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

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Because the 3-limit and the 11-limit are the primes that have all of this foundational functionality, they are naturally very important in musical systems that have their roots in 24edo, and their pivotal role in laying the groundwork for key signatures means that they can be referred as the "navigational primes". In both the traditional and quartertone-based key signatures, 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, and the pure 11-limit is not capable of forming diatonic scales at all. 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. For the sake of simplicity, it can generally be assumed that higher primes are ill-suited for serving as anything other than accidentals.

== Parachromatic, Paradiatonic, and the 11-limit's Importance ==

Now, some may have additional questions as to good reasons for elevating the 11-limit to the status of "navigational prime", and why 11-limit quartertones in particular are better than others in this department. Well, perhaps we should start with the fact that quartertones are the most readily accessible among microtones. Furthermore, research seems to show that quartertones are the smallest musical intervals that can be regularly used in musical capacities without being considered a variation of one of the surrounding pitches.

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, 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 by drawing a distinction between the terms "parachromatic" and "paradiatonic" for purposes of classifying intervals. For starters, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as subminor seconds, while parachromatic quartertones are denoted as superprimes. 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.

This is where we look at the 7-limit, the 11-limit, the 13-limit, along with the 17-limit and the 19-limit, and compare quartertones in 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. Therefore, the 11-limit is the most suitable p-limit for representing quartertones, meaning that it is the best candidate after the 3-limit to be considered a navigational prime. While I didn't initially choose the 11-limit on this exact basis- 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.

== Measuring EDO Approximation Quality ==

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 Because the 3-limit and the 11-limit are the primes that have all of this foundational functionality, they are naturally very important in musical systems that have their roots in 24edo, and their pivotal role in laying the groundwork for key signatures means that they can be referred as the "navigational primes".  In both the traditional and quartertone-based key signatures, 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, and the pure 11-limit is not capable of forming diatonic scales at all.  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.  For the sake of simplicity, it can generally be assumed that higher primes are ill-suited for serving as anything other than accidentals.
+== Parachromatic, Paradiatonic, and the 11-limit's Importance ==
+Now, some may have additional questions as to good reasons for elevating the 11-limit to the status of "navigational prime", and why 11-limit quartertones in particular are better than others in this department.  Well, perhaps we should start with the fact that quartertones are the most readily accessible among microtones.  Furthermore, research seems to show that quartertones are the smallest musical intervals that can be regularly used in musical capacities without being considered a variation of one of the surrounding pitches.
+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, 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 by drawing a distinction between the terms "parachromatic" and "paradiatonic" for purposes of classifying intervals.  For starters, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as subminor seconds, while parachromatic quartertones are denoted as superprimes.  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.
+This is where we look at the 7-limit, the 11-limit, the 13-limit, along with the 17-limit and the 19-limit, and compare quartertones in 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.  Therefore, the 11-limit is the most suitable p-limit for representing quartertones, meaning that it is the best candidate after the 3-limit to be considered a navigational prime.  While I didn't initially choose the 11-limit on this exact basis- 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.
 == Measuring EDO Approximation Quality ==