Beat It

Last weekend was Western Region Oireachtas, or Irish Dance regionals. In my dance school, our tradition it you point on the 4th beat of the music, raise onto your toes on the 8th beat, and start dancing on the 1st. This means that for team dancing, everyone moves at the same time. When practicing our dance, and even on stage, I started wondering how every person knows when the same beat is. Chances are, if you start an Irish dance song on a random count, I will be able to sing the correct next beats after only hearing that first bar of music. Why is that?

Well, to answer this, I first had to break down sound to what it is: vibrations that propegate through some media. While most people think of sound as a puresly sinodal graphy, it is actually caused by bumpin of air molecules together, like the image below:

Luckily, this can be simplified into sine graphs with amplitudes and frequencies to properly answer the question of musical beats. Amplitude determines the volume of a sound. While this is useful in distinguishing bars in Irish Dance as the song naturally rises and falls in volume, it is only a partial explanation. The real determiner lies in frequency. Frequency, measured in Hertz, is how many cycles a wave makes per second. 1 Hertz would mean that the wave goes from amplitude to amplitude in one second. Higher pitch noises have larger frequencies than lower ones because the wave is more “squished“ in comparison.

One simple part to the role of frequencies in beats is the overlap of multiple frequencies. Each musical note (A, B, C, D…) has a different corresponding frequency. When two notes are played at the same time, their corresponding sine waves “overlap.” Look at the picture below: the top wave has a large frequency and the bottom one has a smaller one. Note, that the factor between the two frequencies is not an integer number. This will be explained later:

There is a “net sum” wave that is the amplitudes of each wave added together. This is where musical beats arrive from. Your brain can separate where the waves experience constructive interference (the amplitudes add) or deconstructive interference (the amplitudes are net 0). This is seen in the “bubbles“ between points of 0 amplitude, also noted in the image. With increasingly more variances between waves or a greater number of waves, these beats become more advanced. There might be one spot with total constructive interference, one with total deconstructive, but three other places with a combination of constructive between two waves and deconstructive with others. This would correspond with weaker beats, or in music, beats other than the first or last one.

There is a little more subtle mathematical occurrence happening, however. If you have heard of people with perfect pitch, they can determine the note from any tone. This means that every C note must have the same frequency. For notes that are an active higher, it turns out that the frequency is double that of the original note. But what about all the notes in between, how do they relate?

In total, there are 12 different notes in a full scale, including sharps. It turns out that the difference in eaches frequency from the other is 1.0595, or the 12th root of 2. By nature of frequency interference, one can simplify the resulting frequency of playing two notes together by dividing one frequency by another. Lower value fractions, it turns out, are considered more pleasant in music, and are therefore found in more music. There is also a concept of harmonics important to these sound frequencies. Harmonics are simply integer multipliers/dividers of one frequency. Using the ratio above, the first harmonic of a note (C for example) is just that note, but one octave higher. But because the difference in frequencies is reliant on a square root, the next harmonic is not that same note. It is instead a note with 2/3 the frequency or 3/2 the wavelength. For the note C, its first harmonics are C, C, G.

This relationship between notes and their harmonics is also utilized in music to create beats. For some notes, their harmonics are the harmonics of other notes. When played together, there will be no distinguishable beat. Each of these factors together and the frequency of air molecules “bumping“ against the human ear, helps us determine beats and bars of music, and all start dancing at the same time.

References:

https://letstalkscience.ca/educational-resources/backgrounders/what-sound-and-how-do-we-hear-it

https://www.simplifyingtheory.com/math-in-music/