Talk:Meantone

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Hypopental EDO's

I've added an article for "hyperpent" a.k.a. hyperpental edo's and it's antithesis, "hypopent," or hyperpental edo's to cover classifications of different equal divisions of the octave that temper the fifth either sharp or flat, descriptively, rather than as a matter of intent or purpose.

I'm not sure if this is virtually redundant with "meantone" and "superpyth", but, if it is, I'd be happy to delete those articles or merge any useful information here.

--Bozu (talk) 15:42, 9 July 2020 (UTC)

Definitely not redundant, but I remember seeing these concepts somewhere, in different terms. FloraC (talk) 13:25, 13 July 2020 (UTC)

Interesting. If you come across it again, please update. --Bozu (talk) 14:59, 13 July 2020 (UTC)

Mothra as a 7-limit extension

Seems that Mothra should be included under the 7-limit extensions of Meantone. After all, it IS included in meantone family, even if it doesn't get the organizational prominence of the standard Septimal Meantone and other competitors. Since it includes meantones with a fifth divisible by 3 all the way from 5EDO to 21cEDO, while the other 7-limit extensions are either more restricted in range (standard Septimal Meantone, Flattone, Dominant) or apply to only a very few equal temperaments in their range (Sharptone), it seems that Mothra should get more attention. Lucius Chiaraviglio (talk) 05:34, 1 August 2024 (UTC)

I think this article should mostly focus on septimal meantone and briefly cover other strong extensions (tho each septimal extension deserves its own article). Weak extensions aren't as closely connected to the original temp as strong extensions. I'd expect the discussion on weak extensions to be addressed in meantone family. FloraC (talk) 09:01, 2 August 2024 (UTC)

Shouldn't Mothra be considered a strong extension given its range of coverage? (I could see Cloudy as a weak 7-limit extension since it has a limited useful range.) Or am I misunderstanding what divides strong extensions from weak extensions? Lucius Chiaraviglio (talk) 14:57, 6 August 2024 (UTC)

There's indeed a misunderstanding. The exact meaning of a strong extension is an extension that doesn't split the period and generators. FloraC (talk) 15:23, 6 August 2024 (UTC)

“Meantone” ≠ “Septimal meantone”

I've noticed that in various parts of the wiki, “meantone” is sometimes used without qualification to refer to both conventional 5-limit meantone, which would include tunings across the spectrum, such as 12edo, 19edo, 31edo, 43edo, 50edo, and so on‌; and to septimal meantone, which is rather poorly tuned compared to the 5-limit in most of those (31edo and quarter-comma meantone being the exceptions).

In some cases, this is a mere abuse of terminology, intended to convey some observation about quarter-comma meantone in particular in a succinct way. However, in other cases it is a bait and switch: “meantone” is first introduced as a historical basis of music theory, and then it is claimed that in “meantone” some chord is an essentially tempered chord, but on closer inspection it turns out that the two occurrences of “meantone” actually refer to different things (5-limit meantone and septimal meantone respectively).

To avoid this kind of ambiguity, I suggest that wiki pages should consistently use the phrase “septimal meantone”, rather than just “meantone”, when referring to concepts that require the septimal extension, including chords of septimal meantone, and should use “meantone” on its own only when referring to the conventional 5-limit form.

--Bcmills (talk) 05:48, 24 August 2024 (UTC)

See Temperament naming for the basics of how temps are named. In most tuning between 11\19 and 7\12 of meantone, you get septimal intervals for free, which is why septimal meantone inherits the name. Historically, septimal intervals were known no later than Helmholz and Ellis (1875), tho they didn't use them to analyse meantone. But that's already one and a half centuries. It doesn't do any benefit to stop there when we can apply the knowledge of septimal intervals to reveal what meantone is actually up to.

