User talk:Godtone: Difference between revisions

(41 intermediate revisions by 7 users not shown)

Line 25:

== Comment on your proposal on EDO subgroups ==

For reference, [[~~Xenharmonic_Wiki~~:Things_to_do#~~Set_a_semi~~-~~objective_standard_for_classifying_edos_as_subgroup_temperaments~~|here is the proposal]]. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 04:59, 22 January 2021 (UTC)

For reference, [[Xenharmonic_Wiki_talk:Things_to_do#13-Limit.2C_17-Limit_and_19-Limit_Comma_Pages|here is the proposal]].--[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 04:59, 22 January 2021 (UTC)

: Edit: Fixed link, also, to find the beginning of my proposal, Ctrl+F "Start of suggestion/reply by Godtone" once there (or scroll down a bit). --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]])

: Yes, I read that. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 05:40, 22 January 2021 (UTC)

Line 46:

Line 48:

: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.

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

:(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)

:--[[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 counted if the telos prime is something other than 2, 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 chains 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 prime chains, 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 to and including the point of connection.

:: Just looking at 3-to-2 telicity, which, by definition, involves a circle of fifths as the 2-prime is the only available telos for the 3 prime chain, 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 both 29edo and 87edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 21:30, 22 January 2021 (UTC)

::: "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 chains up to the point of connection must also agree with the connection." Yes, I understood that part. I never said that the circles must accumulate less than half an EDOstep of error in their full/completed chains in an EDO or sub-EDO. I don't think a "circle" of an interval has to necessarily close on (a multiple of) the octave within half an EDOstep to be used as a "circle" because the interval could still be very or sufficiently accurate, although in the case of larger EDOs, having ''some'' strong circles that fulfill that condition is important for orientation. I now see my definition is technically not specific enough and would require that the error of generators don't accumulate so much as to cause inconsistency at any point in the chain up to the connection, but I was mainly intent on confirming understanding rather than restating the definition exactly. Also, I never claimed that EDOs 80, 29 or 87 succeed telicity in the 2.3 subgroup. That doesn't mean their circle of fifths or circles of other intervals can't be useful, interesting or equally valid as a method of organising them, for example the "circle of fifths" in meantone does not necessarily close within 50c to the octave, depending on tuning. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 22:21, 22 January 2021 (UTC)

:::: I must admit that I wanted to make sure your understanding was completely correct before I confirmed it, as you can never completely tell online. Yes, it's true that there are other EDOs with other circles of fifths, but if they don't succeed at having telicity, I find them to be less than ideal, since the 3-prime is the most commonly used prime outside the octave. That said, you did more or less hit the nail on the head when you mentioned that large EDOs need some strong circles that fulfill the telicity condition for the sake of orientation- in fact, even something like the 11-to-3 telicity of 159edo and 24edo is very useful for navigation, though making the best use of this kind of telicity involves building on good 3-to-2 telicity. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 00:38, 23 January 2021 (UTC)

== 159edo and Composition ==

Hey, I'm curious as to whether or not you've heard my songs "[[:File:Space Tour.mp3|Space Tour]]" and "[[:File:Welcome to Dystopia.mp3|Welcome to Dystopia]]" and what your impressions of those songs are, especially since between the two of them, they show off some of 159edo's tricks pretty well. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 20:04, 7 February 2021 (UTC)

== Corrections ==

Hi Godtone,

Sometimes it can be a bit confusing to get the cent values right ([https://en.xen.wiki/index.php?title=25/18&diff=next&oldid=69725], [https://en.xen.wiki/index.php?title=18/13&diff=next&oldid=69726]) 😊 but I prefer these kinds of changes to accepting things we feel are wrong just because they are written there. Have a nice day! --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 07:07, 12 May 2021 (UTC)

: Ah yeah, there was a mistake that confused me where it said 5.3c flat rather than 5.3c sharp so that confused me, it was odd because when doing edits I always try to double-check especially if its editing things like cent values which need to be exactly correct. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 18:09, 12 May 2021 (UTC)

== Meaning of a sentence ==

Hi Godtone, in [[161edo]], you added a sentence, of which the part “in the 100 to 200 range” is unclear to me. Can you clarify this a bit? Thanks in advance. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 14:30, 6 March 2022 (UTC)

: I think they meant edos between 100 and 200. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 14:36, 6 March 2022 (UTC)

:: So a copy and paste error of some sort? I found [https://en.xen.wiki/index.php?title=181edo&diff=prev&oldid=66283 a similar addition to 181edo] (surely your recent edits were the reason for your comment and ultimately my question). In that case, the part in question should be removed. But let's wait and see what Godtone answers. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 15:55, 6 March 2022 (UTC)

::: Yes sorry, I thought it was clear but in retrospect it could be talking about harmonics not patent val EDOs. The reason I wrote it in such a way is because its concise and seemed unambiguous to me, but also because I'm not sure how to make "from 100 EDO to 200 EDO" sound right... is it "in the range of EDOs 100 to 200" or "from 100 to 200 EDO" or something else? --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 19:17, 6 March 2022 (UTC)

::: Oh and the reason it appears in the edit history of 181 EDO is because I accidentally added the sentence to the wrong EDO as I had misremembered the number of the EDO, but I removed that sentence from the page for 181 EDO when I realised. I am sure that the one here is right: [[183edo#Prime_harmonics]]. 161edo is also quite a strong system so I am pretty sure that one is right as it came second to 183 EDO a lot. Note 161 EDO's great strength as a no-9's 21-odd-limit system. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 19:31, 6 March 2022 (UTC)

:::: I think it's that you compare the given EDO ''to'' a range of EDOs. This was not clear (from my POV). I have tried to increase readability by placing the information regarding context at the beginning of the sentence. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 21:23, 6 March 2022 (UTC)

