1 – Why There are Twelve Notes in Music


Hello guys and welcome to Music Minute, the
hot theory guide to learn those extra concepts the right way, brought to you by stevenjacks.com. My name is Steve and today we’re talking
about why there are twelve pitches in the musical language. First, let’s talk about how we get a pitch,
okay? Where do pitches come from? Okay, well let’s take a piece of string.
Let’s take this one on the right hand side here. And let’s pluck it and we get a certain
note. Okay? So that’s how you get a pitch. If
you were to cut that in half, if you were to take your string, whatever this string
length this is, and you cut it in half, you get this one right here. As you can see, we
have a two-to-one ratio here. The just cuts this one in half, okay? So if you do this, and you play this note,
you get a very interesting relationship. Listen to this one again, and then this one. Okay? If we cut it in half again, we get this
pitch. This is a very interesting relationship because if you were to play all these notes
in a normal scale, these would all be considered “Do” or they would all be “Re” or
“Mi” or whatever they are. So you’d go “Do”, “Re”, “Mi”, “Fa”,
“Sol”, “La”, “Ti”, “Do”,“Re”, “Mi”, “Fa”, “Sol”, “La”, “Ti”,
“Do”. Okay? So this two-to-one relationship, or, in this case, four-to-one, or eight-to-one,
or… (you just keep going in powers of two), it will always result in the same Pitch Class,
or “Do” “Do” “Do” “Do” “Do”. The interesting thing about the one-to-two
relationship is that it’s been found in nearly every single civilization. People that
had no contact with one another were able to take a string, cut it in half, and say
“Oh! These things are equal!” So the octave is kind of a universal concept
now. But for the sake of this video, and this lesson,
we will actually consider this one to be the lowest pitch. This one will be the middle
pitch, and this one will be the highest pitch. These are inverted. The way that we can make sense of this is
that this is the lowest frequency. This is twice as high in frequency, and this is four
times as high in frequency. So our notes are actually this one, this one, and this one.
Okay? So remember, this is now low, and this is
now high. Which is nice to know, because this is only a very low string, it only got about
there, and this one goes really high on the screen, right? Okay so the question then is “How do you
get other pitches to appear?” Well we just go around with a certain mathematical formula. We get this one by cutting this one in half,
and then multiplying by three. So here’s one, here’s two, and here’s three. This is called a two-to-three relationship.
A two-to-three ratio, if you will. Two-to-three, and this is two-to-three. And just like all the red ones are one-half
of each other going down or twice as high going up, the same happens with this blue
string. This is now half of this one. Therefor this
is twice as high as this one. And this continues for all octaves. Right now, we’re looking
at just two octaves, this “Do” to this “Do”, and this “Do” to that “Do”. But everything is consistent all the way through,
forever, extending in either direction. We’re simply going to see why we have twelve notes
per octave. The reason we use a two-to-three ratio is
because it’s the next simplest interval after one-to-two. So if you were to continue
this, your next one would be here. Now, you might be thinking well, shouldn’t this one
be on this side, or shouldn’t this one be where this one is? And you’re absolutely
right. But the reason that we’re able to put them here is we simply cut them in half.
So this one will also show up on the right hand side here, but we put it here.
And we put it here. Ta da. Thus we can fill in our octave. So if we keep going with this relationship,
we get these notes. If we stop at twelve, it looks really nice. We actually have pretty equidistant – I know
that this looks really short, this looks really long and that’s actually correct, that it
should be a little bit long, a little bit short in some places, because this system
is not perfect and there’s a reason why. The reason it’s not perfect is because you’ll
never actually end up coming back to this starting pitch. It’s impossible. The reason
it’s impossible is because powers of two, that we’re multiplying by, and dividing
by three, or multiplying by three and dividing by two, the powers of two and the powers of
three will never ever equal each other. The easiest way to argue this is that a power
of two will always be an even number and a power of three will always be an odd number. You can get really close, sure. If you continue
this pattern, you will actually be able to get really close to where we started. But
you’ll never end up exactly. So this is what twelve looks like. If we keep going, we see that our next pitch
is really close to our starting pitch, right there. And yes, if we went further, we’d
see one right here as well. Okay but if we keep going, you can see everything kind of
gets doubled for a little bit. And if we stop at twenty-four it looks kind of nice because
you have twenty-four different pitches each in pairs of two. But the problem is that nothing is really
equidistant anymore. You see the tiny gap between this one and the large gap between
this? That’s not really equal anymore, okay? If we keep going, we’ll eventually even
it out. And as I approach fifty three, you can see that it’s getting pretty good. If
we stop at fifty-three, it looks pretty fleshed out. This system does technically work. You have
equidistant pitches, you have fifty-three of them, but they are equidistant, they do
work out, it does look pretty nice. The problem is now you have fifty-three notes
in one octave. Arguably, fifty-three is a little too many
per octave. I mean, twelve is nice, fifty-three is nice, but it’s kind of a little too much,
don’t you think? If we keep going, we will see that this never,
ever stops. This pattern continues forever. At 306, it looks pretty good, but again you
have 306 notes per octave, so that’s kind of crazy. I’ll go back down to 300 just
so we can jump by hundreds easier. And you can see that this pattern completely
