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. 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~~.

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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For example, I measure representation quality not only by taking the absolute error between just intervals and their tempered counterparts as per Hunt's system, but also the absolute errors in cents as they accumulate when tempered p-limit intervals are stacked, and the number of such intervals I can stack without the absolute error exceeding an unnoticeable comma's distance of 3.5 cents determines the quality of representation, and thus the portions of the harmonic lattice that can be sufficiently represented by any given EDO. Furthermore, when the error accumulation between just intervals and their and tempered counterparts exceeds half an EDO step, contortion is considered to come into play, thus terminating the sequence of viable intervals for any given p-limit and limiting the portions of the harmonic lattice that can be considered viable in any given EDO. Because the sequence of intervals in any given p-limit extends to infinity, it would be wise to use the odd-limit as way of limiting the amount of intervals used in grading the approximation quality of various EDOs. Furthermore, as the octave is foundational in so many respects, we should set our odd-limit by way of selecting octaves of the Harmonic and Subharmonic systems as guides. As the 3-limit accounts for every pitch in 12edo using intervals with odd-limits less than 1024, we shall use 1024 as the cutoff for how high odd-limits can go- yes, 1024 is an even number, but it is also a power of 2, thus rendering suitable as a divider between different categories of odd-limits. This results in the following interval selections for representing p-limits up to 17- 3/2, 9/8, 27/16, 81/64, 243/128, and 729/512 for representing the 3-limit; 5/4, 25/16, 125/64, and 625/512 for representing the 5-limit; 7/4, 49/32 and 343/256 for representing the 7-limit; 11/8 and 121/64 for representing the 11-limit; 13/8 and 169/128 for representing the 13-limit; 17/16 and 289/256 for representing the 17-limit.

When two EDOs are both given a "P" rating in the Hunt ~~system~~ for representation quality, the tie between them is broken by which EDO-tempered version has the smaller absolute error. Furthermore, when the best representation of an interval in a given p-limit sequence cannot be reached by stacking tempered versions of the preceding intervals in that same p-limit sequence, the disconnected interval and any intervals following it in the same sequence are disqualified under my standards, no matter how good their representation rating in the Hunt system is.

When two EDOs are both given a "P" rating in the Hunt System for representation quality, the tie between them is broken by which EDO-tempered version has the smaller absolute error. Furthermore, when the best representation of an interval in a given p-limit sequence cannot be reached by stacking tempered versions of the preceding intervals in that same p-limit sequence, the disconnected interval and any intervals following it in the same sequence are disqualified under my standards, no matter how good their representation rating in the Hunt system is.

== Choice of EDO for Microtonal Systems ==

While Hunt's microtonal system is based on [[205edo]], my microtonal system is built on [[159edo]]. Why this difference? Well, even though 205edo has better interval representation in a number of cases, the step size of 205edo is too small, as half of the distance between individual steps- that is, the distance between the center of a given step and the edge of that same step- is less than 3.5 cents, which is less than the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception. This results in individual steps blending into one another and thus being hard to tell apart- a problem which all EDOs higher than 171 have, and a significant deterrent for me. Secondly, while [[171edo]] itself also has better interval representation in a number of cases, the comma created by one of 159edo's three circles of fifths is smaller than that created by one of 205edo's five circles of fifths, or even that created by the 171edo circle of fifths, leading to 159edo generally having better representations of the 3-limit and 11-limit in general- something which is especially important in light of the aforementioned highly important functions of these two prime limits in particular.

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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.  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.
+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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 For example, I measure representation quality not only by taking the absolute error between just intervals and their tempered counterparts as per Hunt's system, but also the absolute errors in cents as they accumulate when tempered p-limit intervals are stacked, and the number of such intervals I can stack without the absolute error exceeding an unnoticeable comma's distance of 3.5 cents determines the quality of representation, and thus the portions of the harmonic lattice that can be sufficiently represented by any given EDO.  Furthermore, when the error accumulation between just intervals and their and tempered counterparts exceeds half an EDO step, contortion is considered to come into play, thus terminating the sequence of viable intervals for any given p-limit and limiting the portions of the harmonic lattice that can be considered viable in any given EDO.  Because the sequence of intervals in any given p-limit extends to infinity, it would be wise to use the odd-limit as way of limiting the amount of intervals used in grading the approximation quality of various EDOs.  Furthermore, as the octave is foundational in so many respects, we should set our odd-limit by way of selecting octaves of the Harmonic and Subharmonic systems as guides.  As the 3-limit accounts for every pitch in 12edo using intervals with odd-limits less than 1024, we shall use 1024 as the cutoff for how high odd-limits can go- yes, 1024 is an even number, but it is also a power of 2, thus rendering suitable as a divider between different categories of odd-limits.  This results in the following interval selections for representing p-limits up to 17- 3/2, 9/8, 27/16, 81/64, 243/128, and 729/512 for representing the 3-limit; 5/4, 25/16, 125/64, and 625/512 for representing the 5-limit; 7/4, 49/32 and 343/256 for representing the 7-limit; 11/8 and 121/64 for representing the 11-limit; 13/8 and 169/128 for representing the 13-limit; 17/16 and 289/256 for representing the 17-limit.
-When two EDOs are both given a "P" rating in the Hunt system for representation quality, the tie between them is broken by which EDO-tempered version has the smaller absolute error.  Furthermore, when the best representation of an interval in a given p-limit sequence cannot be reached by stacking tempered versions of the preceding intervals in that same p-limit sequence, the disconnected interval and any intervals following it in the same sequence are disqualified under my standards, no matter how good their representation rating in the Hunt system is.
+When two EDOs are both given a "P" rating in the Hunt System for representation quality, the tie between them is broken by which EDO-tempered version has the smaller absolute error.  Furthermore, when the best representation of an interval in a given p-limit sequence cannot be reached by stacking tempered versions of the preceding intervals in that same p-limit sequence, the disconnected interval and any intervals following it in the same sequence are disqualified under my standards, no matter how good their representation rating in the Hunt system is.
 == Choice of EDO for Microtonal Systems ==
 While Hunt's microtonal system is based on [[205edo]], my microtonal system is built on [[159edo]].  Why this difference?  Well, even though 205edo has better interval representation in a number of cases, the step size of 205edo is too small, as half of the distance between individual steps- that is, the distance between the center of a given step and the edge of that same step- is less than 3.5 cents, which is less than the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception.  This results in individual steps blending into one another and thus being hard to tell apart- a problem which all EDOs higher than 171 have, and a significant deterrent for me.  Secondly, while [[171edo]] itself also has better interval representation in a number of cases, the comma created by one of 159edo's three circles of fifths is smaller than that created by one of 205edo's five circles of fifths, or even that created by the 171edo circle of fifths, leading to 159edo generally having better representations of the 3-limit and 11-limit in general- something which is especially important in light of the aforementioned highly important functions of these two prime limits in particular.