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

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Now, 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]]. This even extends to the fact that the standard sharp and flat accidentals modify the base note by an [[2187/2048|apotome]], and how 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 that have their roots in 12edo, and its pivotal role in laying the groundwork for key signatures means that it can be referred as a "navigational prime".

Meanwhile, when one comes from a background in 24edo as I have, and has even used quartertone-based keys signatures as I have, a second p-limit seems to join together with the 3-limit in defining the standard location and structure of the various notes and quartertone-based key signatures that one would see in 24edo~~. Judging on my findings- which I shall cover in the next section-~~ the 11-limit seems to be the best candidate for this second navigational prime despite the fact that the pure 11-limit is not capable of forming diatonic scales at all. Now, some may question the musical grounds for using quartertones in light of their dissonance, as well as the idea that there is any merit to the idea of the 11-limit being considered a navigational prime. Well, we should start with the reasons for considering quartertones musically important in the first place- namely the fact that quartertones are the most readily accessible among microtones, and that current research seems to show that quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches. On this basis, we can proceed to look at the musical functions of semitones, and then go on to define the musical functions of the quartertones themselves.

Meanwhile, when one comes from a background in 24edo as I have, and has even used quartertone-based keys signatures as I have, one sees that a second p-limit seems to join together with the 3-limit in defining the standard location and structure of the various notes and quartertone-based key signatures that one would see in 24edo, and the 11-limit seems to be the best candidate for this second navigational prime despite the fact that the pure 11-limit is not capable of forming diatonic scales at all. Now, some may question the musical grounds for using quartertones in light of their dissonance, as well as the idea that there is any merit to the idea of the 11-limit being considered a navigational prime. Well, we should start with the reasons for considering quartertones musically important in the first place- namely the fact that quartertones are the most readily accessible among microtones, and that current research seems to show that quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches. On this basis, we can proceed to look at the musical functions of semitones, and then go on to define the musical functions of the quartertones themselves.

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. The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo. As mentioned to me by [[KiteGiedraitis | Kite Giedraitis]] in [[Talk:159edo_notation#My_Second_Idea_for_a_Notation System|a conversation]] about this topic, there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and a chromatic semitone that is otherwise known as the apotome- which, when added together, add 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, Kite 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.

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 Now, 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]].  This even extends to the fact that the standard sharp and flat accidentals modify the base note by an [[2187/2048|apotome]], and how 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 that have their roots in 12edo, and its pivotal role in laying the groundwork for key signatures means that it can be referred as a "navigational prime".
-Meanwhile, when one comes from a background in 24edo as I have, and has even used quartertone-based keys signatures as I have, a second p-limit seems to join together with the 3-limit in defining the standard location and structure of the various notes and quartertone-based key signatures that one would see in 24edo.  Judging on my findings- which I shall cover in the next section- the 11-limit seems to be the best candidate for this second navigational prime despite the fact that the pure 11-limit is not capable of forming diatonic scales at all.  Now, some may question the musical grounds for using quartertones in light of their dissonance, as well as the idea that there is any merit to the idea of the 11-limit being considered a navigational prime.  Well, we should start with the reasons for considering quartertones musically important in the first place- namely the fact that quartertones are the most readily accessible among microtones, and that current research seems to show that quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches.  On this basis, we can proceed to look at the musical functions of semitones, and then go on to define the musical functions of the quartertones themselves.
+Meanwhile, when one comes from a background in 24edo as I have, and has even used quartertone-based keys signatures as I have, one sees that a second p-limit seems to join together with the 3-limit in defining the standard location and structure of the various notes and quartertone-based key signatures that one would see in 24edo, and the 11-limit seems to be the best candidate for this second navigational prime despite the fact that the pure 11-limit is not capable of forming diatonic scales at all.  Now, some may question the musical grounds for using quartertones in light of their dissonance, as well as the idea that there is any merit to the idea of the 11-limit being considered a navigational prime.  Well, we should start with the reasons for considering quartertones musically important in the first place- namely the fact that quartertones are the most readily accessible among microtones, and that current research seems to show that quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches.  On this basis, we can proceed to look at the musical functions of semitones, and then go on to define the musical functions of the quartertones themselves.
 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.  The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo.  As mentioned to me by [[KiteGiedraitis | Kite Giedraitis]] in [[Talk:159edo_notation#My_Second_Idea_for_a_Notation System|a conversation]] about this topic, there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and a chromatic semitone that is otherwise known as the apotome- which, when added together, add 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, Kite 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.