The most important corollary of accepting meantone as a 7-limit temp is just that: introducing a number of essentially tempered chords, and to realize most if not all chords in common practice were 9-odd-limit concords, either essentially tempered or essentially just. While many consider 9-odd-limit as concordance and thus will rule out discordance in the analysis of common practice music, septimal intervals and the related essentially tempered chords are still more complex, and contextually later, than say 5-limit triads, so that treating them as discordance is still possible. Since they are basically free to be laid on or taken out, the distinction of classical and septimal meantone seems practically pointless. I can only see the need of it in very technical contexts such as tuning optimization. FloraC (talk) 08:32, 24 August 2024 (UTC)

It may be true that “you get septimal intervals for free” in the RTT mapping, but in many common tunings of meantone the septimal intervals aren't actually tuned well enough to have concordant septimal intervals: pretending that they're always there is “revealing what meantone could be up to if tuned a specific way”, not what it “is actually up to” as performed. Assuming that a piece composed in or influenced by meantone will be tuned in a way that supports a septimal interpretation is too much of a stretch — especially considering how much of the common-practice era applied meantone logic to well-tempered scales, which tend to optimize for better 5-limit intervals.

The septimal bait-and-switch pushes a deeply ahistorical interpretation in order to further the idea that ~every common chord should be interpreted as consonant. It's fine to present that idea as a possibility where it applies, but it shouldn't be taken as a given — especially when that obscures alternatives that don't require ahistorical assumptions.

--Bcmills (talk) 12:59, 24 August 2024 (UTC)

Whether a specific meantone tuning is good enough for septimal intervals is a subjective matter. The interest in septimal intervals is idiomatic from the last century (including the early days of the community such as A meantone tuning with six 7-limit tetrads) thanks to musicians like Harry Partch, and most would agree 11\19 to 7\12 is a reasonable assessment of meantone's range, in which septimal meantone rules. I don't think there's pretention involved in interpreting augmented and diminished intervals as septimal since it's there in any chromatic keyboard. If historical analysis didn't recognize their byproduct presence then it was seriously out of date, and we are here to set them straight since we develop temp theories. Moreoever you can't be 100% sure that literally everyone of common practice was ignorant about septimal intervals to say they're "ahistorical". A quick skim of the archive showed Mozart was suspected to have used them as a means of expression (TUNING digest 1370: Meantone).

Septimal meantone isn't the only evidence that common practice music was mostly concordant (note: I distinguish consonance and concordance and hope you do too; otherwise your summary of the idea would be inaccurate). It's in the existence of meantone in the first place: to eliminate the wolf fourths and fifths. FloraC (talk) 15:04, 24 August 2024 (UTC)

That is strong evidence for concordance in the 5-limit, not the 7-limit. Again, there is nothing wrong with presenting the theory of septimal concordance when it applies; it's just important to distinguish that theory from the mainstream interpretation of meantone as a (conceptual and practical) tuning oriented specifically toward 5-limit concordance. Common practice included many meantone-based well-temperaments and meantones other than quarter-comma, so it is not appropriate to construe interpretations that rely on near-quarter-comma tuning as representing a canonical interpretation of the entirety of meantone-based harmony.

--Bcmills (talk) 15:28, 24 August 2024 (UTC)

Evidence of consideration of septimal intervals in the Renaissance and Baroque seems to be merely hard to find, not impossible. For instance, "Fundamental Principles of Just Intonation and Microtonal Composition"(*) by Thomas Nicholson and Marc Sabat at the Universität der Künste Berlin (no date given but must be at least as recent as 2012) cites people of the times at least thinking about septimal intervals: Giuseppe Tartini and Nicola Vicentino.

(*)Indirect link: Goes to a page that requires clicking another link to download the article PDF.

Other Google search results for septimal intervals in Renaissance and Baroque music (and music of Nicola Vicentino and Giuseppe Tartini) give further tantalizing clues of the consideration of septimal intervals in these times, and the Wikipedia page on 7-limit tuning lists some composers going back to that era who considered septimal intervals to include consonant members, but gives little additional details. Overall, the gestalt impression that I get is that the Renaissance and Baroque music communities mostly swept the 7-limit under the rug, but a few people at least poked at it.

Lucius Chiaraviglio (talk) 06:21, 25 August 2024 (UTC)

There is a difference between presenting septimal meantone as a valid framework for interpreting meantone, and presenting it as the canonical framework for interpreting meantone. The former seems appropriate and historically justified; but taking the term “meantone” to mean “septimal meantone” by default instead does the latter, which is not consistent with how meantone has been presented in most (not all) writing on conventional music theory.

