Can there be more than 12 musical notes?
Portia: “Human beings are terrible messes.”
Andrew: “I’ll grant you that.”
From the movie Bicentennial Man.
A while ago, I was trying to figure out a song to play on my guitar. There was a nice guitar solo in it. There was a part in that solo, where the guitarist bends a note. As I was trying to figure out the bend, I noticed a very surprising thing. I can do half-step bend (one note bend) and a whole step bend (two note bend). So I tried bending half step, but it kinda felt slightly lower than the original bend. Then I tried bending a full step, but that felt slightly higher than the original bend. (Before anyone asks, no I don’t have perfect pitch. That’s why I have a very hard time figuring out songs by ear.) Then I googled for the song’s tab. And I found that the bend was a 3/4 step bend, halfway between half step and whole step. My first reaction was:
Then I looked it up, and I found about microtones and microtonal music :0 By that time I started to question reality xD Jokes apart, I started to question why there are exactly 12 notes in music. And I found something interesting. Today’s article is about that interesting thing. (In case you haven’t understood the musical vocabularies I used in this article so far, no worries. I’m gonna explain the necessary vocabularies to answer the question, “Why are there exactly 12 notes in music?”)
So, what exactly is a “note” ?
A note in music is a sound with a specific frequency. For example, the sound with frequency 220Hz is a note, and it’s called A. Now a natural question arises, there are infinitely many frequencies possible. So why do we use only 12 of them to make music?
Actually we don’t use only 12 of them. If you look at a piano, there are a lot more than just 12 keys, and each key generates sound of different frequency. So I can’t say there are only 12 notes, right? Well, this needs a little bit more explanation. I want you to demonstrate this with me.
Play any key on a piano (if you don’t own a piano, play on some piano app on your phone or pc). Start counting the keys after that key, play the 12th key. These two keys sound kinda similar, right? They are so self similar that we call them by the same name, but different pitch class. The ratio of frequencies of these two notes is exactly 2:1. This ratio of 2:1 is called an octave. So whenever two notes are an octave apart, we just consider them to be essentially the same note.
A note has frequency 220Hz, one octave higher A note has frequency 440Hz. So the essential question is, why is there only 12 notes between A and this higher A? Let’s construct which notes should exist in between these two A’s. For this, we need to answer a deeper and more fundamental question.
Why do some notes sound good together and some don’t?
Human brains are a great searcher of patterns. And it is the very reason why some notes sound good together. It might seem vague at this moment, but I hope it’s gonna be clear after some moments.
First of all, the notes that are one octave apart (notes with frequency ratio 2:1), they sound melodious together. I’m gonna explain why. Name the first note X, and the note with its twice frequency Y. Notice that, the wavelength of a sound wave is inversely proportional to it’s frequency. So X’s wavelength is exactly twice the wavelength of Y. When X and Y played together, it kinda looks like this:
Now, we know that waves tend to “add up”, or superposition if you prefer technical terms. The resultant wave looks like this:
The pink wave is the resultant wave here. It’s not quite a sine wave, but it’s periodic. So it’s a nice pattern. That’s why these two notes sound so good together.
Now let’s take two waves that has frequency ratio π/e.
The resultant pink wave doesn’t have any fixed pattern. That’s because the ratio is irrational. So we can guess that, there exists a pattern when the ratio is rational. In fact it can be proved easily.
Let X has frequency x and Y has frequency y, where x and y are integers, in other words x/y is rational. We can treat single-frequency sound waves as sine waves. So the equation for X and Y are:
We assumed the amplitude to be 1, because amplitude doesn’t really matter here. Let R be the resultant wave. So the equation for R becomes:
If we choose T=1s, then you’ll see that
That means, the resultant wave repeats after each second. But the problem arises when x/y is irrational. For sin(2πxt+2πxT)=sin(2πxt), we need T to be an integer multiple of 1/x, or T = m/x. Similarly, we need T to be an integer multiple of 1/y, or T=n/y. Here m and n are integers.
which contradicts the fact that x/y is irrational. Therefore, we have proved that there does not exist any T such that R(t+T)=R(t) when x/y is irrational. So R is not periodic if x/y is irrational. That’s why rational ratios sound pleasant to our ears and irrationals don’t.
But that’s just theory. Is that the whole picture?
Our ears might not find a combination of notes even if their frequency is rational. If the rational number is a very “weird” one, such as 367/259, the resultant wave is still periodic. But then our ears have a hard time finding the pattern. Because the pattern is not really apparent.
So, we can say that, two notes sound good to our ear if their frequency ratio is a “non-weird” rational number.
All right, it’s music construction time.
So, A has frequency 220Hz, one octave higher A has frequency 440Hz. Hence, the notes in between have frequencies 220r Hz, where 1<r<2. It won’t be wrong to say that music is all about ratios. The musical term for ratio is “interval”.
Now, what’s the simplest ratio between 1 and 2. I know the definition for simplest ratio should vary from person to person. But I think many of you would agree that 1+1/2=3/2 is a nice and simple ratio, and most importantly it doesn’t look like a weird ratio. So firstly we pick r to be 3/2. The musical term for 3/2 ratio is called a “perfect fifth”.
