User talk:Godtone: Difference between revisions

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For the record, part of the reason I'm limiting myself to chains of prime intervals at the moment is because judging from my own exploration of Alpharabian tuning, pure prime chains seem to have a way of acting as the borders for the tuning space of the various combinations of the primes in question. When two primes come together via telicity, the tuning space for combinations of those two primes seems to be finite, and thus, more manageable- on one corner is the unison, and on the other corner is the place where the two primes come together. Aside from this, the other part of the reason I'm limiting myself to pure prime chains is that in some respects, I haven't gotten around to those combinations yet- after all, I need to start with the basics of the concept first. It is true that there are less-straight paths available in the harmonic lattice, but when you want to return to the initial Tonic, as I myself often do, those less-straight paths are often more difficult to navigate, especially when you're dealing with higher primes in higher EDOs- I know this from experience, as I really like working in 159edo. Telicity gives easier-to-navigate paths for modulation, and sometimes, those paths are quite unexpected. For example, suppose you want to modulate down by a 32/27 minor third from your initial Tonic, but you know that the most expected way to get there is by chains of 3/2 fifths- well, it turns out that the nexus comma, which is unnoticeable and thus has a pretty high telicity range, joins the 11/8 prime chain together with the 3/2 prime chain at just that particular point, thus, going up by a chain of six 11/8 intervals allows you to reach the note at 32/27 below your original tonic by unexpected means. From there, you can simply modulate by a chain of perfect 3/2 fifths back to your original Tonic. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 04:32, 22 January 2021 (UTC)

Hi. To my understanding, the idea of "telicity" is very related to the idea of a "circle of n'ths" where "n" is some interval of interest. Specifically, it's connectivity between two such circles where both circles are of primes, and where the patent val for an EDO agrees with this connection. Is that correct?

:Hi. To my understanding, the idea of "telicity" is very related to the idea of a "circle of n'ths" where "n" is some interval of interest. Specifically, it's connectivity between two such circles where both circles are of primes, and where the patent val for an EDO agrees with this connection. Is that correct?

In terms of circles of intervals, my current favourite EDO is [[80edo|80 EDO]] which has a lot of amazingly strong circles that close at the unison when octave-reduced, and where some of these generate all of 80 EDO while others generate sub-EDOs of 80 EDO, although that's just one reason I like 80 EDO. The intervals of significance that generate the entirety of 80 EDO - with less than half a step of error left over - are [[11/10]], [[39/38]], [[17/16]] and [[9/7]] (in order of increasing error). (116/115 is a very good and consistent approximation of 1\80, but it accrues a little too much error to be included in that restriction.) Remarkable commas tempered involving these intervals are (9/7)^3/(17/8) and (9/7)/(11/10)^2/(17/16), with 39/38 instead being linked to the 10 EDO subset being a circle of [[16/13]]'s through (39/38)(17/16)^2/(16/13) and providing a high accuracy "skeleton" for the 19-prime-limit. As you seem to be interested in [[[159edo|159 EDO]], I did notice that it is almost exactly half of that, due to 3\80 being very close in size to 2\53 to the extent that you can use 80 ED8 as an alternative tuning of 53 ED4, with both representing the 2.3.5.13.19 subgroup.

:In terms of circles of intervals, my current favourite EDO is [[80edo|80 EDO]] which has a lot of amazingly strong circles that close at the unison when octave-reduced, and where some of these generate all of 80 EDO while others generate sub-EDOs of 80 EDO, although that's just one reason I like 80 EDO. The intervals of significance that generate the entirety of 80 EDO - with less than half a step of error left over - are [[11/10]], [[39/38]], [[17/16]] and [[9/7]] (in order of increasing error). (116/115 is a very good and consistent approximation of 1\80, but it accrues a little too much error to be included in that restriction.) Remarkable commas tempered involving these intervals are (9/7)^3/(17/8) and (9/7)/(11/10)^2/(17/16), with 39/38 instead being linked to the 10 EDO subset being a circle of [[16/13]]'s through (39/38)(17/16)^2/(16/13) and providing a high accuracy "skeleton" for the 19-prime-limit. As you seem to be interested in [[[159edo|159 EDO]], I did notice that it is almost exactly half of that, due to 3\80 being very close in size to 2\53 to the extent that you can use 80 ED8 as an alternative tuning of 53 ED4, with both representing the 2.3.5.13.19 subgroup.

I will also mention that [[87edo|87 EDO]] is very related to 80 EDO, but emphasizes accuracy in the 5- and 13-prime-limit as opposed to the 19-prime-limit of 80 EDO (and I'd argue 80 EDO deals generally well with the 29- (or at least 23-)prime-limit for its size), as both are tunings of the [[Tolermic family]] and its extensions up to the 17-prime-limit, and it may be interesting to you too as it has a [[29edo|29 EDO]] circle of fifths, but all primes up to and including 13 are one step flat of the nearest 29 EDO note, creating a very simple and elegant model of connectivity. 87 is (IMO) very recommendable if you want approximations of the 13-limit but still want all of the intervals to be musically meaningful to distinguish in the senses of [[User:Godtone#Colourful_EDOs|colour]] and melody.

