VI HART: It's justįundamentally logarithmic. Listening to octaves, we feel like the differenceīetween this octave shouldn't be half as much as VI HART: More like, Iĭon't know, let's say 300. VI HART: C is- SAL KHAN: Well we'll call it x. Right? If this is C- SAL KHAN: Call it x. Well, we could- VI HART: Well I'll give you anĮxample I can give numbers to, which is maybe the differenceīetween this octave. This was maybe a badįrequency of D, but- SAL KHAN: No. VI HART: But if you lookĪt the actual frequencies, the distances are not the same. Here between here and here and here and there. Can I have the pen? SAL KHAN: Here you go. The logarithmic scale in so many things in real life. And so that's whyĪlmost the multiple matters more than the absoluteĭistance between the numbers. Whereas the differenceīetween 1 and 10 is huge. Because the differenceīetween 5 gazillion and 5 gazillion and 10 is- SAL KHAN: Is nothing. Make sense, usually, for how we think about things. Of our mathematics, we plot lines and stuff like that. That's what we're taught, and that's what most We want to say this is 1, and then maybe this We're taught is these linear scales, where Is that we, as humans, even though everything Think on a logarithmic scale is what it is. But the differenceīetween 1 and 2 is huge. VI HART: But when we're thinkingĪbout real-life things, well, the differenceīetween 9 and 10 isn't so big in any real When you're looking at it at the usual scale. VI HART: And theĭifference between 9 and 10 is the same distance The difference between 1 and 2, or the difference betweenĢ and 3, or 1 and 10, we think, 1 and 2. We're not so usedīetween 1,000 and a million. Why do we think of 1,000 asīeing much closer to a million than it is? And we do this,Īctually, all the time. What is this about? Why did I do that? VI HART: Yeah. You couldn't even see theĭifference in thousandths. You barely notice theĭifference between that and- VI HART: Yeah. Because you know there is aĬorrect answer to this problem. Reaction was to put 1,000 like right over here. VI HART: So you can thinkĪbout this logically. Pen now, and I'm going to ask, where is 1,000? SAL KHAN: Where is 1,000? Where is 1,000? I see. Going to start at one, and we're going to goĪll the way to a million. All right, can I borrow the pen? SAL KHAN: Yes. How we think about numbers and what is the mostĮveryday lives. However, the fact that our thinking is not quite like that becomes apparent when we're dealing with large numbers. As a result, we base all our mathematics on this. The reason you feel that you think linearly is because we naturally count objects as 1, 2, 3, 4, 5, 6, etc and since we think of each thing we count as distinct, we think of there being an equal distance between each number on the number line. A German physician by the name of Ernst Weber, who is considered one of the principle founders of experimental psychology was the first to notice this. When it comes to a certain force pressing up on our skin it's the same way. With each octave we actually have a doubling of the sound frequency, however, when we are listening to the notes being played one after another we think of them incrementing in equally spaced intervals because that's how we're taught to think of numbers for the purposes of counting and basic math. All humans naturally think of pitch in a logarithmic scale without realizing it.
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