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

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Of course, there are more examples of things that need to be reevaluated in light of the existence of both Bass-Up and Treble-Down tonality, however, I cannot begin to cover all of these things here on this page. Suffice to say, however, that when one looks at the big picture, one will see that Treble-Down tonality is the exact mirror image of the more conventional Bass-Up tonality, a fact which lends to interesting and unexpected musical possibilities that are not present in more conventional systems like those of Hunt.

== Measuring EDO Approximation Quality ==

Hunt's system measures EDO approximation quality by taking the absolute errors between just intervals and their tempered counterparts, rounding them off, and assigning them ratings based on how many cents they differ once the error values have been rounded to the nearest whole number of cents, then averaging the values together in order to rate how a given EDO represents a random set of intervals. However, while there is some merit to most of Hunt's method, averaging the values together in order to rate how a given EDO represents a random set of intervals is not the best option, as not all possible intervals are good representations of any given p-limit, and averaging everything together is not the best option for grading a given EDO's representation of any given p-limit, especially as this system fails to take contortion into account. Rather, the standards for measuring EDO approximation quality need to be more strict.

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.

== 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, first of all, 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.

While Hunt's microtonal system is based on [[205edo]], my microtonal system is built on [[159edo]]. Why this difference? Well, first of all, 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 common to all EDOs higher than 171edo.

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 Of course, there are more examples of things that need to be reevaluated in light of the existence of both Bass-Up and Treble-Down tonality, however, I cannot begin to cover all of these things here on this page.  Suffice to say, however, that when one looks at the big picture, one will see that Treble-Down tonality is the exact mirror image of the more conventional Bass-Up tonality, a fact which lends to interesting and unexpected musical possibilities that are not present in more conventional systems like those of Hunt.
+== Measuring EDO Approximation Quality ==
+Hunt's system measures EDO approximation quality by taking the absolute errors between just intervals and their tempered counterparts, rounding them off, and assigning them ratings based on how many cents they differ once the error values have been rounded to the nearest whole number of cents, then averaging the values together in order to rate how a given EDO represents a random set of intervals.  However, while there is some merit to most of Hunt's method, averaging the values together in order to rate how a given EDO represents a random set of intervals is not the best option, as not all possible intervals are good representations of any given p-limit, and averaging everything together is not the best option for grading a given EDO's representation of any given p-limit, especially as this system fails to take contortion into account.  Rather, the standards for measuring EDO approximation quality need to be more strict.
+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.
 == 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, first of all, 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.
+While Hunt's microtonal system is based on [[205edo]], my microtonal system is built on [[159edo]].  Why this difference?  Well, first of all, 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 common to all EDOs higher than 171edo.