fleshes itself out into a full color spectrum. Okay if you look at this – it looks gorgeous,
right? But look. You have 7600 notes per octave. That’s a little too much. Okay let’s go onto another visual representation
of why we have twelve notes per octave. Okay so here we have the same thing. But instead
of a continuous line that we can only see two octaves of, this circle represents one
octave, so we can keep going around and around and around and the notes will just stack. So if we were to go to the next one, it would
be about here – about two-thirds trough. Again, these are the same two pitches we saw previously.
From there, our next stop will be about here, and then we can keep going, adding the next
notes. And again, we can get to twelve and it looks
pretty fleshed out. If you have a sharp eye, you can see that
these two notes are a little closer together than these two notes, and that’s because
the system is not perfect. It will keep going in this pattern forever, filling this entire
circle. Let’s continue and you can see all the notes
filling out. Again, if we stop here, it looks pretty good.
Everything is really quite equidistant, and it does work out. Again you have that fifty-three
notes per octave. Remember that this works out because we’re
comparing the octave relationship of one-to-two with the next ratio of two-to-three. If we were to continue, we’d get these notes. Notice that these are, again, really close
each other and everything will start pairing up. Here’s its pair, there’s a pair, there’s
a pair, and it keeps going. Okay, if we continue forever, we will never ever get back to where
we started, but we will fill out the circle indefinitely. Let’s go back to fifty-three. There are
certain problems to be found with fifty three notes per octave. The first problem is that
two notes might seem to almost sound the same. When you play one note, and you play another
note, they might be too close together to actually tell the difference. For a visual representation of that, if you
look at these two right here, you might argue that they are the same color. But based on
the algorithm, they are actually different in color. It’s very slight, and it’s very hard to
tell the difference, but they are actually different colors. If you only have twelve, you’re able to
see the colors quite vividly. Another problem with fifty-three is that the
possibilities for dissonance is incredible. When you only have twelve notes, you usually
don’t have too many problems for dissonant possibilities. But with fifty-three notes,
you usually will have a bigger problem to deal with when it comes to dissonance. The third and probably most practical reason
why fifty-three notes is so difficult to work with is because you would then have to make
instruments that were capable of playing fifty three notes in one octave. You would also then have to make music software
that would be able to support fifty-three notes per octave. These kind of reasons make it a little difficult
to make this a plausible system. So let’s go back to twelve. This system is great but if you were start
in the wrong place, certain things – for example, this note to this note would sound terrible.
This interval is called a Perfect Fifth, which sounds like this. And this interval is called a Wolf Fifth,
which sounds like this. This is because this not wants to go to the next note here. Because the pattern can never finish, we always
have to start in a certain place to make the notes we’re playing sound great. This is the problem found with Pythagorean
tuning, which is what this is, where you take a ratio of one-to-two and you compare it with
a ratio of two-to-three. To fix this problem, you can simply do something
called Equal Temperament, which was big in the 1600’s. The theorists of the 1600’s took all the
notes and made them equally spaced, like this. As you can see, our first note starts in the
same place. But now, when we go to our next stops, they get slightly more off tune. Our final stop looks really off right here,
but the nice part is all of these notes on the outside ring here are now equidistant.
They are evenly spaced, just as the numbers found on a clock. The pros found with this is that we can now
start anywhere and any relationship from any one note to any note following in its pattern
is the same everywhere. This is completely uniform. Thus you can change your keys, you can play
different scales, and everything and it all works, regardless of where you start. The
problem with this is that our perfect fifth of three-to-two is no longer correct. This
is slightly off to compensate. It’s okay, though, because human ears have
a hard time detecting the difference. After all, if we look at these two pitches right
here, these are very similar, just like we had the problem with the fifty-three note
system. These are too close together to actually hear
a difference, especially when played in context. Thus, our perfect system of twelve appears. So, overall, it’s because of that two-to-three
relationship mixed with that two to three relationship that has yielded twelve notes
per octave. If you wanted to make another system that compared different intervals,
you could do it with another ratio instead of that three-to-two. Instead of using a three-to-two or two-to-three
ratio, you could use the next step. The next simplest ratio is one-to-five. The reason the next step is one-to-five is
because one-to-four is just like doing one-to-two. Remember that all powers of two are equal. Instead of one-to-five, you could do four-to-five,
or even eight-to-five, these are all okay. The next steps after one-to-five are one-to-seven,
one-to-eleven, one-to-thirteen, et cetera. But the problem with using more and more complex
ratios like this, is that your initial interval doesn’t sound very good. Therefor all the
notes in this system will be based off that one really strange interval, and it might
make a really interesting sound. It certainly might be a great way to explore new territories
of music, but it might not sound very good. And that, my friends, is why we have twelve
notes in the musical system.

