Frequency, Pitch, Overtones, Harmonics and Timbre

One of the eureka moments I had studying music was when I began to understand the harmonic series and how in turn this relates to pitch, intervals, harmony, tuning, waveforms, frequency and latterly timbre.

The topic was broached by my piano teacher when I was trying to transcribe some Charles Mingus, and struggling with it. To get around the problem he introduced me to a program to slow the audio down. At this juncture it’s worth noting that we didn’t use Ableton Live or other DAWs that would have preserved the pitch information.

The program was called the Amazing Slow Downer and it worked much like a vinyl or tape would; if you slow something down the pitch drops, if you speed it up the pitch would increase.

We found (or rather I found, because Pete already knew) that reducing the speed by half caused the pitch to drop by one octave, and doubling the speed caused the pitch to increase by one octave; this was the first example I’d seen of pitch relating to speed.

In the article I want to discuss a few different topics that are all interconnected in one way or another. First frequency and its relation to pitch, our tuning system(s) and how we came by them, what overtones are and how they differ from harmonics, what is timbre and (hopefully) what the point of knowing all this stuff.

Frequency

I’m going to leave that story there and cover our next steps after covering a few basics. Depending on how familiar you are with looking at waveform displays or using such things as LFOs or filters, you may already have come across the term frequency in music technology. We think of this word as the occurrence or the amount of times something happens, and in short they mean the same thing.

Sound is vibrations or fluctuations in the air pressure that are perceived by a listener (so who knows about the tree that falls in an empty forest?). Any sort of acoustic phenomenon causes air molecules to be pushed up and down. This creates a (sometimes) audible event which our ears can detect and infer certain information from, like distance, pitch and relative location.

The number of times these air molecules complete a cycle determines its frequency. This means that if something is vibrating faster, the pitch is higher, and if it’s slower then the pitch is lower, you get the idea. Imagine a motorbike or car driving along – as it accelerates the sound the engine makes increases in pitch.

Everybody Hertz

We measure frequency in Hertz (named after Heinrich Rudolf Hertz), which is abbreviated to Hz. 1 Hz is one cycle per second. You might see graphs like the one below, where the x-axis represents time, and the y-axis measures amplitude (which you can understand as volume for now, but there is a difference which we won’t go into yet).

If in the above example our time is 1 second, then this is a 2 Hz wave, i.e it has completed 2 cycles in one second.

You may also notice that amplitude is measured in both positive and negative, a bipolar measurement. We probably think of volume as unipolar or unidirectional, i.e goes from quiet to loud – I don’t have enough time to fully go into why this is, but if you’re interested, you can read more about amplitude on the wiki. I understand amplitude as describing the strength of a wave. We can think of non audio waves such as gravitational, electromagnetic, seismic etc. Amplitude is the peak to peak value of these waves. Someone on Yahoo Answers can explain it better than me:

Volume is used to express (audible) sound and it is usually based on a logarithmic scale, decibels. Amplitude could be based on any scale with a magnitude such as displacement or power/voltage/current, et al. Therefore, volume is an amplitude but not vice versa.

Moving on, in the below we will double the frequency:

In the same amount of time as the above image, the wave completes 4 cycles, making this a frequency of 4 Hz.

Pitch

We should note that all pitches are frequencies, but not all frequencies are pitches. Older readers may remember AM and FM radio, being described in kHz (kiloHertz, 1000 Hertz), or those of you familiar with LFOs may know their range to be (typically) 0.01 Hz to 20 Hz.

Our hearing ranges from 20 Hz to 20 kHz, so anything outside of this is inaudible. In the case of LFOs, they tend to be subsonic (lower than human hearing) and radio frequencies supersonic (higher than human hearing). That’s why spectral analysers display ranges between 20 Hz and 20 kHz.

Let’s move away from abstract physics and think about an acoustic instrument, as this is perhaps easier to understand. A nylon string guitar is a good place to start.

You may know that the distance between the nut and the bridge on a guitar is a carefully measured length, and 12th fret is exactly halfway between these points. Plucking the open A string will give you the lowest A note possible in standard tuning. Playing that same string with 12th fret depressed will give you a note twice the frequency; one octave up = double frequency.

