Talk:Meantone: Difference between revisions

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:::: 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). [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk: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 [[User:Recentlymaterialized/Miscellaneous_rank-2_temperaments#Worsetone_.2812f_.26_67.29|worsetone]], thank you! I disagree, however, that prime 13 is badly represented by 12edo and 55edo. Mapped at +39 fifths, it is supported by [https://x31eq.com/pyscript/rt.html?ets=12f&limit=13 12f] and [https://x31eq.com/pyscript/rt.html?ets=55f&limit=13 55f] vals (and patent 67edo, of course), both of which have lower 13-limit error than their patent counterparts [https://x31eq.com/pyscript/rt.html?ets=12p&limit=13 12p] and [https://x31eq.com/pyscript/rt.html?ets=55p&limit=13 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: [https://sintel.pythonanywhere.com/result?edos=55f+%26+67&submit_edo=submit&subgroup=2.3.7.11.13.17.19.23 55f & 67] and [https://sintel.pythonanywhere.com/result?edos=55fi+67&submit_edo=submit&subgroup=2.3.7.11.13.17.19.23 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). [[User:Recentlymaterialized|Recentlymaterialized]] ([[User talk: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.

:::::: [[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 67edo 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♯.) [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:18, 17 June 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)

@@ Line 294: / Line 294: @@
 :::: 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).  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk: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 [[User:Recentlymaterialized/Miscellaneous_rank-2_temperaments#Worsetone_.2812f_.26_67.29|worsetone]], thank you! I disagree, however, that prime 13 is badly represented by 12edo and 55edo. Mapped at +39 fifths, it is supported by [https://x31eq.com/pyscript/rt.html?ets=12f&limit=13 12f] and [https://x31eq.com/pyscript/rt.html?ets=55f&limit=13 55f] vals (and patent 67edo, of course), both of which have lower 13-limit error than their patent counterparts [https://x31eq.com/pyscript/rt.html?ets=12p&limit=13 12p] and [https://x31eq.com/pyscript/rt.html?ets=55p&limit=13 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: [https://sintel.pythonanywhere.com/result?edos=55f+%26+67&submit_edo=submit&subgroup=2.3.7.11.13.17.19.23 55f & 67] and [https://sintel.pythonanywhere.com/result?edos=55fi+67&submit_edo=submit&subgroup=2.3.7.11.13.17.19.23 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). [[User:Recentlymaterialized|Recentlymaterialized]] ([[User talk: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.
+:::::: [[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 67edo 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♯.)   [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:18, 17 June 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)