Talk:Meantone: Difference between revisions

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:::::: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk: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. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 10:03, 9 January 2025 (UTC)

== Unlisted(?) strong 7-limit Meantone extension (maybe related to Catasyc or Laruquadbiyoti?) ==

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 more flat than 50EDO. For a few of these, the next closer approximation equates [[7/4]] to C-G𝄪𝄪(*) (fifthspan +29), tempering out the comma |-43 29 0 -1⟩ = 68630377364883/61572651155456. This starts with [[69edo]], for which a [[Lumatone mapping for 69edo|Lumatone mapping]] has recently been demonstrated [https://www.youtube.com/watch?v=ZAqPonAHuUM&lc=Ugy-2hiGCW-YSngIvmZ4AaABAg.AIVXd3HGZlYAIZiyHbHV7H for playing music that sounds pretty good] (edit: [[https://www.youtube.com/shorts/4XBELeySMPk another example]). While I am not a keyboard player myself, the reach does look like it might be a bit awkward in combination with [[Devichromic chords]] on the layout for that, but very easily reached on the sharp Diatonic layout as long as it doesn't fall into the crack of missed notes, and somewhat awkward on the flat Diatonic layout if the crack of missed notes can be avoided for both it and its root note. 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. Unfortunately I haven't thought of a good placeholder name for this yet (unlike Mildtone in the section above), so temporarily I'll have to make do without.

(*)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 68630377364883/61572651155456 into Graham Breed's x31eq Temperament Finder Unison Vector Search ([https://x31eq.com/pyscript/uv.html?uvs=81%3A80+%5B-43%2C29%2C0%2C-1%3E&page=0&limit=7 quick link for initial search results]) yields (with my best effort to restore line breaks in the output):

{BEGIN x31eq OUTPUT}

Meantone extension with an extra dimension (19 & 69p)

Equal Temperament Mappings

2 3 5 7

[ ⟨ 19 30 44 53 ]

⟨ 69 109 160 194 ] ⟩

Reduced Mapping

2 3 5 7

[ ⟨ 1 2 4 15 ]

⟨ 0 -1 -4 -29 ] ⟩

TE Generator Tunings (cents)

⟨1202.0930, 505.5912]

TE Step Tunings (cents)

⟨25.09362, 10.51180]

TE Tuning Map (cents)

⟨1202.093, 1898.595, 2786.007, 3369.251]

TE Mistunings (cents)

⟨2.093, -3.360, -0.306, 0.425]

Show POTE tunings

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

Complexity 4.171718

Adjusted Error 4.191225 cents

TE Error 1.492944 cents/octave

Unison Vectors

[-4, 4, -1, 0⟩ (81:80)

[-15, 1, 7, -1⟩ (234375:229376)

[-11, -3, 8, -1⟩ (390625:387072)

[7, 7, -9, 1⟩ (1959552:1953125)

{END x31eq OUTPUT}

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

Searching for 234375/229376 on here does not turn up a temperament, but does turn up a few EDO values tempering it out, of which 107b fits with this extension.

Searching for 390625/387072 on here turns up "Catasyc or Laruquadbiyoti" temperaments in [[Tour of regular temperaments]], but does not link to an article, so I have no idea how closely these are related to Meantone

Searching for 1959552/1953125 on here turns up [[Parkleiness temperaments]], but these do not have expected EDO values 69, 88, or 107b.

For Subsets, x31eq outputs:

{BEGIN x31eq OUTPUT}

Equal Temperaments

19, 38d, 57dd, 69p, 50d, 88p, 31d, 107b, 76dd, 12dd

Rank 2 Temperament

complexity error

(cent)

Meantone+ 4.172 4.191 19 & 69p

{END x31eq OUTPUT}

Looking for applicable equal temperaments on my own (and including a d and/or c val at each end to show where it quits working), I found: 50d, 69, 88, 107b, 126bcd (and no secondary series since this always has c and/or d warts).

I have not yet done any work to see what higher-limit extensions would go with Mildtone beyond clicking on the "11-limit" and "13-limit" buttons in x31eq to see what output it would yield; with "13-limit", it starts to come up with rather odd output, so I stopped there.

Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:41, 27 May 2025 (UTC)

Last modifed: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:30, 11 June 2025 (UTC) (added another supporting sound sample)

: Another way to think of this extension is by starting with [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.23&reduce=on&weights=weil&target=&edos=&commas=81%2F80%0D%0A625%2F624%0D%0A576%2F575&submit_comma=submit tunbarsmic meantone] and stacking 13/12 twice to reach 7/6 (tempering out [[169/168]]); since all octave-reduced harmonics are now flat, the error can then be mitigated somewhat by octave stretching. [[User:Recentlymaterialized|recentlymaterialized]] ([[User_talk:Recentlymaterialized|talk]]) 20:13, 16 June 2025 (UTC)

:: Indeed, if I put [-4, 4, -1, 0, 0, 0⟩ (81:80), [-3, -1, 0, -1, 2, 0⟩ (169:168), [6, 2, -2, 0, 0, -1⟩ (576:575), [-4, -1, 4, 0, -1, 0⟩ (625:624) into x31eq temperament finder, I get 19, 38df, 69p, 50d, 88f, 31d, 57ddf, 107bf, 12ddf, 76ddff — way too many warts in most cases (and an 'f' wart is rather nasty to foist upon 107b since the patent 13th harmonic is near-just), but 19edo and 69edo have their patent vals. (And I see that I should have left in 19 in my manually-generated tuning series, since in 19edo G𝄪𝄪 is the same as A♯.)

:: Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:18, 17 June 2025 (UTC)

:: Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 09:40, 20 December 2025 (UTC) (fix typo)

: Another further extension of the above is to the 2.3.5.7.19 subgroup for which I have the placeholder name Devichromic Octacot, for which I can give a link to the [[Devichromic chords]] article — no general article currently exists for Devichromic temperament, and this particular incarnation of Devichromic temperament is likely to receive a different permanent name. The temperament is [https://x31eq.com/pyscript/uv.html?uvs=81%3A80+400%3A399+%5B17%2C-1%2C8%2C0%2C-8%3E&page=0&limit=2_3_5_7_19 14c & 55d], specified by the 2.3.5.7.19 subgroup unison vector list [-4, 4, -1, 0, 0⟩ (81/80), [4, -1, 2, -1, -1⟩ ([[400/399]]), [17, -1, 8, 0, -8⟩ (51200000000/50950689123), the last of which currently has no article, but equates a stack of eight [[20/19]]s to [[3/2]].

: For subsets:

: {BEGIN x31eq OUTPUT}

: Equal Temperaments

: 14c, 55d, 69p, 28ccdh, 41cdh, 83bch, 27ccddhh, 42bcccdhh, 97bcchh, 124bd

: {END x31eq OUTPUT}

: Most of those aren't what you'd want to count as optimal, but the subset 14c (anchor point), 55d (intersection with septimal meantone), 69 (see sound samples), 83bch (intersection with flattone) is reasonable.

: And [https://www.youtube.com/shorts/GOjgAm6uwOY here's a third sound sample] — this one and the first one above both use the Devichromic Octacot Lumatone mapping, but the diatonic mapping (middle sound sample) also works as long as you don't run out of gamut.

: Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 09:40, 20 December 2025 (UTC)

: Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 09:46, 20 December 2025 (UTC) (diatonic Lumatone mapping also works as long as you don't run out of notes)

:: Related to the above, if you relax the requirement to be able to divide the fifth into 8 parts and instead 4 parts like Tetracot, you get white might be the [https://x31eq.com/pyscript/uv.html?uvs=81%2F80%0D%0A130321%2F129654&limit=2.3.5.7.19 Meantone relative of Tetracot] like Superpine is the Meantone relative of Porcupine: 2.3.5.7.19 unison vectors [-4, 4, -1, 0, 0⟩ (81/80), [-1, -3, 0, -4, 4⟩ (130321/129654) (the last comma currently has no article, but it equates a stack of 4 [[21/19]]s with [[3/2]] without colliding with the 2.3.5 subgroup temperament specifications laid down by the normal Tetracot comma).

:: {BEGIN x31eq OUTPUT}

:: Equal Temperaments

:: 14c, 7dh, 7p, 62, 48c, 55h, 55d, 76h, 69dhh, 21cdhh

:: {END x31eq OUTPUT}

:: Most of those aren't what you would want to count as optimal, and I need to figure out which other comma to put in to whittle down the list of rank-2 temperaments it also gives, but the subset 7 (anchor point), 14c, 55d, 55h (need to figure out which one of these to trim out bu only 55d fits with the above so seems like 55h should go), 62 seems reasonable (although it seems like I should be able to get a 76 without a wart — must be missing something, probably whatever would choose between 55d and 55h).

:: Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 10:11, 20 December 2025 (UTC)

== Meantone and being regular/rank-2 ==

Are historical well temperaments not a form of meantone temperament? -- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 19:24, 17 June 2025 (UTC)

: If I understand correctly (including from my reading of the article here about [[well temperament]]s), most well temperaments (historical and otherwise) would be derivatives of meantone temperament rather than meantone itself. They could be viewed as finite note subsets of rank-3 or higher (depending upon the number of different fifth sizes) temperaments derived from meantone, which is rank-2 (having all fifths the same).

: The exception would be something like 12 note 1/6-comma meantone that has a single wolf fifth that is just what is left over to close a finite section of the helix of fifths into a circle (and technically it's a diminished sixth rather than a real fifth), with the proviso that it isn't so far off from a real fifth as to put it out of diatonic range. Since this is sufficient for a circulating temperament, that kind of well temperament would qualify as a finite note subset of a rank-2 temperament (meantone) while being usable (even if not very good) as a circulating temperament.

: The same would be true if anybody had come up with an even less-flattened meantone to use as-is with a likewise finite subset of the helix of fifths being shoehorned into circle (and no purposely-resized second size of fifth). I don't know of anything other than 1/6-comma meantone being used that way in actual practice, but the [[historical temperaments]] page here mentions Romieu and 1/7-comma meantone (of which the 12 note subset would be more circulation-friendly than 12 note 1/6-comma meantone), so depending upon whether that got into actual performance, that might be another example.

: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 21:08, 17 June 2025 (UTC)

== Synch-meantone ==

I took a listen to the major triad of Synch-meantone (fifth 695.632{{c}}), and it sounds ''much'' more in tune than [[quarter-comma meantone]], as well as tunings like [[55edo]]. This tuning deserves to be mentioned in this page.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 00:48, 19 January 2026 (UTC)

I did a bit more fine-tuning, and I found the optimum to actually be a bit sharp of this (fifth 695.8{{c}}), close to 2/7-comma. I likely overlooked this because even [[50edo]] didn't replicate it well enough. And yes, I notice such differences; the harmonic JND is ''tiny''.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 00:54, 19 January 2026 (UTC)

IDK, such tiny differences are perceptible, but I'm not sure. I think 50edo actually ''is'' good enough, and the "optima" I described are just the results of a few tests that might just influence ''each other''. Still, there should be more info in the tunings section, like there is on the page of [[superpyth]].--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 01:03, 19 January 2026 (UTC)

I just realized that the program has a precision limit of 1 cent. Nevermind much of this. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 01:10, 19 January 2026 (UTC)

: For what it's worth, the [[69edo]] page says that 69edo is a very close approximation of synch-meantone.

: From the point of view of Duke Ellington's rule about music (as relayed by Peter Schickele), "If it sounds good, it is good". To this end, you can find amazing sound samples on both sides of the 2/7-comma meantone divide, [https://www.youtube.com/watch?v=r7RmbC-hRFE in 50edo] and [https://www.youtube.com/watch?v=ZAqPonAHuUM in 69edo]. For [[50edo]] this is understandable since it just misses being consistent all the way to the 21-odd-limit (11/9 and 18/11 come out slightly inconsistent)), but apparently the [[devichromic chords]] of 69edo at least partly make up for its consistency shortcomings. You seem to be looking for something in between, but I don't know of any sound samples for [[119edo]].

: I am on the cusp of wanting to propose a meantone extension for meantone EDOs that have a corresponding c val on tetracot the way superpine tunings have a corresponding c val on porcupine, but the series 55*, 62, 69*, 76 ... 14c* does weird things with some of the mappings.

: *These are also octacot, and temper out the [[400/399|devichroma]], although in some cases needing one or more warts.