The open A string has a frequency of 110 Hz, this means that in one second, that string will have oscillated 110 times. We can infer that the A at 12th fret on the same string has a frequency of 220 Hz, twice that of our open string. Here’s someone playing Eric Clapton’s Tears in Heaven, filming from underneath the strings with an iPhone 4. You can quite clearly see how different strings oscillate at different frequencies depending on their pitches.

It’s not possible for a human’s (or phone’s, for that matter) visual system to actually “see” the vibrations on a guitar string (or even a double-bass, for that matter) as clearly as that video claims to demonstrate. At best, a person can see that the string is moving rapidly, but cannot make out the “waves” the string describes in space. What is being shown is what’s known as beat frequencies – the perceived difference between (or multiple of) two frequencies occurring together. Most consumer digital video capture systems capture at 24 frames per second (fps). What the video displays is the string’s vibrating frequency divided by the camera’s capture rate. If you’ve seen the video of water flow being altered by sound, then you’ve seen the same effect, just using a different medium.

Jim Harrison via Facebook

The Harmonic Series

We know that doubling our frequency (f) leaves us with a note one octave higher than where we started (2f). If we multiply our frequency by 3 (3f) we get another harmonious note, and we can continue up all the whole numbers (4, 5, 6 etc) – this is known as the harmonic series. Put differently, harmonics are integer multiples of our fundamental frequency (starting place).

These harmonics are probably more familiar to you than you might first think, as they are naturally occurring. The higher these harmonics ascend the closer the gaps between them are in pitch. The gaps between these harmonics are what informs our western harmony model (more on this later). Certain groups of harmonics at a given amplitude also make up the common waveforms we might find in a subtractive synthesiser (sine, triangle, pulse and saw). Here’s the first eight harmonics starting on the frequency 110 Hz (which is an A note):

Certainly my first experience of these was hearing the various resonances coming from a Whirly Tube (is this really the name for this toy?), it never crossed my mind that these notes had some sort of mathematical relationship with each other nor did it cross my mind to questions why it didn’t produce a single continuously rising tone instead of the stepped or quantised notes.

Here’s an interesting piece from a pal of mine Oliver Thurley on what he calls multiphonics, but importantly for me has a neat map of the natural harmonics found on a stringed instrument, such as a guitar or in the example below, a double bass:

Since we’ve loosely covered frequency and how that can relate to pitch, the next thing we need to look at is tuning, and how different pitches relate to others. Some basic understanding of intervals is useful but not essential. Below (in pink) I’ve denoted the intervals between each of the harmonics from the Wikipedia diagram:

Hopefully some of the above makes some sense to you, but don’t worry if it doesn’t. An interval describes the gap between two notes. There are probably some that laypeople are quite familiar with, such as an octave or semitone, and others you may be less familiar with, eg.  tritones, major sixths etc. Intervals are useful for making up scales and by proxy chords, but to give order to pitches and intervals we need a tuning system, which is what we’re going to discuss next.

If you’re interested in reading more about writing harmonics for stringed instruments (particularly from the violin family) you should have a read of this short piece.

Disclaimer

I’m going to tread carefully here because some people can get quite tetchy about tuning systems, what is deemed correct and why we’re using certain methods. Something to point out to start with is that I am of course approaching this from a Eurocentric perspective; I am aware that these tuning systems are commonly used in Europe (and by proxy North America, Canada and Australia etc) and not in the rest of the world.

Parts of Africa, the Middle East, South and East Asia, the South Americas and others have all developed varied tuning systems different from our own, but for simplicity, (and being European) I am going to be talking about what I’ve been brought up to understand as correct (not wanting to start a flame war).

In addition, even within the Western World, there isn’t total agreement on tuning systems, with Just Intonation and 432 Hz types and others claiming the benefits of one practice over another. This article just serves as a reference or guide, and for that reason I sometimes have to be reductive or simplistic in order to not make this too long and digressional.

