Wikipedia talk:WikiProject Tunings, Temperaments, and Scales

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[edit] Gratuitous assumption of 12 equal temperament

These are articles which can, or have been, cleaned up to make them more general. I just did pitch class.

Pitch class

Everywhere went today I found this gratutious assumption. People writing these articles are pig-ignorant and don't even know it. Barf!

Here's something I put on the talk page of Diatonic scale.

=== Balzano stuff ===

I think this very controversial claim:

These unique relationships are as follows: Only certain divisions of the octave, 12 and 20 included, allow uniqueness, coherence, and transpositional simplicity, and that only the diatonic and pentatonic subsets of the 12-tone chromatic set follow these constraints (Balzano, 1980, 1982).

belongs in a Balzano article, not a diatonic scale article. It's hardly NPOV to put it here. If no one objects, I'll remove it. Gene Ward Smith 08:41, 20 May 2006 (UTC)

Hahah. Yeah, I would definitely disagree with the quote. You should remove it. Rainwarrior 19:17, 20 May 2006 (UTC)

[edit] Keys

Does this project encompass the articles in Category:Musical keys? — ßott e siηi ^(talk) 21:01, 3 June 2006 (UTC)

Not inherently. There is a property of many tuning systems (equal temperament being the big exception) that causes different keys to have distinct sounds, but these properties change from temperament to temperament (other key differences stem mostly from timbral aspects of individual instruments). We could, I suppose, start adding information about this to each Key page... but I don't think it's really that worthwhile. If it is interesting to you, though, go for it. - Rainwarrior 22:10, 3 June 2006 (UTC)

[edit] 88 Equal Temperament

Didn't someone write a 55 equal temperament page? If so, what happened to it? Gene Ward Smith 08:01, 9 June 2006 (UTC)

Maybe you're thinking of 88 equal temperament? —Keenan Pepper 14:22, 9 June 2006 (UTC)

That was it! Thanks. Gene Ward Smith 19:07, 9 June 2006 (UTC)

[edit] Tuning theory

A Tuning theory article just popped up. I'm not sure what I think about its existence... it seems to me like if it takes off it should overlap too much with stuff that belongs on Musical tuning... any thougts? (My tought at the moment is that it probably deserves a category... but not another content-filled page.) - Rainwarrior 13:13, 20 June 2006 (UTC)

I agree, seems like it should be a section of Musical tuning. —Keenan Pepper 00:44, 21 June 2006 (UTC)

[edit] Intun MIDI tuning program

I finally put my MIDI tuning program called Intun back up on a webpage. I don't know how many of you use Windows, but I thought it might be of interest to some of you. I wrote it four years ago but it still serves me well. (It's freeware, by the way.)

Available here.

- Rainwarrior 04:01, 25 June 2006 (UTC)

[edit] name of project?

Must all project names start with "Wikipedia:"? --CKL

The name of the project is WikiProject Tunings, Temperaments, and Scales, but this project page is in the Wikipedia: namespace because it is not an encyclopedia article. —Keenan Pepper 23:19, 1 August 2006 (UTC)

[edit] More keys

Ortgies' list has about 50 organs, Wraight about 25 Italian harpsichords, and I found about the same in reed organs, compiling an editable chronological list would be useful but could it be trouble deciding what makes it or not so it doesn't become uselessly big? Mireut 19:09, 27 August 2006 (UTC)

[edit] Undertone series

I added a page on the undertone series. I know it wasn't on the list but I still think that it's still important musically. I have put everything about the undertone series that I know and to my knowledge it's all correct. If anyone would like to go over it, some more stuff to, and just make sure it's correct I would appreciate it. Thanks. SN122787 22:05, 2 September 2006 (UTC)

That's fine, you can start an article on any subject you want. The only major problem I see is that you didn't cite any sources for the information. I'll try to find some sources and add them so you can see how it's done. —Keenan Pepper 23:03, 2 September 2006 (UTC)

[edit] External links to music

Many of our articles about tuning systems have external links to recordings of compositions in them. This is not explicitely forbidden by WP:EL, but it's not an ideal situation either: people reading Wikipedia on a CD because they have no Internet access are out of luck. Ideally we should have freely licensed examples uploaded to Wikipedia, but sometimes external links are necessary.

