Talk:159edo: Difference between revisions

(51 intermediate revisions by 4 users not shown)

Line 1:

== Approximate errors ==

Okay... I have a list of the approximate errors in cents for 159edo's approximations of certain prime intervals:

Line 65:

Line 67:

:: I don't know how well my response to Flora manages to solve the problem you just stated, but here's to hoping... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 18:00, 7 January 2021 (UTC)

:: Is it me, or can it be said that "~~Complete~~ Boolean Consistency" means being able to go from the unison ~~all the way around~~ a set of nodes ~~and back~~ to the unison without the relative error reaching above the 50% marker? If my speculation is correct, then we're talking about a different type of "consistency" than the kind that Flora's talking about. It's like comparing apples and oranges in a way- apples and oranges are both fruit but have a lot of differences between them. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 18:11, 7 January 2021 (UTC)

:: Is it me, or can it be said that "Boolean Consistency" means being able to go from the unison through a set of nodes in one p-limit to connect with an interval of a lower p-limit without the relative error reaching above the 50% marker? If so, then "Boolean Consistency" for the 3-limit means being able to connect with the pitch class used as the [[unison]] and [[octave]] a second time after going around a complete set of nodes without the relative error reaching above the 50% marker. If my speculation is correct, then we're talking about a different type of "consistency" than the kind that Flora's talking about. It's like comparing apples and oranges in a way- apples and oranges are both fruit but have a lot of differences between them. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 18:11, 7 January 2021 (UTC)

:: The consistency is defined on "an interval set S". There's not a rule against prime limit but that doesn't make sense since it simply can't be consistent. I remember reading about an "n-consistent" somewhere, in which 53edo is hundreds-consistent in the 3-limit as you can stack hundreds of 3's without relative error reaching over 50%. That might be what you look for. Somebody in the FB group also proposed another "n-consistent", in which the n is something substituting 50%, similar to relative error. Another fascinating idea is the ''pepper ambiguity'' (forgive me for saving links in talk pages) – its definition is not completely clear to me and I hope to work on it soon. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 18:31, 7 January 2021 (UTC)

::: It looks like there are multiple types of n-consistency being proposed even within the Facebook group, so yes, we need a discussion on this. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 19:01, 7 January 2021 (UTC)

::: I must point out that the degree of n-consistency that I look for on "an interval set S" in the 3-limit has everything to do with whether or not you can go around a complete circle of fifths in a given EDO without accumulating a relative error of 50% or more. That's the specific type of n-consistency that I think I can regard as "complete". --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 19:07, 7 January 2021 (UTC)

:::: This kind of consistency ("complete circle of fifths") seems problematic to me: How will you generalize these rings to other prime intervals? Also, aren't you interested in combinations of multiple prime dimensions (besides 2, of course)? --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 20:22, 7 January 2021 (UTC)

::::: The kind of n-consistency I'm alluding to involves being able to go from the unison through a set of nodes in one p-limit to connect with an interval of a lower p-limit without the relative error reaching or exceeding the 50% marker. Since the 3-prime can only connect with the 2-prime in this fashion, and since the 2-prime simply results in manifestations of the unison at different registers- meaning that the unison is the only available target- that means that the 3-prime requires a complete circle of fifths without accumulating 50% relative error or more to achieve a form of "complete consistency". However, higher primes have more options for a form of "complete consistency". For instance, the 11-prime in 159edo connects with the 3-prime easily without breaching the 50% relative error marker by means of tempering out the nexus comma, and similarly, the 5-prime connects with the 3-prime by means of tempering out the schisma. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:41, 7 January 2021 (UTC)

::::: As to combinations of multiple prime dimensions, I find these to be largely of secondary importance, but to be fair, they are subject to the same constraints- they must be able to connect to a p-limit lower than the lowest p-limit that is directly involved in the combination in question. For example, stacks of 14/13 trienthirds connect with the 5-prime by means of tempering out the cantonisma- the factor of 2 in 14 is trivial for this since the 2-prime simply results in manifestations of the unison at different registers. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:36, 7 January 2021 (UTC)