100 comments

This is beautifully explained up until step 3. From there the explanation goes off track on a couple of fronts.

In step 2 you explain that you cut the first length in the octave in half and multiply that result by 3 to get the next note. You also appear to place the 2:3 length equidistant between the first and second string. (Likewise the second and third up to that point in the video) But when you get to step 3, the formula for the 3rd length (the new 2nd note) is never given, and suddenly the distance between them doesn’t follow the apparent logic of step 2.

And suddenly you draw attention to which octave your placing the new second note in, which really has nothing to do with understanding the logic of the length of the string and the distance between them. From there on you’ve lost the viewer, and in the end the only explanation for why we have 12 notes that I got out of this is that it looks nice.

The circle isn’t well explained either and doesn’t dispel the confusion accrued at that point, though I see what you are trying to explain about equal tempered scales.

Lastly, there is absolutely no need to invert the visual length of the strings so the lowest is the shortest string – it adds to the cognitive dissonance!

I think you have a great idea and should redo this video and address these deficiencies in you explanation. When you get to the circular explanation, likewise be explicit on how you arrive at the distances between notes. Also sound intervals together so we can hear the dissonances for multiple sets of notes – you only did that once.

Hey Steven, here's how you teach musical concepts and deliver them in a manner easily understood by your audience: https://www.youtube.com/watch?v=mdEcLQ_RQPY

Excellent ! Now I understand equal temperament…….or I did 5 minutes ago when I watched this, now its gone again 🙄

?? STEVEN — The LONTEST string has the LOWEST note and the SHORTEST string has the HIGHEST note.!. You have it backward!! Wavelengths bear and inverse relationship to frequency. The longer the wave the lower the frequency! A string vibrates is just a standing wave, so the formula holds for vibrating strings. On a guitar when you shorten the string by putting your finger behind a fret you shorten the portion of the string that will vibrate when you pluck it and the pitch is higher than if you allow the entire string from nut to bridge to vibrate. I am sure know this but you goofed!.

I never watched beyond the point where you had this glaring error in your explanation. Anyway, the way I teach this is to show the student the relationship between the harmonic that creates the interval we call the FIFTH and the harmonic that creates the interval we call the OCITAVE. and I show they are approximately "commensurate". I demonstrate how the ratio of frequencies between a note and one that is a fifth higher is 2:3 and the ratio of the frequencies between a note and one that is an octave higher is 2:4 (same as 1:2). Then I explain what "stacking" means and I demenstrate that a stack of 12 fifths is almost exactly the same as a a stack of 8 octaves. It is very close (though not exact because powers of 3 not being commensurate with powers of 2. Besides showing this on the piano starting at F (and cheating when you run out of keys on the right end) after 12 fifth intervals you get back to the F that is 8 octaves above the first F (if the piano had 8 octaves instead of 7). Also you can show them mathematically that frequencies involved with the stacking of octavs is base on a powers of 2 is an even number. But powers of 3/2 will alway have a fraction.

