How do ratios work within the harmonic series?

Question

Good day stack family,

So in the harmonic series, we have a fundamental (pitch/note) and a succession of harmonics that stem from the fundamental which also hold other pitches within themselves.

My question is, let's say we have a ratio of 3:2 which is a 5th in a scale pertaining to the fundamental. Where does first of all 3 and 2 stem from exactly, what do those numbers represent?

My second question is why does the third harmonic (3) get put in a ratio with the second harmonic (2) aka 3:2 instead of the 3rd harmonic being put in a ratio with the fundamental which would be 3:1?

I see this pattern a lot where ratios will be created from a harmonics previous harmonic like for e.g. 5:4,3:2 and so on, instead of the ratio being created with a harmonic and the fundamental each time which only exists in unison 1:1 and an octave 2:1.

My last question is, I've seen ratios that are something ridiculous like 54:72 like a tritone I believe, how do these massive jumps in ratio come about?

EDIT: Based on the Plomp and Levelt curve for musical tones, here is a list of an octave in chronological order of the interval with most consonance, straight down to the interval of most dissonance (in order of consonance to dissonance) using the key of C..

• Unison - C to C - 1/1
• Octave - C to C1 - 2/1
• Perfect Fifth - C to G - 3/2
• Perfect Fourth - C to F - 4/3
• Major Third - C to E - 5/4
• Major Sixth - C to A - 5/3
• Minor Third - C to Eb - 6/5
• Minor Sixth - C to Ab - 8/5
• Tritone - C to F# - 7/5
• Minor Seventh - C to Bb - 9/5
• Whole Tone - C to D - 9/8
• Major Seventh - C to B - 15/8
• Semitone - C to C# - 16/15

Is this statement and list correct in its order of consonance to dissonance? If not could you correct the order of the intervals please and rectify the statement as this would finalise my query?

Answer 1

My question is, let's say we have a ratio of 3:2 which is a 5th in a scale pertaining to the fundamental. Where does first of all 3 and 2 stem from exactly, what do those numbers represent?

If we're thinking in terms of the frequency domain, they represent the simplified ratio of the (fundamental) frequencies - for example, if we have a note with frequency 150 Hz, it is a fifth above a note of frequency 100 Hz - the ratio is 3:2.

As to why we'd be interested in the ratio 3:2 - it's basically because it's a consonant interval (simple ratio) and sounds sweet.

This is because the human ear is a machine that is designed to identify harmonic series - sine waves that are multiples of some fundamental - and playing notes that have simple ratios between them creates an interesting effect as there will be many groups of harmonics that are each multiples of some other frequency (even ones that may not themselves be present in any of the notes).

For example, if we have a note with fundamental at 100 Hz and two ovetones at 200 and 300 Hz, and a note with fundamental at 150 Hz and two ovetones at 300 and 450 Hz, we get a series of 100, 150, 200, 300, and 450 Hz, which is a bit like the overtones of a 50Hz note, without the fundamental. So this kind of harmony is a kind of 'aural illusion', giving a sweet sensation that most people appreciate.

My second question is why does the third harmonic (3) get put in a ratio with the second harmonic (2) aka 3:2 instead of the 3rd harmonic being put in a ratio with the fundamental which would be 3:1?

Both those ratios are correct, and might be referred to in different situations. If you're talking about how the frequencies of harmonics can be used to generate notes in a scale, people are usually interested in generating octave-repeating scales - I'll take the liberty of quoting Uncle Bob's comment, which explains it well:

The distance from the fundamental to the third harmonic is an octave and a fifth (or 19 semi-tones), and the frequency ratio is 3:1. To derive the fifth (or 7 semi-tones) from that, we skip the octave and compare the third harmonic to the octave above the fundamental, which is the second harmonic. So the frequency ratio of a fifth (or 7 semi-tones) is 3:2.

My last question is, I've seen ratios that are something ridiculous like 54:72 like a tritone I believe, how do these massive jumps in ratio come about?

Well, 54:72 is the same as 3:4, so that's quite simple and consonant! But you're right - there are some 'big' ratios that can be found in a scale. A couple of reasons for this:

• In music, we often don't only want to have musical intervals that are very consonant - we want to take the listener through a journey of consonance and dissonance, of tension and resolution.