::::: Now this is fascinating... According to my calculations, subtracting the [[symbiosma]] from [[7/4]] results in an interval with the prime factorization of (3^9)/(2^10*11), so it looks like the symbiosma bridges the 7-prime and a combination of 3 and 11. Perhaps I should fix my definition of "complete consistency" by adding the following condition- if one is able to go from the unison through a set of nodes in one p-limit to connect with an interval made purely from a combination of two other primes, complete consistency is only achieved when the highest prime directly involved in the combination in question connects to the lowest prime in that same combination without breaching the 50% relative error marker once octave equivalence is accounted for. This would mean that in 159edo, the connection between the 7-prime on one hand and a combination of 11 and 3 on the other can only be regarded as "complete consistency" because the 11-prime connects to the 3-prime without breaching the 50% relative error marker on account of the nexus comma being tempered out. I still need to work out the details regarding more complicated combinations, but other than that, do you have any thoughts on this idea, Xenwolf? --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 20:20, 17 January 2021 (UTC)

:::::: I can't say anything about that. Considering the precision of 3.7 cents with which any interval is hit in 159edo and the generally accepted detuning degree of 13.7 cents of the major third in 12edo, considerations regarding consistency seem rather remote to me. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 20:47, 17 January 2021 (UTC)

::::::: Actually, if you think about it, the generally accepted detuning of the major third in [[12edo]] still follows the same rules that I'm laying down, as the Syntonic comma ([[81/80]]), which is responsible for that detuning, is smaller than half a step in 12edo, and it's still smaller than half a step in [[24edo]]. In fact, the [[Pythagorean comma]] is also less than half of a step in 24edo, and thus, the 3-prime and the 5-prime can both be regarded as having "complete consistency" in 24edo as well as in 12edo. However, when you start looking at [[36edo]], [[48edo]] and [[72edo]], suddenly, things don't turn out as good on this front, as the relative error percentage in these EDOs- especially for the Pythagorean comma- exceeds 50%. This is why I moved on from the larger 12-based EDOs and was finally open to detwelvulating. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 21:41, 17 January 2021 (UTC)

::::::: If you apply this same type of thinking to smaller EDOs, you see that 2edo is the first to have complete consistency in the 3-limit, but 3edo and 4edo both fail this test as the commas produced by their respective circles of fifths are larger than half of their respective step size. After that, the next EDO to have complete consistency in the 3-limit is 5edo, which accomplishes a completely consistent representation of the 3-prime as 256/243, the interval produced from a single circle of fifths in 5edo, is smaller than half of a step in 5edo. After that, the next EDO to have complete consistency in the 3-prime is 12edo itself, as 6edo, 7edo, 8edo, 9edo, 10edo, and 11edo all fail the test- of course, as I said, 24edo, which is related to 12edo, also passes this test. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:50, 17 January 2021 (UTC)

::::::: So, what about the EDOs between 12edo and 24edo? Well, according to my calculations, literally none of the EDOs from 13edo to 23edo demonstrate complete consistency in the 3-limit. Even the well known [[22edo]] fails this test- looks like I've found one of that EDO's significant weaknesses, and a good enough reason for me not to use it. Anyhow, I'll continue my calculations to see what other EDOs demonstrate the kind of complete 3-prime consistency, and I'll let y'all know about the first dozen or so members of the sequence that emerges from this. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 23:12, 17 January 2021 (UTC)

::::::: I just got to thinking, and, the term "complete consistency" seems like a misleading term for the type of consistency I'm after- perhaps "telic consistency" or even "telicity" are a better terms for this, since this type of consistency means that stacking intervals of one prime will eventually reach an interval of a lower prime without reaching or exceeding 50% relative error, and "telic" is related to "telos" meaning "end" or "goal". Since "telicity" is the noun used to refer to the property of being "telic", I think I'll use the term "telicity" for this type of n-consistency from now on. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 03:30, 18 January 2021 (UTC)

