Love your math & music posts. I’m a math guy who loves music (your fan, too!) but actually knows little and has even less personal ability. Here’s a mathy observation I find interesting, though, dealing with string length. While I can barely discern when something is off-key, I’m told that for professional musicians with ears far, far finer than mine, there are issues that can arise when it comes to playing certain instruments together. A violinist has a continuous set of string length options and can thus adapt to any key, but a piano is tuned to a discrete pre-determined set of string lengths. Under normal circumstances, the piano would be tuned to the well-tempered scale, allowing the pianist to change keys readily. But that scale only approximates the true string lengths required – close enough for most of us, but for high-end musicians a piano must be tuned to a specific key to make the two instrumentalists happy together. The reason is because the actual string lengths needed musically are all rational numbers (fractions like ½, 2/3, 5/8, ¾). Changing keys is like shortening all the strings; for example, make them all ¾ as long (what happens when one moves the capo on a guitar), and ¾ of 5/8 would require a 15/32 length for the new key. Not all the possible fractions of fractions show up in the 12-tone scale, a problem that required instruments to be built for each key back in the day before the 16th Century adoption of the well-tempered scale. Mathematically, that’s a geometric sequence with a common ratio = 12th root of 2 (denoted here as r), making lengths be 1, r, r^2, r^3, …, r^12 = 2. Those are all irrational numbers, so close to the actual fractions needed that only the most discerning ear can hear the difference. It allows instruments to play in any key, because when one shortens all the strings by one of the r-factors all the new lengths are still powers of r, so the notes are all available. That’s clever, and cool! But for us math nerds, the coolest part of this is that it’s the only place where people use irrational numbers to approximate rationals! Almost exclusively it’s the other way around, as in the rational 3.14 approximation for the irrational π. Not here. Cool, indeed! (And please excuse any musical ignorance I put on display…) -- Dave
Hey Jonathan, hoping the show went well and great that you turned that 6 right side up in time😁 I was really enjoying the numbers game you were playing with A-440 and it's relationships, half's and doubles and trebles and on up the line when a tiny explosion went off in the back of my foggy old mind!! It suddenly occurred to me that even though I had spent most of my adult life as a surveyor and a guitar picker, measuring and laying out distances to nth degrees and in my leisure time being totally focused on getting that A-440 tuned in just right. ( from pitch pipes and tuning forks on up to my electronic Snark...I had failed to make the simple connection that 440 yds. is a quarter mile (my favorite race to run in high school track) 440 x 4 = 5280 ft = 1 mile, 8 furlongs = 660 ft. or 10 chains, 1 chain = 66 ft. 10 square chains = 1 acre... and on and on we go...music and arbitrary units of quantifying distance suddenly merging into minor chords. Is this why I shiver when hearing a well voiced minor progression and tend write dark songs about the miles I've measured and the miles I've traveled?? As always Jonathan, we love to hear these stories and thoughts of yours, keep them coming...Gordon and Suzanne, Hornby Island. B.C.
This made me think of Bobby McFerrin's wonderful demonstration of pentatonics, showing that the audience could automatically sing notes they hadn't yet heard if they were placed in a sequence. The mental maths behind predictions about such a series seem innate. I'm not sure if we're carrying out some sort of frequency conversion, but if we are it suggests that pitch recognition might not just be a matter of training and practice. He observes that this demonstration works all over the world, even in countries with microtonal systems.
I think this is my favorite so far. Very you, JB 😁
Thank you, JG.
Love your math & music posts. I’m a math guy who loves music (your fan, too!) but actually knows little and has even less personal ability. Here’s a mathy observation I find interesting, though, dealing with string length. While I can barely discern when something is off-key, I’m told that for professional musicians with ears far, far finer than mine, there are issues that can arise when it comes to playing certain instruments together. A violinist has a continuous set of string length options and can thus adapt to any key, but a piano is tuned to a discrete pre-determined set of string lengths. Under normal circumstances, the piano would be tuned to the well-tempered scale, allowing the pianist to change keys readily. But that scale only approximates the true string lengths required – close enough for most of us, but for high-end musicians a piano must be tuned to a specific key to make the two instrumentalists happy together. The reason is because the actual string lengths needed musically are all rational numbers (fractions like ½, 2/3, 5/8, ¾). Changing keys is like shortening all the strings; for example, make them all ¾ as long (what happens when one moves the capo on a guitar), and ¾ of 5/8 would require a 15/32 length for the new key. Not all the possible fractions of fractions show up in the 12-tone scale, a problem that required instruments to be built for each key back in the day before the 16th Century adoption of the well-tempered scale. Mathematically, that’s a geometric sequence with a common ratio = 12th root of 2 (denoted here as r), making lengths be 1, r, r^2, r^3, …, r^12 = 2. Those are all irrational numbers, so close to the actual fractions needed that only the most discerning ear can hear the difference. It allows instruments to play in any key, because when one shortens all the strings by one of the r-factors all the new lengths are still powers of r, so the notes are all available. That’s clever, and cool! But for us math nerds, the coolest part of this is that it’s the only place where people use irrational numbers to approximate rationals! Almost exclusively it’s the other way around, as in the rational 3.14 approximation for the irrational π. Not here. Cool, indeed! (And please excuse any musical ignorance I put on display…) -- Dave
This is all fascinating stuff! Brass players used to have different parts they’d swap out for different keys.
Hey Jonathan, hoping the show went well and great that you turned that 6 right side up in time😁 I was really enjoying the numbers game you were playing with A-440 and it's relationships, half's and doubles and trebles and on up the line when a tiny explosion went off in the back of my foggy old mind!! It suddenly occurred to me that even though I had spent most of my adult life as a surveyor and a guitar picker, measuring and laying out distances to nth degrees and in my leisure time being totally focused on getting that A-440 tuned in just right. ( from pitch pipes and tuning forks on up to my electronic Snark...I had failed to make the simple connection that 440 yds. is a quarter mile (my favorite race to run in high school track) 440 x 4 = 5280 ft = 1 mile, 8 furlongs = 660 ft. or 10 chains, 1 chain = 66 ft. 10 square chains = 1 acre... and on and on we go...music and arbitrary units of quantifying distance suddenly merging into minor chords. Is this why I shiver when hearing a well voiced minor progression and tend write dark songs about the miles I've measured and the miles I've traveled?? As always Jonathan, we love to hear these stories and thoughts of yours, keep them coming...Gordon and Suzanne, Hornby Island. B.C.
Wow, Gordon! What a fantastic connection. Math was almost entirely geometrical until the invention of the printing press.
Thank you. Always a pleasure to hear from you.
This made me think of Bobby McFerrin's wonderful demonstration of pentatonics, showing that the audience could automatically sing notes they hadn't yet heard if they were placed in a sequence. The mental maths behind predictions about such a series seem innate. I'm not sure if we're carrying out some sort of frequency conversion, but if we are it suggests that pitch recognition might not just be a matter of training and practice. He observes that this demonstration works all over the world, even in countries with microtonal systems.
I love that demonstration too. The human relationship with sound is fascinating.