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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.
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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
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we usually test idealized electronic sounds but we can test natural sounds and compare the results; due to the different harmonics vs inharmonics, the comparison is impossible between different things; one way to fix it is to conduct even more tests with electronic inharmonicity of various degrees
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if you play the same interval in series you distort its octavic reoccurrence and that causes a bias; one solution is to create octaves by using in series the interval and its inverted but used in the same direction of the progression to complete full octaves (the inverted interval has a compatible
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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 *
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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
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I think the chaos literature is pretty consisent in using "circle map" to refer to only this map, although I've seen the more general definition on occasion. The only support I have for this at the moment is the MathWorld article, and the refs that it cites.
699:(you can create usually 5 more tests of intermediate stages of inharmonicity... very hard job to make it unbiased with similar quality sounds. If the sounds are very different you might measure your own artefact / erroneously designed experiment)
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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.
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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.
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I agree with Nitse. A "circle map" can be any map of the circle. The confusion might come from the fact that the standard maps described by Arnol'd essentially model the behavior of many circle maps, up to topological
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you need people with a music ear; even these people have biases, but their biases have patterns (so you might not focus on what specifics they say, but the hypernymous meaning of how they organize their
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I would encourage you to make the change(s) you think is/are necessary. No one has been checking up on this page in quite a while. Your explanation for what you want to do sounds right to me.
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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?
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is happening here. It might probably help to create a connection between the text and the pictures. Doubly so because the terms used in the pictures and the text are still different. --
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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).
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Can't understand the article... where to go to get the basics first... can't
Knowledge have this section "textbook / prerequisites" for all mathematical/technical articles.
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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
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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:
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I support the notion that it is not entirely clear what happens here - I do not understand it myself but it seems that something similiar to the creation of the
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Arnold tongues for each ideal (electronic music) interval as used the objective characterization of each subjective emotion people hearing it give.
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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 --: -->
466:? If not, who's it named after? It is unconscionable that that question is not answered. It should be in the first line.
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Could we remove the redirect from circle map to this? It doesn't explain what a circle map is.
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clearly says the singular should be used except when there's a reason to use the plural.
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or something like this, but I believe Arnold tongues is the main subject of the paper.
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If you only play one interval and not a series, some listeners confuse some emotions
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You need some support to make a good treatise. It is totally necessary!!!
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How does it not explain what a circle map is? It's the first section.
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in the banner shell. Please resolve this conflict if possible.
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