:I will also mention that [[87edo|87 EDO]] is very related to 80 EDO, but emphasizes accuracy in the 5- and 13-prime-limit as opposed to the 19-prime-limit of 80 EDO (and I'd argue 80 EDO deals generally well with the 29- (or at least 23-)prime-limit for its size), as both are tunings of the [[Tolermic family]] and its extensions up to the 17-prime-limit, and it may be interesting to you too as it has a [[29edo|29 EDO]] circle of fifths, but all primes up to and including 13 are one step flat of the nearest 29 EDO note, creating a very simple and elegant model of connectivity. 87 is (IMO) very recommendable if you want approximations of the 13-limit but still want all of the intervals to be musically meaningful to distinguish in the senses of [[User:Godtone#Colourful_EDOs|colour]] and melody.

(Note: like 80, unfortunately, 87's worst prime is 7, but the error and relative error is less and in the opposite direction.)

--[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 08:45, 22 January 2021 (UTC)

:: I'm afraid your understanding of the concept of telicity is an oversimplification. While the concept of telicity does in fact include the idea of a "circle of n'ths" where "n" is some interval of interest, incomplete circles are still counted, hence the term "chains", and while the concept of telicity not only involves connectivity between multiple chains- specifically of primes- and the the patent val for an EDO agreeing with the connection, the fact remains that the [[direct mapping]] for every interval in both circles up to the point of connection must also agree with the connection. Stated more mathematically, where "N" is the number of steps in a given EDO, "r" is the ratio of an interval in one of the two circles, and "M" is the monzo of "r", the equation {N, round(log2(3)*N), round(log2(5)*N), round(log2(7)*N), round(log2(11)*N), ...}.{M} = round(log2(r)*N) ''must'' hold true along ''both'' prime chains up until the point of connection.

:: Just looking at 3-to-2 telicity, which, by definition, involves circles of fifths, the first seven EDOs that pass the test for this telicity are 2, 5, 12, 24, 53, 106, and 159. 80edo, despite being almost half of 159edo, fails the test, which is why I'm not interested in it, the same is true of 29edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 15:27, 22 January 2021 (UTC)

