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Frequency is a measure of vibration or oscillation, related to how quickly an object or a signal moves backwards and forwards between two positions. Sound is itself another type of vibration, caused as air molecules vibrate and collide, and pass on energy. So vibrating objects – such as musical instruments – cause sound vibrations that our ears can easily interpret. The measurement unit for vibration frequency is known as Hertz or Hz. So, if a drum is tuned to 100 Hz, its drumhead will vibrate up and down 100 times in a single second, similar applies to guitar strings and glockenspiel bars. Conveniently, some frequencies and frequency relationships are much more ‘musical’ than others, which provides a basis for music as we know it, and the tuning of musical instruments.

When a piano string vibrates, it vibrates at a very specific frequency related to its length and tension, and by changing the tension we can choose exactly what frequency each string vibrates at – the same principle applies to all musical instruments, including drums. So, every note on the piano keyboard has a corresponding frequency. For example, 98 Hz is note G2 on the piano, 110 Hz is A2 on the musical scale, C3 is at 130.8 Hz. Figure 3.2 shows every note on the piano within 50 – 400 Hz, giving the associated frequency value for each musical note.

Music is all about the organisation of different sounds and frequencies to make something pleasant, arranged, composed and interesting for our ears. Every musical note has a fundamental frequency (or pitch), but, in reality, it’s virtually impossible to create a single frequency at any one time. A single piano note is made up of different related frequencies all occurring at the same time, many of these being ‘harmonics’ which we’ll describe in more detail in the next section. If you hit three, five or eight piano notes at the same time, then you are generating perhaps hundreds of frequencies all at once. And when you scale this up to a full band or orchestra, we see that thousands of different frequencies are being created at any single moment, and if these frequencies are controlled and organised by skilled and knowledgeable musicians, the resultant sounds can create incredible and fascinating experiences for us to hear. For this reason, if analyzing music with a frequency chart – known as a frequency spectrum – it’s possible to identify which instruments contribute most to different frequency ranges. As humans, we can only hear sound vibrations between 20 Hz and 20,000 Hz, but that’s enough to contain all of the different sounds in the most common music forms. We generally call frequencies down in the 20-200 Hz range ‘bass frequencies’, those up in around the 4,000-20,000 Hz range ‘treble frequencies’ and those in between (200-4,000 Hz) ‘midrange frequencies’. The diagram below shows a frequency spectrum of a musical piece and some indication of which instruments generate their most powerful frequencies in which areas.

Note that each instrument also generates harmonics, which also cause additional frequencies to be evident in the frequency range above the fundamental pitch range of each instrument.

There are a number of mathematical associations between musical pitch and frequency, which define the tuning and arrangement of most western music. Understanding these relationships can be helpful when learning to perform as a musician and when recording, mixing or producing music too.

One of the simplest mathematical relationships to identify is that octave notes are always seen when the frequency doubles. Looking back at the piano chart image, we see that the C2 note has a frequency of 65.4 Hz and this frequency doubles at the next C note (C3) which is at 130.8 Hz (65.4 * 2 = 130.8). The frequency doubles again at C4 which is 261.6 Hz, and you can see that all octaves of all notes occur when the frequency doubles. The diagram below shows all the A note octaves from A1 to A6.

It’s an incredible phenomenon of physics, but strings and bars vibrate with perfect harmonic overtones. This means that the main fundamental frequency of a string or bar is joined by many other frequencies that are harmonically related, which results in a beautiful rich tone that is much more ‘musical’ than a single frequency all on its own. The additional overtones are, by a chance of physics, at perfect multiples of the fundamental frequency, so a string tuned to A at 110 Hz also vibrates at harmonics of 220 Hz, 330 Hz, 440 Hz and so on, as shown in the image below – the same principle applies to glockenspiel bars too. This acoustics fact is what makes string and tuned percussion instruments so musical sounding. In fact, the same principle applies to the vibration frequencies on woodwind and brass instruments too!

You may ask yourself how are musical frequencies calculated or decided? We’ll this is all down to mathematics and musical tuning systems, which differ around the world. The most common musical tuning system is the ‘equal temperament’ system which is used mostly in classical and western pop music. Equal temperament’ defines that there are 12 notes in each octave, as we can see on the standard keyboard above.

First with the equal temperament scale, we need to set a ‘standard relative pitch’ which defines the frequency that a particular note will be chosen to have. Usually, we say that note A4 has a frequency of exactly 440.0 Hz, which allows all other musical frequencies to be defined relative to that particular note. Some composers choose to move this datum pitch a little and ask their musicians to tune their instruments with A4 set to, say, 432 Hz, but most popular and classical music stick to the A4=440 Hz norm.

