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

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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.

== Delving into the 11-Limit ==

With the 11-limit established as a second navigational prime, we need to look more closely at its properties- something which most microtonal systems at the time of this writing have yet to do. First off, one will notice that a stack of two 33/32 quartertones falls short of an apotome by a comma called the [[243/242|rastma]], which means that the rastma, when not tempered out, is a very important "parasubchroma"- a term derived from [[Diatonic, Chromatic, Enharmonic, Subchromatic|subchroma]].

Because two parachromatic quartertones equals a chromatic semitone, and because two 33/32 quartertones fall short of an apotome by a rastma, we can deduce that other parachromatic quartertones- or, more generally "parachromas"- can be derived from 33/32 by adding or subtracting the rastma. When one adds the rastma to 33/32, one arrives at [[729/704]]- the parachromatic quartertone that adds up together with 33/32 to form an apotome. Since 33/32 is smaller than 729/704, and both can be described as "undecimal quartertones", we need to disambiguate them somehow, not to mention acknowledge the 11-limit's newfound status as a navigational prime. Thus, for purposes of continuing this discussion, we'll start referring to 33/32 as the "primary parachromatic quartertone", and from here on, we'll refer to [[729/704]] as the "secondary parachromatic quartertone". However, the rastma doesn't only derive other parachromatic intervals from 33/32, but also derives other paradiatonic intervals from [[4096/3993]]- which we shall refer to here as the "primary paradiatonic quartertone". For instance, if one subtracts the rastma from the primary paradiatonic quartertone, we get [[8192/8019]], which, when added to 33/32, yields [[256/243]]- the Pythagorean diatonic semitone. Because of this, and because 8192/8019 is derived from the primary paradiatonic quartertone, we shall refer to 8192/8019 as the "secondary paradiatonic quartertone".

== Measuring EDO Approximation Quality ==

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 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.
+== Delving into the 11-Limit ==
+With the 11-limit established as a second navigational prime, we need to look more closely at its properties- something which most microtonal systems at the time of this writing have yet to do.  First off, one will notice that a stack of two 33/32 quartertones falls short of an apotome by a comma called the [[243/242|rastma]], which means that the rastma, when not tempered out, is a very important "parasubchroma"- a term derived from [[Diatonic, Chromatic, Enharmonic, Subchromatic|subchroma]].
+Because two parachromatic quartertones equals a chromatic semitone, and because two 33/32 quartertones fall short of an apotome by a rastma, we can deduce that other parachromatic quartertones- or, more generally "parachromas"- can be derived from 33/32 by adding or subtracting the rastma.  When one adds the rastma to 33/32, one arrives at [[729/704]]- the parachromatic quartertone that adds up together with 33/32 to form an apotome.  Since 33/32 is smaller than 729/704, and both can be described as "undecimal quartertones", we need to disambiguate them somehow, not to mention acknowledge the 11-limit's newfound status as a navigational prime.  Thus, for purposes of continuing this discussion, we'll start referring to 33/32 as the "primary parachromatic quartertone", and from here on, we'll refer to [[729/704]] as the "secondary parachromatic quartertone".  However, the rastma doesn't only derive other parachromatic intervals from 33/32, but also derives other paradiatonic intervals from [[4096/3993]]- which we shall refer to here as the "primary paradiatonic quartertone".  For instance, if one subtracts the rastma from the primary paradiatonic quartertone, we get [[8192/8019]], which, when added to 33/32, yields [[256/243]]- the Pythagorean diatonic semitone.  Because of this, and because 8192/8019 is derived from the primary paradiatonic quartertone, we shall refer to 8192/8019 as the "secondary paradiatonic quartertone".
 == Measuring EDO Approximation Quality ==