I have walked my students through this article to help advance their knowledge as to how the violin works and how they can think about optimizing the desired tone they wish. The moving diagrams were very interesting to my students. From this , I ask them what they think a certain sound may need to be more beautiful or ... I ask them how is it that the violin just made such a such a sound... What was going on? So gradually they are applying this practical knowledge to their notions of what they think is beautiful. Thank you for going to the effort. I am not a 'math whiz' but I will visiting your site again for my amusement. Linda Ford, a violin teacher

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