== Linking 159edo Songs to This Page ==

Hey, Xenwolf, since I've written like three songs in 159edo now, I'm wondering how to link these songs of mine to this page. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 18:37, 26 February 2021 (UTC)

: I started the [[159edo #Music|''Music'']] section, please feel free to add what you like there. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 19:04, 26 February 2021 (UTC)

== Gentle comma (364/363) and region, also tempering out 352/351 ==

Please note that while this article correctly notes that 159-ed2 tempers out364/363, the gentle region and grntle temperament also involves tempering out 352/351. In other words, -3 fifths represents 13/11 or 33/28; and +4 fifths represents 14/11 or 33/28. Thus there sre two genle commas: 159-ed2 tempers out 364/363, but not 352/351; compare 38\159 for 13/11 or 33/28 with 39\159 (-3 fifths) for 32/27. In gentle temperament as I described it in 2002, 32;27 and 13/11 or 33/28 map to -3 fifths.

[[User:Mschulter1325|Mschulter1325]] 01:18, 11 November 2022 (UTC)

: Would you say the gentle comma should refer to either 352/351 or 364/363? And that gentle temperament is the 13-limit temperament tempering out both 352/351 and 364/363? In that case we'll need to come up with another name for 364/363 cuz right now it's known specifically as ''the'' gentle comma. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 05:14, 11 November 2022 (UTC)

I eould say it's important that any change or updating of terms be graceful and as backward-compatible as possible. Maybe the larger minthma/gentle comma for 352/351 (old minthma) and smaller minthma/gentle comma for 364/363 (old gentle comma). I know that people have relied on the old names, and developed temperaments that, unlike my gentle but just as validly, temper out one but not the other. So this kind of collegiality and consultation is very helpful in seeking out, if you'll forgive the pun, the kindest and most gentle solution. [[User:Mschulter1325|Mschulter1325]] 02:46, 13 November 2022 (UTC)