But WHY THE FIFTH? If you used an interval with a frequency ratio of 3:4 or 3:5 or 4:5 you would get some other number other than 12 that approximately equal to 8 ocatves — but a stacking of FIFTHS COMES MUCH CLOSER THAN ANY OTHER SMALL RATIO. Granted if you are willing to have "taller stack" you might find match with perhaps 20 octaves more. But harmony is best when the ration are involving small number. The first and second harmonic (creating the octave and the fifths) are Golden — and the frequency range of 12 fifths being so close to the frequency range of 8 octaves gives the basis for their being 12 notes in an octave. If you tune and instrument by 5th rather than by octaves you will accumulate an error of .0133. Dividing this error among 8 octaves gives and an error of .0017 per octave and dividing it among all the notes in 8 octaves each note interval would be too large by .00014. Reduce the interval between notes by this much and you would have what is called Pythagorean Tuning. But Octaves would not be exactly in a ratio of 1:2 with such tuning and there would be dissonance on insturments like a piano that has more than 3 octaves. Current tuning makes the intervals between notes equal to the 12rh root of 2. This guarantees that octaves will all be in a ratio of powers of 2 and makes the fifth slightly smaller than 3:2.

I think the confusion came int because in the fist part you used the vertical lines as the length of strings and in the second part you used them like lines on a bar graph with the vertical scale being a frequency rather than a length as they were in the first part. Kind of a "no no" from a teaching point of view (in my opinion). Another flaw I see in this presentation is your assumption (it seems) that the person viewing your explanation is familiar with "the solfa" (do re mi … etc.) It may be reasonable to assume this as anyone interested in knowing why there are 12 notes rather then 10 or 15 will probably have studied music and would have been introduced to the "solfa" and perhaps you have an earlier presentation of the solfa and didn't intend this explanation to stand on its own. But there is nothing in the introduction that indicates a prerequisite. I stopped your video temporarily at 3:50 to make this first part of my comment. On the other hand I guess in my explanation I assume that my reader know the definition of the term "commensurate". So I should probably preface my explanation with the definition of that word.

At marker 5:00 "Powers of 2 will always be a an even number and powers of 3 will always be an odd number" — TRUE but, more importantly, multiplying most numbers by consecutive powers of 3/2 will become fractions well before you reach the 12rh power. But if you keep the arithmetic in pure fractional form then you will have a fraction with an even number on the top and and odd number on the bottom.

On a single viewing it was not perfectly clear to me how you equated the colors with the ptitches — but I must say that your the sound and video of your explanation was VERY "impressive" and your use of the audio with the video was very well done. Your expalantion was far more comprehensive than my explanation — I had never been introduced to the 53 notes per octave. Frankly, I think that your expalation was far more complex than need be to explain the reason there are 12 notes per octave. HOWEVER, I learned something from watching this with regards to spacing. I had not actually been aware of the lack of uniformity of the spacing that your video made clear. I know that in "just intonation" there is a lack of uniformity, but i did NOT realize that there were non uniform spacing in the Pathorean Tuning only that you can't retain perfect harmony for bot the fifth and the octave. You can only have perfect for one or the other … or … you can comprimize and sacrifice both and arrive at system that will work for more than two octaves, but not for more than four.

I VERY MUCH appreciate the effort you have gone to and found your explanation "educational". … but I think for the average viewer it was "overkill" to the extent that a simpler explanation might have given the average person a better grasp. I could be wrong! It is just my honest opinion! Your presentation is valuable enough that I intend to watch it again and I will recommend it to some other people I know who already understand the reason there are 12 notes in an octave, but who can learn from your very comprehensive approach the same as I did. And I will check out other videos you have made if there are any. Peace out!

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What about a 3:4 ratio? I see 2:3 as a great idea, but wouldn't 3:4 be equally as good?

