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

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== Navigational Primes and the Parachromatic-Paradiatonic Contrast ==

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]]. Not only are the traditional key signatures all related to each other along an 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 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".

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]]. Not only are the traditional key signatures all related to each other along an 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 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"- specifically, we can refer to it as the "Diatonic 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, 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.

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When we look at the 11-limit, we again need to draw the line as to how far we go as continuing beyond a certain point is impractical. However, we need to take note of the fact that while the 3-limit works with whole tones and semitones, the 11-limit works with quartertones and semitones. Therefore, the most direct place of comparison between the 3-limit and the 11-limit is in their handling of semitones. However, since the fundamental interval of the 11-limit is 11/8, we should not go into our examination of the 11-limit expecting the same results as for the 3-limit. Since we have already established that is 11/8 a paramajor fourth, we should treat it as such here, and furthermore, we should recall that a stack of two 11/8 paramajor fourths is equal to a 121/64 major seventh, as per the way paradiatonic intervals and parachromatic intervals relate to diatonic and chromatic intervals. So, if we take things a step further and continue on stacking paramajor fourths, what happens? Well, stacking a third paramajor fourth onto 121/64 and octave-reducing the reulting interval yields 1331/1024, a supermajor third in terms of the classification system we have established earlier, and adding a fourth paramajor fourth yields 14641/8192- a type of augmented sixth. Adding a fifth paramajor fourth to the stack and octave reducing the resulting interval yields the complicated 161051/131072- a type of second that differs from 9/8 by 161051/147456, which, in turn, is smaller than a stack of three 33/32 Parachromatic Quartertones by a rastma. Thus, for the purposes of this discussion, we will refer to 161051/131072 as a type of "Sesquiaugmented Second". Finally, stacking six paramajor fourths and octave-reducing the resulting interval yields 1771561/1048576- a type of double augmented fifth- which differs from 27/16 by only a nexuma.

Judging from this, it seems that there is indeed a clear sequence of intervals along the 11-limit, with every other member in this sequence being the octave complement of a stack of Alpharabian Diatonic Semitones. As the 11-limit handles stacks of Alpharabian Diatonic Semitones in much the same way that the 3-limit handles stacks of Pythagorean diatonic semitones, it can thus be argued that the 11-limit meets the standards set by the 3-limit in this respect. In fact, since the Alpharabian and Betarabian Semitones are actually closer to half of a whole tone that either one of the 3-limit semitones- and especially since the 11-limit's version of a double sharp fifth is only an unnoticeable comma's distance away from the 3-limit's major sixth- it can be argued that the 11-limit passes the semitone test with flying colors. With this information in hand, we can now safely assume that the 11-limit ''does'' is fact, carry the function of a navigational axis. Between this, and the previous mathematical confirmation of the 11-limit as being the best p-limit for representing quartertones, we can now safely say that there is indeed sufficient merit to the idea of the 11-limit being considered a 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.

Judging from this, it seems that there is indeed a clear sequence of intervals along the 11-limit, with every other member in this sequence being the octave complement of a stack of Alpharabian Diatonic Semitones. As the 11-limit handles stacks of Alpharabian Diatonic Semitones in much the same way that the 3-limit handles stacks of Pythagorean diatonic semitones, it can thus be argued that the 11-limit meets the standards set by the 3-limit in this respect. In fact, since the Alpharabian and Betarabian Semitones are actually closer to half of a whole tone that either one of the 3-limit semitones- and especially since the 11-limit's version of a double sharp fifth is only an unnoticeable comma's distance away from the 3-limit's major sixth- it can be argued that the 11-limit passes the semitone test with flying colors. With this information in hand, we can now safely assume that the 11-limit ''does'' is fact, carry the function of a navigational axis. Between this, and the previous mathematical confirmation of the 11-limit as being the best p-limit for representing quartertones, we can now safely say that there is indeed sufficient merit to the idea of the 11-limit being considered a navigational prim- in fact, we can now refer to it as the "Paradiatonic 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.