== Comma name proposal for 256/255: "charisma" vs "charsma" ==

Hi Godtone,

I never accept your name proposal for [[256/255]] ("charisma" and "charismic", formerly used for [[Magic family #Horcrux|horcrux extensions]]), but its associated temperament name "charic" (2.3.5.17 subgroup rank-3). For disambiguation, 256/255 should be renamed "char comma", "charic comma" or "charsma" (no-''i'' spelling). Then its associated temperament name for full 17-limit rank-6 would be called "charsmic". Do you deal it? --[[User:Xenllium|Xenllium]] ([[User talk:Xenllium|talk]]) 10:34, 6 January 2024 (UTC)

== Tempering vs tempering out ==

Please make sure not to use the generic ''temper'' to refer to ''temper out'' or less commonly ''temper away''. I notice you've done this a number of times. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:49, 24 June 2024 (UTC)

: Oh hmm, it didn't occur to me that it was ambiguous, sorry. Please feel free to correct any instances of it when you see them. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 13:16, 25 June 2024 (UTC)

== Crediting ==

As part of using your copyleft works, would you like to be thanked to as Godtone, Osmium(ic), or some other name? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 16:39, 18 January 2025 (UTC)

: The copyleft stuff I submitted have all been under User:Godtone so if you want to attach a name I would suggest referring to my xen wiki account (mainly so that the original code can be found on the user page). --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 22:02, 18 January 2025 (UTC)

== User page cleanup ==

I'd like to read your user page more but it's getting quite long. If you're okay, might I suggest organizing them into subpages? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 16:39, 18 January 2025 (UTC)

: Hmm, I like having everything in one place. The navigation/contents/glossary at the top serves the same purpose and I don't want to obscure stuff thru links. The more pressing concern though is that I don't have the energy/interest to do a big cleanup of the user page for a variety of reasons, so I'd rather just make small improvements to how it currently is. Therefore in that spirit, let me know if anything strikes you as particularly messy/in need of cleanup, cuz I agree that some of it could do with some cleaning, I'm just not quite sure what needs it the most. Maybe moving some more frivolous stuff out would make sense so that the code isn't obscured, but IDK about moving ''everything'' except the code cuz it kinda defeats the point of a user page to me. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 22:02, 18 January 2025 (UTC)

== "Augmented–chromatic equivalence continuum" ==

I'm gonna move that article to a subpage for now. First, the title is inappropriate cuz when I see augmented and chromatic I think about 128/125 and 2187/2048, which is a comma basis of 21et. If someone create an equivalence continuum about 21et that will be the title. Second, I don't get at all what's bad about the father--3 continuum and what's not bad about your augmented--dicot continuum. Father is the simplest comma of 3et with order 1 in the prime 5 and is marked against the 3-limit comma 32/27, just like any other continua. Your choice of commas looks arbitrary and unjustifiable to me. Even if you have valid reasons to justify it, it's prolly too deep and niche to quickly grasp compared to the obvious way to organize them. Third, there's the change of parameter used in some continua and you can do that in the father--3 article too. Just use a different variable than ''n'', ''m'', and ''k'', and give your reasons! [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 11:03, 27 February 2025 (UTC)

: Sorry, my mistake with "chroma", I was thinking that it refers to 25/24 based on the [[syntonic–chromatic equivalence continuum]] and based on me intuitively thinking of a chroma as being strictly smaller than a semitone so that I often feel that 2187/2048 is the odd one out. But in my defence, it was used before in this way on the [[Father–3 equivalence continuum]] page:

:: "[...] such that temperaments satisfy ('''25/24''')k = '''16/15'''. This gives rise to the name '''chromatic'''–'''diatonic''' equivalence continuum, where both chromatic and diatonic refer to the classical versions of semitones."

: Anyways, to explain the choice of commas:

: I'm pretty certain* that 128/125 and 25/24 is the best way to organize this continuum because 1\3 is ''relatively speaking'' an extremely good approximation of 5/4, so that practically all the most notable and useful 5-limit temperaments supported by 3edo are defined principally in terms of how the genchain of ~5/4's finds prime 3 (which I'll explain in a moment), as if I'm not mistaken, integer ''n'' (for (128/125)n / (25/24)) always corresponds to using ~5/4 as a gen, so that generally speaking (tho I'm unsure if this is always true) non-integer ''n'' corresponds to using a gen that splits 5/4, 8/5, 2/1 or an octave-equivalent correspondingly. That is, one reason for 128/125 is that 3edo is notable not for its "5-limit" but for its 2.5 subgroup which is also partly why having 5^2 present in the other comma is acceptable IMO because it will cancel with 5^3 (though I discuss an alternative at the end that you might prefer, though I'd want to see it first cuz I worry it'll lead to many weird temperaments at simple points drowning out the interesting ones).

: Let me quote from the page to explain the IMO very straightforward musical significance/application of describing the continuum in this way:

:: "This formulation has a specific reason: 128/125 is significantly smaller than 25/24, so that it makes sense to equate some number of 128/125's with 25/24, but because {{nowrap|25/24 {{=}} ([[25/16]])/([[3/2]])}}, this has the consequence of clearly relating the ''n'' in {{nowrap|(128/125)n {{=}} 25/24}} with how many 5/4's are used to reach 3/2 (when octave-reduced):

:: If ''n'' = 0, then it takes no 128/125's to reach 25/24, implying 25/24's size is 0 (so that it's tempered out), meaning that 3/2 is reached via (5/4)2.

:: For integer ''n'' > 0, we always reach 25/24 via (25/16)/(128/125)''n'' because of (128/125)''n'' ~ 25/24 by definition, meaning that we reach 3/2 at 3''n'' + 2 generators of ~5/4, octave-reduced."

