This week we discuss the merging, bridging and connections of the two worlds (the arts and the sciences). As an artist, musician, craftsman, athlete, mathematician and scientist I try to merge all I know and have experienced to develop understandings of each of these 'worlds'. I imagine my knowledge in these facets like tools to be used when the time is right regardless of its initial designated discipline much like using a fork to comb ones hair (shoutout to Ariel). Each skill is transferable and needs to be acquired with this mindset in place - one of infinite applications - rather than for a very specific context.
This week Vi Hart's video on using a mobius strip to invert a musical tune was beautiful and intriguing. If only it were this easy to do while playing music on a piano, I would love to hear what would happen to a few classics by Chopin (Nocturne in E Flat Major Op.9 No.2, or Nocturne in C Sharp Minor No. 20, for example). I imagine that over time, the surprising effect would wane as I would better able to simulate in my brain what the song would sound like before the tune would become inverted - but to get to that point of familiarity with song inversion, it would take some serious effort!!
If I were to translate this effect into mathematical concepts, based close inspection of the special music box paper, I think it would be like taking C5 as the y=0 line (...sorry to impose a grid upon an already grid-like structure of the musical staff...we've talked about liberation from this cage last week) and creating a reflection about the x-axis, essentially assigning all musical values a positive and negative equivalent, then multiply the entirety by negative one.
How could I trick my brain into doing this automatically? Perhaps using technology... Start with the original (this arrangement was by Rowena Young and contains no key signature unlike the original in E Flat Major):
Grab only the melody line, and exclude the clef so as not to trick oneself:
Then invert with a drag of the mouse:
Try it yourself before you hear my sight-reading performance!
Here's the results:
Vi Hart represents sounds using various pen strokes, then transposes the visual and audio to different harmonic keys overlaying them to develop a unique performance art project. Their art is quite fascinating!
For my assigned Bridges activity, I got excited about representing integers using different base systems. Inspired by Melissa Schumacher's Celtic knot artwork, I wanted to combine the two ideas myself using a dot and two types of lines system, but expand to a based four system to see how it would flow. However, what I realized through experimentation is that the base 3 system is directly related to the three line types, but not the number of digits in the row.
So, while playing around with different base numbers, I created a new number system instead using base 3.
A few wonders I have before I start...
- Could there be a different stroke for Celtic Knot systems that would allow me to draw one beyond base 3? (0=cross, 1=vertical impediment, 2=horizontal impediment, 3=both horizontal + vertical impediment?)
- Does even number base systems work with Celtic knot patterns?
- Would the numbers in which (4^4)x1 also resemble a loose strand of knot similar to Schumacher's?
And an expanded version to 53 triangles (with associated tone)!
I was curious about this expanding relationship between using the side lengths to build onto the next shapes, but I didn't want to re-use the triangle... so I needed a shape that would have a face that would expand depending on the other two faces.
A few questions I had about relationships between the expanded shapes:
1. As we generate an expanding trapezoid, what number does the ratio of a/b approach?
2. At what distance do the sequential pieces exactly cover the previous piece? Is this distance the same for each sequential pair?
Video links referenced this week:
Vi Hart: Mobius music box, (1:51) Sound Braid, (4:13), and the related Doodle music (3:53)
Gerofsky: Bach 333rd canon: recording, sheet music, explanation
Hi Oliver, I like the traffic light integer counting system that you created. This is simple and easy to follow, which I believe is really accessible to students in the math class. To answer your question, I think even numbers also work with Celtic knot patterns (like 2m x 2n, where m and n are integers). The structure will be a square grid.
ReplyDeleteUnfortunately, I don’t have any background in music, so it took me a hard time to understand how math and music notes are connected. However, I appreciate the video that you attached to show the actual music from the piano. The comparison is so obvious, and I can immediately tell the difference after I heard it (although I have zero music background!). This truly connects with the idea of learning with the senses that we discussed in the previous weeks. Listening is a good way of learning and experiencing math. Our students may not be experts in art or music, but they will never be blocked from learning since we all experience things through our senses. We may not know the professional concepts behind it, but we can identify the difference and recognize the pattern. This is an excellent example of how to transfer math learning from a piece of paper to something around us. Without your video, I can never understand the music notes and the pattern.
Wonderful work, Oliver -- and so many fascinating experiments here! As you might guess, I am MOST fascinated by your Wurzelschnecke tunings!! The 53 triangles version is absolutely riveting, and I can listen to it again and again, to start to get a feeling for the ways the square roots get closer and closer together, without ever being identical (i.e., even when it sounds almost like a rational number differential like a semi-tone or quarter-tone... it really isn't!) Very cool to relate different bases (base 3 or 4) with Celtic knotwork, and I would love to see how it plays out in creating knotwork patterns too. Great stuff in every aspect!
ReplyDelete