== Measuring EDO Approximation Quality ==

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 == Navigational Primes and the Parachromatic-Paradiatonic Contrast ==
-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]].  Not only are the traditional key signatures all related to each other along an 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 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".
+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]].  Not only are the traditional key signatures all related to each other along an 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 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"- specifically, we can refer to it as the "Diatonic 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, 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.
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 When we look at the 11-limit, we again need to draw the line as to how far we go as continuing beyond a certain point is impractical.  However, we need to take note of the fact that while the 3-limit works with whole tones and semitones, the 11-limit works with quartertones and semitones.  Therefore, the most direct place of comparison between the 3-limit and the 11-limit is in their handling of semitones.  However, since the fundamental interval of the 11-limit is 11/8, we should not go into our examination of the 11-limit expecting the same results as for the 3-limit.  Since we have already established that is 11/8 a paramajor fourth, we should treat it as such here, and furthermore, we should recall that a stack of two 11/8 paramajor fourths is equal to a 121/64 major seventh, as per the way paradiatonic intervals and parachromatic intervals relate to diatonic and chromatic intervals.  So, if we take things a step further and continue on stacking paramajor fourths, what happens?  Well, stacking a third paramajor fourth onto 121/64 and octave-reducing the reulting interval yields 1331/1024, a supermajor third in terms of the classification system we have established earlier, and adding a fourth paramajor fourth yields 14641/8192- a type of augmented sixth.  Adding a fifth paramajor fourth to the stack and octave reducing the resulting interval yields the complicated 161051/131072- a type of second that differs from 9/8 by 161051/147456, which, in turn, is smaller than a stack of three 33/32 Parachromatic Quartertones by a rastma.  Thus, for the purposes of this discussion, we will refer to 161051/131072 as a type of "Sesquiaugmented Second".  Finally, stacking six paramajor fourths and octave-reducing the resulting interval yields 1771561/1048576- a type of double augmented fifth- which differs from 27/16 by only a nexuma.
-Judging from this, it seems that there is indeed a clear sequence of intervals along the 11-limit, with every other member in this sequence being the octave complement of a stack of Alpharabian Diatonic Semitones.  As the 11-limit handles stacks of Alpharabian Diatonic Semitones in much the same way that the 3-limit handles stacks of Pythagorean diatonic semitones, it can thus be argued that the 11-limit meets the standards set by the 3-limit in this respect.  In fact, since the Alpharabian and Betarabian Semitones are actually closer to half of a whole tone that either one of the 3-limit semitones- and especially since the 11-limit's version of a double sharp fifth is only an unnoticeable comma's distance away from the 3-limit's major sixth- it can be argued that the 11-limit passes the semitone test with flying colors.  With this information in hand, we can now safely assume that the 11-limit ''does'' is fact, carry the function of a navigational axis.  Between this, and the previous mathematical confirmation of the 11-limit as being the best p-limit for representing quartertones, we can now safely say that there is indeed sufficient merit to the idea of the 11-limit being considered a 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.
+Judging from this, it seems that there is indeed a clear sequence of intervals along the 11-limit, with every other member in this sequence being the octave complement of a stack of Alpharabian Diatonic Semitones.  As the 11-limit handles stacks of Alpharabian Diatonic Semitones in much the same way that the 3-limit handles stacks of Pythagorean diatonic semitones, it can thus be argued that the 11-limit meets the standards set by the 3-limit in this respect.  In fact, since the Alpharabian and Betarabian Semitones are actually closer to half of a whole tone that either one of the 3-limit semitones- and especially since the 11-limit's version of a double sharp fifth is only an unnoticeable comma's distance away from the 3-limit's major sixth- it can be argued that the 11-limit passes the semitone test with flying colors.  With this information in hand, we can now safely assume that the 11-limit ''does'' is fact, carry the function of a navigational axis.  Between this, and the previous mathematical confirmation of the 11-limit as being the best p-limit for representing quartertones, we can now safely say that there is indeed sufficient merit to the idea of the 11-limit being considered a navigational prim- in fact, we can now refer to it as the "Paradiatonic 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.
 == Measuring EDO Approximation Quality ==