Here are my recommendations:

I strongly encourage everyone to write a little ditty in their favorite tuning system and upload it to Wikipedia in Ogg Vorbis format under a free license. If you want to do this but have technical problems, talk to me and we'll work it out.
Per WP:EL, external links should definitely not link directly to audio files. They should link to HTML web pages with links and preferably descriptions.

Anything else? Comments? —Keenan Pepper 19:44, 3 September 2006 (UTC)

My thought is that these are all good links. Being able to listen to them definitely aids understanding of the temperament. They improve the article with their presence; this, I think is reason enough to keep them, against the WP:EL caution. Right now what we have is pretty good, it says "MP3" or "OGG" or "MID" next to all of them at least, so the reader should not be surprised when the click opens a program. They are definitely external, though, which is why when replacing them I moved them all into an "external links" section. It would be nice if we had some of this content stored at the wikimedia commons, but that's maybe a longer term goal. If at some point in the future so much good material has appeared there that the external links are outnumbered, maybe then they should be removed. (Also, I have invited User:Gmaxwell to join this discussion, as he was the one who initially removed all of these links.) - Rainwarrior 20:44, 3 September 2006 (UTC)

For the record, I found your 'invitation' rude and condescending.. it was all the worse because you made a mistaken allegation about the talk pages, this is no way to invite someone to help. What is currently in the articles, thanks to your discussion-less reverting, is quite simply unacceptable. Please read our policy on external linking WP:External links. I'd be glad to help out creating recordings for these articles, the equal temperaments are easy at least... but for more complex temperaments I'd need scala files. I am pretty sure I can program one of my synths with arbitrary temperaments, although some of them only support 12 tone temperaments. --Gmaxwell 20:50, 3 September 2006 (UTC)

Rude and condescending? I... I'm genuinely shocked by this reaction. I'm sorry to have caused it, though it was by no means intentional. You removed a lot of material from a lot of articles at once, and I thought it such a sudden change was unwarranted without prior discussion, and said as much, but I didn't realize saying this was rude or condescending. Again, sorry if it was. As for WP:EL, it is covered in "Links normally to be avoided", or "Rich media", neither of which forbids these links, only discourages them, meaning that there is room for discussion. - Rainwarrior 21:15, 3 September 2006 (UTC)

If you'd like to not cause a negative reaction you could avoid mass reverting before discussing, and you could avoid making incorrect accusations about reading the talk page. In any case, why are you opposed to including the content in Wikipedia rather than linking to it externally? Our external linking policy is written to be general so it can be applied rationally, it is not written softly to encourage people to use it as a loophole to avoid our requirements on copyright and formats. --Gmaxwell 22:52, 3 September 2006 (UTC)

I reverted a mass deletion (I could say the same about mass deletions before discussion... isn't any deletion technically a reversion?). At any rate, we're discussing it now, at least. As I said, I think these links are a benefit to their articles. I am not opposed to including their content in Wikipedia (I think I mentioned this above), and I do think that would be better than linking it externally. I also think it would be better to link to HTML pages that describe these pieces. However, I don't think it's better to just delete them all right now.

I did not add any of these links originally, so I do not know their copyright status, but many of them were created by Wikipedia contributors (e.g. Gene Ward Smith, or Joe Monzo), so at least in those cases a migration to the commons may be possible. If you wish to argue that they are copyright violations, I cannot defend them, and on that basis I would understand their mass deletion.

I think I have a solution: I will remove all of these links from their articles, but place them on their respective talk pages instead. That will leave the authors of these articles a list of links to clean up, rather than have all this content just disappear. - Rainwarrior 04:47, 4 September 2006 (UTC)

I've checked out the claim that Wikipedia does not allow such links, and it seems to be false. Unless I hear a reason not to I am reverting. Unless [User:Gmaxwell|Gmaxwell]] can support his claim, he may have to be blocked from reverting back. Gene Ward Smith 23:00, 14 September 2006 (UTC)

WP:EL#Rich media doesn't specifically say "don't link directly to audio files", but the spirit of the guideline is clear. A simple, helpful thing you could do is to make an HTML page on your website with links to all these files (and ideally descriptions of some kind, with copyright information). Then no one would have a case for removing the links. —Keenan Pepper 01:58, 15 September 2006 (UTC)

To delete such links without any discussion as non notable and self-promotional is the Rainwarrior's favourite work. Commator 11:52, 13 September 2007 (UTC)

[edit] 'Musical Color' Temperament?