I am ok with presenting septimal meantone as a separate interpretation on par with conventional 5-limit meantone when it applies. I am not ok with conflating the two, as though users of meantone should be assumed to always intend the 7-limit interpretation.

--Bcmills (talk) 18:06, 25 August 2024 (UTC)

I agree with that. I am just saying that septimal meantone has not been completely absent historically. Lucius Chiaraviglio (talk) 23:26, 25 August 2024 (UTC)

Wrong Septimal Comma in tuning spectrum table?

I scratched my head for a while about the lines having septimal comma fractions in the tuning spectrum table. For a while I thought they were just there to have a spectrum of what some commnn foreign commas do when used to flatten the fifth, but then it occurred to me that whoever put them there was probably really trying to do the right thing for septimal meantone, but didn't understand which 7-limit comma to use. As observed in the current table, flattening the fifth by fractions of an Archytas septimal comma gives the Eigenmonzos listed (apart from one which was off by a typo that I fixed earlier), but they aren't very good Eigenmonzos, for the most part except as educational examples of why one should flatten fifths by fractions of a syntonic comma but sharpen fifths by fractions of an Arcyhtas comma and not the other way around (admittedly, they serve that purpose quite well). (For an additional use, see below.)

The comma the author was probably after is actually Harrison's comma = [-13 10 0 -1⟩ (as listed in Meantone family, which does not have a "septimal meantone comma" alternate name (maybe it should?), although the page text does mention septimal meantone, and currently does not get any mention in the Septimal comma disambiguation page (maybe it should?). So if I put in some examples in the form of a hypothetical set of rows of the table (many lines from the actual table omitted to save space here):

Edo Generator	Eigenmonzo (Unchanged-interval)	Generator (¢)	Comments
	27/20	680.449	Full comma (syntonic comma; from here onwards "comma" without an adjective refers to syntonic comma)
4\7		685.714	Lower bound of 5-odd-limit diamond monotone
	567/512	688.323	1/2 septimal comma
	896/729	689.274	1/4 Harrison's comma
	[16 -10⟩	690.225	1/2 Pythagorean comma, as M2.
	51/38	690.603	As P4.
19\33		690.909	33cddd val
	[-19 9 0 2⟩	691.049	2/5 septimal comma
	9/5	691.202	1/2 comma, tunings flatter than this do not fit the original sense of meantone, since their whole tones are no longer between 9/8 and 10/9, a.k.a. lower bound of 9-odd-limit diamond tradeoff
	243/224	691.810	1/5 Harrison's comma
	112/81	693.501	1/6 Harrison's comma
	28/27	694.709	1/7 Harrison's comma
11\19		694.737	Lower bound of 7- and 9-odd-limit diamond monotone
	5/3	694.786	1/3 comma
	9/7	695.614	1/8 Harrison's comma
	7/6	696.319	1/9 Harrison's comma
	7/4	696.883	1/10 Harrison's comma
	21/16	697.344	1/11 Harrison's comma
	64/63	697.728	1/12 Harrison's comma

As you can see, only the first Harrison's comma entry has a bad Eigenmonzo (that also does not currently have a Xenharmonic Wiki page); all the rest have Xenharmonic Wiki pages, and they keep getting simpler until 1/10 Harrison's comma, after which they get more complex again, with a pass through 64/63 as an Eigenmonzo rather than a tempering comma fraction (but not the same as the line with the 64/63 Eigenmonzo at a fifth of 702.272¢ that was removed on 2024-07-30T10:10:27 -- still haven't figured out where that one came from). Indeed, the actual table appears to contain a subset of the above entries, but with no explanation in the comments column.

While checking the above, I noticed that flattening the fifth by 1/7 of the Archytas septimal comma does have the additional use of generating as its Eigenmonzo the flattone comma [-17 9 0 1⟩ (does this comma have an official name? -- it doesn't seem to have its own page). Inspired by this, I also looked to see if other listed Archytas septimal comma fractions produced as their Eigenmonzos the Harrison's comma [-13 10 0 -1⟩ or the flattertone comma [-24 17 0 -1⟩ (does this have an official name? -- it doesn't seem to have its own page either), but I did not see either of these, so those must require Archytas comma fractions that are not listed (I have not yet tried to the math to figure out what they are).

