“Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity” ― G.H. Hardy, A Mathematician's Apology
The visual of the guitar is incorrect. It’s easier to think about if we have 2 strings, where one string is twice the length of the other. The string that is twice the length is the lower octave. With a guitar, you cut the length of the string in half to get one octave, since that is effectively creating a second string that is half the length of the first. Or creating a 2:1 ratio.
Why was the student confused when he went from English class to math class? Because he was taught that a double negative in English is bad, but in math, it’s a positive.
There's a mistake with the picture of the guitar supposedly representing an octave at 0.55. When we talk about these ratios with strings, the larger number is the whole string-length (the open string); the smaller is the playing length once you've stopped the string (i.e. when you've put a finger down on a fret). The picture at 0.55 divides the neck at the fifth fret, so it's dividing the string at 2/3 its length and the interval is 3 parts to 2 parts, 3:2, a fifth, not an octave (on the sixth string, if you put your finger down where the image shows, you'd play a B note, i.e. a 5th above the open E note). The octave on a guitar has two dots (so it's easy to find). That spot is half-way along the length of the whole, open string. Thus the ratio at the double dots is 2:1 open string to 1/2 the length of the open string.
You're totally correct. However, the image is correct in that the note played above the finger position is an octave above the note played below the same finger position. You're comparing the whole open string to the shorter fretted string, which of course, is usually how we play any stringed instrument. And yes again, dividing a single string as in the picture, is most likely what Pathagoras meant by 3:2, that we call today the perfect fifth. But that's not certain, if you simply imagine what they were doing with a lyre (not lute, guitar, nor violin), basically a tiny harp with no finger board.
It was actually a great example because the guitar is played by dividing a string into ratios. He was just wrong. An octave cuts the string into halves and a fifth is to cut it into thirds. What he showed was a fifth.
The first and simplest example on the is guitar wrong. 2/1 means the full length of the string in relation to half the string, which is at the 12th fret and will sound an octave or octave harmonic higher, the resulting pitch being double the frequency of the open string. For example, if your open A string on the guitar is A2 at 110 cycles per second, playing the same string at the 12th fret halves the string and doubles the frequency, giving you A3 at 220 cps. 220/110 reduces to 2/1 = an octave.
The third was not dissonant. It's that we stacked perfect fifths to generate all notes (Pythagorean tuning) in which the approximate thirds sounded like crap. Meantone tempered the fifth to get better thirds. Twelve tone equal temperament distorts everything so that all notes are acceptably bad, which is what we generally use today.
While you correctly state that the interval of 5:3 is a major sixth, your chart shows it as a minor 3rd, which is the inversion of the major sixth. 1:1 (unison), 2:1 (octave), 5:3 (major sixth), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third)
In twelve tone equal temperament, all intervals (shapes on a guitar) are equally bad. The tonic changes (bass frequency), but otherwise, all the math is the same. Any other tuning system is more complicated, for better or worse, often better.
0:17 "Pitch and rhythm are basically the same thing" Can you clarify that please. Last I heard the pitch describes highness and lowness of a note while rhythm is like the pattern of the sound.
i didn't understand the octave explanation on the guitar - 00:44 it shows the octave being on the 7th fret but the first octave is on the 12th fret ( one-half ( 1/2 ) of the strings length ) and the second octave is on the 5th fret ( half the length of the nut to the 12th fret or one-quarter ( 1/4 ) of the full string length ). the interval at the 7th fret is a perfect fifth ( one-third ( 1/3 ) of the strings length ). a place where one side is 2 times the length of the other side = one-third
t0msan perhaps an instrument with frets wasn’t a great example. But if you put your finger down on the 7th fret (where one side of the string is twice as long as the other) and then pluck the string on either side of the 7th fret, the side closer to the body will be an octave higher than the side closer to the nut. Again, it might not work precisely because of the fret being in the way, but it would definitely work on a fretless instrument.
When you give the example of the guitar and playing an octave, I believe you are incorrect in saying that the the octave lies one third of the way through the string. You can clearly see that the 12th fret (the octave) is half way between the bridge and the nut. This is a well put together vide, so thank you! I just want to share what I noticed
It's not your fault. He talks too quickly and doesn't break things down enough. This only makes sense if you already know and understand all the terms he is using.
well you know what?, keep on watching stuff that makes you feel dumb (because the thing you're watching is genius-level smarts)..... I think that over time you'll be driven to become smarter. I.e, it could be a good method to learn. If what I said doesn't make sense to you, ............yup, keep trying.
I agree, the picture isn't right. That ratio would make a perfect fifth. This is verified by counting frets, starting from the nut, as half steps up from the open string frequency.
Irrational numbers are essentially like infinitely complex ratios. Since the golden ratio is the most irrational number there is, would it make up the most dissonant interval there is?
omg you just did 440 x 3/2 which is what I just did having paused the video. Also, assuming the speed of sound in air is 340 metres per second, (sound travels 340m in 1 second) there will be 440 waves in 1 second for A. So my one second is 340m long, theres been 440 waves of A in that 1 second length of 340m. 340metres divide by 440 waves is a wave 0.772m long. If E is 660 waver per second, 340 metres divide by 660 waves (in 1 second) gives a wave 0.515m long. So the A wave is 0.772m long and the E wave is 0.515m long. So you say this 5th interval is ratio o 3 to 2. I assume you mean there are 3 waves of E for 2 waves of A. 3 of 0.515m is 1.545m long. 2 of 0.772m is 1.545m. So it checks out [never feel I understand this]. You can fit 3 waves of E in the length taken up by 2 waves of A. 3 Es is 1.545m long, 2 As is 1.545 m long.
57 seconds in and that graph overlaying the guitar about the 2:1 ratio is just such a poor way to intuitively introduce the idea of the octave... You should instead put a hold in the middle of the string, which doubles the resonant frequency of a free string.
How are some ways this information can be helpful to, lets say, the process of composing? I found everything you said enticing and informative. You even made me want to get better at math (you don't understand. math is my enemy).
I'm trying to find out scientifically/mathematically what determines the root note of a cord can anyone help me understand this? Like is it some sort of derivative of the frequency or a fraction of the frequency or any other summation from the frequencies put together?
@@twominutemusictheory I did in fact watch that it’s what led me here. It was a very good video and I learned a lot. But I’m still wondering if there’s some sort of mathematical way to understanding route notes. As in like an algebraic equation. Like word the frequencies add up or divide or multiply upon each other and the summation would be some sort of derivative or factor of the route note or something??
While the picture is incorrect showing the octave, which would be a string being divided in half? I must say, noticing what he DOES have - where a string is divided into 2 sections where one section is 2 times the distance of the other? I never looked at a third this way. So - I learned something unintended here ! heh REAL nice videos though...
Perhaps I was unclear in my explanation, and perhaps the guitar fretboard wasn't the best example. If you take a string and divide it so that one portion is twice as long as the other portion, the twice as long side is an octave lower than the shorter side. So if you take a fretless guitar, for example, and press down on where the 5th fret would be, and pluck on the right side of your finger and then the left of your finger, that interval will be an octave. By comparison, if you do the same thing on the 12th fret, either side of your finger will be a unison.
@@twominutemusictheory Plucking on both sides of the 12th fret gives you the same string length for both sides of your finger position. That is why it will be a unison because your ratio in that case is 1/1 = unison.
