we've covered the basic Alpharabian intervals, it's time to continue our journey into the 11-limit. However, because the intervals in this section 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 | Now that we've covered the basic Alpharabian intervals, it's time to continue our journey into the 11-limit. However, because the intervals in this section 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'' basic Alpharabian intervals yet only differ from these same 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 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". | 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". | ||
| Line 249: | Line 362: | ||
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. | 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 | 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 resulting 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 larger than the Alpharabian Chromatic Semitone by 1331/1296. Thus, for the purposes of this discussion, we will refer to 161051/131072 as the "Alpharabian Superaugmented 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- in fact, we can now refer to it as the "Paradiatonic 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. | 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- in fact, we can now refer to it as the "Paradiatonic 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. | ||