I recall reading that Leonardo da Vinci formulated a tuning system based on the frequencies of the visible light portion of the electromagnetic spectrum. I tried looking stuff up on this online but couldn't find much. Anyone know what I am talking about or perhaps what this type of scale was called? SN122787 18:19, 1 October 2006 (UTC)

I don't know about da Vinci's (I am curious, though), but I am familiar with more recent attemps at this sort of thing. The problem with them (in my opinion) is that vision and hearing work in very different ways and in very different domains. For example, the visible light spectrum is less than an 'octave', additive/subtractive color does not work like harmony, etc. -CKL

[edit] Project Directory

Hello. The WikiProject Council is currently in the process of developing a master directory of the existing WikiProjects to replace and update the existing Wikipedia:WikiProject Council/Directory. These WikiProjects are of vital importance in helping wikipedia achieve its goal of becoming truly encyclopedic. Please review the following pages:

User:Badbilltucker/Culture Directory,
User:Badbilltucker/Culture Directory 2,
User:Badbilltucker/Philosophy and religion Directory,
User:Badbilltucker/Sports Directory,
User:Badbilltucker/Geographical Directory,
User:Badbilltucker/Geographical Directory/United States, (note: This page will be retitled to more accurately reflect its contents)
User:Badbilltucker/History and society directory, and
User:Badbilltucker/Science directory

and make any changes to the entries for your project that you see fit. There is also a directory of portals, at User:B2T2/Portal, listing all the existing portals. Feel free to add any of them to the portals or comments section of your entries in the directory. The three columns regarding assessment, peer review, and collaboration are included in the directory for both the use of the projects themselves and for that of others. Having such departments will allow a project to more quickly and easily identify its most important articles and its articles in greatest need of improvement. If you have not already done so, please consider whether your project would benefit from having departments which deal in these matters. It is my hope to have the existing directory replaced by the updated and corrected version of the directory above by November 1. Please feel free to make any changes you see fit to the entries for your project before then. If you should have any questions regarding this matter, please do not hesitate to contact me. Thank you. B2T2 21:26, 23 October 2006 (UTC)

Sorry if you tried to update it before, and the corrections were gone. I have now put the new draft in the old directory pages, so the links should work better. My apologies for any confusion this may have caused you. B2T2 00:15, 24 October 2006 (UTC)

[edit] Wikipedia Day Awards

Hello, all. It was initially my hope to try to have this done as part of Esperanza's proposal for an appreciation week to end on Wikipedia Day, January 15. However, several people have once again proposed the entirety of Esperanza for deletion, so that might not work. It was the intention of the Appreciation Week proposal to set aside a given time when the various individuals who have made significant, valuable contributions to the encyclopedia would be recognized and honored. I believe that, with some effort, this could still be done. My proposal is to, with luck, try to organize the various WikiProjects and other entities of wikipedia to take part in a larger celebrartion of its contributors to take place in January, probably beginning January 15, 2007. I have created yet another new subpage for myself (a weakness of mine, I'm afraid) at User talk:Badbilltucker/Appreciation Week where I would greatly appreciate any indications from the members of this project as to whether and how they might be willing and/or able to assist in recognizing the contributions of our editors. Thank you for your attention. Badbilltucker 17:46, 29 December 2006 (UTC)

[edit] Cris Forster

I don't know if everyone saw this article is nominated for deletion - Mireut 14:27, 31 December 2006 (UTC)

[edit] Regular numbers

As I'm not a music theorist, I'd appreciate it if someone from this project could look over the music-theoretic material I've included in the regular number article (related to just intonation) and make sure that I haven't said anything stupid there. —David Eppstein 05:19, 8 April 2007 (UTC)

[edit] Diatonic and chromatic

I have added this article to the list in this project page. The aim of the article is to show the many meanings that diatonic and chromatic have accumulated (in connection with scales, etc.), and to show something of their evolution and current applications. This seemed like a good idea, especially since editors themselves use these terms in different and incompatible ways in Wikipedia music theory articles. Much has been done; more sections need to be added. Members of this project may like to take a look.

– Noetica^♬♩ Talk 07:55, 29 April 2007 (UTC)

[edit] Does anyone know anthing about this instrument?

A multi-keyboard instrument at the Musical Instrument Museum in Vienna. I believe it was designed so that you could play perfect intervals in any key.