: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:41, 19 January 2026 (UTC)

@@ Line 305: / Line 305: @@
 :::::: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk: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.  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 10:03, 9 January 2025 (UTC)
+== Unlisted(?) strong 7-limit Meantone extension (maybe related to Catasyc or Laruquadbiyoti?) ==
+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 more flat than 50EDO.  For a few of these, the next closer approximation equates [[7/4]] to C-G𝄪𝄪(*) (fifthspan +29), tempering out the comma |-43 29 0 -1⟩ = 68630377364883/61572651155456.  This starts with [[69edo]], for which a [[Lumatone mapping for 69edo|Lumatone mapping]] has recently been demonstrated [https://www.youtube.com/watch?v=ZAqPonAHuUM&lc=Ugy-2hiGCW-YSngIvmZ4AaABAg.AIVXd3HGZlYAIZiyHbHV7H for playing music that sounds pretty good] (edit:  [[https://www.youtube.com/shorts/4XBELeySMPk another example]).  While I am not a keyboard player myself, the reach does look like it might be a bit awkward in combination with [[Devichromic chords]] on the layout for that, but very easily reached on the sharp Diatonic layout as long as it doesn't fall into the crack of missed notes, and somewhat awkward on the flat Diatonic layout if the crack of missed notes can be avoided for both it and its root note.  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.  Unfortunately I haven't thought of a good placeholder name for this yet (unlike Mildtone in the section above), so temporarily I'll have to make do without.
+(*)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 68630377364883/61572651155456 into Graham Breed's x31eq Temperament Finder Unison Vector Search ([https://x31eq.com/pyscript/uv.html?uvs=81%3A80+%5B-43%2C29%2C0%2C-1%3E&page=0&limit=7 quick link for initial search results]) yields (with my best effort to restore line breaks in the output):
+{BEGIN x31eq OUTPUT}
+Meantone extension with an extra dimension (19 & 69p)<br>
+Equal Temperament Mappings<br>
+	3	5	7	<br>
+[ ⟨	19	30	44	53	]<br>
+⟨	69	109	160	194	] ⟩<br>
+Reduced Mapping<br>
+	3	5	7	<br>
+[ ⟨	1	2	4	15	]<br>
+⟨	0	-1	-4	-29	] ⟩<br>
+TE Generator Tunings (cents)<br>
+⟨1202.0930, 505.5912]<br>
+TE Step Tunings (cents)<br>
+⟨25.09362, 10.51180]<br>
+TE Tuning Map (cents)<br>
+⟨1202.093, 1898.595, 2786.007, 3369.251]<br>
+TE Mistunings (cents)<br>
+⟨2.093, -3.360, -0.306, 0.425]<br>
+Show POTE tunings<br>
+This is a trivial subgroup of the rational numbers so TE is TE is TE.<br>
+Complexity 	4.171718<br>
+Adjusted Error 	4.191225 cents<br>
+TE Error 	1.492944 cents/octave<br>
+Unison Vectors
+    [-4, 4, -1, 0⟩ (81:80)<br>
+    [-15, 1, 7, -1⟩ (234375:229376)<br>
+    [-11, -3, 8, -1⟩ (390625:387072)<br>
+    [7, 7, -9, 1⟩ (1959552:1953125)<br>
+{END x31eq OUTPUT}
+I tried searching on the Xenharmonic Wiki for the Unison Vectors other than 81/80 that it gave me:
+Searching for 234375/229376 on here does not turn up a temperament, but does turn up a few EDO values tempering it out, of which 107b fits with this extension.
+Searching for 390625/387072 on here turns up "Catasyc or Laruquadbiyoti" temperaments in [[Tour of regular temperaments]], but does not link to an article, so I have no idea how closely these are related to Meantone
+Searching for 1959552/1953125 on here turns up [[Parkleiness temperaments]], but these do not have expected EDO values 69, 88, or 107b.
+For Subsets, x31eq outputs:
+{BEGIN x31eq OUTPUT}
+Equal Temperaments<br>
+, 38d, 57dd, 69p, 50d, 88p, 31d, 107b, 76dd, 12dd<br>
+Rank 2 Temperament<br>
+	complexity 	error<br>
+(cent) 	<br>
+Meantone+ 	4.172 	4.191 	    19 & 69p<br>
+{END x31eq OUTPUT}
+Looking for applicable equal temperaments on my own (and including a d and/or c val at each end to show where it quits working), I found:  50d, 69, 88, 107b, 126bcd (and no secondary series since this always has c and/or d warts).
+I have not yet done any work to see what higher-limit extensions would go with Mildtone beyond clicking on the "11-limit" and "13-limit" buttons in x31eq to see what output it would yield; with "13-limit", it starts to come up with rather odd output, so I stopped there.
+Added:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:41, 27 May 2025 (UTC)<br>
+Last modifed:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:30, 11 June 2025 (UTC) (added another supporting sound sample)
