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

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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".

However, this just covers the 11-limit's quartertones. Since two parachromatic quartertones add up to a chromatic quartertone, and since 33/32 is the primary parachromatic quartertone, we can safely assume that two 33/32 intervals adds up to a chromatic semitone of some sort, and indeed, [[1089/1024]] ''is'' a chromatic semitone under this definition, but as it differs from the apotome by the rastma, and we don't have the benefit of the "primary" versus "secondary" distinction here due to the apotome being a 3-limit interval, we should instead look to another source for a name. Since 33/32 is also called the "Al-Farabi Quartertone", and since Al-Farabi himself was also referred to as "Alpharabius" according to [https://en.wikipedia.org/wiki/Al-Farabi Wikipedia's article on him], we can use the term "Alpharabian" to refer the 11-limit semitones in the same way that we use "Pythagorean" to refer to the 3-limit semitones. Thus, just as the apotome can also be referred to as the "Pythagorean Chromatic Semitone", we can refer to 1089/1024 as the "Alpharabian Chromatic Semitone". Furthermore, just as there's an Alpharabian Chromatic Semitone, there is an "Alpharabian Diatonic Semitone" which adds up together with the Alpharabian Chromatic Semitone to create a 9/8 whole tone, and, when you take 9/8 and subtract 1089/1024, you arrive at [[128/121]] as the ratio for the Alpharabian Diatonic Semitone. Not only that, but just as there's the [[Pythagorean comma]] which forms the difference between a stack of two Pythagorean Diatonic Semitones and a 9/8 whole tone, so there is an "[[Alpharabian Comma]]" which forms the difference between a stack of two Alpharabian Diatonic Semitones and a 9/8 whole tone, and doing the math yields 131769/131072 as the ratio for the Alpharabian Comma.

However, this just covers the 11-limit's quartertones. Since two parachromatic quartertones add up to a chromatic quartertone, and since 33/32 is the primary parachromatic quartertone, we can safely assume that two 33/32 intervals adds up to a chromatic semitone of some sort, and indeed, [[1089/1024]] ''is'' a chromatic semitone under this definition, but as it differs from the apotome by the rastma, and we don't have the benefit of the "primary" versus "secondary" distinction here due to the apotome being a 3-limit interval, we should instead look to another source for a name. Since 33/32 is also called the "Al-Farabi Quartertone", and since Al-Farabi himself was also referred to as "Alpharabius" according to [https://en.wikipedia.org/wiki/Al-Farabi Wikipedia's article on him], we can use the term "Alpharabian" to refer the 11-limit semitones in the same way that we use "Pythagorean" to refer to the 3-limit semitones. Thus, just as the apotome can also be referred to as the "Pythagorean Chromatic Semitone", we can refer to 1089/1024 as the "Alpharabian Chromatic Semitone". Furthermore, just as there's an Alpharabian Chromatic Semitone, there is an "Alpharabian Diatonic Semitone" which adds up together with the Alpharabian Chromatic Semitone to create a 9/8 whole tone, and, when you take 9/8 and subtract 1089/1024, you arrive at [[128/121]] as the ratio for the Alpharabian Diatonic Semitone. Not only that, but just as there's the [[Pythagorean comma]] which forms the difference between a stack of two Pythagorean Diatonic Semitones and a 9/8 whole tone, so there is an "[[Alpharabian comma|Alpharabian Comma]]" which forms the difference between a stack of two Alpharabian Diatonic Semitones and a 9/8 whole tone, and doing the math yields 131769/131072 as the ratio for the Alpharabian Comma.

== Measuring EDO Approximation Quality ==

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 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".
-However, this just covers the 11-limit's quartertones.  Since two parachromatic quartertones add up to a chromatic quartertone, and since 33/32 is the primary parachromatic quartertone, we can safely assume that two 33/32 intervals adds up to a chromatic semitone of some sort, and indeed, [[1089/1024]] ''is'' a chromatic semitone under this definition, but as it differs from the apotome by the rastma, and we don't have the benefit of the "primary" versus "secondary" distinction here due to the apotome being a 3-limit interval, we should instead look to another source for a name.  Since 33/32 is also called the "Al-Farabi Quartertone", and since Al-Farabi himself was also referred to as "Alpharabius" according to [https://en.wikipedia.org/wiki/Al-Farabi Wikipedia's article on him], we can use the term "Alpharabian" to refer the 11-limit semitones in the same way that we use "Pythagorean" to refer to the 3-limit semitones.  Thus, just as the apotome can also be referred to as the "Pythagorean Chromatic Semitone", we can refer to 1089/1024 as the "Alpharabian Chromatic Semitone". Furthermore, just as there's an Alpharabian Chromatic Semitone, there is an "Alpharabian Diatonic Semitone" which adds up together with the Alpharabian Chromatic Semitone to create a 9/8 whole tone, and, when you take 9/8 and subtract 1089/1024, you arrive at [[128/121]] as the ratio for the Alpharabian Diatonic Semitone.  Not only that, but just as there's the [[Pythagorean comma]] which forms the difference between a stack of two Pythagorean Diatonic Semitones and a 9/8 whole tone, so there is an "[[Alpharabian Comma]]" which forms the difference between a stack of two Alpharabian Diatonic Semitones and a 9/8 whole tone, and doing the math yields 131769/131072 as the ratio for the Alpharabian Comma.
+However, this just covers the 11-limit's quartertones.  Since two parachromatic quartertones add up to a chromatic quartertone, and since 33/32 is the primary parachromatic quartertone, we can safely assume that two 33/32 intervals adds up to a chromatic semitone of some sort, and indeed, [[1089/1024]] ''is'' a chromatic semitone under this definition, but as it differs from the apotome by the rastma, and we don't have the benefit of the "primary" versus "secondary" distinction here due to the apotome being a 3-limit interval, we should instead look to another source for a name.  Since 33/32 is also called the "Al-Farabi Quartertone", and since Al-Farabi himself was also referred to as "Alpharabius" according to [https://en.wikipedia.org/wiki/Al-Farabi Wikipedia's article on him], we can use the term "Alpharabian" to refer the 11-limit semitones in the same way that we use "Pythagorean" to refer to the 3-limit semitones.  Thus, just as the apotome can also be referred to as the "Pythagorean Chromatic Semitone", we can refer to 1089/1024 as the "Alpharabian Chromatic Semitone". Furthermore, just as there's an Alpharabian Chromatic Semitone, there is an "Alpharabian Diatonic Semitone" which adds up together with the Alpharabian Chromatic Semitone to create a 9/8 whole tone, and, when you take 9/8 and subtract 1089/1024, you arrive at [[128/121]] as the ratio for the Alpharabian Diatonic Semitone.  Not only that, but just as there's the [[Pythagorean comma]] which forms the difference between a stack of two Pythagorean Diatonic Semitones and a 9/8 whole tone, so there is an "[[Alpharabian comma|Alpharabian Comma]]" which forms the difference between a stack of two Alpharabian Diatonic Semitones and a 9/8 whole tone, and doing the math yields 131769/131072 as the ratio for the Alpharabian Comma.
 == Measuring EDO Approximation Quality ==