So I would recommend integrating the above table into the septimal meantone tuning table, since Harrison's comma is the comma other than the syntonic comma that really fits in with the septimal meantone tuning spectrum, but maybe leaving in Archytas septimal comma fractional flattenings that produce as their Eigenmonzos commas that are used in 7-limit meantone extensions.

As an alternative, I could certainly go along with the idea of splitting septimal meantone into its own article with its own tuning spectrum table (as Bcmills has recommended) that has the 7-limit comma fractions (as has been done for flattone, and reserving the tuning spectrum table in the main meantone article for 3-limit and 5-limit comma fractions and the EDO tunings in their vicinity. This table could reference the other tables, for instance noting at 19EDO that tunings flatter than this use the flattone 7-limit extension, while tunings sharper than this use septimal meantone (and like wise for 12EDO as the intersection between septimal meantone and dominant, and likewise for 26EDO as the intersection between flattone and flattertone -- and even likewise for 5EDO as the intersection between dominant and sharptone, if one accepts tunings with a negative size of diatonic semitone in the way-out northwest wilderness of the grand unified 5-limit meantone spectrum).

Lucius Chiaraviglio (talk) 06:28, 29 August 2024 (UTC)

Note that septimal comma here refers specifically to 64/63. I think the original author simply intended to show flattening the fifths by a fraction of a syntonic comma and then subsitituting the septimal and/or Pythagorean comma for it to give slightly detuned results. That's why I said these septimal-comma tunings were useless and proposed to remove them, as no one actively looked for them after anything than recreational-math purposes.

I think the author was aware of the harrison comma. The harrison comma is never called a septimal comma, and it shouldn't, except in the non-idiomatic sense where a septimal comma is any 7-limit comma. Idiomatically the septimal comma always refers to 64/63.

Adding the harrison comma ofc is a reasonable idea, tho I'm not sure how helpful it is. Basically every eigenmonzo tuning is a fractional-comma tuning. FloraC (talk) 13:16, 29 August 2024 (UTC)

P.s. these septimal- and Pythagorean-comma tunings were added by MMTM which means I'll simply remove them.

The Pythagorean comma tunings do have some legitimacy for the historical use of some of them. Lucius Chiaraviglio (talk) 21:10, 29 August 2024 (UTC)

A use for Sharptone?

Finally thought of a potential use for Sharptone beyond using it as a septimal extension for 5EDO and even sharper tunings: Since the 12EDO approximation of 7/4 starts out sharp, and since the most common guitars (and related instruments) in the Western world are fretted for 12EDO, and since pitch bending on such instruments needs to be upwards, Sharptone gives you a way to play notes that have a septimal numerator by playing the next note down and bending it up to the right values. Of course, for notes that have the 7 in the numerator, you wouldn't want to use Sharptone, because then you would have to bend the pitch down; however, at least in jazz and blues having the 7 in the numerator relative to the root note seems to be the most common septimal interval. Lucius Chiaraviglio (talk) 06:10, 14 September 2024 (UTC)

Unlisted strong 7-limit Meantone extension (provisional name Mildtone)?

