Talk:Consistency

This page also contains archived Wikispaces discussion.

About Terminology

Hey, Inthar, I must admit that I'm having trouble seeing terminology such as "minimal consistent EDOs" and "maximal consistent set" as being correct, seeing as I'm a native English speaker. I would think such terms would be better written as "minimally consistent EDOs" and "maximally consistent set". In addition, I'd sooner see the term "move" rather than "walk away" in the phrase "a chord is consistent to distance d in an edo, if the chord is consistent and you can "walk away" up to distance d from the chord consistently," seeing as "walk away" doesn't sound quite right. The trouble is that fixing some of these involves moving a page, and I don't have the permission to do that. --Aura (talk) 20:53, 20 January 2021 (UTC)

I changed the wording to move. I wish regular users could move pages without leaving redirects. Inthar (talk) 21:05, 20 January 2021 (UTC)

Thanks. I must point out, however, that this permissions business seems to have a lot to to with preventing vandalism. --Aura (talk) 21:06, 20 January 2021 (UTC)

I don't see what would be bad about leaving redirects when renaming. This way users who have the pages saved in their bookmarks can find them even after a rename. --Xenwolf (talk) 23:32, 20 January 2021 (UTC)

On the definition of consistency to distance d

I really like the concept and it's something I was thinking about before in my theoretical investigations into the approximative capabilities of EDOs of interest to me. However, I don't understand why the definition of consistency to distance d isn't just the maximum step error of all intervals being less than 1/(2d), because this would allow picking a multiset of any d intervals within a chord and multiplying them together and having the result be consistent, including picking one interval d times. I strongly suggest this alternative definition for the sake of simplicity, intuitiveness and ease of understanding.
(It also means "consistency to distance 1/2" can be seen as guaranteeing - at worst - a second-best mapping of an interval, and that consistency to distance k as k approaches 0 implies infinite inconsistency, thus representing ever-weaker consistency, and ultimately, no consistency, as you can't move anywhere without being inconsistent, AKA you can move 0 distance while being consistent.) --Godtone (talk) 04:20, 22 January 2021 (UTC)

Not going to lie, I'm somewhat against this proposed alternative definition at this point because I'm not settling for second-best mappings in when it comes to the definition of things like telicity. Perhaps if this definition can meet the strict requirements behind my idea of telicity, then I can get behind this idea. --Aura (talk) 06:11, 22 January 2021 (UTC)

This definition is not about telicity, although it is strongly related to it and can be used to define it. We are talking about the correct definition of a specific concept of consistency for chords and the subgroups they represent and more generally for a set of intervals of interest. If you don't like spreading out error between primes to get an approximate chord and would prefer increased accuracy on a small number of primes, that's fine, you can use this definition to specify that too through a higher distance requirement. My proposed definition alteration means that if an interval r is consistent to distance d, all integer powers of r from -d to +d are mapped consistently. Telicity (from my understanding) builds on this concept by having 2 prime pintervals meet at some point in their chains, and thus the smaller of the two distances is the consistent distance of the pair of prime intervals taken together. My confusion and thus suggestion for alteration is I don't see any strong reason to complicate this definition that gives such an easy and intuitive conceptualisation. --Godtone (talk) 23:56, 22 January 2021 (UTC)

Alright. I just wanted to make sure that the concepts still fit together. --Aura (talk) 00:41, 23 January 2021 (UTC)

Assimilate or Occupy?

I have the impression that this article originally tried to explain consistency in odd-limit-diamonds. I'm not sure if it's a good idea to occupy the term for a different concept. Would it be possible to share the term and turn the article into an overview of different concepts of consistency? If the answer is yes, then this approach should also be reflected in the introduction. As soon as these concepts need more space than a section, we can add dedicated articles with unique names. --Xenwolf (talk) 20:51, 22 January 2021 (UTC)

From what I gather, Inthar's second concept integrated the original concept into it. That said, I think it really is possible to share the term and turn the article into an overview of different concepts of consistency. --Aura (talk) 20:53, 22 January 2021 (UTC)

Now that I'm thinking about it, I think you should talk to Inthar to see just how these two concepts of consistency are related, and if it really turns out that I was right about Inthar's second concept being an expansion on the original, well, I'll let you two decide how to handle things from there. That said, I should mention that the different concepts of consistency need to be disambiguated in some way, even though they share this same article. --Aura (talk) 21:02, 22 January 2021 (UTC)

