October 2003 - mathfun@mailman.xmission.com

[math-fun] Matrix Arithmetic Compendium
by Schroeppel, Richard 28 Oct '03

28 Oct '03

Does anyone know of a compendium of matrix arithmetic? I'm imagining a collection of numerical & simple algebraic facts about matrices-- things like the determinant of a Vandermonde matrix, or the inverse of a finite section of Pascal's Triangle. Rich rcs(a)cs.arizona.edu

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Re: [math-fun] RE: Prentice Hall strikes again
by asimovd＠aol.com 28 Oct '03

28 Oct '03

<< Lest you find six-dimensional money frivolous, meet nine dimensional pollution: In a Greepeace magazine I saw a photo of a vandalized sewer pipe with a caption crowing that before their sabotage, it had carried "thousands of cubic liters" of pharmaceutical waste to the ocean. >> About 15 years ago, a news article in the S.F. Chronicle described water pollution (of a kind I've long forgotten) as occurring in so many "parts per liter". I called up the reporter and explained that she must have meant something else, and I was curious what the actual extent of this polllution was. Although I was trying not to sound critical of her, but rather just wanting the information (which was the case),. she immediately got all defensive and refused to discuss it further. (She said I could speak with her editor if I wanted.) --Dan

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Re: [math-fun] The Pythagorean Comma
by asimovd＠aol.com 28 Oct '03

28 Oct '03

Thane writes: << The frequency ratio of 3^12 divided by 2^19 = 1.0136... is called the Pythagorean Comma, at least by music people. ... if you up 12 pure fifths from a note then go down 7 octaves, the resulting tone is little bit higher than the one you started out with. ... I don't know how the Shepard thing works but it might be related >> I don't believe there is any relation at all between the Pythagorean comma and the Shepard tones phenomenon. The Shepard tones phenomenon occurs simply because if you play a note and simultaneously all audible notes that are an integer number of exact octaves above it, then as you travel (say) up the scale doing the same thing at each note, a listener gets the sense of a rising tone, but after an octave you reach exactly the same set of audible notes played at the start. --Dan

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Re: [math-fun] The Pythagorean Comma
by Dan Hoey 28 Oct '03

28 Oct '03

Richard Guy <rkg(a)cpsc.ucalgary.ca> writes: > It IS known by some mathematicians. See p.257 > of The Book of Numbers. But, lack-a-day, we > define it as `the difference between 7 fifths > and 4 octaves' whereas it should be `12 and 7'. According to Margo Schulter's believable-sounding Usenet post (citing Boethius's _De Institutione Musica_ as translated by Calvin Bower in _Fundamentals of Music_), 2^8:3^5 is a limma or diesis or Pythagorean minor semitone, 3^7:2^11 is an apotome, 3^12:2^19 is the Pythagorean comma, 3^6:2^9.5 is a schisma, and 2^4:3^2.5 is a diaschisma. Dan Hoey(a)AIC.NRL.Navy.Mil

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[math-fun] Re: cyclic quadrilaterals
by James Propp 28 Oct '03

28 Oct '03

I wrote: >What I ideally want is a set of three parameters such that the >distances between all the points can be expressed as rational >functions of the parameters and the original lengths AB, AC, BC. >Is this too much to hope for? I should say that if I can't have this, I'd settle for a k-parameter sub-family of the three-parameter family with the property that all distances (for configurations within the sub-family) can be expressed as rational functions of the k parameters and the lengths AB, AC, BC. I'd even be okay with k=0. That is, I'd settle for knowing ONE way of choosing P,D,E,F so that all the distances are rational functions of the lengths AB, AC, BC. (But this seems even less likely to me.) As you might guess from the odd specificity of my question, there's some context for this problem. The answer might be very useful to me as a source of analogy for some research I'm doing in algebraic combinatorics, or it might turn out to be a dead-end, but in any case the context isn't very math-fun-ish, so I'll spare you the details (except to say that it has to do with things called "cluster algebras" that have been turning up lately in a bunch of different fields of mathematics). Jim Propp

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[math-fun] The Pythagorean Comma
by Thane Plambeck 28 Oct '03

28 Oct '03

The frequency ratio of 3^12 divided by 2^19 = 1.0136... is called the Pythagorean Comma, at least by music people. That's a constant name that doesn't seem to be known much to math people, not that it should be, I suppose. Anyway, if you up 12 pure fifths from a note then go down 7 octaves, the resulting tone is little bit higher than the one you started out with and the difference is the pythagorean comma. I don't know how the shepard thing works but it might be related Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm

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[math-fun] New Math
by Thane Plambeck 28 Oct '03

28 Oct '03

My new math experience (early 1970s in school) was characterized by endless rehashings of the distributive and associative laws for * and +. Bases other than 10 were paid their respects, but always as second class citizens. I remember pondering whether a number would still be a prime number when written in some other base. I decided eventually that it wouldn't matter how you wrote it. I tried to explain my great discovery and was encouraged to return to my distributive law and associative law worksheet. Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm

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Re: [math-fun] The New Math
by asimovd＠aol.com 27 Oct '03

27 Oct '03

Cal writes: << i took the first class of the new math in pasadena in 1960 (7th grade) - my class had an advantage over most of the other people i know who encountered it then or later; my teacher didn't know anything about it, just like everyone else's, but it DIDN'T bother him ... >> I believe Cal is exactly right. The main reason for the downfall of the New Math in that era was that a large number of teachers felt they were being asked to teach something they had never been taught. Since math has always had a problem of being taught, esp. in public school, by too many teachers who are already prejudiced against it, this new externally-imposed curriculum just exacerbated their negative feelings. --Dan

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[math-fun] The New Math
by Chris Landauer 27 Oct '03

27 Oct '03

hihi, all - i took the first class of the new math in pasadena in 1960 (7th grade) - my class had an advantage over most of the other people i know who encountered it then or later; my teacher didn't know anything about it, just like everyone else's, but it DIDN'T bother him ("ok, what are we going to learn today?") - as i recall, we did base 7 arithmetic and lotsa other cool stuff i loved it; it made me decide that math was the right subject for me more later, cal

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Re: [math-fun] Shepard's tones
by asimovd＠aol.com 27 Oct '03

27 Oct '03

It may be of interest that the ever-rising tone phenomenon springs can be viewed as an application of the fact that a typical helix curve in R^3 (e.g., h(theta) = (cos(theta), sin(theta), theta) ) can be thought of as a fibre bundle whose total space is the helix, base space is the circle, and fibre is the group Z of integers. The bundle projection, from the total space to the baseb is just p( (cos(theta), sin(theta), theta)) = theta.mod (2pi). I think the idea such "instrument" would play a "chord" of 11 or 12 octaves of a given tone simultaneously at equal volumes -- since the human range of hearing is often cited as between 20 and 20,000 cps, a factor of 1000 ~ 2^10. It might be fun to hear music written specifically for this circular range of tones. --Dan A.

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