I’m also going to try and avoid too much hardcore maths where possible because a) I’m not a mathematician and it could be easy for me to slip into unfamiliar territory for me b) most people could find it pretty dull and c) too much abstraction can lead to us forgetting we’re looking at music. That said, some really basic maths is necessary.

Equal Temperament vs Just Intonation

There are many tuning systems in use, including Pythagorean tuning, Meantone temperament, Well temperament, Syntonic temperament and others, but the two we’re going to look at is Equal temperament and first, Just Intonation.

Just Intonation uses the harmonic series verbatim. As the system isn’t widely used these days, special instruments have to be constructed for playing music in Just Intonation. Various softwares and programming environments such as Reaktor, Max/MSP, Csound and others easily allow this sort of composition to be possible too.

Using the harmonic series discussed earlier, Just Intonation can describe intervals with nice easy ratios of the string length. Here’s a scale in C major for example.

C (1:1), D (9:8), E (5:4), F (4:3), G (3:2), A (5:3), B (15:8), C (2:1).

Our first note is our root, C. This has a ratio of 1:1 and is our starting point. I’m not going to define it with Hz or similar just yet, we’ll stick to the ratios. The octave above has a ratio twice that, 2:1, and another octave would be 4:1. The octave below our starting note could be described as 1:2 (half). A Perfect 5th (G) is a ratio of three halves (3:2), E is five quarters (5:4) and so on.

I found this article helpful in breaking down the maths behind Just Intonation. Not being a wizard at school means I’ve had to learn a lot of this stuff of the bat when I’ve encountered it in music, and whilst these ratios are quite simple it’s still nice to find plain English explanations.

Just Intonation (and other aforementioned tuning systems) have been widely used for a number of years but have largely been superseded by more flexible approaches.

A common misconception is that Johann Sebastian Bach’s The Well-Tempered Clavier was composed using Equal Temperament. But, as the name suggests, it’s actually not. It’s likely that measuring frequency as accurately as needed by Equal Temperament was not possible until much later. There is still a scene of composers using Just Intonation to make some interesting music. Here’s some examples of Just Intonation music provided to me by Zeroes and Ones editor Kieran Jones.

It might sound a bit out of tune to you, and in some ways you’re right, and in some ways not so much. It is out of tune when we compare it to the tuning system we’ve been brought up to accept as correct; Equal Temperament. However, the Just Intonation folk would probably argue that we’ve been conditioned to hear music in this way, or similar.

Historically it has been problematic to tune some polyphonic instruments, and it’s here in 1917 where Equal Temperament makes an appearance. YouTuber Adam Neeley nicely explains the difference between Just Intonation and Equal Temperament in this recent Q&A video.

While Just Intonation uses simple mathematic ratios, Equal Temperament uses a logarithmic division of an octave into twelve equal steps. More of which we will delve into into the next sections.

Just Intonation may make more for mathematically coherent structure, but practically it limits what we can do musically. Equal Temperament may technically be ever so slightly out of tune with harmonics, but it allows us to change keys and modulate more freely. This is far better explained by the awesome guys at Minute Physics who made this great video with diagrams and everything.

Don’t worry if you didn’t swallow all of that, some of the maths is on the edge of my understanding too, but it’s important to understand the principle: Equal Temperament was created to more evenly tune a piano, deliberately accepting that certain intervals are out of tune with each other.

As a result, we are left with the below relationship between pitch and frequency. Most are decimal numbers (A and G are whole integers, which makes them easier for using as examples).

The ratios are no longer simplistic, but what we’ve gained is the ability to move through different keys with ease; something not afforded to other tuning systems that rely on purer mathematics. I have recently run into this video by Two Minute Music Theory, which neatly explains some of the nuance in why other tuning systems can be problematic:

Cents

As we’re left with imperfect integers for our pitches, we are led to start thinking about what is called cents. People who have dabbled with subtractive synthesis may be familiar with this. A cent is the smallest division into which we split musical frequencies.

There are 100 cents in the semitone, usually displayed as +/- 50, either side of the note. With this in mind, let’s look at the previous table, using the example of A. The reason we’re using A is that the maths uses whole integers, 55, 110, 220, 440 etc. This is much easier than using other notes, such as C, which has multiples of 65.4 Hz. Not so easy!

