The brain is a wonderful thing, isn't it? How do we manage to make sense out of the passage below?
Aoccrnidg to an Elnisgh uniesvitry sutdy, the oredr of letetrs in a wrod dnose’t metatr, the olny thnig thta’s iopmrantt is that the frsit and lsat ltteer of eevry wrod is in the cerocrt ptoision. The rset can be jmbueld and one is slitl albe to raed the txet wiohtut dclftfuiiy.
I don't know about you, but I read through it quickly, without a hitch. No wonder we need computers to detect our spelling errors for us.
A very clever 18th century guy named Euler discovered an interesting mathematical relation that others have since called the most beautiful equation in mathematics.
The equation says, somewhat less beautiful without superscript:
e^(i𝜋) + 1 = 0 (or e raised to the power of i times 𝜋, plus one equals 0).
To explain the terms:
e is the base of the natural logarithm. By definition, e is the limit (as x approaches infinity) of (1+(1/x))^x. A special property of e is that the value of the function e^x at any point, x, is equal to the slope of the function at that point. People who know maths a lot better than I do can do cool things with this.
i is the imaginary unit, equal to the square root of -1. Neither this number nor any number multiplied by it is in the set of real numbers, but it is very useful nonetheless, especially in electrical engineering where it changes name to j.
𝜋 is a geometric constant, the ratio of a circle's circumference to its diameter. It is an irrational number, inexpressible as a ratio of two integers, and is therefore an unending, non-repeating decimal roughly equal to 3.14. It plays an important role in trigonometry.
The number 1 itself is special, since any other number multiplied or divided by 1 or raised to the power of 1 equals itself. It is the unit of identity for these operations.
The number 0 is also special, in that any number plus or minus 0 equals itself. It is the unit of identity for addition and subtraction. It has other special properties as well: any number raised to the power of 0 equals 1, and any number multiplied by 0 equals 0.
Relating all of these special numbers in one equation, using the operations of addition, multiplication and exponentiation, gives us hope that reality (or our particular way of looking at it) has an elegant simplicity beneath its chaotic surface.
The proof of this relationship is well beyond me, but you can find out more about it by researching “Euler's Identity”.
I love music. I love singing. I love playing the piano. I love listening to music. Certainly not most important, but also among my loves, is understanding the science behind music.
Among the most basic understandings for western music is how harmony works. Harmonics is the physics (and, by necessity, the maths) behind the sounds. If you pick a note, say C, and designate its frequency (the 'speed' with which its sound wave oscillates) as 1, then the C exactly one octave above that will have a frequency of 2. Moving up an octave is defined as doubling the frequency. This higher C is called the second harmonic of the lower C, which itself is called the fundamental.
(By the way, the norm is to define the A above middle C on a piano as 440 Hz and to build all of the other notes off of that, but we'll stick to a C-to-C scale, which more people are used to dealing with).
The third harmonic is the note whose frequency is three times that of the fundamental. That note is one octave plus a fifth above the original C. In this case, it is the G above the upper of the two Cs we've defined so far. To get the G that resides in the octave between our two Cs, we have to divide the frequency of the third harmonic by two to bring it down an octave.
Having built the two Cs and the G, we can move in both forward and backward directions in the 'circle of fifths' to build the rest of the scale. For instance, D is a fifth up from G. A is a fifth up from D, etc. Similarly, F is a fifth down from C. B♭ is a fifth down from F, etc.
The entire sequence of fifths (with C in the middle) is:
Since G♭ is the same as F#, the sequence becomes a circle. But wait, although we're used to G♭ and F# being the same note, that is only because most of us learn music with a piano or guitar - fixed pitch instruments that use the modern even-tempered scale, which we'll come to in a bit.
Actually, building the scale based on pure harmonics does NOT give us a G♭ equal to F#. Without boring you (perhaps I'm too late) with the steps that get us there, the notes and their relative frequencies end up working like this:
C - 1; D♭ - 256/243; D - 9/8; E♭ - 32/27; E - 81/64; F - 4/3
F# - 729/512; G♭ - 1024/729
G - 3/2; A♭ - 128/81; A - 27/16; B♭ - 16/9; B - 243/128; C - 2
You can use your calculator (!) to confirm that those F# and G♭ fractions are not equal.
Despite the dizzying array of fractions, the progression from the lower C to the higher one is relatively, if not perfectly, smooth. The percentage increase in frequency for any semi-tone move up the scale is either 5.35% (for C-D♭, D-E♭, E-F, F-G♭, F#-G, G-A♭, A-B♭ and B-C) or 6.79% (for D♭-D, E♭-E, F-F#, G♭-G, A♭-A and B♭-B).
Because the steps are not of uniform size, a song can sound quite different in different keys (i.e. depending on what note the song starts on). Different musical eras have used different approaches to get round this 'imperfection'. Bach would have written his pieces for a specific key because of the 'feel' that key had.
Now, back to the piano and guitar (and all other fixed pitch instruments) of today. For some time now, the music world has been using what is called an even-tempered scale. One in which the sizes of the steps from the lower C to the higher one are all the same. As it turns out, to double the frequency over the course of twelve uniform steps, each note needs to have a frequency 5.95% greater than the note below it. Unsurprisingly, this interval lies between the sizes of the two intervals (5.35% and 6.79%) in the 'pure' or 'perfect' harmonic scale.
There is a cost to this simplicity, but one only borne by the most sensitive of ears. None of the intervals / harmonies produced by the even-tempered scale is perfect. Each is slightly off what physics calls for. These days, nearly all of us are attuned to this minor blemish, but we should all recognise that the harmonies we hear, while undeniably beautiful, are sadly not perfect.
I'm curious. I like looking beneath and behind the obvious, also looking for what is between me and the obvious, obscuring or distorting my view.