My new definition of consistency (to distance 0) just reformulates the term to be applicable to chords, rather than sets of notes not viewed as a chord. By my definition of chord consistency, an edo approximates chord C consistently iff it is consistent in S (according to the old definition) where S = diamond(C) where C is viewed as a set of notes from tonic. It's not a new definition except that you have to be careful that you're talking about a chord viewed as musically meaningful vs just a set of notes. Inthar (talk) 22:51, 22 January 2021 (UTC)

Terminology split: "distinctly consistent" vs "uniquely consistent"

Steve Martin recently did some clarifications and clean-up across several wiki pages related to consistency. Over private email with me, he'd broached the question of "distinctly consistent" vs "uniquely consistent" and I pointed out that the former appeared on 41 wiki pages and the latter on 36 wiki pages, so it was worst-case scenario of a popularly used concept with about a 50/50 split in terminological choice. I wasn't sure what he'd do with that information, but it looks like he's gone with "distinctly" over "uniquely" in these latest changes of his. This is fine by me.

So my question is: I wonder how people would feel about standardizing to "distinctly" more widely, then (i.e. a note on the main page that many use "uniquely", but otherwise change "uniquely" to "distinctly"). The value of standardization I hope should be self-evident; less confusion (I'd prefer we don't have half of people using one term and half of people the other, but for the exact same concept) and clunkiness (I'd prefer not to see "distinctly/uniquely" in every position).

As for historical precedence, I find only a single occurrence of "distinctly consistent" in Yahoo from Gene Ward Smith (here: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_15640#15640), and no occurrences of "uniquely consistent". Due to technical limitations, it's impractical to search Facebook records for this sort of problem, unfortunately, but as for the Discord server, it appears people have somewhat of a preference for "distinctly" as well. So that's not strong evidence either way. Let me know if anyone has any other sources.

I personally think "distinctly" is slightly clearer. But I haven't thought about it that much, so I welcome arguments either way.

Let me know what we're thinking. I'd be happy to make the changes if we decide they should be made. --Cmloegcmluin (talk) 16:26, 29 July 2022 (UTC)

I think a note on the main page would be a bit overkill for such a specific vocabulary issue. We do have a Conventions page, but I think it should be limited to the elements that are used all over the wiki, like number precision, cents as standard logarithmic measure and so on. Specific vocabulary issues should be addressed in their own articles. In general, the preferred term is the first that appears in the relevant article, while the alternative terms should be in parentheses after the preferred term, sometimes with a word to qualify the rarity of the alternative term (e.g. see Dyad). See also Wikipedia:MOS:BOLD. That rule, in itself, could be mentioned in the Conventions page, if that helps.

As for the term discussed here, for some reason, I came across "uniquely consistent" more often, but it's probably just a matter of which pages I've browsed. Anyway, I also think that "distinctly consistent" is clearer. --Fredg999 (talk) 19:13, 29 July 2022 (UTC)

Sorry for the confusion: I meant this page, the main page for consistency, not the main page of the entire wiki, in the manner you've described above. --Cmloegcmluin (talk) 20:07, 29 July 2022 (UTC)

Right, that makes sense. I'll ponder over the idea I mentioned about the Conventions page, since this gave me the idea anyway. --Fredg999 (talk) 22:16, 29 July 2022 (UTC)

Differences and Sums

The main definition (with q-odd limit) only says "difference"; the definition in terms of a chord says "interval between"; I guess these make sense because some sums would stray outside the limit or the chord. But the mathematical definition is in terms of a set, and uses a sum (the first example also uses a sum, but the example is clearly valid and could be re-ordered into a difference). Is it correct to exclude sums?

PS:

• distinct vs unique: covered by cmloegcmluin above, thanks.
• closest vs nearest: no difference in meaning, I think; I chose "closest".
• best vs closest: I argue that "best" can mean different things depending on your purpose, whereas "closest" is simply factual.

--Martins (talk) 20:14, 1 August 2022 (UTC)

Hi Steve! Thanks for posting here. I corrected the spelling of my username in your post, hope you don't mind (I admit it's a tough one to spell, haha). Also, FYI, people tend to sign their posts here. The way you do that is to type four tildes in a row. I usually type it with two dashes in the front because that's what I was taught by a fellow user here, like this: --~~~~. Then the wiki transforms the ~~~~ into your username with a link to your user page, plus a timestamp. Sorry if you knew that already and just forgot; I forget sometimes myself.

