Describing the Relationship Between Two Notes: Harmonics as Decimals

This was originally published in December 2017. I am giving the graphics and writing an update.

I was walking along the canal a while back with my good pal discussing saxophone and how his work rota had allowed him more time at home during the day, meaning he can practise the (often) unwelcoming sound of overtones. To cut a long story short we began discussing the language of overtones and harmonics – or rather, what qualifies an overtone being harmonic. We hypothesised that an overtone is harmonic if it appears in the harmonic series, but then I thought to myself… how well do I really know the harmonic series?

I roughly know up to the 16th harmonic, I know that the distance between the 8th and 9th harmonic is about a tone and between the 15th and 16th harmonic is approximately a semitone. However, the harmonic series carries on forever, and we only measure our equally tempered scale as dividing the octave into twelve (12TET). So from an infinite series of numbers, we extrapolate twelve discreet pitches. There are other ways of dividing the octave up (more on some of these later), but 12TET is by far the most popular in Western culture and for that reason I am most familiar with it.

From this we can infer that not only does every note, diatonic or not, appear in the harmonic series, but even the notes in between notes and everything down to the smallest scale. It would be spurious to claim a quarter tone between C and C# is harmonic in a traditional, Euro-classical sense, but it led me to thinking about expressing notes, diatonic or chromatic, in different ways.

I figured my original conjecture was inadequate. I thought back to an excellent video by YouTuber 3Blue1Brown which I’ve linked to elsewhere in this blog in my article on Frequency, Pitch, Overtones, Harmonics and Timbre (I’ll embed his video at the bottom of this post so as to not disrupt the flow). The video is aimed at mathematicians (of which I am not one) but asks an interesting question: can you intuitively tell if two sets of numbers have a harmonic relationship?

After pondering the idea for a while I tried to describe how we can think about intervals in numerous different ways but my verbal skills let me down, so I’ve decided to write this article to explore the concept. This is certainly not to be taken in isolation and I would recommend either being on speaking terms with the harmonic series (maybe you’ve come across them in high school maths or physics) and how they fit into music. This article by me is a good starting place.

Some Basics

The most natural signpost we have dealing with frequency and pitch is the octave. I would hazard a guess that all musicians reading this and even most producers would be familiar with this term. Before we clarify what an octave is, let’s discuss some useful terminologies to better understand the octave.

To measure frequency we use Hertz (Hz), which is the number of cycles per second (see diagram, right).

If the frequency is in the audible part of the spectrum, we can infer pitch. All pitches are frequencies but not all frequencies are pitches.

We can denote the pitch and position within the audible spectrum of a note by ascribing a letter and number to it. There are twelve different pitches, A through to G#, and their octave (or position on the piano) can is labelled by numbers counting from 0 to 8. For example, A0 is the lowest A on a grand piano, A1 would be the octave above it and so on.

*The notes over two octaves starting from C. In 12TET, the ‘black notes’ can either be sharp (#) or flat (b) depending on their relationship to a reference note.*

In MIDI we count from C-2 to G8, giving us a total of 128 notes, although this is technically greater than the range of most instruments. The reason for this is that MIDI is an 8 bit number.

But what is an Octave?

To understand where I want to go with this, it’s crucial to understand the concept of an octave in some degree of technical terminology. An octave is an intervals where the frequency is twice or half the frequency of our reference note. The note A2 (110 Hz) is an octave lower than A3 (220 Hz). The next octave above A3 would be A4 (440 Hz), and A5 would be 880 Hz.

The human ear can hear roughly ten octaves, from 20 Hz right the way up to 20 kHz (a logarithmic scale). Not all of these octaves have a musical usage to us but they’re necessary for sub bass and higher harmonics and overtones at the higher end of the spectrum. Let’s plot out the integer multiples starting at A2. At each stage we’ll note the harmonic relationship to the root note we’re starting on:

Multiple	MIDI Note	Hertz	Distance to fundamental
1	A2	110	N/A
2	A3	220	One octave above fundamental
3	?	?	?
4	A4	440	Two octaves above fundamental
5	?	?	?
6	?	?	?
7	?	?	?
8	A5	880	Three octaves above fundamental
etc.	etc.	etc.	etc

We’ve deciphered that doubling a frequency gives us octaves, 1, 2, 4, 8, 16, 32, 64 and so on. But what are the numbers in between those? These are the other notes in the harmonic series and all of this information is covered in detail in this article.

