Where does the 12-tone scale come from?

This video explains the origin of the 12-tone scale: why it takes the form it does, and the considerations that shape any tonal system.

CORRECTION: at 15:07 I say that you have to go to 29-TET to beat 12-TET on average proximity to the consonant ratios. In fact, 19-TET beats it on the ratios shown (which is clear from the image on screen), and also beats it (barely, depending on how you count) over 11 common Just Intonation intervals. However 19-TET does include some pretty dissonant intervals, so the guidelines as presented would still lean towards 12-TET. Similarly, 24-TET, having exactly twice the notes of 12-TET, contains all the same intervals as 12-TET, so it is obviously a tie in terms of proximity to consonant intervals, but has extra dissonant intervals.

0:00 intro
0:50 basics (pitch, octaves, intervals, and consonance)
3:02 the most consonant intervals
4:39 consonant intervals among overtones
6:05 consonant intervals in human speech
7:54 building to 12 tones
8:28 guidelines for our tonal system
11:15 other systems
11:39 inconsistency among equivalent intervals
13:01 equal temperament
14:22 other equal temperament options
15:17 discussion of compromises
16:47 channel announcement

Nuances of consonance:

Further reading:

[RobertMilesAI]

This is the best explanation of this subject I've ever seen, thank you

[sayhello]

I'm writing a math report for my high school IB report that's due tomorrow, and you just saved my ass.

[kongqianfu]

Best music theory education channel, hands down. Every explanation is well motivated and intuitive, which is a rarity in music education. I've seen no one else explain the "why" of music theory so clearly

[amj.composer]

I actually wrote a paper for my musicology final, discussing why the perfect fifth is responsible for 12 tone. I've basically touched a lot on the universal nature of the perfect fifth, lol.

[flowjee]

"It´s not instruction, it´s insight" This is so good. I always looked for the underlying Physics of Music explained like this. You explained it extremly well, on the point, clever and efficent(Also just the right amount of foreseeing possible Confusion of the viewer as well as things he already could know and skip)

[Gtr]

this video is what made me realize how special non-equal tempered scales are. Thank you. You've inspired me (along with a math discovery i just made) to make my own!

TLDR: I learned about the harmonic mean, and realized how you can use it to construct scales using the 'reflection' concept discussed in the video, along with the arithmetic mean (average). These scales are very very close to 12-TET.



So I discovered how if you take 4 different means of an octave interval, you can get very close to the 5-8 chromatic scale degrees. Eg, we have 440 Hz (A5), and 880 Hz (A6).
If you take the harmonic mean, you get a perfect 4th. (D5)
The geometric mean, you get the 6th chromatic note (Eb5), in perfect 12-TET tuning.
The arithmetic mean (aka, the average), yields a perfect 5th (E5).
And the Root mean squared seems coincidental, but it actually supports this hypothesis i have. It yields a very close approximation to (F5). (440*sqrt(2.5) = 695.70 Hz).

In this case, we did these 4 means for f vs 2f (440 & 880Hz). But if you do this for 4f, 8f, and 16f, you can fill out most of the chromatic scale. Just use the reflection concept like in the video and keep dividing by 2. You will see a shocking number of candidates.

Strangely, RMS almost always yeilds a 'hit'. Ie, you always get a ratio very close to the 12-TET scale ratios compared to the root. While this seems obvious for powers of 2, the same thing occurs for most integers.

So, most of the chromatic scale can be found with taking these different means using f vs f*2^n. Many yeild very nice rational ratios. Most of the scale can be constructed using only the harmonic and arithmetic means. But seeing the geometric and RMS constantly get soooo close to others is what I'm curious about.

I haven't found a satisfying way to get the 'missing spots' (ie, 3rd and 9th chromatic note). The 9th one is close to the harmonic mean of f vs 5f. The 3rd note (harmonic minor) eludes me. For now, i have justified it as a 'quarter octave', which is the same definition in 12-TET. Which feels like cheating, except further 'so close'-ratios are found when doing means of f vs 2^(n+0.5) (where n is an integer). Very interesting.


Welp, that's my book report. More exploration is needed. Making scales using that reflection concept is very liberating. Really makes you wanna explore the possibilities. 🤔

[rileyandrew]

I’ve been looking for this video ever since I started to learn the piano 15 years ago, but never knew it. Thank you!

[frankhovis]

I'm impressed by how long you managed to hold that G note.

[cyborg]

In this video and others like it by other YouTube creators such as 12tone or David Bennett piano, you excellently explain why we have a chromatic scale as well as the pros and cons.

One question I hadn't thought of until recently is somewhat phrased "Why these 12 notes?" For example we know that 440 Hz is "A" and then everything else is a relative frequency from that. But how did "A" become standardized at 440? It's only in relatively recent times that oscilloscopes, frequency analyzers or other such equipment could determine that frequency.

Traditionally at the beginning of an orchestra performance, the first violin plays an "A" and everyone else in the orchestra tunes to that pitch. A cappella singers often use a pitch pipe to get them on key at the beginning of a song.

But again… How do we know that that first violin or the pitch pipe is actually accurate. When were the standards for notes established so that if you heard a concert in one venue or another or even between performances of the same artists, how did you know that they were on pitch according to some established standard?

[therealwhite]

Okay! Now this makes more sense :)
Thank you so much for making these videos buddy!

[_Niko]

Finally helped me understand Just Intonation, and there was so much I never even thought to ask. This is easily the best educational channel on youtube, you should be a teacher. Seriously, I wish there was someone like you for every subject.

[raykay]

Your breakdowns of something that normally isn't easy to explain, is amazing! You've taken a difficult topic and made it clear in a fun, fast and brilliant way. Love the "simplified" animations too. Thank you so much. I'm donating for sure! (Just did a test donation, will send more if it worked)

[Murrlin]

9:40 The most concise and well layed-out 12-tet (and just int.) explanation I've ever seen!

[Eidolon]

In band our teacher would always tell us to bend the pitch of the third down because it's so sharp in equal temperament. It makes the I chord ring so purely

[telamaes]

As a music theory teacher myself, I'll recommend this to my students. Amazing work in this channel!

[simondemeule]

Amazingly well put together! Thank you for making this!

[johnhricko]

You should be able to get a grant from NEA, or some music teachers group. Very well done, and I would recommend it to anyone who challenges or questions the 12 tone system.

[guillaumebrooking]

You're an absolute reference for me, not only on understanding of music theory, but also on how to explain it. I recommend your videos as much as I can. I also watch them often to memorize how you explain the different subjects.

I really admire your work and hope it will get as famous as it should.

[DJPastaYaY]

Very good video. I really like your way of explaining things!

[belowaverageasian]

Sir, this was an excellent explanation of a topic that has rattled my brain since I was a teen. Thank you so much for such great insight!