Archive for January, 2023

Monday Morning Math: More Math and Music

January 30, 2023

Good morning! This week we have a bit more info about music (thanks TwoPi!):   Last week I mentioned a “just fifth”, which is when the frequencies of two notes are in a 3:2 ratio: A is 440 Hz (on a piano at least) and the note E above that is at 660Hz.  These two notes are a perfect fifth, and the ratio 660:440 is a 3:2 ratio.

The frequencies in an octave, on the other hand, are in a 2:1 ratio:  The A above the one with 440 Hz has 880 Hz.  So the work that Jing Fang did, showing that 53 fifths was almost exactly 31 octaves, amounted to showing that \left(\frac32\right)^{53} (which is approximately 2,151,972,563.22) is very close to  2^{31} (which is 2,147,483,648).  If you want to avoid fractions, this means that 3^{53} is almost the same as 2^{84}.  They won’t be exactly the same, but it’s pretty close.

But let’s go a little more in depth!  The savvy reader (that would be you) might realize that we can’t ever have a power of 3 exactly equal to a power of 2, because they are different prime factors.  This means we’ll never have a power of 3/2 equal to a power of 2.  This is a fundamental problem with piano tuning:  if you prioritize fifths then the octaves don’t exactly match up, and if you prioritize the octaves then the fifths don’t exactly match up.  Piano tuning today prioritizes the octaves, and uses the fact that 12 fifths is approximately the same as 7 octaves.  That is, \left(\frac32\right)^{12} (which is approximately 129.7) is very close to 2^7, which is 128.  

Isn’t that cool?  If you want to hear some of it, here’s a 3-minute video of mathematician and musician Eugenia Cheng giving more detail and connecting it to Bach and to Category Theory:

And if you want even more music, here’s some music played on a Mobius strip, just for fun:

  (And there are more videos on the AMS page on Math and Music.)

Monday Morning Math: Happy New Year!

January 23, 2023

Good morning everyone! Happy 2023!    Today’s post is in honor of two new years.

The first is the new calendar year: 2023.  If you want to do something fun (which of course you do) then you can see if you can use exactly the digits 2, 0, 2 and 3  and different math configurations to write the numbers 1-100.  For example: 1=\frac22+0\cdot3.  Or  1=2\cdot0+3-2.  (There are examples all over the internet, so a quick search reveals many solutions should you wish.)

The other is the Lunar New Year, which began yesterday.  This is also known as the Spring Festival, and is observed around the world, including China, Indonesia, Japan, Malaysia, the Philippines, Singapore, South Korea, Taiwan, Thailand, and Vietnam: we are now in the Year of the Water Rabbit in many countries, and the Year of the Cat in Vietnam.

In honor of the New Year we’ll talk about the mathematician Jing Fang  京房.  He was born in China 2100 years ago (78 BCE, during the Han Dynasty).  He was a mathematician and, appropriately enough for this post, he described astronomy – the solar and lunar eclipses.

But these weren’t his only accomplishments.  He was also very good at making predictions using the Yijing, or I Ching. There are 64 hexagrams, each made up of 6 rows that have one long or two short marks in each row.  While this isn’t mathematics, it does lead to the math question to ponder: namely, can you explain why there are exactly 64 configurations?

Picture from Wikimedia showing a diagram that Gottfried Wilhelm Leibniz owned.

And his math-connection doesn’t end there either – he used mathematics to describe music theory, particularly that 53 fifths (technically “just fifths”, which may or may not be the same as a perfect fifth – but you can here one here) was almost exactly 31 octaves.  It took more than 1600 years for anyone (said anyone being Nicholas Mercator) to caculate the difference between the two more exactly than Jing Fang. 

Math Moons and Music – a good way to start the year.

Sources: Wikipedia and…just that, because the few other sources I found had the same information.