Alpharabian tuning: Difference between revisions

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== Basis ==

Many if not most musicians who are not microtonalists are acquainted with standard music notation, with its clefs and staves, key signatures and time signatures. However, when you take all of this into the microtonal realm, it becomes readily apparent that- in all of the most intuitive systems- it is the 3-limit that defines both the standard location and structure of the various standard notes and key signatures that one finds in [[12edo]]. Not only are the traditional key signatures all related to each other along a navigational axis formed by the 3-limit, but the standard sharp and flat accidentals modify the base note by an [[2187/2048|apotome]], and the double sharp and double flat accidentals modify the base note by two apotomes. Furthermore, it is the Pythagorean Diatonic Scales that arise as the standard variants for the various key signatures, as these are the simplest diatonic scales that can be formed with the 3-limit. Because the 3-limit is a prime that has all of this foundational functionality, it is naturally very important in musical systems, and its pivotal role in laying the groundwork for key signatures means that its significance is widely accepted. In addition to all this, most music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime. They also know that a diatonic semitone and a chromatic semitone add up to a whole tone. In 3-limit tuning, the diatonic semitone has a ratio of [[256/243]], and the corresponding chromatic semitone is the apotome- two intervals adding up to a 9/8 whole tone. Furthermore, in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by [[81/80]]. On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields 27/25- two additional diatonic semitones. On the other hand, subtracting 81/80 from the apotome yields 135/128, and subtracting another 81/80 yields 25/24- two additional chromatic semitones. When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone. In light of all this, it has been argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.

Many if not most musicians who are not microtonalists are acquainted with standard music notation, with its clefs and staves, key signatures and time signatures. However, when you take all of this into the microtonal realm, it becomes readily apparent that- in all of the most intuitive systems- it is the 3-limit that defines both the standard location and structure of the various standard notes and key signatures that one finds in [[12edo]]. Not only are the traditional key signatures all related to each other along a navigational axis formed by the 3-limit, but the standard sharp and flat accidentals modify the base note by an [[2187/2048|apotome]], and the double sharp and double flat accidentals modify the base note by two apotomes. Furthermore, it is the Pythagorean Diatonic Scales that arise as the standard variants for the various key signatures, as these are the simplest diatonic scales that can be formed with the 3-limit. Because the 3-limit is a prime that has all of this foundational functionality, it is naturally very important in musical systems, and its pivotal role in laying the groundwork for key signatures means that its significance is widely accepted.

In addition to all this, most music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime. They also know that a diatonic semitone and a chromatic semitone add up to a whole tone. In 3-limit tuning, the diatonic semitone has a ratio of [[256/243]], and the corresponding chromatic semitone is the apotome- two intervals adding up to a 9/8 whole tone. Furthermore, in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by [[81/80]]. On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields 27/25- two additional diatonic semitones. On the other hand, subtracting 81/80 from the apotome yields 135/128, and subtracting another 81/80 yields 25/24- two additional chromatic semitones. When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone. In light of all this, it has been argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.

Building on this 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 superprimes of some sort. 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.

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 == Basis ==
-Many if not most musicians who are not microtonalists are acquainted with standard music notation, with its clefs and staves, key signatures and time signatures.  However, when you take all of this into the microtonal realm, it becomes readily apparent that- in all of the most intuitive systems- it is the 3-limit that defines both the standard location and structure of the various standard notes and key signatures that one finds in [[12edo]].  Not only are the traditional key signatures all related to each other along a navigational axis formed by the 3-limit, but the standard sharp and flat accidentals modify the base note by an [[2187/2048|apotome]], and the double sharp and double flat accidentals modify the base note by two apotomes.  Furthermore, it is the Pythagorean Diatonic Scales that arise as the standard variants for the various key signatures, as these are the simplest diatonic scales that can be formed with the 3-limit.  Because the 3-limit is a prime that has all of this foundational functionality, it is naturally very important in musical systems, and its pivotal role in laying the groundwork for key signatures means that its significance is widely accepted.  In addition to all this, most music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime.  They also know that a diatonic semitone and a chromatic semitone add up to a whole tone.  In 3-limit tuning, the diatonic semitone has a ratio of [[256/243]], and the corresponding chromatic semitone is the apotome- two intervals adding up to a 9/8 whole tone.  Furthermore, in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by [[81/80]].  On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields 27/25- two additional diatonic semitones.  On the other hand, subtracting 81/80 from the apotome yields 135/128, and subtracting another 81/80 yields 25/24- two additional chromatic semitones.  When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone.  In light of all this, it has been argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.
+Many if not most musicians who are not microtonalists are acquainted with standard music notation, with its clefs and staves, key signatures and time signatures.  However, when you take all of this into the microtonal realm, it becomes readily apparent that- in all of the most intuitive systems- it is the 3-limit that defines both the standard location and structure of the various standard notes and key signatures that one finds in [[12edo]].  Not only are the traditional key signatures all related to each other along a navigational axis formed by the 3-limit, but the standard sharp and flat accidentals modify the base note by an [[2187/2048|apotome]], and the double sharp and double flat accidentals modify the base note by two apotomes.  Furthermore, it is the Pythagorean Diatonic Scales that arise as the standard variants for the various key signatures, as these are the simplest diatonic scales that can be formed with the 3-limit.  Because the 3-limit is a prime that has all of this foundational functionality, it is naturally very important in musical systems, and its pivotal role in laying the groundwork for key signatures means that its significance is widely accepted.
+In addition to all this, most music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime.  They also know that a diatonic semitone and a chromatic semitone add up to a whole tone.  In 3-limit tuning, the diatonic semitone has a ratio of [[256/243]], and the corresponding chromatic semitone is the apotome- two intervals adding up to a 9/8 whole tone.  Furthermore, in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by [[81/80]].  On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields 27/25- two additional diatonic semitones.  On the other hand, subtracting 81/80 from the apotome yields 135/128, and subtracting another 81/80 yields 25/24- two additional chromatic semitones.  When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone.  In light of all this, it has been argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second.
 Building on this 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 superprimes of some sort.  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.