Example: you have a slow Stevie Nicks or Bowie song that you want to speed up (and change the pitch of subsequently). You can speed it up 33.3% according to the 2:3 ratio, and that would sound great but why not speed it up 25% wouldn't that be properly in tune as well? Maybe I should be more specific?

Very good explanation. If you really want to understand it, this is it, very logic. If you're too stupid to deal with it, maybe you should ask easier questions 😀

Great visuals! Dear Haters, if you think you can make a better video – for f#@%'s sake MAKE IT and we'll all watch it ^_^

What doe the number in the upper left represent? it was a ratio now it is just a number. Looks like you skipped a step in communication

Maybe this video is just for people who already understand basic things about music because I understood this explanation perfectly despite not knowing before exactly why there were 12 notes

I am a little puzzled. I just used the log calculator (https://www.calculator.net/log-calculator.html?xv=32&base=2&yv=&x=51&y=20) and if you use 1.4983 (rather than 1.5) as the frequency ratio for the FIFTH and divide by 2 when the value "escapes" the octave the starting note is in, and do this 12 times you will get a number of frequencies which have larger and larger differences as you go toward the higher pitches — but when if you take the inverse log base 2 of the freqencies, all of the ptiches are equally spaced!? It would seem to me that that if I used 1.5, I should just get larger frequecies within the octave but applying the inverse log would still give me pitchies with equal increments between them. So I don't know how you came up with unequal increments. Just as you said, Log base 2 of the frequencies yeilds the pitches, but just as 128 goes to 7, 64 goes to 6 and 32 goes to 5 you still have 5 to 6 = 1 and 6 to 7 = 1 and as long as you are raising a constant number (or fraction) to consectuive powers, the difference between one and the next will be the exactly the same. "like the numbers on a clock" So uncorrected Pythagorean tuning will result in "bloated" octaves, but if you subtract the inverse log of one from the inverse log of the next, the difference in pitch from one to the next will be uniform.

Both corrected (1.4983) fifths and uncorrected (1.5) fifths will give you TET tuning. For example, if I use the fifth fret (or an electronic tuner) to tune my guitar (which makes a fifth 1.4983 because the frets are postitioned so that the middle of the string is precisely over the 12rh fret) the 1/3 point node is NOT precisely over the 7th fret and the 1/4 node is not precisely over the 5th fret, so I will get "beats" when I use nodes to try to mach the harmonics. If I count the beats and memorize the number of beats for each pair then I can use beats to tune my guitar, assume I memorize the beats per second for each pair when the guitar is precisely tuned to standard pitch. If I were to move the bridge slightly towards the nut. then the middle of the string would be displaced slightly down from the 12th fret rather than above it but with proper adjustment of the bridge the 1/3 node would be precisely over the 7th fret and I could create Pythagorean tuning by eliminating all beats using harmonics, but it would still be TET because of the logarithmic placement of the frets have not changed. That is how I see it anyway. I could be wrong!

After reading some of the other comments I was relieved to see I wasn't the only one confused by this video. I give Steven Jacks credit for presenting some interesting ideas but he would do well to redo this video addressing the issues many of us had (eg, @ 3:52 where you move the vertical lines around for no good reason?!?).

It's called creation, intelligent design. One of God's great numbers is twelve, 12 apostles, 12 tribes of Israel 12 foundations of the new Jerusalem (in Revelation) etc. 12 is one number that God has made, but then there is 7 tones and 5 semi tones, 7 the number of perfection and 5 the number of grace or the Holy Spirit
God gave man certain things that are almost infinite. Math, music, art, even maybe more but also the fact that man and woman could create life by having a child.

Sounds interesting. Read somewhere that triangle guy, Pythagoras,w as working on the octave. The video is dark on my phone.. it may be a glitch on my phone. Otherwise good video. I used to love to do calculus. This appeals to me.

I liked your method of illustrating the concepts Thanks for clarifying something that has confused me for a long while. Somebody has to build the 53 note instrument soon.

Fyi, your 1st ratio example is backwards, you stated a 3:1 three to one ratio but the is 1:3 one to three.

Just stop! You don't know anything about math or music. I hope no one believes what you are saying. This is the biggest bunch of nothing I have ever heard anyone try to explain something.

Your so-called reason for twelve note system isn't even close to the real reason. Man did not invent music. Music existed a long time before man existed. The angels from heaven introduced music to man. Also, the very first human was Adam. The creator designed his physical ear structure to be perfectly tuned to heaven's absolutely perfectly designed twelve tone system. Ever since, mankind has nothing other than the perfect system given to us by God. The beauty of this system is recognized by every human, because we are designed that way. Here is the truth: we humans recognize and use the twelve tone system because the Creator designed that system into our very mechanical design and we know of nothing else. It is perfect. That is the declaration of the Creator when he finished his work of all creation. It was perfect good and lovely. All was in harmony. As he gave us the same perfect musical foundation and system, it had to be perfect. God knows nothing other than perfection when he creates anything. You can blather on about fractions and such, but you still know nothing of the foundation upon which perfection is designed. God designed perfect music into our physical structure and into our mental perception and programming inside our brains. That is the real truth of our musical system.

Music, like all electromagnetic waves are curves, not linear, trying to fit a curve into a linear arrangement causes alignment errors. like PI, you cannot measure a curve with a line you just end up
breaking the curve into infinite line segments never arriving at an accurate measurement.

You should have started out by saying, "….twelve tones in European Music". Some other cultures, like Oriental and Persian music, have pitched instruments that can play multiple tones in between C and C#, for example. Western music calls these quarter tones.

If you want to get super technical there are an infinite number of notes in the music of the universe, and there are an infinite number of notes between any two notes. Going up in pitch from, say, middle C note to the following D note, there are an infinite number of pitches ascending from C to D. I don't know why nobody has ever experimented with non-traditional notes. I bet you could make some cool and strange alien sounding music with the notes in between notes.

Thanks to the previous comments, down below, I will save some valuable minutes of my precious time.  So, thank you all.

Very well done and I see with approval all the work you have done in the past several years. You MIGHT like to add the underlying physics of sound (acoustics) which shows how octaves, harmonies etc., are actual physical properties of matter. Nothing about the octaves fifths, etc. are arbitrary except to simplify them for convenience. A 'true' octave with all the pitches corresponding to each key (another concept you have in your site) is a massive number of frequencies, over 400 if memory is still right. Thanks and keep up the intelligent work.

There are an infinite number of notes in music. The octave is the ONLY interval in standard 12-Tone tuning that is in tune, ALL other intervals have been mistuned, or ‘tempered’. So most folks have never heard or played a true in-tune chord.
But the infinite variety of pitches in the Harmonic Series can be used to play any type of music. There’s even a company, FreeNote Music, that makes guitars with the frets in the proper place. And there are groups like Fretless Brothers who play jazz in pure Harmonic tuning.

Wow. Thanks! I've probably seen a half-dozen different explanations of these ideas, but this was the first that made sense instantly! I think your graphics helped a great deal. Nice work.

The craziest part of this whole thing is that it's based on taking two strings and "demonstrating" that making one half the length of the other, you get an octave. Well you do, after you tune it. Think about a guitar with multiple octaves using strings that are all the same (!) length. If you were using the bars on a xylophone, the physical relationship of length at least has some validity. This is the most convoluted explanation that I've ever heard — tossing off terms like "powers of two" that shows a complete lack of mathematical background, and in the end totally fails to justify why we have twelve notes. How very sad.

I had to watch a couple of parts a few times over to really get it, but on the whole I found this really helpful for understanding the 12 note octave. Thank you!

If you had a 53-note octave, wouldn't that be a 53-tave? A fifty-threave? 53-ve? You know, since octave refers to the "eighth note" of a traditional 7-note scale.

So, when the 12 note system (or equal temperament) was invented, it was to make it easier for music software? Hmm.

Why the fuck you inverted the strings length with high and low frecuencies? You will only confuse beginners. Also the idea of pitch do not come from strings vibrations, were around much before. I could go on if I continue to watch this video, but i cant.

This is horrible for someone new to theory. The people that it would help are those that don’t need help

did anyone notice ow the three notes in the beginning are the same as 2001: a space odyssey's intro?

I like your atempt to explain something that is literally out of this world. But truely your information gives we humans an idea of what true music is in the spiritual realm.

