Talk:Twelfth root of two

message: Good edit by Scythe33 as that it makes the article easier to read. hydnjo talk 01:17, 8 August 2005 (UTC)
thread_title:  Good edit! 

message: I made a few changes to the page. I found a lot of the information to be inaccurate or untrue, so a good deal of it was removed. I did try to add an explanation for the use of this particular number. If someone else can think of a more concise way of explaining it, please edit it. Here's a list of the other things I removed or changed:
* "Growling" should have been "howling". Growling was not the word used, historically, and it doesn't describe anything close to the sound of a wolf fifth.
* Harmonic tuning does not cause "errors" in temperament. The use of the word temperament refers to a tempering (modification) of harmonic tunings. In other words, tempering is the error, albeit an intentional and purposeful one. Furthermore harmonic tuning doesn't cause dissonance especially not in the harmonic content of strings. (I'm not sure what harmonic content meant, but inharmonicity in strings is caused by tension and thickness, and doesn't have much to do with tuning, except the fact that temperaments are by nature out of tune with the harmonics of the string.)
* Meantone tuning was favoured in the Renaissance, which was very much not an equal temperament. In fact, other solutions to the modulation problem were preferred, ie. Nicola Vicentino's Archicembalo with an extra manual and divided keys to provide just intonation in many more keys, or even in the Baroque, it was not uncommon for the black keys of an organ to be divded in two so that sharp and flat keys could be tuned differently.
* While the black keys and white keys on a piano to correspond to pentatonic and diatonic scales, these two things have nothing to do with equal temperament, and equal temperament was certainly not any kind of attempt to combine the two. It's an interesting observation, but really not relevant to the subject of this page.
Rainwarrior 22:51, 31 March 2006 (UTC)
replies: text: Vicentino's Archicembalo (article needs expanding BTW) was not designed for just intonation, but extended meantone temperament, or the very similar 31 equal temperament. Other than that, good comments. —Keenan Pepper 23:24, 31 March 2006 (UTC)
replies: text: I thought I might mention that you should be careful about quoting Partch on historical matters. His account in Genesis isn't entirely accurate (you might already be aware of this). This particular factoid I think is correct, but personally I haven't checked up on it. I did change "string lengths" to "pipes" for you, not only because I'm sure it was pipes in the example, but because strings also depend on tension for their pitch. Pipe pitch doesn't entirely depend on length either, but at least it's a slightly better model instrument for the experiment. (As for Vicentino, I tend to refer to meantone as a form of just intonation, which is a pretty broad term for me. Pay no mind...) Rainwarrior 00:15, 1 April 2006 (UTC)
replies: text: Yeah, Partch is great, but he's not always the most reliable source. I'll try to find a different one. —Keenan Pepper 00:32, 1 April 2006 (UTC)
thread_title:  Changes 03/31/2006 

message: How were these ratios chosen?  I whipped up a little program to find the "best" rational approximations using the method of continued fractions, and got: 1/1, 17/16, 18/17, 89/84, 196/185, 1461/1379, 1657/1564, 3118/2943, 7893/7450, 18904/17843, 140221/132351, 579788/547247, 720009/679598, 2019806/1906443, 2739815/2586041. (The sequence of partial quotients is 1, 16, 1, 4, 2, 7, 1, 1, 2, 2, 7, 4, 1, 2, 1.) —Tamfang (talk) 07:35, 13 October 2011 (UTC)
replies: text: You're quite correct! I have updated the list using the fractions I boldfaced in your remark. You picked a great place to stop, the next partial quotient is 60. — Glenn L (talk) 09:23, 13 October 2011 (UTC)

text: I have to agree. —Tamfang (talk) 22:41, 14 July 2023 (UTC)
thread_title:  Choice of rationalizations