As for your actual questions, well, I'm afraid I don't know for sure whether to exclude sums. But I would say that where you've replaced "direct approximation" with "closest approximation", I would at least preserve a link to the page for "direct approximation", since that's the name under which that concept is discussed elsewhere on the wiki at this time. I hope that makes sense, if not, just let me know. --Cmloegcmluin (talk) 23:50, 31 July 2022 (UTC)

I didn't know about the username thing (or had forgotten) so thanks. I have not replaced "direct approximation" except in my failed definition in two daughter pages. I hope someone can assist with my question. --Martins (talk) 20:14, 1 August 2022 (UTC)

WP:NOUN

The page should be titled "Consistency" per WP:NOUN. I cannot move it myself because Consistency has a non-empty edit history. The previous versions of that page are soft redirects and no significant information will be lost if the page is overwritten. If there are no objections, I would like to request that an admin moves the page. --Fredg999 (talk) 05:20, 21 February 2023 (UTC)

q-odd-limit harmonics? prime harmonics?

The current way of writing, not covered the case that direct mapping of up-to q-th harmonics is already inconsistent, e. g. Dual-fifth system.

Now I'm rewriting the article "一貫性", I think the story we approximate all q-limit interval at first, then we check consistency... --Dummy index (talk) 14:37, 28 April 2023 (UTC)

If I understand correctly, your concern is that consistency is defined on edos assuming an infinitely long map (val) based specifically on all prime harmonics, therefore excluding maps with composite or fractional subgroups. It is good to keep in mind that the choice of the map affects the consistency; for example, 18edo is consistent in the 7-odd-limit, but 18b (second-best mapping for prime 3) is only consistent in the 3-odd-limit, because then 5/3 wouldn't be mapped to its direct approximation. And so we find that a dual-fifth system such as 35edo has a well-defined consistency if we assume its integer uniform map (patent val) — in this case 7-odd-limit, because 9/1 is not mapped to its direct approximation —, but if we coupled 35edo with a 2.9.5.7.11 subgroup map, we would indeed have no way to reach the interval 3/1 at all. Since the goal of consistency is to check if a given equal temperament (edo + map) can map all intervals in an odd-limit to their direct approximation, it wouldn't make sense to allow subgroups that skip over certain intervals. Therefore, a uniform map, or at least a map based on a full prime limit, should be used to evaluate an equal temperament's consistency. It would be interesting to check which equal temperaments have a higher consistency limit with a "warted" map compared to the standard one; for example, 11edo is only consistent in the 3-odd-limit, but 11b is consistent in the 7-odd-limit (but not 9-odd-limit because 9/5 is not mapped to its direct approximation). --Fredg999 (talk) 16:20, 29 April 2023 (UTC)

Sorry, I don't talk about we should/shoudn't use subgroups. I talk about the Opening clause of the article. The mathematical definition section is fine (but we should add the wording that determines the range of the r, odd-limit and subgroup or any other). Opening clause says... Oh, sorry I see it's no problem. 18edo in the 9-odd-limit maps 3/1 to 29 steps, 9/1 to 57 steps, therefore difference between 9/1 and 3/1 doesn't give closest approximation of 3/1.

Hence, I think 18b doesn't map 3/1 as closest approximation. Difference between 3/1 and 1/1 doesn't give closest approximation of 3/1. Wouldn't the premise of the discussion be different? --Dummy index (talk) 06:52, 30 April 2023 (UTC)

All edos are consistent in the 3-odd-limit since there's only 1, 3/2 and 4/3 to consider. 1 is by definition pure and closest. Take the closest 3/2 and that guarantees its octave complement 4/3 is closest too. The consistency is enabled by the patent val and not any other vals, so you're right that 18b is "inconsistent" tho consistency is defined for equal tunings rather than equal temperaments. FloraC (talk) 08:45, 30 April 2023 (UTC)

I have noticed that 23edo is consistent in the no-1's 7-odd-limit ({1/1(3/3 and 5/5 and 7/7), 5/3, 3/5, 7/3, 3/7, 7/5, 5/7}). (Oh, you must have said this on 11edo.) But this is not to say that we consider 23d, it was just almost consistent and similar to 23d. Direct approximation on 18edo gives 9/1 to odd-number steps, it can't be p-limit 18 nor 18b. Wait, now we have to derive the mapping of up-to 7 harmonics from no-1's 7-odd-limit intervals, but can't determine uniquely... (⟨23 36 53 64] or ⟨23 37 54 65] or ⟨23 38 55 66] or ...) --Dummy index (talk) 13:57, 7 June 2023 (UTC)

Under consideration. By the way,

For instance, 12edo is consistent in the no-11's, no-13's 19-odd-limit, ...

Do I need the apostrophe? --Dummy index (talk) 15:50, 9 June 2023 (UTC)