+: Another way to think of this extension is by starting with [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.23&reduce=on&weights=weil&target=&edos=&commas=81%2F80%0D%0A625%2F624%0D%0A576%2F575&submit_comma=submit tunbarsmic meantone] and stacking 13/12 twice to reach 7/6 (tempering out [[169/168]]); since all octave-reduced harmonics are now flat, the error can then be mitigated somewhat by octave stretching. [[User:Recentlymaterialized|recentlymaterialized]] ([[User_talk:Recentlymaterialized|talk]]) 20:13, 16 June 2025 (UTC)
+:: Indeed, if I put [-4, 4, -1, 0, 0, 0⟩ (81:80), [-3, -1, 0, -1, 2, 0⟩ (169:168), [6, 2, -2, 0, 0, -1⟩ (576:575), [-4, -1, 4, 0, -1, 0⟩ (625:624) into x31eq temperament finder, I get 19, 38df, 69p, 50d, 88f, 31d, 57ddf, 107bf, 12ddf, 76ddff &mdash; way too many warts in most cases (and an 'f' wart is rather nasty to foist upon 107b since the patent 13th harmonic is near-just), but 19edo and 69edo have their patent vals.  (And I see that I should have left in 19 in my manually-generated tuning series, since in 19edo G𝄪𝄪 is the same as A♯.)
+:: Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:18, 17 June 2025 (UTC)<br>
+:: Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 09:40, 20 December 2025 (UTC) (fix typo)
+: Another further extension of the above is to the 2.3.5.7.19 subgroup for which I have the placeholder name Devichromic Octacot, for which I can give a link to the [[Devichromic chords]] article &mdash; no general article currently exists for Devichromic temperament, and this particular incarnation of Devichromic temperament is likely to receive a different permanent name. The temperament is [https://x31eq.com/pyscript/uv.html?uvs=81%3A80+400%3A399+%5B17%2C-1%2C8%2C0%2C-8%3E&page=0&limit=2_3_5_7_19 14c & 55d], specified by the 2.3.5.7.19 subgroup unison vector list [-4, 4, -1, 0, 0⟩ (81/80), [4, -1, 2, -1, -1⟩ ([[400/399]]), [17, -1, 8, 0, -8⟩ (51200000000/50950689123), the last of which currently has no article, but equates a stack of eight [[20/19]]s to [[3/2]].
+: For subsets:
+: {BEGIN x31eq OUTPUT}
+: Equal Temperaments
+: 14c, 55d, 69p, 28ccdh, 41cdh, 83bch, 27ccddhh, 42bcccdhh, 97bcchh, 124bd
+: {END x31eq OUTPUT}
+: Most of those aren't what you'd want to count as optimal, but the subset 14c (anchor point), 55d (intersection with septimal meantone), 69 (see sound samples), 83bch (intersection with flattone) is reasonable.
+: And [https://www.youtube.com/shorts/GOjgAm6uwOY here's a third sound sample] &mdash; this one and the first one above both use the Devichromic Octacot Lumatone mapping, but the diatonic mapping (middle sound sample) also works as long as you don't run out of gamut.
+: Added:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 09:40, 20 December 2025 (UTC)<br>
+: Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 09:46, 20 December 2025 (UTC) (diatonic Lumatone mapping also works as long as you don't run out of notes)
+:: Related to the above, if you relax the requirement to be able to divide the fifth into 8 parts and instead 4 parts like Tetracot, you get white might be the [https://x31eq.com/pyscript/uv.html?uvs=81%2F80%0D%0A130321%2F129654&limit=2.3.5.7.19 Meantone relative of Tetracot] like Superpine is the Meantone relative of Porcupine:  2.3.5.7.19 unison vectors [-4, 4, -1, 0, 0⟩ (81/80), [-1, -3, 0, -4, 4⟩ (130321/129654) (the last comma currently has no article, but it equates a stack of 4 [[21/19]]s with [[3/2]] without colliding with the 2.3.5 subgroup temperament specifications laid down by the normal Tetracot comma).
+:: {BEGIN x31eq OUTPUT}
+:: Equal Temperaments
+:: 14c, 7dh, 7p, 62, 48c, 55h, 55d, 76h, 69dhh, 21cdhh
+:: {END x31eq OUTPUT}
+:: Most of those aren't what you would want to count as optimal, and I need to figure out which other comma to put in to whittle down the list of rank-2 temperaments it also gives, but the subset 7 (anchor point), 14c, 55d, 55h (need to figure out which one of these to trim out bu only 55d fits with the above so seems like 55h should go), 62 seems reasonable (although it seems like I should be able to get a 76 without a wart &mdash; must be missing something, probably whatever would choose between 55d and 55h).
+:: Added:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 10:11, 20 December 2025 (UTC)
+== Meantone and being regular/rank-2 ==
