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3 Blue 1 Brown – Music And Measure Theory

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A connection between a classical puzzle about rational numbers and what makes music harmonious.

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  1. I just came across this video. I was fascinated from the point of view of "temperament" and the ultimate clash of "equal" vs "well" climaxing with J S Bach and his students. The business of √12 of course sounds precise, but it isn't as you have pointed out. This means each step in the 12 step set of notes is not constant. For the musician (in me), I am fascinated by the well tempered scales. C Major is not the same in scale or sound as D Major but even better, The scale of F# Major is playable in a well tempered tuning with flexible steps but not so in Equal tempering.
    Each scale and it's attendant tonality, then has a different "feel" to the ear and brain and results in a choice of tonality on the part of the musician. That is until we get to digitization which can produce intervals so close that they fall below the threshold of most listeners to discern difference. And this is the sadness for musicians and serious composers or should be.
    Thank you for bringing this to front of mind (albeit years later) and for perhaps reading this long comment

  2. Your explanation of why some intervals are "harmonious" and others "cacaphonous" is not correct. You do not hear a pure sinusoid as a series of "clicks" occurring at the times of the peaks in the waveform. In fact, you don't hear the individual vibrations at all; your ear is a spectrum analyzer, and what you generally perceive is the frequency spectrum.

    Pitch perception is related to the fact that many sources of sound, such as vibrating string, vibrating column of air in a cylindrical pipe, or a person pronouncing a vowel, have a harmonic frequency spectrum; that is, the frequencies produced are integral multiples of the lowest frequency, which is called the fundamental, (Actually, the frequency spectrum is a series of lines of nonzero width centered around the harmonics of the fundamental.) A sound with a harmonic frequency spectrum is perceived to have a pitch corresponding to a pure sinusoid at the fundamental frequency; a sound with a different frequency spectrum (such as the sound produced when you strike a snare drum) is not perceived to have a pitch.

    If the ratio of the fundamentals of two pitched sounds is close to a rational number with a small denominator, many of the lines in the spectra will overlap. This is effect likely contributes to determining whether an interval is "harmonious" or not.

  3. 1:35 – (this interval = 5/3, not 8/5.) Synchronistic sidebar: Grant, when you speak "eight-fifths," your voice closely mimics the given keyboard pitches 220 (A) & 366.2/3 (f#). Your two words "eight-fifths" is somewhere between 7/4 and 15/8 – only a tad higher than f#. Note too that while A to F# (220 to 366 +2/3) is a Major 6th, that interval ratio is 5/3, not 8/5.
    Thanks for this and all your outstanding videos, Grant!

  4. kinda stupid but i never thought about fractional powers other than 1/2 being referred to as the root of a number. so hearing u call 2 ^1/12 the twelfth root of two was a little bit of a trip for me. but if i think about it for more than 2 seconds it makes sense

  5. For the second challenge. I tried using the fact that approximations of reimans sum ( for example 12 rectangles) the ends of each rectangle is x=k/12 where k is an integer. When you approach infinitly many rectangles you get every rational. The problem is is you also get every irrational number. So dx is approximately 1/n-epsilon. I couldn’t get it to work. Let me know if that was confusing

  6. I am a musician and i am interested in math and of course math related to music. For me it es EXTREMELY disturbing having to listen to complete unrelated pianopatterns, (like dudidadi-dudidadi….) while am trying to understand concepts that are quite complicated (at least for me) even when you talk about two related frequencies (meanwhile dudidadidudidadi) even in your other videos, which are not music-related, and which i love because of their Stunning Graphic presentation and representations, that really helps understan—dudidadidudidadi— ITS DISTRACTING!!!! WHY???

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