You may well have heard of ‘A 440’ – the frequency to which orchestras tune to. In addition many analog synthesisers have an ‘A 440’ reference tone as a reference to tune the oscillator(s) to.

Below charts the first 16 harmonics (there are an infinite number of harmonics, regardless of how many we can hear an measure on DAWs), the note they are closest to, their frequency in Hz and their error compensation in cents. The last column is their interval. This relates to the note’s relationship with the original note, in this case A. For example our 13th harmonic is F#, and F# is the sixth note in an A major scale, so its interval is a major sixth.

nf	Note	Freq. (Hz)	Cents +/-	Interval
1f	A	110	0.00	Root
2f	A	220	0.00	Root
3f	E	330	−1.96	Perfect Fifth
4f	A	440	0.00	Root
5f	C#	550	+13.69	Major Third
6f	E	660	−1.96	Perfect Fifth
7f	G	770	+3.91	Minor Seventh
8f	A	880	0.00	Root
9f	B	990	−3.91	Major Second
10f	C#	1110	+13.69	Major Third
11f	D#	1210	+17.49	Tritone
12f	E	1320	−1.96	Perfect Fifth
13f	F#	1430	-13.69	Major Sixth
14f	G	1540	+3.91	Minor Seventh
15f	G#	1650	+11.73	Major Seventh
16f	A	1760	0.00	Root

A few things to note here. Firstly, as the multiple of f increases, the harmonics become closer and closer together. The nf column would continue ad infinitum, with the the harmonics gradually becoming smaller and smaller, closer and closer together.

Again our 7th harmonic is troubling us, this is where the difference between Just Intonation and Equal Temperament can be heard most obviously; the note sits in between the intervals we know as a minor and major seventh, or 10 and 11 semitones from the root.

Waveforms

Now we have harmonics under our belt let’s quickly peer at some basic waveforms we come across in subtractive synthesis. I’ve already written at length about this this topic, which you can read about here, but if you’ve not read it you can read on.

Many analog and analog emulation synthesisers use basic waveforms to generate the tones we shape with filters and amplifiers. Even more advanced synths such as Massive include ole’ reliables sine, triangle, pulse and sawtooth waveforms.

These are waveforms constructed with simple harmonics, and those harmonics’ amplitude relationship to the fundamental (or first harmonic). Confused? It’s not that difficult really. Let’s start off with our sine wave, as it’s dead easy.

Sine waves are solely one harmonic. Let’s hear one and see what  it looks like on a spectral analyser (mapping frequency on the x-axis against amplitude on the y-axis) and an oscilloscope (time against amplitude). I’m using an A at 110 Hz to make the maths simple.

We can clearly see the sinusoidal pattern (reflecting the shape of the air pressure change); this creates the single harmonic we see in SPAN.

Our triangle wave contains only odd harmonics, so 110, 330, 550, 770, 990 Hz etc. Notice the sharp drop off in amplitude from the first to third harmonic, and so on.

There is some complicated maths involved in constructing a triangle wave through additive synthesis if that’s your thing, but a cursory understanding will be sufficient.

Unsurprisingly, the display on the oscilloscope is that of a triangle.

Like our triangle wave, the pulse (or square) wave also contains odd harmonics, however their relative amplitudes are much closer together:

It’s important to note that square waves has a symmetrical balance of positive and negative energy in their cycle. Offsetting this leads to rectangle waves, and the harmonic composition is different.

Lastly let’s have a look at a sawtooth. These contain all the harmonics (110, 220, 330, 440, 550 Hz etc).

The oscilloscope display looks much like that of the teeth of a saw, so you can imagine how the name came about.

All of the above wave forms can have their harmonic makeup described as simple integer multiples, but what happens if we have other tones, not so easily described?

Overtones

My definition of an overtone is any frequency that appears above our fundamental. By this definition we can include harmonics as types of overtones. However some frequencies can be inharmonic or simply unrelated.

How could we describe this sound for example? Certainly harmonics are present but I couldn’t guarantee they’re all simple fractions and ratios.