The Harmonic Series

To start filling the gaps in the above table, let’s start with multiply by 3. Three times a frequency is a note that’s one octave plus a perfect fifth above it. A perfect fifth is an interval that’s seven semitones above a root note and in Western music it’s hugely important in understanding functional harmony. A perfect fifth above A would be E, above C would be G, above D would be A and so on.

Where I want to get to, is understanding all of the chromatic notes between two octaves as multiples, so how can we understand the third harmonic as a decimal number in the first octave?

We’ve established that the third harmonic is an octave plus a fifth above the root, and we know that doubling or halving a frequency gives us +/- one octave, so we can half the third harmonic (330 Hz), resulting in 165 Hz, and to express the third harmonic one octave down we can simply half that to 1.5, therefore, 1.5 times a given frequency is a perfect fifth above it.

Let’s look at the fifth harmonic, or multiply by 5. This is a note that’s (close to) two octaves plus a major third above our fundamental. A major third interval is four semitones above the root and again is key to outlining tonality. A major third above A is C#, above C is E and above D is F#.

The fifth harmonic in relation to A is a frequency of 550 Hz, so we divide this down twice to get 137.5 Hz – the major third directly above 110 Hz. Why twice? Because once gives us a major third one octave above the root, and twice gives us the major third in the same octave as our starting position.

If we continue up the harmonic series, we can work out all of the diatonic notes between A2 and A3. Let’s map out A major. In order to fill in our chromatic scale, we just need to find the relevant note in the harmonic series and keep dividing in half until it appears in the correct octave, right? Well, sort of.

Nearest Note	A2	B2	C#3	D3	E3	F#3	G#3	A3
Freq (Hz)	110	123.75	137.5	146.4	165	185.625*	206.25	220

The reality is, the frequencies in the bottom row of the table are not what you’d get if you compared the notes on a keyboard with an oscilloscope or other device for measuring audio frequency. There’s something we haven’t discussed yet, equal temperament.

*This table has come under some flack from people more learned than me in regards to tuning and Just Intonation. I am aware that a more common understanding of the major sixth degree is 5/3 (183 Hz). I have amended it to some degree but what I want to make clear is that I’m not claiming the diatonic scale is extrapolated directly from the harmonic series, merely that some intervals are and from the next section, hopefully I explain why this is the case.

Equal Temperament

The piano is not in tune with itself. And no contemporary music is perfectly tuned. The octaves on a piano are in tune with each other but each note between two octaves drifts slightly sharp of flat depending on its relation to the root.

This isn’t accidental and is in fact a design feature of all polyphonic instruments. The detune (measured in cents – 1/100th of a semitone) is mostly negligible. Furthermore the amount of detune differs depending on the harmonic importance of that note relating to the root.

Let’s make a practical example. If we played F#3 on a piano rather than getting 183 Hz (as stipulated in the table above), we would get 185 Hz.

183 is a multiple of 110, (the 27th harmonic) but it is a harmonic none the less. 185 Hz is not.

You might think that this is dreadful and there certainly are many people who look down on equal temperament but it has huge advantages and I’m not going into them right now. Instead I would watch this video by minutephysics on why it’s impossible to tune a piano:

To summarise: equal temperament divides the octave into twelve equal steps (called semitones) allowing modulation and changing key to be achieved much more easily.

What is the Most Dissonant Interval?

There are two ways we can think about this. Firstly let’s look at intervals in an equally tempered system. In 12TET each interval has an opposite, i.e an interval diametrically opposed to it.

If we consider the perfect fifth, this interval is 1.96 cents flat compared to the just tuning system derived using harmonics. Additionally, the perfect fourth is 1.96 cents sharp. We can think of these intervals like inversions around the root: a perfect fifth is the same number of semitones above the root as the perfect fourth is below it.

*The circle of fifths (clockwise) or fourths (counter clockwise). Using A as the start point we get the interval’s relation to the root and the margin of error (in cents) from the perfect interval.*

From the above we can ascertain that the tritone (augmented fourth) is the most dissonant interval. In terms of western harmony there is a long history of the tritone being considered problematic in counterpoint and there are some great videos made about this (Adam Neely’s being one of the better ones), but in addition to that it has the most drift from a just interval (17.49 cents sharp).

It has been pointed out to me that the tritone’s dissonance is not solely because of it’s deviation from Just Intonation.

Quickly though, what is Harmony?

If we take take two copies of the same note we hear their amplitude doubled (ignoring issues regarding phase and such). Detuning one or both of these intervals causes beating – a small amount of phase cancelation caused by the inharmonic relationship between the two intervals. This causes them to clash slightly. An example of this is more clearly heard when tuning two adjacent strings on a guitar.