: That is, ''n'' is the number of dieses you need to flatten ~25/16 by to reach ~3/2, which seems musically very clear to me, as this explanation tells you how many gens of ~5/4 you need to move to reach ~3/2 (up to octave-equivalence), as each diesis is three ~5/4's, and thereby tells you the intuitive logic of pumping the comma of the associated temp for integer ''n'' as well as gives you crucial information about how many intervals are guaranteed to exist between ~6/5 and ~5/4 in a nontrivial tuning (which is why the integer part is important; ''n'' - 1 is the number of intervals between 6/5 and 5/4, which is equivalent to saying ''n'' is how many pieces we split 25/24 into).

: Having said that, it'd be good to also include a change of basis ''k'' = 3''n'' + 2 so that we don't have to manually and indirectly deduce the number of gens of ~5/4 needed for ~3/2 (because this can be annoying for non-integer ''n''), but I don't know how to calculate the associated comma pair, but I'm pretty sure it doesn't result in any of the coordinate schemes currently documented in [[Father–3 equivalence continuum]]. Like, are you really okay with magic being fractional in both ''n'' and ''m'' there? Also, IMO, 25/24 being at 0 is super natural; other than large integer values of ''n'', it's the only natural number value of ''n'' that corresponds to an exotemp, and is meaningfully trivial compared to integer ''n'' > 0. Also notice how as ''n'' grows, ~5/4 becomes sharper and approaches 1\3, which makes taking the limit of ''n'' to infinity to reach ~5/4 = 1\3 very intuitive as well. This also makes ''n'' = 1 (magic) notable as the simplest nontrivial tuning (one that admits a reasonable amount of structure and as a result isn't an exotemp), and if you accept the approximation ~5/4 = 1\3 (which even I accept given the right tuning of the fifth), then every integer ''n'' > 0 is non-exo in terms of the tuning of the 5-odd-limit (even if the edo and mapping complexity implied is absurd).

: <nowiki>*</nowiki> Re: "pretty certain", initially I was completely certain, hence the boldness of my edit cuz it seemed like a very unambiguous and significant improvement, but the consideration about 3''n'' + 2 maybe being a little more work than is reasonable to ask of the user makes me unsure given I haven't seen what the continuum looks like if we apply this transformation. The alternative transformation would have dicot at 2, magic at 5, mutt at 7, würschmidt at 8, etc. but already we see this might not be desirable because the info that mutt splits something (the octave and the diesis) into 3 is lost.

== [[Bird's eye view of temperaments by accuracy]] ==

I've expanded this page a bit and added some more temperaments. This page could be useful, but it seems like no one contributes to it anymore.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 22:45, 21 November 2025 (UTC)

Hey man I noticed your changes in the bird's eye page, I partially reversed them, before we begin an edit war I'd like to give some justifications and also hear your opinion so we can come to a truce:

* Garibaldi is the simplest 7-limit effective extension of schismic: -14 fifths versus -26 fifths in gracecordial or whatver is called (kinda alright but more complex), and -2 fifths in schism (bad and very innacurate extension)

* When I mean bad, I mean ''temperament'' bad. As in high badness. This also makes garibaldi and cassandra the ''best'' compared to helenus and andromeda which are comparatively worse.

* I removed the s-expressions because I believe we don't need them. I don't know the target audience of this, but if it's a less temperament informed reader, the s-expressions will only clutter as the comma ratios or names, whatever is shortest, are already succint.

* I have a huge spot for cassandra and became one with it. Sorry if I come of as defensive or glaze it too much. Can't help myself.

That's all --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 09:48, 7 May 2026 (UTC)

: It appears "minicomma" is an idiosyncratic term, so I removed it. I moved the discussion of cassandra to the bottom to avoid the feeling of repetition of information as that way it feels more like a summary of what 41edo, 53edo and 94edo have in common. I have no issue with describing garibaldi as "arguably the best extension to schismic for bestowing prime 7 effectively", and I doubt any other fans of it do either. Generally, the issue I take with your writing is it's written very similarly to general exposition for temperaments on the xen wiki, where I'd instead recommend to try to write in a way that explains more clearly what the temperament is actually doing in terms of tuning without simply stating commas. For example, "inflating the rastma to a single step" is only meaningful if you already both know what the rastma is and why you would care about it, so I removed it in favour of a different wording that essentially says the same thing. The spirit of this concern is also reflected in the next paragraph. Also, fwiw, the info of why 53edo's 5-limit is tuned better is already kind of implicit in the initial explanation, but I do agree it could be clearer, though I'm not sure how to word it.

: I am confident all the info for 53edo is reasonably useful cuz it's specifically how it handles interseptimals that explains how its handling of primes 5 and 13 is better, but I'm not sure how to improve the paragraph to feel less verbose.

