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)