+Are historical well temperaments not a form of meantone temperament? -- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 19:24, 17 June 2025 (UTC)
+: If I understand correctly (including from my reading of the article here about [[well temperament]]s), most well temperaments (historical and otherwise) would be derivatives of meantone temperament rather than meantone itself.  They could be viewed as finite note subsets of rank-3 or higher (depending upon the number of different fifth sizes) temperaments derived from meantone, which is rank-2 (having all fifths the same).
+: The exception would be something like 12 note 1/6-comma meantone that has a single wolf fifth that is just what is left over to close a finite section of the helix of fifths into a circle (and technically it's a diminished sixth rather than a real fifth), with the proviso that it isn't so far off from a real fifth as to put it out of diatonic range.  Since this is sufficient for a circulating temperament, that kind of well temperament would qualify as a finite note subset of a rank-2 temperament (meantone) while being usable (even if not very good) as a circulating temperament.
+: The same would be true if anybody had come up with an even less-flattened meantone to use as-is with a likewise finite subset of the helix of fifths being shoehorned into circle (and no purposely-resized second size of fifth).  I don't know of anything other than 1/6-comma meantone being used that way in actual practice, but the [[historical temperaments]] page here mentions Romieu and 1/7-comma meantone (of which the 12 note subset would be more circulation-friendly than 12 note 1/6-comma meantone), so depending upon whether that got into actual performance, that might be another example.
+: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 21:08, 17 June 2025 (UTC)
+== Synch-meantone ==
+I took a listen to the major triad of Synch-meantone (fifth 695.632{{c}}), and it sounds ''much'' more in tune than [[quarter-comma meantone]], as well as tunings like [[55edo]]. This tuning deserves to be mentioned in this page.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 00:48, 19 January 2026 (UTC)
+I did a bit more fine-tuning, and I found the optimum to actually be a bit sharp of this (fifth 695.8{{c}}), close to 2/7-comma. I likely overlooked this because even [[50edo]] didn't replicate it well enough. And yes, I notice such differences; the harmonic JND is ''tiny''.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 00:54, 19 January 2026 (UTC)
+IDK, such tiny differences are perceptible, but I'm not sure. I think 50edo actually ''is'' good enough, and the "optima" I described are just the results of a few tests that might just influence ''each other''. Still, there should be more info in the tunings section, like there is on the page of [[superpyth]].--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 01:03, 19 January 2026 (UTC)
+I just realized that the program has a precision limit of 1 cent. Nevermind much of this. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 01:10, 19 January 2026 (UTC)
+: For what it's worth, the [[69edo]] page says that 69edo is a very close approximation of synch-meantone.
+: From the point of view of Duke Ellington's rule about music (as relayed by Peter Schickele), "If it sounds good, it is good".  To this end, you can find amazing sound samples on both sides of the 2/7-comma meantone divide, [https://www.youtube.com/watch?v=r7RmbC-hRFE in 50edo] and [https://www.youtube.com/watch?v=ZAqPonAHuUM in 69edo]. For [[50edo]] this is understandable since it just misses being consistent all the way to the 21-odd-limit (11/9 and 18/11 come out slightly inconsistent)), but apparently the [[devichromic chords]] of 69edo at least partly make up for its consistency shortcomings. You seem to be looking for something in between, but I don't know of any sound samples for [[119edo]].
+: I am on the cusp of wanting to propose a meantone extension for meantone EDOs that have a corresponding c val on tetracot the way superpine tunings have a corresponding c val on porcupine, but the series 55<sup>*</sup>, 62, 69<sup>*</sup>, 76 ... 14c<sup>*</sup> does weird things with some of the mappings.
+: <sup>*</sup>These are also octacot, and temper out the [[400/399|devichroma]], although in some cases needing one or more warts.
+: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:41, 19 January 2026 (UTC)