: Regarding S-expressions: I don't see what's wrong with using S-expressions in writing given you can read what they mean from the expression once you learn them, however, to address your concern, I've made it more self-evident what it means. But generally, the target audience is the curious, as once you know how to read them it's often easier to remember than the base 10 ratios. For this same reason, it's recommendable to avoid mentioning commas without explanation - this isn't a rule, but it should probably be added to the style guide. Notice that we have explanations of structures for other temperaments on the page. For example, for schismic, it's mentioned that "[[#Garibaldi]] finds [[~]][[8/7]] as [[9/8]] * [[81/80]] by tempering out [[5120/5103]] = [[64/63|S8]]/[[81/80|S9]], so that it prefers a slightly-sharp or just fifth", where any further explanation would serve to clutter* and decrease the density of the information, while any simpler explanation would probably miss out important information. That's generally the idea of using S-expressions. *Specifically, what I mean is there's a line in writing where you can't reasonably be expected to explain ''everything'' that is unfamiliar to a reader just because your reader might be a complete beginner, because if you assume that, you will need to explain arbitrarily simple concepts. Instead, you should assume that a reader is comfortable with ignoring some (ideally fairly minimal, but often this isn't practical) amount of "new" content to learn. In the case of S-expressions, all we are assuming really in these two examples is knowledge of what S''k'' means. Given this perspective, my perspective from talking to people disliking the xen wiki in the past is that writing a plain ratio will potentially serve to alienate the reader with a sense that the ratios must be memorised or having to click through every ratio to find why that ratio might be worth caring about, so it seems to me that a notation that self-explains the meaning of the comma is a preferable method, especially if we combine with the ratio for reference and especially if we try to give some helpful hints in the writing, where I only suggest this because I do agree that it's ideal to be as self-explaining as reasonably possible without clutter. I included a mention of 325/324's S-expressions at the end as it was explained how the comma was reached so the S-expressions serve as further interpretations of what it does.

: I'd like to note that I took great care in my original edit, deliberately trying to preserve as much of what you wrote as possible while adhering to what I thought was clearer writing in the spirit of compromise, so the rewriting feels a little ad-hoc.

: Anyways, please kindly take a careful read of the edit I did and let me know if you think any key information is still missing.

: Also, if I'm being honest, I find [this version](https://en.xen.wiki/index.php?title=Bird%27s_eye_view_of_temperaments_by_accuracy&oldid=229781#Garibaldi) to be ideal so I'd be curious if you was willing to state what's wrong with that version, as it feels to me that all the important info is there and stated as efficiently as reasonable, which makes for a pleasant reading experience when reviewing temperaments on the page.

: --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 13:20, 7 May 2026 (UTC)

:: I generally did not find missing information, problem is that I found too much of it, specially in 53edo.

:: The key points I believe you'd agree on:

::* Garibaldi is a natural extension of schismic.

::* 10/9 - 9/8 - 8/7 are made equidistant, and the distance is a 81/80 ~ 64/63 ~ pythcomma. 5120/5103 is tempered out.

::* 41edo and 53edo are the coarsest tunings to support it. Its choice depends on whether to optimize for prime 11 or prime 13.

::* 41edo has a hemififth 11/9~27/22, good 11-limit, bad 15/13, 13/10.

::* 53edo has a hemifourth ~15/13, great 2.3.5.13, bad 14/11; 11/9 and 27/22 are made too far apart.

::* Both support cassandra, which extends garibaldi equating 39/32~11/9, and (10/9)^2 = ~16/13. 352/351 and 325/324 are tempered out.

::* 94edo, their sum, is close to optimal for cassandra.

::* 135edo and 147edo also support cassandra but with more inconsistencies.

:: How can we express them with max concision? On the topic of S-expression, I can't agree that S-expressions are easier to remember than ratios... I don't chunk them, I have to parse them. 5120/5103, 225/224, 32805/32768, 531441/524288, all are to me as a single thing, alongside many other commas whose ratios, monzos, names I've memorized. I find ratios, monzos or names much easier to memorize (within reason), and I alternate between the three. --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 16:29, 7 May 2026 (UTC)

::: I reject your argument of "easy to memorise" as there's far too many commas for that (as many could tell you) and because as proof of this, I'm probably the person on XA who is ''most comfortable'' with reading temperament info in the form of the pure ratios. Take a look at how much of my post history is raw comma ratios if you don't believe me ^_^;. I specifically discovered the S-expression-based comma families ''because'' of very common patterns in the kinds of commas I wanted to temper out or equate, like:

::: S''k'' = (''k''/(''k''-1))/((''k''+1)/k) and S''k''/S(''k''+1) = (''k''+2)/(''k''-1) / ((''k''+1)/''k'')3, and I then discovered S(''k''-1)/S(''k''+1) [[S-expression#Sk/S(k_+_2)_(semiparticulars)|had a general meaning too]] as well as (the much more obvious) S''k''*S(''k''+1)*...*S(''k''+''n''-1) = (''k''/(''k''-1)) / ((''k''+''n'')/(''k''+''n''-1)).

::: In other words, my recommendation for practicality is that commas that ''don't'' uniquely fit into an S-expression-based infinite comma family with a clear meaning are ones that should be memorised. (And if there's a comma that fits into multiple families, that's usually notable enough that it's worth bothering to memorise all of its S-expressions, so in that case I like stating all of its known S-expressions.) Basically I'm saying, if you're struggling to remember S-expressions, that's because they're not ''supposed'' to be memorised; you're supposed to ''read'' what they do ''from the expression'', which if you actually know your commas, will easily tell you what comma it corresponds to. Hopefully I explained that well enough.

:: 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.~~

::: --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 17:09, 7 May 2026 (UTC)

:: ~~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)

::: Separately, the key points you believe I'd agree on aren't really all that useful to me because they don't really address anything that I actually said, as it gives me no idea what you actually take issue with in my writing other than S-expressions, and it's already clear in this page that the writing style is supposed to be different from the rest of the xen wiki cuz of being aimed at being a compendium of useful rank 2 temperaments. (I think covering other ranks is going to make things too complicated/confusing probably/realistically, so it's probably best to keep it at rank 2 given a lot of the most useful rank 3 temperaments like marvel have a bunch of rank 2 interpretations anyways.) Anyways, if it's more convenient to you than reading dense writing, I'd be happy to talk on Discord. (I've pinged you there.) --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 17:09, 7 May 2026 (UTC)