The note with 220*3/2=330Hz frequency is named E. And I think most of you would agree on the fact that A and E sound good together. You can demonstrate it with a piano as before. Here is the resultant wave for a 3/2 frequency ratio:
Now, what ratio should we add next. We added 1+1/2 before. So you might think of 1+1/3 now. In fact 4/3 is a good choice. It looks like a nice, simple, non-weird ratio. The note with frequency 220*4/3 is called D. And you can verify that A and D sound good together. The musical term for this 4/3 ratio is “perfect fourth”. Here is the resultant wave:
These two a widely known as the most pleasing-sounding intervals. The next most pleasing sounding intervals and their corresponding ratios are: major third(5/4), minor third(6/5), major sixth(5/3), minor sixth(8/5), minor seventh(7/4). So we add all these notes, and name them C#, C, F#, F, G.
And the other ratios don’t really sound good. So we’ve got 8 Notes now: A, C, C#, D, E, F, F#, G. Is this the complete list? Do we not need some more? The answer is, yes we need some more.
There are several reasons why this list of 8 notes is incomplete. The first one is, music is not only about consonant notes. If someone makes music with these 8 notes, the music would sound pleasant, no doubt. But it would lack drama and tension (Imagine Hans Zimmer composing a film score with consonant notes only).
But I think the second reason is a more coherent reason for adding more notes. I’m gonna explain it now. We’ve seen before that the perfect fifth of A (220Hz) is E (330Hz). But what about the perfect fifth of E? The perfect fifth of E should have frequency 330*(3/2) = 220*(9/4)Hz. It is more than 440Hz, so this note should exist after the higher A. Therefore, if we half the frequency, that should exist in our 8 notes system. But looking back, we didn’t have the ratio 9/8 in our system. So we better add it, otherwise our system would remain incomplete. So the interval with ratio 9/8 is called major second. And the major second of A will be called B. So we get another note in our system.
Similarly, D is the perfect fourth of A. If we wanna play the minor sixth of D, it has frequency 220*(4/3)*(8/5)=220*(32/15)Hz, which is larger than 440 so it should exist after the higher A. Therefore, we need to add the ratio 16/15 in our system. We call this interval minor second, and we name the minor second of A as A#.
Then C is the major third of A, and we wanna play the perfect fifth of C. It should have frequency 220*(6/5)*(3/2)=220*(9/5)Hz. And hence we add the ratio 9/5 in our system and call this interval major seventh. This major seventh of A is named G#.
We are not done yet. G is the minor seventh of A. Then the major sixth of G will have frequency 220*(7/4)*(8/5)=220*(14/5)Hz. It should exist after the higher A, so we add the ratio 7/5 in our system. This interval is widely known as tritone. And the tritone of A is named D#.
So, now we have constructed 12 notes: A, A#, B, C, C#, D, D#, E, F, F#, G, G#. Now the question is: is this system perfect? Well, my friend, the answer is a bit tricky.
No system is perfect. The best we can have is a good enough system.
First of all, can we add new notes in between the current existing notes? Yes, sure we can add notes in between. As I said, there can be infinitely many different intermediate frequencies. But turns out, adding new notes doesn’t significantly help us in making music. Rather it makes the instrument more complex and confusing.
But this system still lacks something. Let’s try to play the perfect fourth of D. This note has frequency 220*(4/3)*(4/3)=220*(16/9). But 16/9 is not in our ratio list. Shouldn’t we add this ratio too? The answer is, no. 16/9 is approximately 1.7777… which is very close to 1.8=9/5. Human ear can’t really distinguish between 16/9 and 9/5, unless you have an extraordinary pair of ears.
Also, if you play with the ratios (1, 16/15, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 9/5, 2) carefully, you would see that the ratio of the two consecutive ratios is almost constant. Almost. So they almost form a geometric series, but not quite a geometric series. The ratio between two consecutive ratios stays pretty close to 1.06.
As this sequence is not quite a geometric series, we might get into troubles. Here we assumed A to be our root note, and based on that root note and different ratios we created other notes. If we took some other root note, the ratios would still be the same, but the frequency of the notes would be different. So our system fails to be perfect. Here’s a minutephysics video about this problem explained from a different perspective: (spoiler alert, it’s gonna spoil out the solution too)
The solution is: we shall preserve the 2:1 ratio between octaves, and make the other ratios a perfect geometric series. A geometric series has first element 1, 13th element 2. So what’s the common ratio of this geometric series? Using the formula for geometric series:
So instead of those rational ratios, we now use these irrational ratios, 2^{1/12} to 2^{11/12}. These are irrationals, so they should sound dissonant to our ears, right? WRONG!
These irrational numbers are, in fact, very close to the corresponding rationals. So close that our brain thinks them as those rationals. Here is a comparison that shows that every irrational number in the sequence lies within 1% of their corresponding rationals:
So after considering all these, turns out that 12 note system is just enough for making music, and it doesn’t make the system unnecessarily complex. Furthermore, it gives us a good enough system to work with. That’s why only 12 notes is used to make most of the music.
However, some musicians use 24-note system to make music. The notes of this 24-note system that lie in between our normal 12-note system, are called microtones. And music that use microtones are called microtonal music. You can watch this David Bennett video to know more about microtonality:
If you’re interested in music and math, this 3blue1brown video might interest you. It’s more about introduction to measure theory.
