To illustrate what is happening, every interval in Pythagorean tuning is generated by the perfect octave (2) and the perfect fifth. In other words, every intervallic ratio is of the form 2^m·(3/2)^n for integers m, n. The problem being highlighted is that there exist no integers p, q such that (3/2)^p = 2^q. As such, there cannot be an unbroken circle of fifths using Pythagorean tuning, mathematically speaking. In particular, if you try to form a duodecatonic scale, you have an unbroken circle of fifths with a gap of size (3/2)^12/2^7 = 3^12/2^19, and this interval is known as the Pythagorean comma, which is an audible interval. However, in practice, the situation is more complicated, and Pythagorean tuning can still be used if you use a different scale. Yes, I did say that there exist no integers p, q such that (3/2)^p = 2^q or (3/2)^p/2^q = 1. However, there are pairs of integers that make (3/2)^p/2^q extremely close to 1, so the equation above is almost true. The Pythagorean comma 3^12/2^19 is very close to 1, but it is still audible and distinguishable from 1, which is the reason why this is considered a problem. There are commas that are so close to 1, though, that they are not audible, which means that they cannot be distinguished from 1 by the human ear in most circumstances, and so we can actually ignore these commas. For example, if you use a quintavigesimaltritonic scale, which is to say, a scale with 53 divisions of the octave, then we can generate said scale using a circle of fifths that, while not mathematically unbroken, is indistinguishable from an unbroken, perfect circle of fifths. The gap in this circle of fifths is equal to (3/2)^53/2^31 = 3^53/2^84, and this interval is called Mercator's comma, a comma much tinier than the Pythagorean comma, and unlike the Pythagorean comma, Mercator's comma is not audible. This fixes the problem with Pythagorean tuning. Therefore, in this circle of fifths, the solution is to not treat 3^12/2^19 as its own note in the scale. This is fascinating. This produces a scale where you start with 1, and end with 3^53/2^84, which is indistinguishable from 1, rather than 3^12/2^19, which is distinguishable from 1. Furthermore, this scale is indistinguishable from 53-equal temperament. This type of tuning and scale are used in Ottoman music, Turkish music, and Chinese music.
Record at The Bins with Mike Britt yes but this tells you why 2^7 doesn't equal (3/2)^12 Because in the equation (3/2)^x=2 x is irrational but I guess people think x is close enough to 12/7. I don't
Hey, I always thought yards did add up to miles. Maybe I was cheated running the mile race when I was younger. Yards do not add up to Kilo-Meters though.
Why, Pythagoras would be spiraling in his grave 😮. Too bad he wasn’t the first to discover it, though. Or musical intervals. Or the Pythagorean Theorem, for that matter. He was only a millennia or two late to the game. They call that a “Pythagorean Country Mile”. Maybe thats why the yard and the mile don't add up!
Why, Pythagoras would be spiraling in his grave 😮. Too bad he wasn’t the first to discover his eponymous comma, though. Or musical intervals. Or the Pythagorean Theorem, for that matter. Or the irrationality of the diagonal. He was only a millennia or two late to the game. They call that a “Pythagorean Country Mile”. Maybe thats why the yard and the mile don't add up!
@@Emrebenkov does that change anything? Musical harmony humanity used for the last centuries have to do with all the integer ratios between pitches, that's what music is that we hear.
To illustrate what is happening, every interval in Pythagorean tuning is generated by the perfect octave (2) and the perfect fifth. In other words, every intervallic ratio is of the form 2^m·(3/2)^n for integers m, n. The problem being highlighted is that there exist no integers p, q such that (3/2)^p = 2^q. As such, there cannot be an unbroken circle of fifths using Pythagorean tuning, mathematically speaking. In particular, if you try to form a duodecatonic scale, you have an unbroken circle of fifths with a gap of size (3/2)^12/2^7 = 3^12/2^19, and this interval is known as the Pythagorean comma, which is an audible interval.
However, in practice, the situation is more complicated, and Pythagorean tuning can still be used if you use a different scale. Yes, I did say that there exist no integers p, q such that (3/2)^p = 2^q or (3/2)^p/2^q = 1. However, there are pairs of integers that make (3/2)^p/2^q extremely close to 1, so the equation above is almost true. The Pythagorean comma 3^12/2^19 is very close to 1, but it is still audible and distinguishable from 1, which is the reason why this is considered a problem. There are commas that are so close to 1, though, that they are not audible, which means that they cannot be distinguished from 1 by the human ear in most circumstances, and so we can actually ignore these commas. For example, if you use a quintavigesimaltritonic scale, which is to say, a scale with 53 divisions of the octave, then we can generate said scale using a circle of fifths that, while not mathematically unbroken, is indistinguishable from an unbroken, perfect circle of fifths. The gap in this circle of fifths is equal to (3/2)^53/2^31 = 3^53/2^84, and this interval is called Mercator's comma, a comma much tinier than the Pythagorean comma, and unlike the Pythagorean comma, Mercator's comma is not audible. This fixes the problem with Pythagorean tuning. Therefore, in this circle of fifths, the solution is to not treat 3^12/2^19 as its own note in the scale. This is fascinating. This produces a scale where you start with 1, and end with 3^53/2^84, which is indistinguishable from 1, rather than 3^12/2^19, which is distinguishable from 1. Furthermore, this scale is indistinguishable from 53-equal temperament. This type of tuning and scale are used in Ottoman music, Turkish music, and Chinese music.
Man wrote an entire essay and got no likes for 8 months
thanks for elaborating :)
This is such a great animation. Good Work!
this is as exciting as learning why yards don't add up with miles
Record at The Bins with Mike Britt yes but this tells you why 2^7 doesn't equal (3/2)^12
Because in the equation
(3/2)^x=2
x is irrational
but I guess people think x is close enough to 12/7. I don't
Hey, I always thought yards did add up to miles. Maybe I was cheated running the mile race when I was younger. Yards do not add up to Kilo-Meters though.
Perfect fifths do not loop into octaves thanks to the fact that 3 and 2 are prime numbers, and 3 and 2 are coprime.
Why, Pythagoras would be spiraling in his grave 😮. Too bad he wasn’t the first to discover it, though. Or musical intervals. Or the Pythagorean Theorem, for that matter. He was only a millennia or two late to the game. They call that a “Pythagorean Country Mile”. Maybe thats why the yard and the mile don't add up!
Why, Pythagoras would be spiraling in his grave 😮. Too bad he wasn’t the first to discover his eponymous comma, though. Or musical intervals. Or the Pythagorean Theorem, for that matter. Or the irrationality of the diagonal. He was only a millennia or two late to the game. They call that a “Pythagorean Country Mile”. Maybe thats why the yard and the mile don't add up!
This kind of spiral makes sense following the awkward deviation of Pythagorean tuning. Perfecto !! ... in an imperfect way.
Excellent. Thank you.
1:36
The intonation at the start of each note is poor.
also proves nothing
FacePalmProductions also stfu
Your comment proves nothing. It is funny that you don't understand that powers of odd numbers will never be even unless the power is irrational.
@@rohmo He maybe thought that this video is about math, but not about music
@@Emrebenkov does that change anything? Musical harmony humanity used for the last centuries have to do with all the integer ratios between pitches, that's what music is that we hear.
This isn't about proof! It's the demonstration of a natural phenomenon.