Activities

So, what is frequency? Frequency is simply how many times a sound wave repeats itself. The single wave that makes up the sound wave is called an oscillation. The lower the frequency, the fewer oscillations we see, so changing the frequency changes the pattern. Lower frequencies tend to be deeper sounds and are low pitched, while high frequencies are squeaky and are high pitched.

Frequency is measured in hertz, which is how many vibrations per second. It also measures different types of radiation, light and soundwaves.

• If we hear a frequency of 125 Hz, is that more or less vibration than a frequency of 250 Hz?
• What do you think are the highest and lowest frequencies a person can hear?
  
• Test your frequency range and learn more about frequency and sound HERE

A musical note’s pitch is determined by its frequency, measured in Hertz. A tuning fork tuned to 440 Hz vibrates 440 times per second and produce the note A. When a musical instrument plays a note, it produces not only one frequency, but also a series of harmonics that are whole-number multiples of that frequency.

• For example, the first harmonic is twice the frequency. What is the first harmonic of the note A in Hz?
  

Pythagoras was an important Greek philosopher and mathematician who lived from 570-495 BC. The real Pythagoras is obscured in the mists of time but he is credited with discovering that musical notes could be expressed in mathematical form and his famous Pythagoras’ Theorem giving the important relationship for right angled triangles.

Among the legends of Pythagoras there is one about his discovery of harmonics and that musical notes could be expressed in mathematical form. He once happened to walk past a blacksmith where some apprentices were at work. Listening to them striking an anvil he realised that different hammers created different notes. Notes on a stringed instrument depend on the length of the string and Pythagoras realised that certain sounds are considered pleasing together when the lengths of the strings are in particular ratios such as 2:1, 3:2 and 4:3. This led many to believe that the universe operated by mathematical laws.

• If a string measures 21 cm, how long should the second string be to respect the 3:2 ratio? (The luthier doesn’t have any longer strings available.)

The beat is basically the unit of time in a piece of music. It’s a steady rhythm that underpins melodies, giving the music its pulse. Just like a text is organised with words, sentences and paragraphs, a piece of music is structured with beats, bars and phrases. Without this structure, reading music is like reading a text without punctuation or paragraphs. It’ll be very difficult to read and understand the text.

The written music always starts with the time signature. The most common is 4/4. The top number shows you how many beats are in a bar. The bottom number shows which note represents a single beat. So, 4/4 means there are four beats in a bar, and a semibreve* (whole note or the longest note) last 4 beats. A time signature of 2/2 means there are two beats in a bar, with the semibreve* representing 2 beats. When musicians say one, two, three, four, they’re basically just announcing the time signature they’re using, at least the number of beats.

• What does the time signature ¾ mean?
• And 6/8?

* the name of the notes and their relationship is explained in the next section ‘Take Notes!’

If the time signature* is 4/4, the longest note (the semibreve) lasts 4 beats. So the next notes are divided by 2. Then:

• A semibreve is an empty circle and lasts four beats. [whole note]
• A minim has a stem in addition and lasts two beats. [half note]
• A crotchet becomes black and lasts one beat. [quarter note]
• A quaver has a flag in addition and lasts half a beat. [eighth note]
• A semiquaver has another flag and lasts quarter a beat. [sixteenth note]

– How many beats do the different notes last if the time signature is 8/8? or 2/2?

* Time signature is explained in the previous part ‘Why do musicians call out one, two, three, four?’

We know 5 different notes that are linked with the times 2 table. But now, if we want a note to last 3 beats we have to write at least 2 notes on the music sheet. To limit the number of notes, they created the dot. If you add a dot, the dotted note worth one-and-a-half.

If the time signature is 4/4*, a dotted crotchet (quarter note) lasts one and a half beat.

• How many beats does the dotted semibreve last in 4/4?
• Is the dotted crochet in 4/4 last longer or shorter than the dotted minim in 2/2?

* the time signature 4/4 means that the longest note (the semibreve) lasts 4 beats. Time signature is explained in the previous part ‘Why do musicians call out one, two, three, four?’

Singers and musicians can’t afford to be out of rhythm, so counting is a must. Music is a team effort. If one singer or musician forgets to count, they might speed up or slow down, which can lead to a messy sound. Counting helps everyone stay together and the sounds will fit perfectly. That way, the music flows nicely and sounds good to the audience. It’s like following a dance step: if you don’t keep the beat, the dance can get mixed up!

Somebody else is counting too. All the time. For everyone.

Conductors are always counting during a performance; they must keep time to make sure the orchestra can run smoothly. Of course, the musicians are counting too but it is up to the conductor to ensure that everyone is on time!

‘Counting remains vital. This is especially true when not playing and you are counting the empty bars until you have to begin again. Let us consider one legendary example. If you are the tuba player in the famous ‘Largo’ from Dvořák’s New World Symphony, you play for five bars, then count – very slowly – for 114 bars, play for another four bars, and you are done. As a conductor, I am always counting, if subconsciously, and it would be expected that I give the tuba player a decisive nod at this crucial moment. But as a player, you may not want to take that chance, so you methodically count the 114 bars of 4/4’, says Dr Ciarán Crilly of UCD’s School of Music – 2023.

• How many beats does the tuba player have to count before he can play again?
• Read more about music and maths in the UCD magazine HERE.

The tempo of a piece is basically just a way of measuring how quickly the beats occur. The tempo is usually measured in beats per minute (BPM). If a piece of music has a tempo of 120 BPM, that’s the average for a pop song. That means there are 120 beats in one minute. To get it down to seconds, just divide 120 by 60, which is 2 beats per second.