--☑ SamuelWantman 05:22, 4 August 2007 (UTC)

Richard Maunder wrote a little about it in Keyboard Instruments in Eighteenth-Century Vienna Oxford University Press, 1998, page 78 - the piano has inventory number SAM 610, and is labelled "Andreas Rebele von/ Monheim in der oberen/ Pfalz dem 22^ten^ Julÿ 1796". It has been attributed to Johann Jacob Könnicke, and apparently it would be the only one by him with a non escapement Stossmechanik action. An article from the Wiener Zeitung October 12 that year reported J. G. Roser, Kapellmeister at Linze Cathedral had played a concert in Vienna on a "pianoforte with complete harmony" made by "Johann Jakob König" (according to Roser's son the audience included Haydn and Beethoven), but does not explain any more or describe the instrument, and Maunder wrote "there seems little reason to think that it was anything like the Rebele piano, although the coincidence of date is curious." He refers to an article by Alfons Huber about it in Der Klangwelt Mozarts, Vienna, 1991. - Mireut 14:19, 4 August 2007 (UTC)

[edit] Preferred Style for Pitch Set Theory

I want to contribute to some articles within the scope of this project, and I'm wondering if there is a preferred way of expressing the names of pitch class sets. Allen Forte has a naming system in his The Structure of Atonal Music, which I believe to be notable, but pitch sets can also be referred to by naming the pitches contained within the set (024579E being either C Major or A minor with no transposition or inversion, or the so called "prime form" of all major and natural minor scales, for example.), which is also notable. I'll work on some articles, perhaps if the manual of style people object to something, we can come up with a standard for when to use either naming convention. --Kento 12:00, 14 August 2007 (UTC)

[edit] Audio file demonstrating diesis comma

At the Diesis page (and other similar pages) I saw a request for an audio file and I thought I might be able to make one with Scala. A quick attempt seems to have worked. Scala makes MIDI files, so I'm not sure it would work correctly on various computers, so I thought I'd ask before trying to add it to the article (and figuring out how best to do that). I know it could sound better, but it is just demonstrating the sound of the diesis comma. I just wonder if people think something like this would be useful for that page, and other pages on commas.

I uploaded the file to the Commons, Image:Diesis-example.mid, or just http://upload.wikimedia.org/wikipedia/commons/6/6c/Diesis-example.mid

Description of what the does: It first plays an octave (2:1), C3 to C4; then, starting from the same C3 pitch, it plays a justly tuned major third (5:4), then another, then another (ie, C, E, G#, B#). Then it plays the 2:1 octave C4 again, followed by the 125:64 B#, followed by both at once. The difference between the C and the B#, 128:125 is the diesis comma. In equal temperament, on a piano for example, B# is the same as C, and three major thirds in a row equal an octave. But three justly-tuned major thirds fall quite a bit flat of an octave. Does this file help understanding that? Thanks! Pfly 08:55, 16 November 2007 (UTC)

Good work, but I don't think MIDI is the appropriate format for this for a number of reasons:

MIDI files contain no information about the timbre of the sound. MIDI data is only sequences of notes and other events. So when people play it back with different software, it could sound totally different, and the timbre might be inappropriate for this example.
MIDI is especially inappropriate for non-12-equal tuning demonstrations, because there are different ways to specify tuning information and AFAIK none are universally supported. The user might not even hear the right pitches!
As you say, some computers have no software installed to play MIDI files, and Wikipedia currently doesn't have anything built in, as it does for Ogg Vorbis.

Therefore I took your MIDI file and rendered it into actual audio using ZynAddSubFX (just because it was a software synthesizer I had handy). The result is Image:Diesis-example.ogg. Does the default timbre sound okay? —Keenan Pepper 01:48, 17 November 2007 (UTC)

Thanks, it sounds good. I'm new to using audio on wikipedia, so am taking some time to read up and look at ways people do it. I'll get around to adding this file to the diesis page sooner or later. I also began looking into ZynAddSubFX, which looks cool for a free softsynth. I might try to make some more soundfiles for pages with requests, but will probably wait until I have my main computer (a mac) back, with all its audio tools I know so well. Also, yea I knew MIDI was non-ideal, but didn't know how to turn it into an ogg on this Windows Vista box I'm using. Pfly (talk) 22:56, 17 November 2007 (UTC)

With the piano timbre that my system uses to play the .mid files, I can clearly hear the chord at the end in which both tones C and B# are played at once. With the pure sine wave of the ogg file, it sounds to my ear more like only the C is being played: the octave drowns out the other note. So while I'm sure ogg is a better format for this purpose, I'm not sure the choice of timbre made in the ogg is optimal for demonstrating the interval clearly. —David Eppstein (talk) 23:05, 17 November 2007 (UTC)