@@ Line 25: / Line 25: @@
 == Comment on your proposal on EDO subgroups ==
-For reference, [[Xenharmonic_Wiki:Things_to_do#Set_a_semi-objective_standard_for_classifying_edos_as_subgroup_temperaments|here is the proposal]]. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 04:59, 22 January 2021 (UTC)
+For reference, [[Xenharmonic_Wiki_talk:Things_to_do#13-Limit.2C_17-Limit_and_19-Limit_Comma_Pages|here is the proposal]].--[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 04:59, 22 January 2021 (UTC)
+: Edit: Fixed link, also, to find the beginning of my proposal, Ctrl+F "Start of suggestion/reply by Godtone" once there (or scroll down a bit). --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]])
 : Yes, I read that. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 05:40, 22 January 2021 (UTC)
@@ Line 46: / Line 48: @@
 :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.
-(Note: like 80, unfortunately, 87's worst prime is 7, but the error and relative error is less and in the opposite direction.)<br/>
+:(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)
+:--[[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 counted if the telos prime is something other than 2, 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 chains 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 prime chains, 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 to and including the point of connection.
+:: Just looking at 3-to-2 telicity, which, by definition, involves a circle of fifths as the 2-prime is the only available telos for the 3 prime chain, 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 both 29edo and 87edo. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 21:30, 22 January 2021 (UTC)
+::: "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 chains up to the point of connection must also agree with the connection."<br/>Yes, I understood that part. I never said that the circles must accumulate less than half an EDOstep of error in their full/completed chains in an EDO or sub-EDO. I don't think a "circle" of an interval has to necessarily close on (a multiple of) the octave within half an EDOstep to be used as a "circle" because the interval could still be very or sufficiently accurate, although in the case of larger EDOs, having ''some'' strong circles that fulfill that condition is important for orientation. I now see my definition is technically not specific enough and would require that the error of generators don't accumulate so much as to cause inconsistency at any point in the chain up to the connection, but I was mainly intent on confirming understanding rather than restating the definition exactly.<br/>Also, I never claimed that EDOs 80, 29 or 87 succeed telicity in the 2.3 subgroup. That doesn't mean their circle of fifths or circles of other intervals can't be useful, interesting or equally valid as a method of organising them, for example the "circle of fifths" in meantone does not necessarily close within 50c to the octave, depending on tuning. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 22:21, 22 January 2021 (UTC)
+:::: I must admit that I wanted to make sure your understanding was completely correct before I confirmed it, as you can never completely tell online.  Yes, it's true that there are other EDOs with other circles of fifths, but if they don't succeed at having telicity, I find them to be less than ideal, since the 3-prime is the most commonly used prime outside the octave.  That said, you did more or less hit the nail on the head when you mentioned that large EDOs need some strong circles that fulfill the telicity condition for the sake of orientation- in fact, even something like the 11-to-3 telicity of 159edo and 24edo is very useful for navigation, though making the best use of this kind of telicity involves building on good 3-to-2 telicity. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 00:38, 23 January 2021 (UTC)
+== 159edo and Composition ==
+Hey, I'm curious as to whether or not you've heard my songs "[[:File:Space Tour.mp3|Space Tour]]" and "[[:File:Welcome to Dystopia.mp3|Welcome to Dystopia]]" and what your impressions of those songs are, especially since between the two of them, they show off some of 159edo's tricks pretty well. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 20:04, 7 February 2021 (UTC)
+== Corrections ==
+Hi Godtone,
+Sometimes it can be a bit confusing to get the cent values right ([https://en.xen.wiki/index.php?title=25/18&diff=next&oldid=69725], [https://en.xen.wiki/index.php?title=18/13&diff=next&oldid=69726]) 😊 but I prefer these kinds of changes to accepting things we feel are wrong just because they are written there. Have a nice day! --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 07:07, 12 May 2021 (UTC)
+: Ah yeah, there was a mistake that confused me where it said 5.3c flat rather than 5.3c sharp so that confused me, it was odd because when doing edits I always try to double-check especially if its editing things like cent values which need to be exactly correct. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 18:09, 12 May 2021 (UTC)
+== Meaning of a sentence ==
+Hi Godtone, in [[161edo]], you added a sentence, of which the part “in the 100 to 200 range” is unclear to me. Can you clarify this a bit? Thanks in advance. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 14:30, 6 March 2022 (UTC)
+: I think they meant edos between 100 and 200. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 14:36, 6 March 2022 (UTC)
+:: So a copy and paste error of some sort? I found [https://en.xen.wiki/index.php?title=181edo&diff=prev&oldid=66283 a similar addition to 181edo] (surely your recent edits were the reason for your comment and ultimately my question). In that case, the part in question should be removed. But let's wait and see what Godtone answers. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 15:55, 6 March 2022 (UTC)
+::: Yes sorry, I thought it was clear but in retrospect it could be talking about harmonics not patent val EDOs. The reason I wrote it in such a way is because its concise and seemed unambiguous to me, but also because I'm not sure how to make "from 100 EDO to 200 EDO" sound right... is it "in the range of EDOs 100 to 200" or "from 100 to 200 EDO" or something else? --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 19:17, 6 March 2022 (UTC)