The more drastic the detune the faster the beating becomes. Once it speeds up beyond a certain point we perceive two discreet pitches and this is the sound of harmony. But harmony can be subjective. We cannot just use the harmonic series to extrapolate definitions of harmony, for example:

But if we played an A and Bb together on a piano most ears wouldn’t call this harmonic. However, adding a lower F and higher D, we now perceive this as a chord (Bbmaj7). We hear each note as a degree of the scale, the A being major seventh of Bb, Bb being the tonic, the F is the perfect fifth and the D is the major third.

If you add an even lower G into the chord, suddenly the chord now appears to be a G-9. Each pitch remains the same but the low G re-contextualises each scale degree. Now the F is the minor seventh, A is the ninth (second degree), the Bb is minor third and the D is the perfect fifth.

Earlier I mentioned there was a second way we could try and extrapolate what “the most dissonant interval” was. The brings us to our next topic.

Missing Intervals?

There are in fact harmonics that appear in the series before some of our equally tempered intervals, in particular the 7th, 11th, 13th and 14th harmonics (although 14th is just twice the frequency of the 7th).

Are these musical or dissonant, or both? The 7th harmonic is similar to the minor seventh interval, although it’s 31 cents flat, so you could be excused for confusing the two.

The 11th harmonic sadly has some nonsense new age/astrological connotations which I wont pay lip service to. Musically speaking it’s between the perfect fourth and tritone, an entire 49 cents flat of the tritone to be precise. This is arguably the closest resemblance to a “tone within two tones”, that appear in the harmonic series this close to the tonic.

Lastly we have the 13th harmonic. This is closest to that of a minor sixth interval but 41 cents sharp. It turns out the cosmologists like to claim this one too, but there we go.

Should we make inclusions for these in our music? Broadly speaking – no. The 12TET system works so well because it has a small margin of error in cents correction to just intervals compared to other systems (under 1% for each note). The octave can be divided up however you want, with 19, 22, 31 and even staggeringly 53 being used by microtonal composers.

As we ascend the harmonic series, the harmonics become closer and closer together and there are some intervals that actually appear closer to the tonic than others we include on the keyboard.

Here’s first 32 harmonics starting on a low A. I’ve colour coded the first 17 harmonics with unique colours, repeating the shade where a frequency is doubled (3, 6, 12 24, for example). After 17 there are not more repeated harmonic within the first 32 (twice 17 is 34). Harmonics 23, 25, 29 and 31 are in between 12TET tones we’ve established (or at least harmonics close to them) so I’ve not ascribed a unique colour to each one.

Intervals as Decimals

The crux of what I want to get to is expressing intervals as a decimal. We touched upon this earlier: our perfect fifth appears as the third harmonic, so dividing 3 by 2 gives us 1.5. Therefore the perfect fifth is a frequency 1.5 times the root.

Using the intervals as they naturally appear in the harmonic series, let’s see how each note in the chromatic scale relates to the root:

1.0667

1.125

1.2

1.25

1.33

1.4

1.5

1.6

1.666

1.777

1.875

1:1

16:15

9:8

6:5

5:4

4:3

7:5

3:2

8:5

5:3

16:9

15:8

Some things to note:

The intervals A#, D, F# and G (minor second, major sixth and minor seventh respectively) are recurring, so I’ve rounded to three or four decimal places.
Some intervals (in particular the minor second, minor and major sevenths) have markedly higher integers are higher than some others. As was pointed out to me this is not to do with their deviation from Just Intonation but their commonly agreed value. I was speculate that their relative complexity of their fraction is explainable by their generally agreed upon dissonant relationship to the tonic (I appreciate that I am using dissonant colloquially here).
These are not the exact values you would get if you played these notes on a piano, these decimals are approximated and/or truncated 12TET values, because equally tempered systems use 12√2 to divide the octave up and 12√2 is an irrational number, meaning it cannot properly be expressed as a decimal.

I’ve taken onboard some criticism here about the inaccuracy of the above table, so I’ve tried to amend where possible. Steve Morris points out:

I think to do what you are trying to do, you need to give the 12edo decimals (powers of 2^(1/12)) and then give the commonly agreed just values for comparison. You can then try to give fractions, but there’s a trade-off – how close do you want to get, and how large can the integers in the fraction get, do you want to remain within a certain prime-limit?

