Talk:Meantone: Difference between revisions

(5 intermediate revisions by 2 users not shown)

Line 310:

== 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.

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𝄪♯~~.

(*)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):

Line 382:

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)

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

Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 17:22, 29 June 2026 (UTC) (fix a couple of typos)

: 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)

Line 401:

Line 402:

: 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).

:: 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 what 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

Line 408:

Line 409:

:: 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)

:: Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 17:22, 29 June 2026 (UTC) (fix where autocorrupt changed "what" to "white")

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

Line 421:

Line 423:

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 310: / Line 310: @@
 == 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.
+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𝄪♯.
+(*)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):
@@ Line 382: / Line 382: @@
 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)
+Previously modifed:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:30, 11 June 2025 (UTC) (added another supporting sound sample)
+Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 17:22, 29 June 2026 (UTC) (fix a couple of typos)
 : 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)
@@ Line 401: / Line 402: @@
 : 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).
+:: 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 what 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
@@ Line 408: / Line 409: @@
 :: 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)
+:: Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 17:22, 29 June 2026 (UTC) (fix where autocorrupt changed "what" to "white")
 == Meantone and being regular/rank-2 ==
@@ Line 421: / Line 423: @@
 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)