• We don't necessarily want our scales to only contain notes that are consonant with the root note of the scale. We also want notes that are consonant with each other, to allow chords to be built on other degrees of the scale, and allow us to move the perceived tonal root. IF the root of our tonality is C, then F# is very dissonant with C, but it is very consonant with D.

Answer 2

The harmonic sequence is very simple. Given a fundamental tone of frequency f0 the harmonics are all n*f0. Doubling a frequency produces an octave, so 2*f0, and 4*f0 are the same note as the fundamental but one and two octaves above f0.

To relate the harmonics to other notes we bump them down as many octaves as we can so that they are within one octave. This is not to say that the harmonic is vibrating at a new lower frequency as it cannot. So the third harmonic is 3*f0. which is more than one octave above the fundamental, dividing by 2 gives us 1.5*f0 which is a Just 5th above the fundamental. So the n = 3 harmonic is an octave and a 5th. Not all the notes in the sequence can be found within the just major scale. There is quite a bit more to the harmonic sequence than what can be found in the diatonic scale, or even the chromatic scale. Not all harmonics are audible under typical circumstances.

The first few can be identified as follows:

n = 1, Fundamental

n = 2, Octave

n = 3, is an octave and a just 5th

n = 4, 2 octaves

n = 5, is 2 octaves and a just third

n = 6, is 2 octaves and a just 5th

n = 7, is 2 octaves and very close to a dominant seventh (sub-minor seventh).

n = 8, three octaves.

As an example, the harmonics (4, 5, 6) are the major triad. This can be played on the guitar using the harmonic at (or near) the 5th fret and the next two which are between the 4th and 3th frets.

The harmonic content is what generates tone of a note, i.e. brightness etc. Bright tones have more and stronger high harmonics. They can be generated by picking a string close to the bridge, or striking a string percussively with something hard.

I would add to this the following. The simple harmonic sequence is a consequence of the vibrating material obeying the second order wave equation with boundary conditions imposed. For strings BC is that the string be under tension and not moving at each end. Most other vibrating systems, including real guitar strings, do not behave this way exactly. Stiff rods and plates obey other equations that lead to non-linear relationships between higher vibration modes and the fundamental. One can hear the deviation from n*f0 in some musical instruments.

Answer 3

On top of topo morto's answer, I think a more didactic approach would be suitable.

Every sound in the real world consists of not one, but several frequencies at the same time. When those frequencies don't have a special relation, we hear them as noises. But for some reason, when these harmonics are multiples of the fundamental, we hear them as harmonic sounds. So, let's keep in mind that every harmonic note consists of it's fundamental, let's say 100Hz, and it's multiples: x2 = 200Hz, x3 = 300Hz, etc..

My question is, let's say we have a ratio of 3:2 which is a 5th in a scale pertaining to the fundamental. Where does first of all 3 and 2 stem from exactly, what do those numbers represent?

As said, it represents the ratio between the frequency of the fifth above the fundamental, and the fundamental itself.

My second question is why does the third harmonic (3) get put in a ratio with the second harmonic (2) aka 3:2 instead of the 3rd harmonic being put in a ratio with the fundamental which would be 3:1?

As you know, when we double the frequency, or divide it by two, we arrive at the same note, one octave higher or lower. The third harmonic is indeed 3:1 (threefold) the frequency of the fundamental (since the ratio is above 2:1, it is more than one octave apart). But, we can freely multiply it or divide it by 2 to fulfill our purposes. So, if we want to know the frequency of the fifth just above the fundamental, we need to bring it one octave down, therefore we divide it by 2, making it 3:2. Dividing by 4, 3:4 would be the fifth in the octave below the fundamental. Multiplying by 2, 6:1 would be the fifth 2 octaves above the fundamental.

My last question is, I've seen ratios that are something ridiculous like 54:72 like a tritone I believe, how do these massive jumps in ratio come about?

When you try to describe every other note in terms of one fundamental, these things can happen. Let's picture the notes of the harmonic series of C4:

1:C4, 2:C5, 3:G5, 4:C6, 5:E6, 6:G6, 7:Bb6, 8:C7...

C#, for example, will only show up in this sequence in the 17th (more than four octaves apart) harmonic. Whats the ratio of C#4, just above C4? Bring it four octaves down, by dividing by 2 four times, resulting 17:16. There you have a weird ratio.

Here's a useful Wikipedia article on harmonic series.