September 2015 - mathfun@mailman.xmission.com

[math-fun] animated music
by Warren D Smith 07 Sep '15

07 Sep '15

Stephen Malinowski has made a large collection of youtube videos "animating", music with the aid of software he wrote; he has many animation techniques, choosing each time one he feels most-reveals the underlying structure of that music. One random (not especially good) example: https://www.youtube.com/watch?v=NJWqKNnA-BU -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] musical chords geometry
by Henry Baker 06 Sep '15

06 Sep '15

Re Tymoczko: As far as I can tell, 2-note chords form a 2-D surface (in this case, a Mobius strip), where "closeness" is defined by mod 12 arithmetic around the band. The problem comes with the edges of the band, where Tymoczko claims that the chords "bounce off" the "singularity" on the edge. 3-note chords form a 3-D volume with similarly weird topology, and 4-note chords form a 4-D "volume" with weird topology. Tymoczko "closenss" is required in order to try to figure out which individual note in each chord goes with which "voice"; the technical musical term apparently being "voice leading" (i.e., really "voice following" by a listener). At 09:21 AM 9/6/2015, Fred Lunnon wrote: > The Hart video is well worth a look! > > It also raises an elementary question which has me scratching my head >--- what is the connection between the Moebius band and the orbifold >described by Tymoczko? > >WFL > >On 9/6/15, Henry Baker <hbaker1(a)pipeline.com> wrote: >> As a lifelong lover of music, I'm amazed that I never knew that 2-note >> chords formed a Mobius strip. The George Hart video shows this the best. >> >> Octaves in Modular Mobius Music - YouTube >> >> https://www.youtube.com/watch?v=Q8moAKBegFg >> >> Perhaps everyone else on this list already knew this, but it was new (& >> cool) to me. >> >> It might be fun to animate Bach's 2-part counterpoint on such a Mobius strip >> from a midi score, although I suspect that others have already done this -- >> perhaps Hart himself? >> >> At 06:05 AM 9/6/2015, Warren D Smith wrote: >>>I found Dmitri Tymoczko's paper >>>THE GEOMETRY OF MUSICAL CHORDS >>>here >>>http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.215.7449&rep=rep1&…

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Re: [math-fun] musical chords geometry
by Henry Baker 06 Sep '15

06 Sep '15

As a lifelong lover of music, I'm amazed that I never knew that 2-note chords formed a Mobius strip. The George Hart video shows this the best. Octaves in Modular Mobius Music - YouTube https://www.youtube.com/watch?v=Q8moAKBegFg Perhaps everyone else on this list already knew this, but it was new (& cool) to me. It might be fun to animate Bach's 2-part counterpoint on such a Mobius strip from a midi score, although I suspect that others have already done this -- perhaps Hart himself? At 06:05 AM 9/6/2015, Warren D Smith wrote: >I found Dmitri Tymoczko's paper >THE GEOMETRY OF MUSICAL CHORDS >here >http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.215.7449&rep=rep1&…

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[math-fun] Dmitri Tymoczko's musical chords geometry
by Warren D Smith 06 Sep '15

06 Sep '15

I found Dmitri Tymoczko's paper THE GEOMETRY OF MUSICAL CHORDS here http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.215.7449&rep=rep1&… I do not see what is the big deal...

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[math-fun] George Hart's Modular Mobius Music
by Henry Baker 05 Sep '15

05 Sep '15

I just discovered Dmitri Tymoczko's work on the topology of musical chords. George Hart did a wonderful ~7 minute video for the Simons Foundation about this idea. Prof. Dmitri Tymoczko (Princeton) "The Shape of Music" https://www.youtube.com/watch?v=XUyx31f-U3M Octaves in Modular Mobius Music - YouTube https://www.youtube.com/watch?v=Q8moAKBegFg http://www.brainmusic.org/EducationalActivitiesFolder/Tymoczko_chords2006.p… http://dmitri.mycpanel.princeton.edu/voiceleading.pdf http://www.sciencemag.org/content/315/5810/330.3.full.pdf http://www.sciencemag.org/content/315/5810/330.2.full.pdf http://www.sciencemag.org/content/suppl/2006/07/07/313.5783.72.DC1/Tymoczko… Dmitri's own web site with his actual program to play & display the chords: http://dmitri.tymoczko.com/ChordGeometries.html