+::: Oh and the reason it appears in the edit history of 181 EDO is because I accidentally added the sentence to the wrong EDO as I had misremembered the number of the EDO, but I removed that sentence from the page for 181 EDO when I realised. I am sure that the one here is right: [[183edo#Prime_harmonics]]. 161edo is also quite a strong system so I am pretty sure that one is right as it came second to 183 EDO a lot. Note 161 EDO's great strength as a no-9's 21-odd-limit system. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 19:31, 6 March 2022 (UTC)
+:::: I think it's that you compare the given EDO ''to'' a range of EDOs. This was not clear (from my POV). I have tried to increase readability by placing the information regarding context at the beginning of the sentence. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 21:23, 6 March 2022 (UTC)
+== Comma name proposal for 256/255: "charisma" vs "charsma" ==
+Hi Godtone,
+I never accept your name proposal for [[256/255]] ("charisma" and "charismic", formerly used for [[Magic family #Horcrux|horcrux extensions]]), but its associated temperament name "charic" (2.3.5.17 subgroup rank-3). For disambiguation, 256/255 should be renamed "char comma", "charic comma" or "charsma" (no-''i'' spelling). Then its associated temperament name for full 17-limit rank-6 would be called "charsmic". Do you deal it? --[[User:Xenllium|Xenllium]] ([[User talk:Xenllium|talk]]) 10:34, 6 January 2024 (UTC)
+== Tempering vs tempering out ==
+Please make sure not to use the generic ''temper'' to refer to ''temper out'' or less commonly ''temper away''. I notice you've done this a number of times. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:49, 24 June 2024 (UTC)
+: Oh hmm, it didn't occur to me that it was ambiguous, sorry. Please feel free to correct any instances of it when you see them. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 13:16, 25 June 2024 (UTC)
+== Crediting ==
+As part of using your copyleft works, would you like to be thanked to as Godtone, Osmium(ic), or some other name? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 16:39, 18 January 2025 (UTC)
+: The copyleft stuff I submitted have all been under User:Godtone so if you want to attach a name I would suggest referring to my xen wiki account (mainly so that the original code can be found on the user page). --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 22:02, 18 January 2025 (UTC)
+== User page cleanup ==
+I'd like to read your user page more but it's getting quite long. If you're okay, might I suggest organizing them into subpages? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 16:39, 18 January 2025 (UTC)
+: Hmm, I like having everything in one place. The navigation/contents/glossary at the top serves the same purpose and I don't want to obscure stuff thru links. The more pressing concern though is that I don't have the energy/interest to do a big cleanup of the user page for a variety of reasons, so I'd rather just make small improvements to how it currently is. Therefore in that spirit, let me know if anything strikes you as particularly messy/in need of cleanup, cuz I agree that some of it could do with some cleaning, I'm just not quite sure what needs it the most. Maybe moving some more frivolous stuff out would make sense so that the code isn't obscured, but IDK about moving ''everything'' except the code cuz it kinda defeats the point of a user page to me. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 22:02, 18 January 2025 (UTC)
+== "Augmented–chromatic equivalence continuum" ==
+I'm gonna move that article to a subpage for now. First, the title is inappropriate cuz when I see augmented and chromatic I think about 128/125 and 2187/2048, which is a comma basis of 21et. If someone create an equivalence continuum about 21et that will be the title. Second, I don't get at all what's bad about the father--3 continuum and what's not bad about your augmented--dicot continuum. Father is the simplest comma of 3et with order 1 in the prime 5 and is marked against the 3-limit comma 32/27, just like any other continua. Your choice of commas looks arbitrary and unjustifiable to me. Even if you have valid reasons to justify it, it's prolly too deep and niche to quickly grasp compared to the obvious way to organize them. Third, there's the change of parameter used in some continua and you can do that in the father--3 article too. Just use a different variable than ''n'', ''m'', and ''k'', and give your reasons! [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 11:03, 27 February 2025 (UTC)
+: Sorry, my mistake with "chroma", I was thinking that it refers to 25/24 based on the [[syntonic–chromatic equivalence continuum]] and based on me intuitively thinking of a chroma as being strictly smaller than a semitone so that I often feel that 2187/2048 is the odd one out. But in my defence, it was used before in this way on the [[Father–3 equivalence continuum]] page:
+:: "[...] such that temperaments satisfy ('''25/24''')<sup>k</sup> = '''16/15'''. This gives rise to the name '''chromatic'''–'''diatonic''' equivalence continuum, where both chromatic and diatonic refer to the classical versions of semitones."
+: Anyways, to explain the choice of commas:
+: I'm pretty certain* that 128/125 and 25/24 is the best way to organize this continuum because 1\3 is ''relatively speaking'' an extremely good approximation of 5/4, so that practically all the most notable and useful 5-limit temperaments supported by 3edo are defined principally in terms of how the genchain of ~5/4's finds prime 3 (which I'll explain in a moment), as if I'm not mistaken, integer ''n'' (for (128/125)<sup>n</sup> / (25/24)) always corresponds to using ~5/4 as a gen, so that generally speaking (tho I'm unsure if this is always true) non-integer ''n'' corresponds to using a gen that splits 5/4, 8/5, 2/1 or an octave-equivalent correspondingly. That is, one reason for 128/125 is that 3edo is notable not for its "5-limit" but for its 2.5 subgroup which is also partly why having 5^2 present in the other comma is acceptable IMO because it will cancel with 5^3 (though I discuss an alternative at the end that you might prefer, though I'd want to see it first cuz I worry it'll lead to many weird temperaments at simple points drowning out the interesting ones).
+: Let me quote from the page to explain the IMO very straightforward musical significance/application of describing the continuum in this way:
+:: "This formulation has a specific reason: 128/125 is significantly smaller than 25/24, so that it makes sense to equate some number of 128/125's with 25/24, but because {{nowrap|25/24 {{=}} ([[25/16]])/([[3/2]])}}, this has the consequence of clearly relating the ''n'' in {{nowrap|(128/125)<sup>n</sup> {{=}} 25/24}} with how many 5/4's are used to reach 3/2 (when octave-reduced):