Dr. Jacob Jolij, a Dutch scientist, came up with an equation to measure how a song makes you feel. The Feel Good Index (FGI) is made up of three things: the total number of positive references in the lyrics, the song’s tempo in BPM* and its key. The higher a song’s FGI, the happier it is likely to make us feel. Simply put, happy lyrics, a fast 150 BPM tempo, and a major musical key all contribute to making music that we feel is full of positive emotion.

According to his FGI, Dr. Jacob Jolij classifies Queen’s ‘Don’t Stop Me Now’ as a great example of a feel-good song!

*BPM is explained in the previous part ‘Beat the speed’.

Space is thought to be completely silent because it’s a vacuum, but there are still particles that vibrate just like sound waves. This means we can capture some pretty eerie but magical sounds from our solar system. These sounds are way too low for us to hear, so we use sonification to magnify them, which helps us hear them much more clearly.

• Listen to sounds of space HERE.

Albert Einstein

We know Albert Einstein mostly for his scientific discoveries, but he was also a well accomplished pianist and violinist. His wife remembers him playing the piano for a few noted while jotting down his notes and he was known to be up on stage playing the violin with well-known artists of the time.

‘Einstein claimed that, had he not pursued science, he would have been a musician.’

Read more about Einstein and Music in this article on theconversation.com

 

 

Brian May – Queen

Brian May is a rock musician known for being in the famous band ‘Queen’, as well as being a rock star he is also an astrophysicist. He studied at Imperial College London and started his PhD when he was then whisked away into stardom in music. That didn’t stop him though, 33 years later in 2006 he went on to restart his PhD which he finished less than a year later. His PhD was based on a different type of rock, a grainy rock cloud called zodiacal dust.

Read more about Brian May and Science in this article on nationalgeographic.com

 

 

Johnny Buckland – Coldplay

 

Johnny Buckland is the lead guitarist in the band ‘Coldplay’, he also studied astronomy and mathematics at University College London where he met his fellow band member Chris Martin.

Koji Kondo – the main composer of the Legend of Zelda and J.S. Bach – the famous 18th century classical composer both used symmetry in their music! However, Bach loved to use maths and symmetry as much as possible in his work, whether it was in long phrases or even the whole piece.

BACH

J.S. Bach was fascinated by maths and symmetry. Bach often used techniques like inversion (horizontal mirror), retrograde (vertical mirror) and transposition (moving up or down in pitch) of themes to create symmetry in his work. He also made good use of mirroring and palindrome techniques in his compositions.

You can see how he handled melodies and harmonies in a symmetrical way, using a melodic line that mirrors itself in reverse or parallel. Bach also came up with pieces called canons, which have phrases and harmonic structures that are symmetrical and give them a sense of completeness and unity. The Crab Canon, which dates from 1747, is an 18-bar theme that is normally played, but then at the end it goes in reverse entirely. You can even play both versions at the same time! It’s a bit like the mathematical Möbius strip, which has two ends that connect so that the strip only has one side.

ZELDA

In ‘the Legend of Zelda’ series, particularly in ‘The Legend of Zelda: Ocarina of Time’, the Song of Time uses retrograde as a key musical element. You can play the song in reverse to unlock certain aspects of the game and move the story along.

In Zelda’s Lullaby, the main theme is played at different pitch levels, which creates a rich harmonic texture. Also, in the trailer for ‘Legend of Zelda: Skyward Sword’, Zelda’s Lullaby is hidden using retrograde on the whole song. Fans can discover it by playing the part of the trailer in reverse.

In ‘The Legend of Zelda: A Link to the Past’, the track Dark World takes the Overworld theme from the same game and plays it in a minor key with different instruments. This effectively inverting and transforming the original melody to convey a sense of darkness and foreboding.

When you think of string instruments like a violin, you imagine them to be perfectly symmetrical. Well, they are on the outside, but inside the violin is hiding a secret – it is completely asymmetrical! The top of the violin varies in thickness and contains a wooden support beam around the third right. This allows for amplification and more frequencies, which means more harmonics.

• Fabulous Frequencies
• A frequency of 125 Hz is going to vibrate less than a frequency of 250 Hz.
• On average, a person can hear between 20 Hz and 20,000 Hz.
• The first harmonic of the note A is 880 Hz.
• Pythagoras Rocks
• The luthier doesn’t have any longer string so the second string should be shorter than 21 cm. The second string should be 2/3 of 21 = 14 cm to respect the 3:2 ratio.
  
• Why do Musicians call out one, two, three, four
• The time signature ¾ means that there are three beats in a bar, and the semibreve lasts 4 beats.
• The time signature 6/8 means that there are six beats in a bar, and the semibreve lasts 8 beats.
• Take Notes
• If the time signature is 8/8, then a semibreve lasts 8 beats. Now we divide by two: a minim lasts 4 beats, a crotchet 2 beats, quaver is a beat and a semi quaver half a beat.
• If the time signature is 2/2, then a semibreve lasts 2 beats. Now we divide by two: a minim is a beat, a crotchet half a beat, quaver is a quarter of a beat and a semi quaver a eighth of a beat.
• Going a Note Further
• The dotted semibreve lasts 6 beats in 4/4?
• The dotted crotchet lasts 1,5 beats in 4/4 (semibreve = 4 so the crotchet = 1. 1 + 1/2 = 1,5). The dotted minim lasts 1,5 beats in 2/2 (semibreve = 2 so the minim = 1. 1 + 1/2 = 1,5). They are equal.
• Counting is essential
• The tuba player has to count 4 x 114 = 456 beats before he can play again!
• Beat the Speed
• ‘Zombie’ with its BPM of 80: 80 / 60 = 1,33 beats per second (you can also think of it as 4 beats in 3 seconds: 80/60 = 4/3)
• ‘Shake it Off’ with its BPM of 165: 165 / 60 = 2,75 beats per second. You need to click your fingers or clap almost 3 times in just one second. This rhythm is fast!