Batch 2 - Class 311 - Music in Math, Topology

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Preclass Exercise:

Imagine folding a 8x8 chessboard into a mobius strip. Explore the paths of a bishop on the same (you may assume that the initial strip was printed back to back). You can rollover the edge of the strip when you encounter it.

Attendance: Anika, Shikhar, Rhea Chadha, Ryan Chadha, Ekagra, Ayush, Dhriti, Advay, Tarush, Kabir, Yatharth, Aneesh, Raghav, Siddhant, Siddharth, Kushagra, Aditya, Aarav

Class Notes:

Math in Music (INSTRUCTOR NOTE: WORKED WELL FOR STUDENTS WHO HAD SOME EXPOSURE TO MUSIC, NOT FOR OTHERS)

Show wave formation in string - use a guitar. Show distance computation between successive Cs, and explain in terms of closed waves
Draw up the theory (high level), that frequency of successive Cs are double and the 12 notes are equally spaced as a geometric sequence
Ask kids if they understand chords, and why some notes sound well together (both different octaves, as well as fifth/harmonic)

String vibrations correspond to chords - following table shows frequency of different notes/octaves, and successive frequencies in a vibrating string (C note). Observe the correspondence to notes highlighted in red

C	C#	D	D#	E	F	F#	G	G#	A	A#	B	C
		2		2	1		2		2		2	1
16.35	17.32	18.35	19.45	20.60	21.83	23.12	24.50	25.96	27.50	29.14	30.87	32.70
32.70	34.65	36.71	38.89	41.20	43.65	46.25	49.00	51.91	55.00	58.27	61.74	65.41
65.41	69.30	73.42	77.78	82.41	87.31	92.50	98.00	103.83	110.00	116.54	123.47	130.81	string vibrations
130.81	138.59	146.83	155.56	164.81	174.61	185.00	196.00	207.65	220.00	233.08	246.94	261.63	130.81	1
261.63	277.18	293.66	311.13	329.63	349.23	369.99	392.00	415.30	440.00	466.16	493.88	523.25	261.63	2
523.25	554.37	587.33	622.25	659.25	698.46	739.99	783.99	830.61	880.00	932.33	987.77	1046.50	392.44	3
1046.50	1108.73	1174.66	1244.51	1318.51	1396.91	1479.98	1567.98	1661.22	1760.00	1864.65	1975.53	2093.00	523.25	4
2093.00	2217.46	2349.32	2489.01	2637.02	2793.82	2959.95	3135.96	3322.44	3520.00	3729.31	3951.06	4186.01	654.06	5
4186.01	4434.92	4698.63	4978.03	5274.04	5587.65	5919.91	6271.92	6644.87	7040.00	7458.62	7902.13		784.88	6

Music on Mobius Strip

Single notes - put the notes on a strip of paper. As we have seen, all the C notes are really the same, so we can roll this strip into a cylinder to align the same notes. This represents the unique single notes
Double notes - what happens when we play two notes together

Illustrate on a graph paper what it means to play two notes together (as two axis). Show paths of chords (eg: parallel third)
What does the main diagonal represent? These are called unison chords
What if we want all Cs to be equivalent - we can roll up the x axis, and then the y axis - so we will get a torus. Mark the notes on this torus; mark the major diagonal

What is wrong with this picture? C,D and D,C are two different points - they should really be the same
How do we do that? On a piece of paper, this is folding on the major diagonal - show on isometric graph paper, and then folding an actual cutout
On a torus?

How would you do three notes (typical of actual musical chords)

Umbilical torus

Homework Problem:

A girl goes to a shop, pays a Rs 10 note and buys a Rs 5 pencil. The shopkeeper doesn't have change, and gets change from the trader next door. When the girl's gone, the trader comes back and informs the shopkeeper that the Rs 10 note he gave is fake. The shopkeeper gives the trader another Rs 10 note, and has to throw away the fake note. How much loss has the shopkeeper had to incur?

Rs 10

Instructor Notes: Surprisingly, all kids said 5 or 15. Useful to have them step back, and make them see that ultimately only Rs 10 have been lost. Also worked to show the whole transaction with real coins.

References:

https://www.youtube.com/watch?v=8bgvRvh88-w (music with mobius strips)