@@ Line 1: / Line 1: @@
+== Approximate errors ==
 Okay...  I have a list of the approximate errors in cents for 159edo's approximations of certain prime intervals:
@@ Line 65: / Line 67: @@
 :: I don't know how well my response to Flora manages to solve the problem you just stated, but here's to hoping... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 18:00, 7 January 2021 (UTC)
-:: Is it me, or can it be said that "Complete Boolean Consistency" means being able to go from the unison all the way around a set of nodes and back to the unison without the relative error reaching above the 50% marker?  If my speculation is correct, then we're talking about a different type of "consistency" than the kind that Flora's talking about.  It's like comparing apples and oranges in a way- apples and oranges are both fruit but have a lot of differences between them. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 18:11, 7 January 2021 (UTC)
+:: Is it me, or can it be said that "Boolean Consistency" means being able to go from the unison through a set of nodes in one p-limit to connect with an interval of a lower p-limit without the relative error reaching above the 50% marker?  If so, then "Boolean Consistency" for the 3-limit means being able to connect with the pitch class used as the [[unison]] and [[octave]] a second time after going around a complete set of nodes without the relative error reaching above the 50% marker.  If my speculation is correct, then we're talking about a different type of "consistency" than the kind that Flora's talking about.  It's like comparing apples and oranges in a way- apples and oranges are both fruit but have a lot of differences between them. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 18:11, 7 January 2021 (UTC)
+:: The consistency is defined on "an interval set S". There's not a rule against prime limit but that doesn't make sense since it simply can't be consistent. I remember reading about an "n-consistent" somewhere, in which 53edo is hundreds-consistent in the 3-limit as you can stack hundreds of 3's without relative error reaching over 50%. That might be what you look for. Somebody in the FB group also proposed another "n-consistent", in which the n is something substituting 50%, similar to relative error. Another fascinating idea is the ''pepper ambiguity'' (forgive me for saving links in talk pages) – its definition is not completely clear to me and I hope to work on it soon. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 18:31, 7 January 2021 (UTC)
+::: It looks like there are multiple types of n-consistency being proposed even within the Facebook group, so yes, we need a discussion on this. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 19:01, 7 January 2021 (UTC)
+::: I must point out that the degree of n-consistency that I look for on "an interval set S" in the 3-limit has everything to do with whether or not you can go around a complete circle of fifths in a given EDO without accumulating a relative error of 50% or more.  That's the specific type of n-consistency that I think I can regard as "complete". --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 19:07, 7 January 2021 (UTC)
+:::: This kind of consistency ("complete circle of fifths") seems problematic to me: How will you generalize these rings to other prime intervals? Also, aren't you interested in combinations of multiple prime dimensions (besides 2, of course)? --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 20:22, 7 January 2021 (UTC)
+::::: The kind of n-consistency I'm alluding to involves being able to go from the unison through a set of nodes in one p-limit to connect with an interval of a lower p-limit without the relative error reaching or exceeding the 50% marker.  Since the 3-prime can only connect with the 2-prime in this fashion, and since the 2-prime simply results in manifestations of the unison at different registers- meaning that the unison is the only available target- that means that the 3-prime requires a complete circle of fifths without accumulating 50% relative error or more to achieve a form of "complete consistency".  However, higher primes have more options for a form of "complete consistency".  For instance, the 11-prime in 159edo connects with the 3-prime easily without breaching the 50% relative error marker by means of tempering out the nexus comma, and similarly, the 5-prime connects with the 3-prime by means of tempering out the schisma. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:41, 7 January 2021 (UTC)
+::::: As to combinations of multiple prime dimensions, I find these to be largely of secondary importance, but to be fair, they are subject to the same constraints- they must be able to connect to a p-limit lower than the lowest p-limit that is directly involved in the combination in question.  For example, stacks of 14/13 trienthirds connect with the 5-prime by means of tempering out the cantonisma- the factor of 2 in 14 is trivial for this since the 2-prime simply results in manifestations of the unison at different registers. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:36, 7 January 2021 (UTC)
+::::: Now this is fascinating...  According to my calculations, subtracting the [[symbiosma]] from [[7/4]] results in an interval with the prime factorization of (3^9)/(2^10*11), so it looks like the symbiosma bridges the 7-prime and a combination of 3 and 11.  Perhaps I should fix my definition of "complete consistency" by adding the following condition- if one is able to go from the unison through a set of nodes in one p-limit to connect with an interval  made purely from a combination of two other primes, complete consistency is only achieved when the highest prime directly involved in the combination in question connects to the lowest prime in that same combination without breaching the 50% relative error marker once octave equivalence is accounted for.  This would mean that in 159edo, the connection between the 7-prime on one hand and a combination of 11 and 3 on the other can only be regarded as "complete consistency" because the 11-prime connects to the 3-prime without breaching the 50% relative error marker on account of the nexus comma being tempered out.  I still need to work out the details regarding more complicated combinations, but other than that, do you have any thoughts on this idea, Xenwolf? --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 20:20, 17 January 2021 (UTC)