On earth, everything is limited, life span, gravity, even music. That is why life on earth is a passage to the real life in eternity. If you have ever heard the angels sing, thier octave range is way beyond what we consider music on earth. If you ever had visions of heaven, music can be visually seen as a spectrum of brilliant colours.

It pains my heart when I hear people ignorantly speak of heaven to be a boring place or a place not to be desired.

The main argument here seems to be that the human ear has difficulty distinguishing the intervals in a 53-tone scale, thus we revert to the simpler 12-tone scale as a matter of convention. Another important point, which seems to get a bit lost in the mathematical explanation here, is that 'tempered' scales were developed so that songs could be played in any key. The intervals of the naturally-occurring "just" scales that arise out of string lengths being divided in half, for example, will be different depending on the differing tonic notes of the keys chosen. Thus, musicians in the late 16th century introduced tempered scales with equal intervals between notes so that songs could be easily transferred from one key to another and still be recognizable in those different keys.

What if instead of looking at it as a circle you look at it as a spiral in 3D? could that solve the relationship problem?

This video has several merits, but how do you reach that interval a 3:50 ?? Is it 3:5, 2:5, or what? It isn't explained at all !!

Of all the problems one may have with having 53 notes in an octave, I'd say software is pretty down the list

When he started, the short string was the higher sound. Then he quickly reversed it, that the long string was the high sound. And that was WRONG. To me, this does NOT explain why we have 12 notes!

Imagine notating with 306 notes per octave…

Composer: "Uh, yes, that's going to be played in BBBBBBBBb or 8Bb, and then do trills between 4A# and 19Hb.

So you showed us how not not calculate notes…
When I was learning c++ coding in 2001, I wanted to make a textfile noteplayer. The information I had was the notes, and 440 herz en 880 herz.
Trying to find out what sounded right, I first tried equal stepsize.
So 440-880=440
440/12= 36.6
just added 36,6666666 when going up one note.
I sounded terrible.
I then tried to find out what the frequency of the next note was. Just by listening to a keyboard and producing it with the PC speaker.
I found out to sound normal, the stepsize should grow. Meaning it's logorithmical.
Now to find out what it was.
By some testing I found a number of 1.594 in growth. This was ofcourse not really exact. But it sounded much better.
I also noticed when reaching the 880 it actually was really close to 880.
It was about 879,3713 herz.
I guesed this could be done more exact.
The formula the calculate it correctly is calculating the 12the power of 2.
I didn't know it was named that way, but I just tried until I found a way in c++.
Try it on your calculator (windows calculator also works)
Press the buttons in order [2] [x^y] [ ( ] [1] [/] [12] [=]
This will actually make 2^(1/12).
It will gues you something similar to 1.0594630943592952645618252949463 this will go on much longer behind the decimal point.
If you want to find the frequency of a note, take the base note C (depends on the octave) multiply it by that number, you get a c# the result can than also be multiplied with that long number for the next note. And so on.
It will reach the 880 almost perfectly. It only is off because the number isn't accurate enough.
You won't hear it, because its about 879,99999999999999999999999999958
Much better then using the 2:3 and 1:2 way. that doesn't even come close

This is incorrect. Stacked 5ths do not underly the tempered scale. E.g, a perfect 4th has the ratio 4/3=1.333. The 6th, 5/3. That is, intervals sound "good" or consonant because the frequency ratios are ratios of small numbers, Instead you are using powers of 3/2, which give 9/4, 27/8, etc. Now note that the number 1.059463 to the 12th power =2. So the ratio of successive frequencies in the 12 tone tempered scale is 1.059463. For the 4th, 1.059463^5=1.335, which is very close to 1.33. In the same way, for most of the other notes, the "tempered" frequency is very close to the "perfect" ratios of 3/2, 4/3, 5/3, 5/4, 7/4, etc. (The minor 2nd ratio is close to 18/17 – big numbers, which is why it sounds dissonant against the root.). So, twelve equally spaces tones allow us to get very close to "pleasing" ratios, independent of key, without needing very many notes. That is why there are 12 notes in the western chromatic scale. You need to change your circle analysis to show dots at the simple ratios, not powers of 3/2.

Weak logic. Broken in fact.

There is NO need for 12 notes designing musical systems. NONE.