+:: If ''n'' = 0, then it takes no 128/125's to reach 25/24, implying 25/24's size is 0 (so that it's tempered out), meaning that 3/2 is reached via (5/4)<sup>2</sup>.
+:: For integer ''n'' > 0, we always reach 25/24 via (25/16)/(128/125)<sup>''n''</sup> because of (128/125)<sup>''n''</sup> ~ 25/24 by definition, meaning that we reach 3/2 at 3''n'' + 2 generators of ~5/4, octave-reduced."
+: That is, ''n'' is the number of dieses you need to flatten ~25/16 by to reach ~3/2, which seems musically very clear to me, as this explanation tells you how many gens of ~5/4 you need to move to reach ~3/2 (up to octave-equivalence), as each diesis is three ~5/4's, and thereby tells you the intuitive logic of pumping the comma of the associated temp for integer ''n'' as well as gives you crucial information about how many intervals are guaranteed to exist between ~6/5 and ~5/4 in a nontrivial tuning (which is why the integer part is important; ''n'' - 1 is the number of intervals between 6/5 and 5/4, which is equivalent to saying ''n'' is how many pieces we split 25/24 into).
+: Having said that, it'd be good to also include a change of basis ''k'' = 3''n'' + 2 so that we don't have to manually and indirectly deduce the number of gens of ~5/4 needed for ~3/2 (because this can be annoying for non-integer ''n''), but I don't know how to calculate the associated comma pair, but I'm pretty sure it doesn't result in any of the coordinate schemes currently documented in [[Father–3 equivalence continuum]]. Like, are you really okay with magic being fractional in both ''n'' and ''m'' there? Also, IMO, 25/24 being at 0 is super natural; other than large integer values of ''n'', it's the only natural number value of ''n'' that corresponds to an exotemp, and is meaningfully trivial compared to integer ''n'' > 0. Also notice how as ''n'' grows, ~5/4 becomes sharper and approaches 1\3, which makes taking the limit of ''n'' to infinity to reach ~5/4 = 1\3 very intuitive as well. This also makes ''n'' = 1 (magic) notable as the simplest nontrivial tuning (one that admits a reasonable amount of structure and as a result isn't an exotemp), and if you accept the approximation ~5/4 = 1\3 (which even I accept given the right tuning of the fifth), then every integer ''n'' > 0 is non-exo in terms of the tuning of the 5-odd-limit (even if the edo and mapping complexity implied is absurd).
+: <nowiki>*</nowiki> Re: "pretty certain", initially I was completely certain, hence the boldness of my edit cuz it seemed like a very unambiguous and significant improvement, but the consideration about 3''n'' + 2 maybe being a little more work than is reasonable to ask of the user makes me unsure given I haven't seen what the continuum looks like if we apply this transformation. The alternative transformation would have dicot at 2, magic at 5, mutt at 7, würschmidt at 8, etc. but already we see this might not be desirable because the info that mutt splits something (the octave and the diesis) into 3 is lost.
+== [[Bird's eye view of temperaments by accuracy]] ==
+I've expanded this page a bit and added some more temperaments. This page could be useful, but it seems like no one contributes to it anymore.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 22:45, 21 November 2025 (UTC)
+Hey man I noticed your changes in the bird's eye page, I partially reversed them, before we begin an edit war I'd like to give some justifications and also hear your opinion so we can come to a truce:
+* Garibaldi is the simplest 7-limit effective extension of schismic: -14 fifths versus -26 fifths in gracecordial or whatver is called (kinda alright but more complex), and -2 fifths in schism (bad and very innacurate extension)
+* When I mean bad, I mean ''temperament'' bad. As in high badness. This also makes garibaldi and cassandra the ''best'' compared to helenus and andromeda which are comparatively worse.
+* I removed the s-expressions because I believe we don't need them. I don't know the target audience of this, but if it's a less temperament informed reader, the s-expressions will only clutter as the comma ratios or names, whatever is shortest, are already succint.
+* I have a huge spot for cassandra and became one with it. Sorry if I come of as defensive or glaze it too much. Can't help myself.
+That's all --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 09:48, 7 May 2026 (UTC)
+: It appears "minicomma" is an idiosyncratic term, so I removed it. I moved the discussion of cassandra to the bottom to avoid the feeling of repetition of information as that way it feels more like a summary of what 41edo, 53edo and 94edo have in common. I have no issue with describing garibaldi as "arguably the best extension to schismic for bestowing prime 7 effectively", and I doubt any other fans of it do either. Generally, the issue I take with your writing is it's written very similarly to general exposition for temperaments on the xen wiki, where I'd instead recommend to try to write in a way that explains more clearly what the temperament is actually doing in terms of tuning without simply stating commas. For example, "inflating the rastma to a single step" is only meaningful if you already both know what the rastma is and why you would care about it, so I removed it in favour of a different wording that essentially says the same thing. The spirit of this concern is also reflected in the next paragraph. Also, fwiw, the info of why 53edo's 5-limit is tuned better is already kind of implicit in the initial explanation, but I do agree it could be clearer, though I'm not sure how to word it.
+: I am confident all the info for 53edo is reasonably useful cuz it's specifically how it handles interseptimals that explains how its handling of primes 5 and 13 is better, but I'm not sure how to improve the paragraph to feel less verbose.