+:::::: I can't say anything about that. Considering the precision of 3.7 cents with which any interval is hit in 159edo and the generally accepted detuning degree of 13.7 cents of the major third in 12edo, considerations regarding consistency seem rather remote to me. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 20:47, 17 January 2021 (UTC)
+::::::: Actually, if you think about it, the generally accepted detuning of the major third in [[12edo]] still follows the same rules that I'm laying down, as the Syntonic comma ([[81/80]]), which is responsible for that detuning, is smaller than half a step in 12edo, and it's still smaller than half a step in [[24edo]].  In fact, the [[Pythagorean comma]] is also less than half of a step in 24edo, and thus, the 3-prime and the 5-prime can both be regarded as having "complete consistency" in 24edo as well as in 12edo.  However, when you start looking at [[36edo]], [[48edo]] and [[72edo]], suddenly, things don't turn out as good on this front, as the relative error percentage in these EDOs- especially for the Pythagorean comma- exceeds 50%.  This is why I moved on from the larger 12-based EDOs and was finally open to detwelvulating. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 21:41, 17 January 2021 (UTC)
+::::::: If you apply this same type of thinking to smaller EDOs, you see that 2edo is the first to have complete consistency in the 3-limit, but 3edo and 4edo both fail this test as the commas produced by their respective circles of fifths are larger than half of their respective step size.  After that, the next EDO to have complete consistency in the 3-limit is 5edo, which accomplishes a completely consistent representation of the 3-prime as 256/243, the interval produced from a single circle of fifths in 5edo, is smaller than half of a step in 5edo.  After that, the next EDO to have complete consistency in the 3-prime is 12edo itself, as 6edo, 7edo, 8edo, 9edo, 10edo, and 11edo all fail the test- of course, as I said, 24edo, which is related to 12edo, also passes this test. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 22:50, 17 January 2021 (UTC)
+::::::: So, what about the EDOs between 12edo and 24edo?  Well, according to my calculations, literally none of the EDOs from 13edo to 23edo demonstrate complete consistency in the 3-limit.  Even the well known [[22edo]] fails this test- looks like I've found one of that EDO's significant weaknesses, and a good enough reason for me not to use it.  Anyhow, I'll continue my calculations to see what other EDOs demonstrate the kind of complete 3-prime consistency, and I'll let y'all know about the first dozen or so members of the sequence that emerges from this. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 23:12, 17 January 2021 (UTC)
+::::::: I just got to thinking, and, the term "complete consistency" seems like a misleading term for the type of consistency I'm after- perhaps "telic consistency" or even "telicity" are a better terms for this, since this type of consistency means that stacking intervals of one prime will eventually reach an interval of a lower prime without reaching or exceeding 50% relative error, and "telic" is related to "telos" meaning "end" or "goal".  Since "telicity" is the noun used to refer to the property of being "telic", I think I'll use the term "telicity" for this type of n-consistency from now on. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 03:30, 18 January 2021 (UTC)
+== Linking 159edo Songs to This Page ==
+Hey, Xenwolf, since I've written like three songs in 159edo now, I'm wondering how to link these songs of mine to this page. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 18:37, 26 February 2021 (UTC)
+: I started the [[159edo #Music|''Music'']] section, please feel free to add what you like there. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 19:04, 26 February 2021 (UTC)
+== Gentle comma (364/363) and region, also tempering out 352/351 ==
+Please note that while this article correctly notes that 159-ed2 tempers out364/363, the gentle region and grntle temperament also involves tempering out 352/351. In other words, -3 fifths represents 13/11 or 33/28; and +4 fifths represents 14/11 or 33/28. Thus there sre two genle commas: 159-ed2 tempers out 364/363, but not 352/351; compare 38\159 for 13/11 or 33/28 with 39\159 (-3 fifths) for 32/27. In gentle temperament as I described it in 2002, 32;27 and 13/11 or 33/28 map to -3 fifths.
+[[User:Mschulter1325|Mschulter1325]] 01:18, 11 November 2022 (UTC)
+: Would you say the gentle comma should refer to either 352/351 or 364/363? And that gentle temperament is the 13-limit temperament tempering out both 352/351 and 364/363? In that case we'll need to come up with another name for 364/363 cuz right now it's known specifically as ''the'' gentle comma. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 05:14, 11 November 2022 (UTC)
+I eould say it's important that any change or updating of terms be graceful and as backward-compatible as possible. Maybe the larger minthma/gentle comma for 352/351 (old minthma) and smaller minthma/gentle comma for 364/363 (old gentle comma). I know that people have relied on the old names, and developed temperaments that, unlike my gentle but just as validly, temper out one but not the other. So this kind of collegiality and consultation is very helpful in seeking out, if you'll forgive the pun, the kindest and most gentle solution. [[User:Mschulter1325|Mschulter1325]] 02:46, 13 November 2022 (UTC)