The Septimal Meantone tempering out Harrison's comma ([-13 10 0 -1⟩ = 59049/57344) thus equating 7/4 to C-A♯ (fifthspan +10) actually does not give the proper closest approximation to the 7th harmonic for some of the higher-numered equal divisions of the octave having fifths less flat than 43EDO. For a decent-sized set of these, the next closer approximation equates 7/4 to C-G𝄪♯(*) (fifthspan +22), tempering out the comma |-32 22 0 -1⟩ = 31381059609/30064771072. This starts with 55EDO, which is still within the range of seemingly practical playing using at least one commercially available MIDI controller; I am not a keyboard player myself, but the reach looks doable (if awkward) on both layouts given in Lumatone mapping for 55edo, and since somebody recently invented at least plasuble Lumatone mappings for 72EDO and 118EDO, it is not out of the question that members of the meantone temperament beyond 55EDO might also be usable with this 7-limit extension. In the case of 55EDO, Septimal Meantone yields 55d, although this is only slightly further off from a just 7th harmonic than the patent val; however, as EDO numbers increment by 12, the discrepancy grows, and this unlisted extension is more urgently needed. I have tried searching for this comma on Xenharmonic Wiki by monzo and ratio, but it does not turn up, and nothing hinting at such an extension is currently in Meantone or Meantone family. For convenience here, I will use the placeholder name Mildtone (not found elsewhere on Xenharmonic Wiki).

(*)For readability, it would be best to find a better way of notating this in an actual score, since otherwise a simple 4:5:6:7 chord will produce overlapping notes with C-G𝄪♯.

Putting 81/80 and 31381059609/30064771072 into x31eq's Unison Vector Search yields (with my best effort to restore line breaks in the output):

{BEGIN x31eq OUTPUT}

Meantone extension with an extra dimension (12 & 43d)
Equal Temperament Mappings
2 3 5 7
[ ⟨ 12 19 28 34 ]
⟨ 43 68 100 120 ] ⟩

Reduced Mapping
2 3 5 7
[ ⟨ 1 2 4 12 ]
⟨ 0 -1 -4 -22 ] ⟩

TE Generator Tunings (cents)
⟨1199.7437, 501.3407]

TE Step Tunings (cents)
⟨37.73668, 17.36985]

TE Tuning Map (cents)
⟨1199.744, 1898.147, 2793.612, 3367.429]

TE Mistunings (cents)
⟨-0.256, -3.808, 7.298, -1.397]

Show POTE tunings
This is a trivial subgroup of the rational numbers so TE is TE is TE.

Complexity 3.115840
Adjusted Error 5.608794 cents
TE Error 1.997893 cents/octave

Unison Vectors

   [-4, 4, -1, 0⟩ (81:80)
   [8, 2, -6, 1⟩ (16128:15625)
   [12, -2, -5, 1⟩ (28672:28125)
   [4, 6, -7, 1⟩ (81648:78125)

{END x31eq OUTPUT}

I tried searching on the Xenharmonic Wiki for the Unison Vectors other than 81/80 that it gave me:

Searching for 16128/15625 on here only turns up the Alphaquarter sub-temperament of Escapade. Searching for 28672/28125 on here turns up a few things, but none of them seem to be related to Meantone. Searching for 81648/78125 on here only turns up the comma list for 55EDO.

For Subsets, x31eq outputs:

{BEGIN x31eq OUTPUT}

Equal Temperaments

12, 24d, 31dd, 43d, 36d, 19dd, 55p, 7ddd, 67p, 48cdd
Rank 2 Temperament
complexity error
(cent)
Meantone+ 3.116 5.609 12 & 31dd

{END x31eq OUTPUT}

Looking for applicable equal temperaments on my own (and including a d or cc val at each end to show where it quits working), I found: 12(*), 43d, 55, 67, 79c, 91cd, 110cd(**), 122c, 134c, 146c, 158cc.

(*)12EDO is the grand junction between Garischismic Meantone, Dominant, Mildtone, and Septimal Meantone — taking the example of going up from C by the closest approximation to 7/4: C𝄫♭, B♭, A♯, and G♯𝄪 are all the same note.

(**)Contorted in the 7-limit -- only potentially useful if enabled by a higher-limit extension.

I have not yet done any work to see what higher-limit extensions would go with Mildtone beyond clicking on the "11-limit", "13-limit", "17-limit", and "19-limit" buttons in x31eq to see what output it would yield; with "19-limit", it goes crazy; since it needed manual checking even for the 7-limit, I would assume that manual checking would be needed for any higher-limit extensions even before the 19-limit blowup.

