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

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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 primes, albeit 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.

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, and thus answer the question as to whether or not there is any merit to the idea of the 11-limit being considered a navigational prime. 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 must confess that 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.

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, and thus answer the question as to whether or not there is any merit to the idea of the 11-limit being considered a navigational prime. 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 must confess that 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. Nevertheless, my decision to consider the 11-limit a navigational prime in my system- with 33/32 being the primary parachroma for this limit- not only sets my system apart from the Hunt System, but also both the Helmholtz-Ellis Notation system and the Functional Just system.

With the 11-limit now reasonably well established as being the best p-limit for representing quartertones, we can safely assume that the 11-limit is therefore the second 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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 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 primes, albeit 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.
-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, and thus answer the question as to whether or not there is any merit to the idea of the 11-limit being considered a navigational prime.  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 must confess that 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.
+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, and thus answer the question as to whether or not there is any merit to the idea of the 11-limit being considered a navigational prime.  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 must confess that 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.  Nevertheless, my decision to consider the 11-limit a navigational prime in my system- with 33/32 being the primary parachroma for this limit- not only sets my system apart from the Hunt System, but also both the Helmholtz-Ellis Notation system and the Functional Just system.
 With the 11-limit now reasonably well established as being the best p-limit for representing quartertones, we can safely assume that the 11-limit is therefore the second 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.