Wikipedia’s article about partials I find helpful here. I’m picking up after the definition of harmonics as we’ve hopefully covered that already:

A harmonic partial is any real partial component of a complex tone that matches (or nearly matches) an ideal harmonic.

An inharmonic partial is any partial that does not match an ideal harmonic. Inharmonicity is a measure of the deviation of a partial from the closest ideal harmonic, typically measured in cents for each partial.

Maths YouTuber 3Blue1Brown has a fantastic video that covers a few related topics here. Some of the numbers are a bit much for the average joe but I found sticking well worth it even if you only get the first half or so.

Timbre

With overtones and harmonics under our belt, we can start to understand timbre, which was something that I only really became familiar with recently. If we take an isolated snapshot in time of a given sound, we cannot understand the sound properly.

Sound requires time to be understood, otherwise it’s just a tone. The sine, triangle, square and sawtooths we looked at previously don’t exist in nature, they are synthetically generated. There are very few (if any) examples of single cycle repeated tones in nature or our acoustic instruments we’ve created.

Timbre is the word used to describe a sound’s sonic signature. It’s how can we tell the difference between a trumpet and a guitar, two different guitars, my voice and your voice or two completely different sounds. Even singing/playing the same note and the same volume we can tell there’s a difference.

Playing the same root, we can analyse the pitch of two different instruments and while they are the same they will contain a different harmonic makeup, but that’s only half the story. It’s how the harmonics amplitude changes over time that give a specific sound.

Let’s take two examples from Logic’s Sampler, a piano and a standup bass. Both are playing the same note (A 110 Hz) in the same octave as the same velocity. There’s no added processing or effects. Let’s start with the piano:

We can see the second and third harmonics are actually stronger than the first. In addition there’s a complex but unique undulating in the upper harmonic portion of the sound and distinct drop off in amplitude after that.

A different piano would have a slightly different harmonic makeup, but they would share certain nuances that allow us to instantly detect that we are listening to a piano. Here’s the oscilloscope capture:

Time is the important factor here, and this is in part why acoustic instruments are so hard to synthesise. If we took a cycle or portion of our waveform and looped it, it would simply not sound like a piano but instead sound like a complex oscillator trying to (badly) impersonate a piano.

Certain additive synthesisers have tried to emulate instruments by giving harmonics their own amplitude envelope by short comings in detail, processing power (I would imagine) and other factors tend to lead to these sounding unconvincing. Let’s look at the standup bass.

We can see two harmonics below our fundamental. These were absent in the piano. The upper harmonics captured by this display are also much lower in amplitude and their drop off (or envelope if you like) more sudden.

This is what timbre is: the change of harmonics amplitude over time. Certain x, y, z graphs capture this nicely. We see examples like this in early sampling/re-synthesis like the Fairlight CMI:

And acoustic acoustic treatment graphs, displaying sounds decay in a given space:

The above graph is not describing timbre, it’s just a helpful way to visualise the three axis, frequency (x), amplitude (y) and time (z).

Further Reading

That more of less concludes everything I wanted to talk about, covering frequency and pitches intrinsic relationship, string lengths, tuning systems, harmonics and overtones, timbre and how they are all in fact related.

I’ve found the follow sites helpful in expanding on what I already knew and sometimes shedding new light on the topic.

• What is the relationship between music and math? (Ethan Hein)
• Can science make a better music theory? (Ethan Hein)
• Pythagoras and the Music of the Future Part I – Timbre
• It Really is a Musical Universe! (Dale Pond)
• Describing the Relationship Between Two Notes: Harmonics as Decimals

Post Script

I contacted Adam Neely to tell him I’ve linked to this video and he let me know he was just about to publish something on intervals and poly rhythms. This sort of expands on (albeit it a slightly more manageable and less maths-heavy way) on the concepts discussed in the 3Blue1Brown video earlier in the article.

It discusses how intervals such as perfect fourths and fifths are almost like really fast polyrhythms of beating cycles, ratios of 4:3 and 3:2 respectively. Of course he goes much further into far more complex rhythms. Enjoy!