– Lucius Chiaraviglio (talk) 09:03, 11 November 2024 (UTC)

Yes, this is the meantone extension supported by 67edo, mapping 7/4 to two dieses below 9/5 instead of the usual 1 diesis. I have the 13-limit extension documented on my user page (currently named "worsetone", though the name isn't final by any means), but it is not really an efficient temperament until you go to very high prime limits. Recentlymaterialized (talk) 19:12, 30 December 2024 (UTC)

Starting to do the work for higher limits -- 11-limit for now. Computing manually, I find that assigning a fifthspan of +30 gets the right 11/8 (without warts) for 55EDO and 67EDO, but not 79EDO, and also gets the right 11/8 (with a c wart but no others) for 122EDO and 134EDO but not 110EDO or 146EDO. Amazingly, it even works for 12EDO without warts. I also tried fifthspan -25, but found that worked only for 55EDO, so fifthspan +30 it is. Fifthspan +30 coresponds to C-D𝄪𝄪, which corresponds to ((3/2)/(81/80)^(1/6))^30 / 2^17 in 1/6-comma meantone (which is more or less in the middle of the target region of meantone), which works out to |-27 10 5⟩, which yields a value of 551.119¢, which is just 0.119¢ flat of just 11/8 (551.318¢).

Have not yet tried to compute badness, since when I go to the page for that, I run into 2 problems currently beyond my knowledge to solve: What the symbol "∧" means in this context, and how wedgies work, for which the article itself says "This page may be difficult to understand to those unfamiliar with the mathematical concepts involved." So I will simply note for now that the 11th harmonic accuracy is extremely high (even in the actual non-warted EDOs other than 134EDO, where it is about to go overboard in relative error), but with very high complexity, and having overall 11-limit accuracy dragged down by drift in the 7th harmonic, the drift and sharpness in the 5th harmonic, and (of course) the classic ~1/6-comma meantone flat 3rd harmonic. Would be interesting to see what the badness comes out to without the 13th harmonic.

Have not yet tried to see if I can get rid of the warts you found in the 13th harmonic -- will have to do this later. I see that the mapping I have not yet counting the 13th harmonic matches what you have for the second mapping you have under Worsetone, but maybe something different needs to be done with the 13th harmonic.

Lucius Chiaraviglio (talk) 10:19, 2 January 2025 (UTC) edited Lucius Chiaraviglio (talk) 10:56, 2 January 2025 (UTC)

Yes, excluding compton extensions, fifthspan +30 is the 11-limit extension with the lowest logflat badness (it's 0.0649609 gene smith tenney-euclidean badness, or 2.1475667 dirichlet badness). Similarly, my 13-limit extension with 3/1 at +39 generators is the 13-limit extension with the lowest logflat badness (0.0494043 smith TE, 2.0414469 dirichlet), excluding compton extensions. This 13-limit extension is supported by 67edo unwarted; however, if you would like to get rid of the warts on 12edo and 55edo, 12 & 55 is an option, though it has much higher complexity and logflat badness because the 13/1 is reached with -28 fifths rather than a positive number of generators (and thus it cannot be played in otonal chords involving 7 and 11 without the use of the 67-note and larger MOS scales). Recentlymaterialized (talk) 16:03, 6 January 2025 (UTC)

Those are some monster-high Badness values, for sure. But I went to the Lumatone mapping for 55edo, and although I am not a Lumatone player, while the finger stretches with the Bosanquet layout look extremely awkward, the finger positionings with the 6L 1s layout actually look quite doable (or only mildly awkward) even for somebody with average hands, although I'll admit to having not yet analyzed more than a couple of chords that would include 11/8 and/or 13/8 (only had time for just a cursory inspection so far). Haven't seen a Lumatone mapping for 67edo yet, but since Bosanquet isn't going to work for this anyway, maybe whatever layout somebody eventually does come up (this won't even be the highest EDO with a Lumatone mapping, by a longshot) with will have those intervals be usable as well Lucius Chiaraviglio (talk) 20:25, 6 January 2025 (UTC)

Looked at this a little bit more, and seeing that the 13th harmonic is very badly represented by the remaining EDOs other than 67EDO (even when considered independently of any rank-2+ temperaments), maybe the thing to do is declare no-13s when checking higher limits? And then Worsetone could be the name for the version that doesn't exclude the 13th harmonic? Lucius Chiaraviglio (talk) 10:04, 8 January 2025 (UTC)

Looked at the 17th harmonic -- that's easy, it's equated to the diatonic semitone, so fifthspan -5. And the 19th harmonic is almost as easy, equated to the minor third, so fifthspan -3, except that if I haven't botched something by being in a hurry and up way too late, it actually doesn't quite work in 122EDO, so we get a ch wart here (EDO list: 12, 55, 67, 122ch -- most are close, but 122ch has high relative error). Lucius Chiaraviglio (talk) 12:00, 8 January 2025 (UTC)

Your mappings of 17 and 19 are much better choices than the ones I used on worsetone, thank you! I disagree, however, that prime 13 is badly represented by 12edo and 55edo. Mapped at +39 fifths, it is supported by 12f and 55f vals (and patent 67edo, of course), both of which have lower 13-limit error than their patent counterparts 12p and 55p. Because this 13 only needs another +9 fifths (and suffers less than a cent of error in the WE optimization), I don't see much reason to go no-13s in this case.

Regardless, if you're searching for higher-limit extensions, it may be helpful to remove the 5 so that the optimizer doesn't obsess over the 5/1 (this is 1/6 comma meantone, we've accepted a sharp 5/4!). With this in mind, I suggest two possible 23-limit extensions using your mappings of 17 and 19: 55f & 67 and 55fi & 67. The former is slightly more complex as it needs 13 extra fifths while the second only needs 10, but the two merge in 67edo anyways so you could probably use either one depending on the situation (if you're using 67edo, that is). Recentlymaterialized (talk) 04:31, 9 January 2025 (UTC)

I agree that using 12f instead of 12 and 55f instead of 55 improves the consistency, but either way, the 13th harmonic is very far off in those in relative error, which translates to huge absolute error for 12EDO and still enough error in 55EDO to stick out like a sore thumb even when sticking out in the same direction as nearby harmonics. (At least in 122cEDO, the absolute error finally gets small enough that you cloud plausibly gloss over it as long as you don't use it too much.) And either way, the huge fifthspan makes the representation of the 13th harmonic very brittle to small changes within the region around 1/6-comma meantone, and potentially inconsistent with the better higher (as well as lower) harmonics, so even if not dumping it permanently, I am still inclined to exclude it at least temporarily while exploring the higher harmonics, and then maybe back-extend to it later (maybe as a dual-13 system).

Although I can see why you might have also wanted different mappings for the 17th and 19th harmonics, since in 55EDO the error in these is in the opposite direction of that of most lower harmonics and/or each other (and in 122cEDO, it's a mixed bag). The problem is that the flat approximation to the 17th harmonic is pretty bad, and the 17th harmonic isn't far off enough in any of these to be a good candidate for a split harmonic.

I get fifthspan -18 for the 23rd harmonic for patent vals (except for the c wart on 122) for 12, 55, 67, and 122c; other than 12EDO, these are not too far off from just (well, 12EDO is, but that's as good as you're going to get for most harmonics with a small EDO like 12). If I instead use fifthspan +49, I get 12i, 55i, 67, and 122ci; only 67 is close to just. And fifthspan +49 is awfully brittle against a slight shift in tuning within the region around 1/6-comma meantone. So I would go with fifthspan -18 even though it is pointing the other way from the fifthspans for most of these harmonics.

Lucius Chiaraviglio (talk) 08:51, 9 January 2025 (UTC)

Decided to look at where the higher harmonics end up on the Lumatone mapping for 55edo. While the harmonics up to 13 fit better with the lowest harmonics in the 6L 1s mapping, the 17th and 19th harmonics fit together best with the lowest harmonics in the Bosanquet mapping (and the 23rd harmonic is awkward with both); of course this is subject to extensive modification if playing in a key signature that causes vertical wrapround. So I have the barest beginnings of an idea for badness rating (nowhere near complete since I don't know how to compute the original types of badness) that takes into account playability, but it is going to be hamstrung by such differences in key layouts. Lucius Chiaraviglio (talk) 10:03, 9 January 2025 (UTC)