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Check copyright status Cite this Title Which way did the bicycle go? Author Konhauser, Joseph D. Other Authors Velleman, Daniel J. Wagon, S. Physical Description xv, p. Series Dolciani mathematical expositions ; no. Mathematics -- Problems, exercises, etc. Contents Machine derived contents note: Preface 1.

The bicycle problem that nearly broke mathematics

Plane geometry 2. Number theory 3. Algebra 4. Combinatorics and graph theory 5. Three-dimensional geometry 6. Miscellaneous Solutions. Notes Includes bibliographical references p. View online Borrow Buy Freely available Show 0 more links Set up My libraries How do I set up "My libraries"? Rockhampton Campus Library. Federation University Australia - Gippsland campus library. Open to the public. Griffith University Library. In support of which theory, he mentions a late, unpublished prose work, called The Way , in which Beckett twice describes a repeating-loop path going up and down a hill.

The title of the first part is the figure 8; the title of the second is the infinity symbol. Which sounds plausible to me. We use cookies to personalise content, target and report on ads, to provide social media features and to analyse our traffic. For more information see our Cookie Policy.

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Your Comments. Octaves of Mathematically this means that the second note in the interval is the first note, multiplied by the first number in the ratio and divided by the second one. This works because adding a 0 is the same as multiplying a frequency by 10, which in music gives the same note as multiplying it by 5. So now you can see that numbers like , and and separated by a major third and 3 octaves.

Because there are 60 seconds in a minute, the bpms are reached by multiplying the Hz frequency of a lower octave of that note by This means that adding a decimal into a number lowers it by a major third and 3 octaves. You can use any Hz frequency as reference pitch first note in the scale from which the others are calculated using the ratios. Because some of these are octaves of each other, you only need 1, 3, 5, 7, 9, 11, 13 and I have color coded all of the octaves of 1, 2, 3, 4, 5, 6, etc so that you can see how these numbers connect harmonically.

If you look at the ratios and the lowest whole number octaves of the Hz frequencies in the above chart, it becomes obvious that the smallest whole numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, etc are the true source of all of this Musically these numbers are of the greatest importance because they represent the harmonic series , which is the true source of all sounds and music.

The way the same numbers represent musical harmony and simple mathematics is quite interesting. It is almost as though confusing mathematics and disturbing musical intervals are one and the same.

If you want to hear the difference between just intonation simple maths and normal equal temperament messy maths more clearly, listen to the examples below. Major chord:. Harmonic C major chord by Indigo Aura. Equal temperament C major chord by Indigo Aura.


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As you can hear, the harmonic chord is smoother and more stable when compared to the more wobbly equal temperament one. The Video below uses amazing visualization software that shows chords as moving patterns. Although these chords are in another key, the intervals between the notes are the same as in the above audio, and so they are actually the same two chords.


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Now you can see and hear that simple maths with nice whole numbers sounds good when played as Hz frequencies, while maths with lots of decimals and confusion does not sound as good. I am tired of writing now, so if you want to delve deeper into this you may want to read my books Follow the links below and go to "look inside" to read the table of contents and first few pages for free.