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Re: [math-fun] Isometric embedding questions, etc.
by Dan Asimov 03 Sep '15

03 Sep '15

Arrrgh, never mind. I feel as if I'm living through the comic strip <http://www.gocomics.com/doonesbury/1981/06/01>. —Dan

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[math-fun] Verticals
by Keith F. Lynch 03 Sep '15

03 Sep '15

There was recent discussion in another forum of the fact that tourists visiting the Royal Observatory in Greenwich are complaining that when they stand on the marked prime meridian their GPS systems are telling them they are not at zero longitude. You can also see this in Google Earth, in which the prime meridian is seen to pass, not through the observatory, but through a tour bus parked in front of it. The explanation they were given, that measurement is more accurate today, made no sense, since what was done in Greenwich wasn't measuring the meridian, but defining it. Saying they got it wrong would seem to be like saying that careful measurements have shown that there are actually 100.000003 centimeters in a meter. The correct explanation appears in various places, but rather poorly phrased. The 19th-century astronomers at Greenwich were defining, not a line on the ground, but a plane through the center of the Earth. It eventually turned out that local distribution of mass caused the local vertical, when projected downwards, to miss Earth's center by about 100 meters. Once this was realized, they had the choice of keeping the line and tilting the plane, or of keeping the plane and moving the line. They wisely chose the latter. (Unfortunately, they didn't literally redraw the line, resulting in annoyed tourists.) (Strictly speaking, it missed the center by a lot more than that, due to centrifugal force and the resulting equatorial bulge. But the north-south tilt is harmless, since they only cared about longitude, not latitude.) If latitudes and longitudes were defined by local verticals, parallels and meridians would be neither straight nor equally spaced. In some cases, they might even split. There could be three closely-spaced equators. (It would have to be an odd number.) If the north pole were defined as the point where Polaris was directly overhead (assuming Polaris was directly to the celestial north, which it isn't) there could be multiple points where that is the case, hence multiple north poles. That's not what we want in a coordinate system. They were aware of these issues at the time, but had no way to measure such small effects. (When Mason and Dixon surveyed their notorious line, they did so twice, first in one direction then in the other, to minimize errors. They noticed that the errors they were getting weren't random, and correctly guessed that their verticals were being deflected by the gravitation of the Appalachian mountains. This was in the 1760s.) Neither did such small errors matter for their purposes. It was sufficient to be able to navigate to within sight of one's intended destination; there was no need to be able to tie a sailing ship to a dock while blindfolded. Several math questions immediately come to mind. Why does the number of local-vertical-defined equators have to be odd? Are there any constraints on the number of north or south poles? If you measure the verticals everywhere, do the deflections necessarily all average to zero? Is the Earth's center well-defined and unique? How is latitude defined? I can think of three ways to deal with the equatorial bulge: * Your latitude is the elevation of Polaris from your location (assuming it was exactly at the celestial north pole, which it isn't). * Locate the pole and equator, and space the lines of latitude evenly between them along the actual (mean sea level) shape of the Earth. * Superimpose a sphere on Earth, and mathematically project the lines of latitude upward or downward onto the actual surface. I don't know which of these three is done. They all have advantages and disadvantages. Or perhaps some fourth method is used. Do any of you know? Thanks. These ideas can of course be extended to lumpier worlds. Could a local vertical ever point directly away from the center? (Assuming that centrifugal force never exceeds gravity.)

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[math-fun] test
by rcs＠xmission.com 02 Sep '15

02 Sep '15

Just a test message. --Rich

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