It sounded to your ear like only the C was being played because that was actually the case. I neglected the second track of the MIDI file. The new version I just uploaded has both tracks, and you can clearly hear the beating. —Keenan Pepper 02:31, 18 November 2007 (UTC)

Much better. There's still an issue where the second tone (high C after a low C) comes in as two separate short notes, rather than a single longer note, some of the times I play it, but I think that's just a software problem with my browser. —David Eppstein (talk) 08:21, 18 November 2007 (UTC)

I added it to the Diesis page, thanks for the help! Pfly (talk) 20:43, 19 November 2007 (UTC)

[edit] Hexany AfD

I just nominated Hexany for deletion. I thought you folks might have a better sense than I if it was worth keeping: I'm a equal-tempered sort. :-)--uɐɔlnʌɟoʞǝɹɐs 19:58, 11 February 2008 (UTC)

[edit] Orwell tetrad and unexplained small intervals

Both the 22 equal temperament and 31 equal temperament claim that the "Orwell tetrad" exists within them, but neither article explains what it is (beyond being a possible sonority), and there is no article. I can't find anything on it anywhere. Does anybody know its significance? Does it have anything to do with the (redlinked, stated as 1728/1715 but otherwise unexplained) Orwell comma?

And speaking of redlinked intervals, what's the gamelan residue? It's mentioned in both the 31-EDO and miracle temperament articles, but nothing beyond the ratio is mentioned. Not even what it has to do with gamelans. — Gwalla | Talk 07:05, 6 March 2008 (UTC)

[edit] Ratios or fractions?

Currently, tuning ratios in Wikipedia articles are inconsistently expressed as ratios (3:2) and sometimes as vulgar fractions (3/2), sometimes both within a single article. Which notation should be preferred? — Gwalla | Talk 17:37, 10 April 2008 (UTC)

I guess I usually see the colon used: 3:2, so that's what I would go with. Jeff Dahl (Talk • contribs) 06:11, 15 April 2008 (UTC)

Sounds good to me. It's easier to make those look good, too. (Fractions look best with the fraction template, but that results in really messy wikicode) — Gwalla | Talk 21:46, 17 April 2008 (UTC)

[edit] The kleisma needs help

The kleisma article, a sad little stub, needs love. (Awww.) Specifically, it needs a citation for that name (even better, the etymology of the term and who gave it that name), and more detail on how it is important to tempering the Bohlen-Pierce scale (the latter article doesn't even mention it). I've been able to find it in online catalogs of microtonal intervals, but nothing beyond simple listings of frequency ratio and size in cents. — Gwalla | Talk 23:00, 5 May 2008 (UTC)

[edit] Trouble at list of meantone intervals

A succession of anonymous IPs keep adding 88-EDO to List of meantone intervals. I keep reverting it, since 88-EDO is not a meantone. It's slow, only an edit war because there isn't any other editing going on, but still a bit frustrating.

On a related note, I would appreciate it if some kind person could add a ¼-comma column to that table. It's kind of silly that a list of meantone intervals doesn't show them in one of the most historically significant meantone temperaments. I'd do it myself but I'm not sure how to calculate the cents for it. A ²⁄₇-comma column wouldn't be out of place either. — Gwalla | Talk 05:57, 10 May 2008 (UTC)

Why do you say 88-EDO is not a meantone? The best approximation to the 3/2 perfect fifth in 88-EDO is 51 steps, or 695.5 cents, which is definitely a meantone fifth. Four of them, less two octaves, make a 28-step interval which is the best approximation to the 5/4 major third, so I don't know why you say it's not meantone. Do you have some more strict criterion in mind?

As for 1/4-comma, that's easy to calculate, and certainly appropriate. I'll add it when I have time. —Keenan Pepper 16:24, 10 May 2008 (UTC)

AIUI, 88-EDO differentiates between the major and minor tones. 15 steps is about 205.55¢, which is about 0.64¢ wider than 9:8 and 22.14¢ wider than 10:9, whereas 14 steps is about 190.91¢, which is about 13¢ narrower than 9:8 and 8.51¢ wider than 10:9, therefore the major and minor tones are most closely approximated by distinct 88-EDO intervals. — Gwalla | Talk 21:41, 10 May 2008 (UTC)

As far as I can tell from reading the articles here, the composers who care about 88-EDO care about it because it resembles Lucy tuning if you set the whole tone to 14 steps (even though 15 steps is much closer to just). With a 14-step whole tone and a 51-step fifth, it's a meantone, no? —David Eppstein (talk) 22:01, 10 May 2008 (UTC)

That subset of it could be considered a meantone, sure. I guess you could say that it contains a meantone. But is that the same as saying that 88-EDO itself is a meantone?

