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Isomorphism between symmetric chords and powers of complex numbers #96

Open rotem15 opened 2 years ago

rotem15 commented 2 years ago

About the author

I'm a university student, first year. I have special interest in maths and music, and the many relations that can be found between them :)

Quick Summary

The target audience is anyone who has basic understanding of maths, and is ready to learn. And if someone happens to have some knowledge in complex numbers or music, why not.

Target medium

I think the best medium for the idea is an animated video, might as well be a very simple one.

More details

The main idea will be explained step by step, going through the following topics in order:

  1. music: 1.1. introduction to music

    • notes
    • semitones
    • octaves 1.2. chords
    • what are chords
    • chords in different inversions 1.3. diminished 7 chords
    • how to build diminished 7 chords
    • the symmetry they have between inversions
    • there are only 3 "unique" diminished chords

  2. multiplying complex numbers: 2.1. multiplication of real numbers as a transformation of space 2.2. expanding the definition to the complex plane 2.3. what is i 2.4. multiplying complex numbers with magnitude 1 is rotating

  3. powers of complex numbers: 3.1. spirals or not

    • magnitude > 1: spiral outwards
    • magnitude < 1: spiral inwards
    • magnitude = 1: rotate 3.2. powers of i
    • gets the same 4 values in a cycle

  4. isomorphism between symmetric chords and powers of complex numbers: 4.1. what is isomorphism 4.2. partial isomorphism

    • the powers of i are partially isomorphic to a diminished chord and it's inversions.
    • no part of the powers sequence match the other 2 possible unique diminished chords. 4.3. augmented chords
    • how to build them
    • symmety between inversions
    • each chord has 4 unique versions 4.4. isomorphism between augmented chords and 1,120 (-½ + √3/2 i) 4.5. different fractions of the circle
    • a fraction of the sircle is like a fraction of an octave

  5. more ideas: 5.1. semitones limit us to 12ths of the circle 5.2. taking an amount of 12ths that doesn't divide 12 makes a chord over more than 1 octave. putting all those notes in the same octave is like rapping around the same circle. 5.3. microtonality 5.4. golden ratio chord 5.5. non-unit radius

additional information

I will gladly explain anything to anyone who thinks about producing the video with me (:

HKTRp commented 2 years ago

Hello! Very interesting idea! Music and math are very similar and both beautiful. I can make some animations with Manim and some music examples for the video.

univalency commented 2 years ago

Nice, do you consider collaborators with no music background ?