Talk:159edo

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

Facts:

There's basically no relationship between contorsion and inconsistency.
There's basically no relationship between comma size and inconsistency.
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" has everything to do with whether or not you can go around a complete circle of fifths without accumulating a relative error of 50%. That's the specific type of n-consistency that I think I can regard as "complete". --Aura (talk) 19:07, 7 January 2021 (UTC)