From Xenharmonic Wiki
Jump to navigation Jump to search

Approximate errors

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

  • 3: -0.068
  • 5: -1.408
  • 7: -2.788
  • 11: -0.374
  • 13: -2.792
  • 17: +0.705
  • 19: -3.173
  • 23: -1.859
  • 29: -3.162
  • 31: +2.134

I'm hoping that someone can make tables for Just Approximation like the ones found on the page for 94edo... --Aura (talk) 07:18, 7 September 2020 (UTC)

Done. FloraC (talk) 09:02, 7 September 2020 (UTC)
Thanks! Once we do a lot more exploring of 159edo, I hope to put our findings here. After all, there's no way I'm just letting an EDO as useful as this just languish anymore. --Aura (talk) 13:08, 7 September 2020 (UTC)
I have 159edo's patent val for primes up to the 19 limit- 159 252 369 446 550 588 650 675]. How consistent is this EDO when it comes to this group of primes? (preceding unsigned comment by Aura (talk))
According to Scala it's only consistent up to 17-odd limit. It might still be consistent when we add some higher odd numbers, though. IlL (talk) 15:36, 7 September 2020 (UTC)
Let's check it out then... let's try 19, 21, 23, 25, 27, 29 and 31... --Aura (talk) 15:39, 7 September 2020 (UTC)
Easy to speculate with an understanding of relative error. It's consistent in 17-limit or no-17 29-limit. FloraC (talk) 15:41, 7 September 2020 (UTC)
Wait... why specifically a no-17 29-limit? Is it consistent in 19-limit or 23-limit? Perhaps I ought to reveal one final patent val for 159edo- that of the 23-prime limit... --Aura (talk) 15:54, 7 September 2020 (UTC)
Okay, so, if 159edo is extended to the 23-prime, 159edo has the patent val of 159 252 369 446 550 588 650 675 719]... --Aura (talk) 16:01, 7 September 2020 (UTC)
I must admit that the main reason I'm interested in whether or not 159edo is consistent up to the 23-limit is because I'm currently compiling a list of Just Intervals corresponding to the various steps in 159edo, and 23 is the highest prime I've had to use so far... --Aura (talk) 16:08, 7 September 2020 (UTC)
Okay, I've managed to confirm that 159edo is not consistent in the 19-odd limit as the difference between the best 17/16 and the best 19/16 is 25 steps, while the best 19/17 is 26 steps... Not good at all... Looks like I need to search for several new values for step sizes --Aura (talk) 16:32, 7 September 2020 (UTC)
159edo has two intervals in 29-limit with >50% relative error —- 19/17 and 29/17. That's why you have to decide, full 17-limit or no-17 29-limit. FloraC (talk) 03:03, 8 September 2020 (UTC)
Afaik no edo between 94 and 282 is fully consistent in 23-limit. There's 111, 149 and 217 fully consistent in 19-limit. 94 is special consistency-wise but it's not superior in accuracy, so not all edos above 94 need to directly compare with it, especially when there's nothing to relate them. FloraC (talk) 02:59, 8 September 2020 (UTC)
I have used 94edo in the past, and the article on 94edo states that it is "a remarkable all-around utility temperament", while 159edo has other strengths, so I figured a comparison was at least somewhat warranted in this case. However, if such a comparison is not really warranted here, I'll remove the comparison altogether. --Aura (talk) 03:35, 8 September 2020 (UTC)

Reverting factually wrong additions


  1. There's basically no relationship between contorsion and inconsistency.
  2. There's basically no relationship between comma size and inconsistency.
  3. There's only one reasonable mapping for 5 and 7 and it's consistent.

FloraC (talk) 17:18, 7 January 2021 (UTC)

How then do you judge inconsistency? I note that 128/125, when approached by way of a chain of 5/4 intervals doesn't match the step that best fits 128/125 directly in terms of absolute error, and I have the same problem with 49/32. I also noted that Mercator's comma is less than half the size of a single step in 159edo, so why is what I said about that entirely wrong? Please do tell. --Aura (talk) 17:26, 7 January 2021 (UTC)
Also, I wasn't talking about odd-limit here, I was talking about prime limit. I agree that there's only one reasonable mapping for 5/4 and 7/4, but once you get beyond the 17-odd-limit, that's where we start to have issues. --Aura (talk) 17:33, 7 January 2021 (UTC)
In your way every edo would be "inconsistent in the 3-limit" because the 3-limit contains an infinity of different intervals and there're always some intervals with error over 50% of step size. FloraC (talk) 17:39, 7 January 2021 (UTC)
Ah. --Aura (talk) 17:40, 7 January 2021 (UTC)
I don't know about you, but to me, that high error rate does affect how the interval in question is actually used from a musical standpoint. --Aura (talk) 17:42, 7 January 2021 (UTC)
The error "rate" of a specific prime is the same as its relative error, I suppose. FloraC (talk) 17:50, 7 January 2021 (UTC)
I would say that's a reasonable conclusion, but only in part. I'm saying that the end of the usable portion of the harmonic lattice for a given prime as represented in a given EDO is marked by the relative error being less than 50%- or at least that's my policy on the matter. --Aura (talk) 17:58, 7 January 2021 (UTC)
I, too, am somewhat unsure about this issue. Is it correct that inconsistency/consistency is only defined in relation to a specific odd limit? Otherwise it would not be in the Boolean domain. I wished we had another measure for consistency, something that does not depend on an odd limit, but tells how many nodes of a (p-1)-dimensional lattice could be (somehow) reached from the unison. (But unfortunately my mathematical skills are not sufficient to comprehend this "somehow".) --Xenwolf (talk) 17:53, 7 January 2021 (UTC)
I don't know how well my response to Flora manages to solve the problem you just stated, but here's to hoping... --Aura (talk) 18:00, 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. --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. 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. --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". --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)? --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. --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. --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? --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. --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. --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. --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. --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. --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. --Aura (talk) 18:37, 26 February 2021 (UTC)

I started the Music section, please feel free to add what you like there. --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.

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. 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. Mschulter1325 02:46, 13 November 2022 (UTC)