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

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== Navigational Primes and Key Signatures ==

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. When one comes from a background in 24edo as I have, and has even used quartertone-based keys signatures as I have, it becomes apparent that the 11-limit joins 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. Because the 3-limit and the 11-limit are the primes that have all of this foundational functionality, they are naturally very important in musical systems that have their roots in 24edo, and their pivotal role in laying the groundwork for key signatures means that they can be referred as the "navigational primes". In both the traditional and quartertone-based key signatures, 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, and the pure 11-limit is not capable of forming diatonic scales. 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 ~~than~~ the standard key signatures themselves.

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. When one comes from a background in 24edo as I have, and has even used quartertone-based keys signatures as I have, it becomes apparent that the 11-limit joins 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.

Because the 3-limit and the 11-limit are the primes that have all of this foundational functionality, they are naturally very important in musical systems that have their roots in 24edo, and their pivotal role in laying the groundwork for key signatures means that they can be referred as the "navigational primes". In both the traditional and quartertone-based key signatures, 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, and the pure 11-limit is not capable of forming diatonic scales. 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. For the sake of simplicity, it can generally be assumed that that higher primes are ill-suited for serving as anything other than accidentals.

== Measuring EDO Approximation Quality ==

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== Choice of EDO for Microtonal Systems ==

While Hunt's microtonal system is based on [[205edo]], my microtonal system is built on [[159edo]]. Why this difference? Well, even though 205edo has better interval representation in a number of cases, the step size of 205edo is too small, as half of the distance between individual steps- that is, the distance between the center of a given step and the edge of that same step- is less than 3.5 cents, which is less than the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception. This results in individual steps blending into one another and thus being hard to tell apart- a problem which all EDOs higher than 171 have, and a significant deterrent for me. Secondly, while [[171edo]] itself also has better representation in a number of cases, the comma created by one of 159edo's three circles of fifths is smaller than that created by one of 205edo's five circles of fifths, or even that created by the 171edo circle of fifths~~- yes~~, ~~closing the circle~~ of fifths with the least amount of error possible was one consideration. There's also the matter of good 11-limit representation in particular, and 159edo surpasses both 171edo and 205edo on this point, and as if that weren't enough, I've since found out that the deal is made even sweeter by the fact that the 3-limit and ~~the~~ 11-limit ~~are joined~~ in ~~this EDO by~~ the ~~tempering out~~ of ~~1771561/1769472- a feature that is absent from both 171edo and 205edo~~.

While Hunt's microtonal system is based on [[205edo]], my microtonal system is built on [[159edo]]. Why this difference? Well, even though 205edo has better interval representation in a number of cases, the step size of 205edo is too small, as half of the distance between individual steps- that is, the distance between the center of a given step and the edge of that same step- is less than 3.5 cents, which is less than the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception. This results in individual steps blending into one another and thus being hard to tell apart- a problem which all EDOs higher than 171 have, and a significant deterrent for me. Secondly, while [[171edo]] itself also has better interval representation in a number of cases, the comma created by one of 159edo's three circles of fifths is smaller than that created by one of 205edo's five circles of fifths, or even that created by the 171edo circle of fifths, leading to 159edo generally having better representations of the 3-limit and 11-limit in general- something which is especially important in light of the aforementioned highly important functions of these two prime limits in particular.

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 == Navigational Primes and Key Signatures ==
-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.  When one comes from a background in 24edo as I have, and has even used quartertone-based keys signatures as I have, it becomes apparent that the 11-limit joins 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.  Because the 3-limit and the 11-limit are the primes that have all of this foundational functionality, they are naturally very important in musical systems that have their roots in 24edo, and their pivotal role in laying the groundwork for key signatures means that they can be referred as the "navigational primes".  In both the traditional and quartertone-based key signatures, 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, and the pure 11-limit is not capable of forming diatonic scales.  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 than the standard key signatures themselves.
+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.  When one comes from a background in 24edo as I have, and has even used quartertone-based keys signatures as I have, it becomes apparent that the 11-limit joins 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.
+Because the 3-limit and the 11-limit are the primes that have all of this foundational functionality, they are naturally very important in musical systems that have their roots in 24edo, and their pivotal role in laying the groundwork for key signatures means that they can be referred as the "navigational primes".  In both the traditional and quartertone-based key signatures, 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, and the pure 11-limit is not capable of forming diatonic scales.  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.  For the sake of simplicity, it can generally be assumed that that higher primes are ill-suited for serving as anything other than accidentals.
 == Measuring EDO Approximation Quality ==
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 == Choice of EDO for Microtonal Systems ==
-While Hunt's microtonal system is based on [[205edo]], my microtonal system is built on [[159edo]].  Why this difference?  Well, even though 205edo has better interval representation in a number of cases, the step size of 205edo is too small, as half of the distance between individual steps- that is, the distance between the center of a given step and the edge of that same step- is less than 3.5 cents, which is less than the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception.  This results in individual steps blending into one another and thus being hard to tell apart- a problem which all EDOs higher than 171 have, and a significant deterrent for me.  Secondly, while [[171edo]] itself also has better representation in a number of cases, the comma created by one of 159edo's three circles of fifths is smaller than that created by one of 205edo's five circles of fifths, or even that created by the 171edo circle of fifths- yes, closing the circle of fifths with the least amount of error possible was one consideration.  There's also the matter of good 11-limit representation in particular, and 159edo surpasses both 171edo and 205edo on this point, and as if that weren't enough, I've since found out that the deal is made even sweeter by the fact that the 3-limit and the 11-limit are joined in this EDO by the tempering out of 1771561/1769472- a feature that is absent from both 171edo and 205edo.
+While Hunt's microtonal system is based on [[205edo]], my microtonal system is built on [[159edo]].  Why this difference?  Well, even though 205edo has better interval representation in a number of cases, the step size of 205edo is too small, as half of the distance between individual steps- that is, the distance between the center of a given step and the edge of that same step- is less than 3.5 cents, which is less than the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 JND] of human pitch perception.  This results in individual steps blending into one another and thus being hard to tell apart- a problem which all EDOs higher than 171 have, and a significant deterrent for me.  Secondly, while [[171edo]] itself also has better interval representation in a number of cases, the comma created by one of 159edo's three circles of fifths is smaller than that created by one of 205edo's five circles of fifths, or even that created by the 171edo circle of fifths, leading to 159edo generally having better representations of the 3-limit and 11-limit in general- something which is especially important in light of the aforementioned highly important functions of these two prime limits in particular.