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

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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.

Most music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime. They also know that a diatonic semitone and a chromatic semitone add up to a whole tone. The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo. As mentioned to me by[[KiteGiedraitis | Kite Giedraitis]] in [[Talk:159edo_notation#My_Second_Idea_for_a_Notation System|a conversation]] about this topic, there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and a chromatic semitone that is otherwise known as the apotome- which, when added together, add up to a 9/8 whole tone. Furthermore, in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by 81/80. On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields 27/25- two additional diatonic semitones. On the other hand, subtracting 81/80 from the apotome yields 135/128, and subtracting another 81/80 yields 25/24- two additional chromatic semitones. When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone. In light of all this, Kite argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.

Most music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime. They also know that a diatonic semitone and a chromatic semitone add up to a whole tone. The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo. As mentioned to me by [[KiteGiedraitis|Kite Giedraitis]] in [[Talk:159edo_notation#My_Second_Idea_for_a_Notation System|a conversation]] about this topic, there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and a chromatic semitone that is otherwise known as the apotome- which, when added together, add up to a 9/8 whole tone. Furthermore, in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by 81/80. On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields 27/25- two additional diatonic semitones. On the other hand, subtracting 81/80 from the apotome yields 135/128, and subtracting another 81/80 yields 25/24- two additional chromatic semitones. When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone. In light of all this, Kite argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.

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.

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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.

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 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.

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 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.
-Most music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime.  They also know that a diatonic semitone and a chromatic semitone add up to a whole tone.  The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo.  As mentioned to me by[[KiteGiedraitis | Kite Giedraitis]] in [[Talk:159edo_notation#My_Second_Idea_for_a_Notation System|a conversation]] about this topic, there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and a chromatic semitone that is otherwise known as the apotome- which, when added together, add up to a 9/8 whole tone.  Furthermore, in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by 81/80.  On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields 27/25- two additional diatonic semitones.  On the other hand, subtracting 81/80 from the apotome yields 135/128, and subtracting another 81/80 yields 25/24- two additional chromatic semitones.  When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone.  In light of all this, Kite argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.
+Most music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime.  They also know that a diatonic semitone and a chromatic semitone add up to a whole tone.  The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo.  As mentioned to me by [[KiteGiedraitis|Kite Giedraitis]] in [[Talk:159edo_notation#My_Second_Idea_for_a_Notation System|a conversation]] about this topic, there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and a chromatic semitone that is otherwise known as the apotome- which, when added together, add up to a 9/8 whole tone.  Furthermore, in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by 81/80.  On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields 27/25- two additional diatonic semitones.  On the other hand, subtracting 81/80 from the apotome yields 135/128, and subtracting another 81/80 yields 25/24- two additional chromatic semitones.  When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone.  In light of all this, Kite argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.
 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.
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 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.
+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 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.