+: Regarding S-expressions: I don't see what's wrong with using S-expressions in writing given you can read what they mean from the expression once you learn them, however, to address your concern, I've made it more self-evident what it means. But generally, the target audience is the curious, as once you know how to read them it's often easier to remember than the base 10 ratios. For this same reason, it's recommendable to avoid mentioning commas without explanation - this isn't a rule, but it should probably be added to the style guide. Notice that we have explanations of structures for other temperaments on the page. For example, for schismic, it's mentioned that "[[#Garibaldi]] finds [[~]][[8/7]] as [[9/8]] * [[81/80]] by tempering out [[5120/5103]] = [[64/63|S8]]/[[81/80|S9]], so that it prefers a slightly-sharp or just fifth", where any further explanation would serve to clutter* and decrease the density of the information, while any simpler explanation would probably miss out important information. That's generally the idea of using S-expressions. *Specifically, what I mean is there's a line in writing where you can't reasonably be expected to explain ''everything'' that is unfamiliar to a reader just because your reader might be a complete beginner, because if you assume that, you will need to explain arbitrarily simple concepts. Instead, you should assume that a reader is comfortable with ignoring some (ideally fairly minimal, but often this isn't practical) amount of "new" content to learn. In the case of S-expressions, all we are assuming really in these two examples is knowledge of what S''k'' means. Given this perspective, my perspective from talking to people disliking the xen wiki in the past is that writing a plain ratio will potentially serve to alienate the reader with a sense that the ratios must be memorised or having to click through every ratio to find why that ratio might be worth caring about, so it seems to me that a notation that self-explains the meaning of the comma is a preferable method, especially if we combine with the ratio for reference and especially if we try to give some helpful hints in the writing, where I only suggest this because I do agree that it's ideal to be as self-explaining as reasonably possible without clutter. I included a mention of 325/324's S-expressions at the end as it was explained how the comma was reached so the S-expressions serve as further interpretations of what it does.
+: I'd like to note that I took great care in my original edit, deliberately trying to preserve as much of what you wrote as possible while adhering to what I thought was clearer writing in the spirit of compromise, so the rewriting feels a little ad-hoc.
+: Anyways, please kindly take a careful read of the edit I did and let me know if you think any key information is still missing.
+: Also, if I'm being honest, I find [this version](https://en.xen.wiki/index.php?title=Bird%27s_eye_view_of_temperaments_by_accuracy&oldid=229781#Garibaldi) to be ideal so I'd be curious if you was willing to state what's wrong with that version, as it feels to me that all the important info is there and stated as efficiently as reasonable, which makes for a pleasant reading experience when reviewing temperaments on the page.
+: --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 13:20, 7 May 2026 (UTC)
+:: I generally did not find missing information, problem is that I found too much of it, specially in 53edo.
+:: The key points I believe you'd agree on:
+::* Garibaldi is a natural extension of schismic.
+::* 10/9 - 9/8 - 8/7 are made equidistant, and the distance is a 81/80 ~ 64/63 ~ pythcomma. 5120/5103 is tempered out.
+::* 41edo and 53edo are the coarsest tunings to support it. Its choice depends on whether to optimize for prime 11 or prime 13.
+::* 41edo has a hemififth 11/9~27/22, good 11-limit, bad 15/13, 13/10.
+::* 53edo has a hemifourth ~15/13, great 2.3.5.13, bad 14/11; 11/9 and 27/22 are made too far apart.
+::* Both support cassandra, which extends garibaldi equating 39/32~11/9, and (10/9)^2 = ~16/13. 352/351 and 325/324 are tempered out.
+::* 94edo, their sum, is close to optimal for cassandra.
+::* 135edo and 147edo also support cassandra but with more inconsistencies.
+:: How can we express them with max concision? On the topic of S-expression, I can't agree that S-expressions are easier to remember than ratios... I don't chunk them, I have to parse them. 5120/5103, 225/224, 32805/32768, 531441/524288, all are to me as a single thing, alongside many other commas whose ratios, monzos, names I've memorized. I find ratios, monzos or names much easier to memorize (within reason), and I alternate between the three. --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 16:29, 7 May 2026 (UTC)
+::: I reject your argument of "easy to memorise" as there's far too many commas for that (as many could tell you) and because as proof of this, I'm probably the person on XA who is ''most comfortable'' with reading temperament info in the form of the pure ratios. Take a look at how much of my post history is raw comma ratios if you don't believe me ^_^;. I specifically discovered the S-expression-based comma families ''because'' of very common patterns in the kinds of commas I wanted to temper out or equate, like:
+::: S''k'' = (''k''/(''k''-1))/((''k''+1)/k) and S''k''/S(''k''+1) = (''k''+2)/(''k''-1) / ((''k''+1)/''k'')<sup>3</sup>, and I then discovered S(''k''-1)/S(''k''+1) [[S-expression#Sk/S(k_+_2)_(semiparticulars)|had a general meaning too]] as well as (the much more obvious) S''k''*S(''k''+1)*...*S(''k''+''n''-1) = (''k''/(''k''-1)) / ((''k''+''n'')/(''k''+''n''-1)).
+::: In other words, my recommendation for practicality is that commas that ''don't'' uniquely fit into an S-expression-based infinite comma family with a clear meaning are ones that should be memorised. (And if there's a comma that fits into multiple families, that's usually notable enough that it's worth bothering to memorise all of its S-expressions, so in that case I like stating all of its known S-expressions.) Basically I'm saying, if you're struggling to remember S-expressions, that's because they're not ''supposed'' to be memorised; you're supposed to ''read'' what they do ''from the expression'', which if you actually know your commas, will easily tell you what comma it corresponds to. Hopefully I explained that well enough.
-:: 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.
+::: --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 17:09, 7 May 2026 (UTC)
-:: 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)
+::: Separately, the key points you believe I'd agree on aren't really all that useful to me because they don't really address anything that I actually said, as it gives me no idea what you actually take issue with in my writing other than S-expressions, and it's already clear in this page that the writing style is supposed to be different from the rest of the xen wiki cuz of being aimed at being a compendium of useful rank 2 temperaments. (I think covering other ranks is going to make things too complicated/confusing probably/realistically, so it's probably best to keep it at rank 2 given a lot of the most useful rank 3 temperaments like marvel have a bunch of rank 2 interpretations anyways.) Anyways, if it's more convenient to you than reading dense writing, I'd be happy to talk on Discord. (I've pinged you there.) --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 17:09, 7 May 2026 (UTC)