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Great article; I learnt that Arnol'd tongues can in fact be described in very simple language. However, I have the distinct impression that "circle map" refers to any map from the circle to itself. I think the map described here is usually called the "sine circle map" (though I've also seen "sine map"). -- Jitse Niesen ( talk) 11:52, 19 January 2006 (UTC)
I renamed article to Arnold tongues per the discussion above. The term circle map can certainly refer to any map of a circle, and there are various researches devoted to study of circle maps, see e.g. Katok-Hasselblat. The family exposed can be named standard circle maps family or something like this, but I believe Arnold tongues is the main subject of the paper. Ilya Voyager ( talk) 15:52, 5 September 2012 (UTC)
How does the circle map correspond to a model of the beating heart? If this could be explained briefly and in language suitable for a layman, possibly in a section of its own (near the end of the article?), that would be excellent. -- Quuxplusone 17:14, 16 April 2006 (UTC)
Is this named after Vladimir Arnold? If not, who's it named after? It is unconscionable that that question is not answered. It should be in the first line. Michael Hardy ( talk) 04:11, 6 September 2012 (UTC)
Can't understand the article... where to go to get the basics first... can't Wikipedia have this section "textbook / prerequisites" for all mathematical/technical articles. 117.198.121.106 ( talk) 08:42, 27 October 2012 (UTC)
In the second image, with the subtitle "Some of the Arnold tongues for the standard circle map, ", shouldn't alpha by Omega on the horizontal axis? And the comma a period in the subtitle? — Preceding unsigned comment added by Mmbosschaert ( talk • contribs) 16:33, 20 January 2013 (UTC)
Could we remove the redirect from circle map to this? It doesn't explain what a circle map is. 98.150.246.242 ( talk) 07:19, 14 February 2014 (UTC)
The rotation number in this article looks strange to me or simply wrong, so to say, though I might be wrong. Compare the expression with the rotation number article: /info/en/?search=Rotation_number Simply put, the rotation number is something like an average rotation angle per iteration. In this article's formula, however, there is a sum over all angles theta_n. For example, if we set K = 0, we should obtain Omega as the rotation number. For K = 0, we should have something like theta_n = n * Omega. If we put this in the sum of the numerator, we get something like 1/2 * n * (n+1) * Omega and thus, the limit n --> infinity does not exist. Without the sum, however, everthing is right, since we immediately get omega = Omega. Without the sum, the formula would also be consistent with the one in the rotation number article if we choose theta_0 = 0 (which we can do, since the number is proven to be independent of the starting angle, see the other article). I also checked some of the sources for this specific expression, but I didn't find anything similar. In fact, all the definitions of the rotation number in the sources are the same as in the Rotation Number article. I therefore suggest to simply remove the sum. But if I am wrong, you may correct me, which is my I didn't change anything yet. 178.20.90.65 ( talk) 19:14, 18 February 2016 (UTC)
Arnold tongues for each ideal (electronic music) interval as used the objective characterization of each subjective emotion people hearing it give.
This is problematic for many reasons.
Arnold tongues aren't the only tool used for the psychoacoustic comparison between the mathematically described objective and the subjective (we ask people to listen) moods of the intervals, but its a good way to start.
It is extremely hard work. Totally necessary to help build better artificial neural networks for composition or high-order compositional recognition (for treatises on great works).
You need some support to make a good treatise. It is totally necessary!!!
Add more data on it.