He is of course correct and this was a trade off I internally debated when writing this. I want to keep things as simple as possible for the beginner without being too inaccurate (I understand this could read like an oxymoron as there’s no degree of inaccuracy).

For a much more in depth explanation I would refer to musicologist Kyle Gann’s site where he lists many more divisions of the octave including ones from other tuning systems and ones of historical importance.

For perhaps a little clearer and more succinct explanation I would point you to the Wikipedia entry on Equal Temperament and particularly the comparison to Just Intonation.

Alternate Tunings in your DAW

In Logic, it is possible to experiment with some of these other tunings, including lots of interesting historical ones too. Head to File > Project Settings > Tuning and there is a dropdown list of fixed tunings. You can even make your own too.

Post Script

I was recently contacted via email by Nathan Powers. He had this to say on the subject:

I just came across your article Describing the Relationship Between Two Notes: Harmonics as Decimals. It was probably the most concise and clear exposition of scales that I’ve come across, and I thought you might be interested in another author’s work that seems to be unfortunately obscure, but is especially relevant in the arena of synthesized music. It is definitely the most interesting thing I’ve learned while looking into music theory (well, maybe not Shepard tones) but it is more fundamental.

To make a long story short, the harmonics based on integers in your article are the modes of vibration of a string, or, in practice, any one-dimensional object, e.g. long narrow tubes (flutes), elongated bars, essentially all western instruments apart from drums and bells. While that is no accident, it does mean that the octave itself is arbitrary. What I mean by that is perhaps best shown by example here:

In this example, the interval from f to 2f is dissonant, while the interval from f to 2.1f is consonant. In other words, there is nothing inherent about the harmonic scale itself that creates consonance, but the harmonic content of the notes themselves establish what is consonant.

Because the sound produced by everyday instruments contains more than the fundamental harmonic, e.g. this great article here http://dalemcgowan.com/every-note-is-a-chord/ the very first note you hear actually sets the scale. Because so much of western music is based on strings, this produced the diatonic scale. This is why cultures that primarily use bells or other nonlinear instruments often use different scales.

To produce that 2.1f octave, Sethares used some old research by Plomp and Levelt, where the authors surveyed subject’s perception of dissonance when listening to two pure sinusoidal tones. From the resulting entirely anthropic dissonance curve, and the spectra of a given instruments sound, the most harmonic scale can readily be constructed. Of course, using the harmonic spectra of strings produces the diatonic scale, but the most consonant scale for say, a xylophone, is actually somewhat different. You can find his website here: http://sethares.engr.wisc.edu/consemi.html.

(I have edited some of the formatting but this is otherwise verbatim).

Additionally, after posting this in The Xenharmonic Alliance – Mathematical Theory Facebook group, Ffred Tronge had this to say:

There are a few ways to arrive at approximately the 12TET chromatic set. You’ve gone with prime5-limit. You can also go via prime3-limit increments, which gives different ratios for most of the degrees. There are other methods of generation too, but I think these are broadly understood as the most basic. The 3-limit ladder-of-fifths method is essentially the most classical and is associated with Pythagoreanism. Much conventional Western music responds well to analysis based on the recursive 3/2 ontology, especially melodic lines. Some of the ratios in that system have large numbers in them, quantitatively speaking, but the underlying prime-structure is very simple. 5-limit is mayhem in that it adds another prime, but it goes very nicely with stacked thirds.

He insisted I include this disclaimer:

I wouldn’t quote me on anything [too late, pal]. Give it a couple of days and there’ll probably be a correction to my comment. Although contingencies and constructions condition every path-node in the library of intonational truths, some matters are of historical fact and I only have a desultory awareness of the field.

Earlier, I added some comments from Steve Morris, I thought this was pertinent in the addendum rather than in the main body:

Note, the circular diagram assumes 7/5 for the tritone – introducing another prime (Ffred talked about 3 and 5, now here’s 7 as well). Of course the interesting thing is that in 12edo it is 600 cents, exactly half an octave, and so represents its own inverse (10/7) as well. Often it is thought of as a 5-limit or even a 3-limit value, e.g. 45/32 or 729/512.

2 Comments

Dennis Rich says:

25/10/2022 at 11:57

Prefer log base 2 of frequency for better graphic and tabular representation and comparison of ratios and tempered values.

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- Ali Jamieson says:
  
  27/10/2022 at 10:59
  
  Hi Dennis. If I’m honest I didn’t do any of this in Python or any maths software, so there are likely some small margins of error. I don’t entirely follow your comment, can you link me to an example of what you mean? Thanks
  
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