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 For the record, part of the reason I'm limiting myself to chains of prime intervals at the moment is because judging from my own exploration of Alpharabian tuning, pure prime chains seem to have a way of acting as the borders for the tuning space of the various combinations of the primes in question. When two primes come together via telicity, the tuning space for combinations of those two primes seems to be finite, and thus, more manageable- on one corner is the unison, and on the other corner is the place where the two primes come together. Aside from this, the other part of the reason I'm limiting myself to pure prime chains is that in some respects, I haven't gotten around to those combinations yet- after all, I need to start with the basics of the concept first. It is true that there are less-straight paths available in the harmonic lattice, but when you want to return to the initial Tonic, as I myself often do, those less-straight paths are often more difficult to navigate, especially when you're dealing with higher primes in higher EDOs- I know this from experience, as I really like working in 159edo. Telicity gives easier-to-navigate paths for modulation, and sometimes, those paths are quite unexpected. For example, suppose you want to modulate down by a 32/27 minor third from your initial Tonic, but you know that the most expected way to get there is by chains of 3/2 fifths- well, it turns out that the nexus comma, which is unnoticeable and thus has a pretty high telicity range, joins the 11/8 prime chain together with the 3/2 prime chain at just that particular point, thus, going up by a chain of six 11/8 intervals allows you to reach the note at 32/27 below your original tonic by unexpected means. From there, you can simply modulate by a chain of perfect 3/2 fifths back to your original Tonic. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 04:32, 22 January 2021 (UTC)
-Hi. To my understanding, the idea of "telicity" is very related to the idea of a "circle of n'ths" where "n" is some interval of interest. Specifically, it's connectivity between two such circles where both circles are of primes, and where the patent val for an EDO agrees with this connection. Is that correct?<br/>
+:Hi. To my understanding, the idea of "telicity" is very related to the idea of a "circle of n'ths" where "n" is some interval of interest. Specifically, it's connectivity between two such circles where both circles are of primes, and where the patent val for an EDO agrees with this connection. Is that correct?<br/>
-In terms of circles of intervals, my current favourite EDO is [[80edo|80 EDO]] which has a lot of amazingly strong circles that close at the unison when octave-reduced, and where some of these generate all of 80 EDO while others generate sub-EDOs of 80 EDO, although that's just one reason I like 80 EDO. The intervals of significance that generate the entirety of 80 EDO - with less than half a step of error left over - are [[11/10]], [[39/38]], [[17/16]] and [[9/7]] (in order of increasing error). (116/115 is a very good and consistent approximation of 1\80, but it accrues a little too much error to be included in that restriction.) Remarkable commas tempered involving these intervals are (9/7)^3/(17/8) and (9/7)/(11/10)^2/(17/16), with 39/38 instead being linked to the 10 EDO subset being a circle of [[16/13]]'s through (39/38)(17/16)^2/(16/13) and providing a high accuracy "skeleton" for the 19-prime-limit.  As you seem to be interested in [[[159edo|159 EDO]], I did notice that it is almost exactly half of that, due to 3\80 being very close in size to 2\53 to the extent that you can use 80 ED8 as an alternative tuning of 53 ED4, with both representing the 2.3.5.13.19 subgroup.<br/>
+:In terms of circles of intervals, my current favourite EDO is [[80edo|80 EDO]] which has a lot of amazingly strong circles that close at the unison when octave-reduced, and where some of these generate all of 80 EDO while others generate sub-EDOs of 80 EDO, although that's just one reason I like 80 EDO. The intervals of significance that generate the entirety of 80 EDO - with less than half a step of error left over - are [[11/10]], [[39/38]], [[17/16]] and [[9/7]] (in order of increasing error). (116/115 is a very good and consistent approximation of 1\80, but it accrues a little too much error to be included in that restriction.) Remarkable commas tempered involving these intervals are (9/7)^3/(17/8) and (9/7)/(11/10)^2/(17/16), with 39/38 instead being linked to the 10 EDO subset being a circle of [[16/13]]'s through (39/38)(17/16)^2/(16/13) and providing a high accuracy "skeleton" for the 19-prime-limit.  As you seem to be interested in [[[159edo|159 EDO]], I did notice that it is almost exactly half of that, due to 3\80 being very close in size to 2\53 to the extent that you can use 80 ED8 as an alternative tuning of 53 ED4, with both representing the 2.3.5.13.19 subgroup.<br/>
-I will also mention that [[87edo|87 EDO]] is very related to 80 EDO, but emphasizes accuracy in the 5- and 13-prime-limit as opposed to the 19-prime-limit of 80 EDO (and I'd argue 80 EDO deals generally well with the 29- (or at least 23-)prime-limit for its size), as both are tunings of the [[Tolermic family]] and its extensions up to the 17-prime-limit, and it may be interesting to you too as it has a [[29edo|29 EDO]] circle of fifths, but all primes up to and including 13 are one step flat of the nearest 29 EDO note, creating a very simple and elegant model of connectivity. 87 is (IMO) very recommendable if you want approximations of the 13-limit but still want all of the intervals to be musically meaningful to distinguish in the senses of [[User:Godtone#Colourful_EDOs|colour]] and melody.
+:I will also mention that [[87edo|87 EDO]] is very related to 80 EDO, but emphasizes accuracy in the 5- and 13-prime-limit as opposed to the 19-prime-limit of 80 EDO (and I'd argue 80 EDO deals generally well with the 29- (or at least 23-)prime-limit for its size), as both are tunings of the [[Tolermic family]] and its extensions up to the 17-prime-limit, and it may be interesting to you too as it has a [[29edo|29 EDO]] circle of fifths, but all primes up to and including 13 are one step flat of the nearest 29 EDO note, creating a very simple and elegant model of connectivity. 87 is (IMO) very recommendable if you want approximations of the 13-limit but still want all of the intervals to be musically meaningful to distinguish in the senses of [[User:Godtone#Colourful_EDOs|colour]] and melody.
 (Note: like 80, unfortunately, 87's worst prime is 7, but the error and relative error is less and in the opposite direction.)<br/>
 --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 08:45, 22 January 2021 (UTC)
+:: I'm afraid your understanding of the concept of telicity is an oversimplification.  While the concept of telicity does in fact include the idea of a "circle of n'ths" where "n" is some interval of interest, incomplete circles are still counted, hence the term "chains", and while the concept of telicity not only involves connectivity between multiple chains- specifically of primes- and the the patent val for an EDO agreeing with the connection, the fact remains that the [[direct mapping]] for every interval in both circles up to the point of connection must also agree with the connection.  Stated more mathematically, where "N" is the number of steps in a given EDO, "r" is the ratio of an interval in one of the two circles, and "M" is the monzo of "r", the equation {N, round(log2(3)*N), round(log2(5)*N), round(log2(7)*N), round(log2(11)*N), ...}.{M} = round(log2(r)*N) ''must'' hold true along ''both'' prime chains up until the point of connection.
+:: Just looking at 3-to-2 telicity, which, by definition, involves circles of fifths, the first seven EDOs that pass the test for this telicity are 2, 5, 12, 24, 53, 106, and 159.  80edo, despite being almost half of 159edo, fails the test, which is why I'm not interested in it, the same is true of 29edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 15:27, 22 January 2021 (UTC)