So… you say 53 bad because:
'Because there are more options for dissonance'.
So what? 53 gives more OPTIONS, that's a GOOD thing for intelligent people.

Your other answer was "because we'd have to design new software" as if that's difficult (it's not). Dude wake up, software has existed for years, microtonal synth patches have existed for years.

Your answers are MISLEADING PEOPLE. It's BS.

Sounds like you have run out of things to do. next … why are there 18 notes in an octave of eastern music? why do we have 5 fingers and not 6? why are the black key black and the white keys white? Why do cars have 4 tires? Why do we have only two eyes?

"but the problem with…" rinse and repeat…
"but the problem with…" rinse and repeat…
"but the problem with…" rinse and repeat

Excellent presentation! This should be required viewing for anyone picking up an instrument for the first time.

your'e are pretty bad at explaining the issue at hand. though, since i know what you are talking about, i must say it is difficult to explain. especially to a non musician.
and the answer to the title question is : it's quite arbitrary. 12 notes is a western standard, a 12 edo ( equal division of scale or octave). there are other divisions around the world, though in modern times, with prevalence of western popular culture, they have became an oddity. there are 17 edo, 19 edo…. so on. meaning -the octave separated into 17 or 19 pitches, not 12.

You lost me specifically at 3:50, I don’t get what ‘carrying this pattern on’ entails or why you did what you did. Frustrating.

Well, if only we use sort of belief historical connection in relation with the 12 notes used in the western modern music – hence we will get 12 half notes with distance 12 root square of 2 which doubled every octave, isn't it? That is even tuning if I still remember, even though historically – tuning is not that simple: Kimberger 1&2, etc. Coincidences? Number of months, Jesus's disciples, Jacob/Israel's tribes, etc. What a mystery, indeed. Let us be aware of the cause of confusion: Gen11 if I still remember again. Anyhow, thank you for sharing sort of another way of approach to introduce the wonder of music. Apart of those frequencies sets of ethnic music scales from all over the world. Yet, not forgetting the debate whether we should use either 432Hz or 440Hz based tunings. Hahaha, music is kind of mystery – but for sure, our bodies are musical instruments lol – not only Lucifer. Oops, sorry – too long.

Hey! I really like your movie. One thing is bothering me. How do you put points on the wheel? Do you arrange them for a fixed angle? (Certainly not 2/3 of the circle, because the points would meet). How much does he exactly? What does it result from? Please reply. Best wishes!

It's is obvious you don't understand music theory.
.
Try picking up a book on the subject.
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Stop confusing people with you ignorance.
.

.

This person is rewarding his own dopaminergic self.A self talk.
Who is the suspect?Indeed its ME!!!!.
Although I am sure some geniuses out there will immediately scream I get it while.me scratching my head suicidal.

3:45 "So if you are to continue this the next one would be here …" – if you continue what? How do you arrive at this third note? You completely fail to explain this crucial step (I'd guess it's 2/3 or 3/2 of something else, but I don't see that), so I still don't know how you end up with 12 notes.

Before 3:49 the blue vs red lines are at 2:3 ratio. But at 3:49 yellow line appears and is 2:3 against which line?

I really found this video to me interesting. I think basic comprehension of music theory helps to understand what he's talking about, and that most people who are confused are feeling so because they may lack simple music theory knowledge (e.g.: octaves, dissonance, wolf notes, etc.)

After pointing out that you are covering two octaves (do- to do- to do-), you start filling in and referring to the added notes as "306 in one octave*" (!) WTH?!? NO! It would be *153 per octave, 306 over the TWO octaves!! Try slowing down and thinking before you speak– as HD Music
commented @ 4 months ago, "If you can’t explain it simply, you don’t understand it well enough." – Albert Einstein….. Back to the woodshed, young Paduwan!!

Steven, this video is an absolute gem. This was the best explanation of the 12 tone octave I have ever seen, and I have seen a lot of them. You also managed to briefly touch on intonation (a very challenging subject), and visually it made so much sense. Thank you!

So can someone explain because this idiot explained it to me (actually all of us) like I know university-level science lmao

Threw me when he changed length of lines to refer to frequency. But persisted
Shows people in 1600s were very clever.

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