Does anybody actually use 88-equal? I've only ever seen it mentioned in reference to Lucy Tuning. The 88 equal temperament article even states that it's a relatively poor match to just intervals compared to some equal temperaments with fewer steps, and that it is studied for its "other theoretic properties" (the only example it gives is its closeness to Lucy Tuning). — Gwalla | Talk 23:37, 10 May 2008 (UTC)

Ah, I see what the confusion is. You (Gwalla) think that in order to be a meantone, the closest approximations to 10/9 and 9/8 should be the same. But I think that 15 steps of 88-EDO is an unacceptable approximation of 9/8 because then the 103-step interval that represents 9/4 cannot be divided into two equal 3/2 intervals. If we insist on having one unique interval that represents 3/2, and on preserving the mathematical identity "3/2 * 3/2 = 9/4" in our temperament system, then 15 steps cannot represent 9/8 and therefore 14 steps is the closest acceptable interval.

You might disagree with that (admittedly subtle) argument, but regardless of whether 88-EDO is a meantone system or not, I agree with you that it doesn't belong in List of meantone intervals. Why should 88 be in there when 50 and 69 aren't? —Keenan Pepper 05:27, 12 May 2008 (UTC)

May I naively ask here why 69, as well as why 88? I thought I understood why 19, 31, 50, but those last two confuse me. The sequence of best rational approximations to the quarter-comma fifth (mathematically, to log₂(5)/4 ≈ 0.580482) begins, if I'm calculating it correctly, 4/7, 7/12, 11/19, 18/31, 29/50. I've written these as fractions rather than ratios because they should not be interpreted as musical intervals but rather as (number of steps in fifth)/(number of steps in octave); with this interpretation, the 7/12, 11/19, and 18/31 approximations are all standard meantones (and 4/7, or at least 7-EDO, is stated in equal temperament to have been used in ancient China); the corresponding whole tones are 1/7, 2/12, 3/19, 5/31, and 8/50. So 50 is right as the next one to look at after 31. But then the next steps do not involve 69 or 88; they are 47/81 and 65/112. —David Eppstein (talk) 06:01, 12 May 2008 (UTC)

[indentation reset] You're assuming there's something really special about quarter-comma. You might as well use third-comma ( $\log_2(10/3)/3 \approx 0.578989$ ), and then the continued fraction sequence goes 7/12, 11/19, 18/31, 29/50, 40/69, 51/88, 62/107... or Lucy tuning ( $1/2 + 1/(4\pi) \approx 0.579577$ ) and then the sequence goes 7/12, 11/19, 18/31, 29/50, 40/69, 51/88, 91/157...

Really it's more like a tree that starts with 7 and 12 as the roots.

 40  59  71  64  69  |81  74  55
   33      45      50|      43
       26            |  31
               19    |
7                    |          12

You're saying the only legitimate meantone EDOs are those right next to the vertical line (which represents quarter-comma). I'm saying that any EDOs in that general area are good for meantone (which definitely includes 69, even though 50 and 81 are closer to the quarter-comma line). —Keenan Pepper 16:43, 13 May 2008 (UTC)

Hmm...I hadn't really thought of that distinction. So it's a meantone if two times the best approximation for 3:2 is equal to an octave plus the best approximation of 10:9? If so, meantone temperament should be corrected. I haven't seen that argument before, though. — Gwalla | Talk 21:36, 12 May 2008 (UTC)

I think it's more or less useless to try to make a strict mathematical definition for a word like meantone that's been around for centuries and used to mean many different things. In any event, I don't have any sources to back up the argument I gave, so I can't really put it in the Meantone temperament article. I don't see anything wrong with that article anyway. In fact,

the fundamental character of a meantone tuning is that all intervals are generated from fifths, and the syntonic comma is tempered to a unison

is a perfect description and I wouldn't change a word. —Keenan Pepper 16:56, 13 May 2008 (UTC)