Once the standard relative pitch has been set, we now need to calculate the frequencies of all 12 notes within an octave, giving each and ‘equal’ spacing. However, because an octave relates to a doubling of the frequency, it’s not possible to use a linear scale and simply divide the octave band by 12 to find the frequencies of each note. Instead, we need to use a logarithmic scale to define the frequency of each note within an octave band. Some simple maths tells us that the interval between two notes (or semitones) needs to be a number which wen multiplied by itself 12 times gives a perfect octave (i.e. 2, or double). This therefore gives us the following equation to find the frequency multiplier for semitone intervals:

And we can now use this value to calculate all of the frequency values for the semitones in a given octave range, as shown in the diagram below for the range A4 – A5. Note that each time we go up a semitone, the frequency increased by 1.0595 times, until we reach the octave which has an overall increase of 2 (since 1.0595 multiplied by itself 12 times equals 2!)

Musical intervals define the relationships between frequencies in a musical scale. Looking at this on the piano keyboard, we see from C to C there are 12 semitones (i.e. 12 piano keys), but a major scale has only 8 notes, those being C-D-E-F-G-A-B-C for the scale of C major. The figure below shows the frequency differences between each of these notes, which are all musically and mathematically related to the root or first note in the scale. For example we call the third note in the major scale the major 3rd and the 5th is the fifth note in the scale, which is G in the scale of C major.

By looking at the frequency ratios (i.e. mathematically dividing one frequency by the other), we can see the multipliers for each note in the scale. For example, we see that the 5th of C3 (130.8 Hz) is G3 at 196.0 Hz, and some simple math shows that 196.0 / 130.8 gives a frequency ratio of 1.50. Similarly, the frequency ratio of the major 3rd is 164.8 / 130.8 = 1.26 and the frequency ratio of an octave is exactly 2. These particular frequency ratios are what define musical intervals of a particular scale. The full list of major musical intervals and their associated frequency ratios is given in the table below.

With this understanding of musical frequencies, it’s possible to tune drums to sound extremely musical, rich and interesting. There are lots of different areas where this knowledge is applicable, firstly for tuning the pitch of each drum in the drum kit, as discussed in detail our tutorial on pitch tuning. This can be a really useful exercise to make sure your drums are setup to best suit the song you are playing and to give you a reference point each time you go to retune or change drumheads.

Secondly, we use the concept of musical intervals to help with tuning the resonant drumhead. We have a tutorial on resonant drumhead tuning too, and we generally recommend trying a musical fifth relationship between the fundamental and overtone frequencies of your drums, resulting in a rich warm sound.

Thirdly, it’s useful to use the concept of musical pitch and intervals to setup the whole drumkit and to ensure each drum gives a unique and interesting tone to the sound of the whole kit. Choosing musical intervals for your toms can be a complex but rewarding task, and we have a detailed tutorial on tuning the kit for specific styles and genres too. Legendary drummer Terry Bozzio (Frank Zappa, Herbie Hancock, Korn) has pioneered approaches towards musical tuning of drums, with his ‘Big Kit’ including 26 toms each tuned to a different musical pitch. Bozzio’s kit incorporates fourteen 8” x 3” piccolo toms tuned chromatically from C#5 (with a fundamental frequency of 554.4 Hz) down to C4, a B3 tuned 12” snare, and a range of other 8” to 14” diameter drums covering lower pitches from A3 down to E2 at 82.4 Hz. You can find out more info on this kit setup and hear examples at Terry Bozzio’s website here.

So, we’ve covered all the essential aspects of musical frequencies and discussed how these apply to music performance, instrument sound and, more specifically, to drum tuning too. If you want to tune your drums consistently and to musical pitches, and to achieve a great rich tone from each drum, and a musical setup of intervals between each drum in the kit, do check out our tutorials on these subjects here. The iDrumTune Pro app is specifically designed to help with learning about accurate drum tuning and ensuring you get the best possible sound out of your kit, every time you play and every time you retune or change drumheads – so equipped with some powerful musical acoustics knowledge and a precision tuning tool like iDrumTune Pro, you’re equipped to become a master of drum tuning!

If you want to know more about the underlying science of drumheads and drum sound, and learn more creative approaches to drum sound and drum tuning, check out the free iDrumTune ‘Drum Sound and Drum Tuning’ course at www.idrumtune.com/learn

Author Professor Rob Toulson is an established musician, sound engineer and music producer who works across a number of different music genres. He is also an expert in musical acoustics and inventor of the iDrumTune Pro mobile app, which can be downloaded from the App Store links below: