I’m an engineer. I started learning guitar and eventually found a book on music theory that was tremendously helpful. I ended up with a spreadsheet with the modes, chords, scales, and a basic chord-builder, etc. I still can’t play guitar.

For those wondering how any of this could possibly be useful: Forming connections. Looking at topics from weird perspectives is fundamental to efficiently learning-by-doing and making personal breakthroughs, both in that topic and in others. As a kid you probably spent dozend of cumulative hours digging through your Lego bin looking for a specific brick and instead finding half the pieces you were looking for *yesterday*. Same deal with mental connections and especially breakthroughs, they appear most consistently when you're focusing on something else and some set of conditions triggers a sense of familiarity.

Once you try to connect music theory to your ears, it's no long 8th grade math; it's years of work.

This only makes sense to those who have studied theory already. If you're a beginner, you're gonna end up a tad miffed by this guy, to put it lightly.

This video really makes the major scale visually understandable. The circles make it look sonic and they hit all the correct frets. A great bit of history in there too.

Hello, I genuine enjoy the perspective you offer. I think anyone who complains that the video is unnecessarily complicated is unnecessary complicated themselves because, clearly, this video is about exploring music through a mathematical lens. It's meant to offer a different perspective to those who already know music theory not teach beginners the fundamentals. There are thousands of videos that explain music theory in "simple" ways on KZfaq and having the choice to explore it this way offers unique insights for me. The diagram was especially appreciated. However, I did note one error worthy of correction: The etymological roots of the term scale in music are not from weights but late middle English from the Latin word "scala" meaning ladder. I'd also appreciate also appreciate recognition of the equal temperament and how, despite it being the standard, does not apply in this context. Thank you.

Terrific video. Thank you 🙏 In fact the very BEST that I’ve found, on where the major scale comes from, and why. The circles do really help!

That last complete image is astoundingly beautiful. It would be great wall art... serving as a functional reference also. I've known of the relation of most of the harmonics on the strings, but never ever seen it laid out so visually stunning. I can see why it all makes sense now. Thank you!!

These visualizations are phenomenal. Hope to see your channel grow more soon!

Thank you so much. I have always shied away from learning music theory because so much of it just seems like "memorize this" without explaining why. I can't tell you how many times my teacher just listed off the intervals, calling some major, some minor, some perfect, and just wanting me to know their names like it meant anything to me. So many things in music theory for me have just been "this is what we call it when this happens" and it feels like no matter how many things I memorize I still don't know how they fit together or what their purpose is. Thank you for explaining it like the science that it is, instead of making me learn it like a language.

The idea that you can "weigh" the notes, hence the name "musical scale" probably only works in the English world. In french, it's "gamme" (series of things of the same nature but with nuances, as in "shades" in english. "different shares of blue"), in Spanish it's "escala" as in scale in the context of size (1:2 scale of a map for example) or ladder; in German it's "Tonleiter", literally "tone ladder".

The last math class I received a passing grade was in the 3rd grade. I had to repeat 8th grade algebra 4 times and the only reason I was not held back a grade was that all my other subject I had 100%, but math my grades were never above D-. In college I chose journalism as my major because it only required the most basic math. I still had to take remedial math six time and ended up dating the last teacher, a graduate assistant and she mercifully passed me because I simply never got it. Now in my 50s I never use any form of math beyond basic addition etc and always with a calculator. I've been a musician for 42 years and have never been able to grasp music theory. I had to stop this video halfway through because I was literally gasping for breath, sweating and with tears in my eyes.

The geometric visuals are great. It has helped me understand the relationship of the frets to the strings. I always wondered why a guitar seemed so different than a piano. I wondered why it's layed out like it is. Its the way it has to be for western music. This was enlightening.

The History was interesting, but all the circles on the fret board got confusing. Especially when they went behind the 12th fret Colors might help, a different one for each interval.

wow, welcome back! I just found your channel about a week ago and it has been a revelation. Just when I thought this had become a legacy channel and then this upload shows up. Please keep it up

THE LEGEND HAS RETURNED!

Thank you so much for this content. I really know a lot about music theory and had even studied about these measures before but it was the first time I realized the importance of the perfect intervals and their relation mathematically to compose the major and minor keys. You just helped me so much to organize all the content in my head. Thank you so much!

Pythagoras just adapted his maths to existing musical scales. He did not invented them as widely reported. The picture next my username shows a statue of a man playing a harp. This statue is dated 2800 BC. Do you think a harp can be constructed and played without the use of scales? Now, Pythagoras lived some 2200 years after that statue was made. So little we know about history

As a music nerd.... This hits the spot 😂

Amazing. The ends of the circles are also the natural harmonics, by touching strings there you set the string into another vibration mode. The visuals are beautiful too nice job!

(2) in "Slicex" you could create New Drum Loops and softs in any order from different slices. And keep the loop slices playing in and

This is absolutely golden. Thank you so much for this. That last picture I actually have done when working with geometry a couple of years back. This makes so much sense. Huge thank you

I saw a Luthier do something similar once with a compass and protractor ... he was working out the spacing of the frets so i'd imagine he was just working out step one of this same system. Very interesting video. Seeing it all visually explained like that is mindblowing. Well done.

Per Lydian chromatic theory, "the" major scale built on perfect fifths would be the Lydian scale. (And therefore, as Miles Davis once said, the center note of the piano keyboard should be "middle F" rather than "middle C"...!)

There is so much deeper things than this. Bach actually used set theory for his last unfinished fugue and the most simple think about the main theme is it's symmetrical. The work is called The Art of Fugue BWV 1080

This is absolutely golden. Thank you so much for this. That last picture I actually have done when working with geometry a couple of years back. This makes so much sense. Huge thank you 🙏

Your series is my favorite music theory on youtube

I have previously only seen this ratio stuff presented in scattershot form, a bit here and a bit there. Nothing systematic like this with someone working out a whole scale from fundamentals. So this was revelatory! Thank you! :)

I just stumbled across this channel and it's just awesome. For us who need to know why things are the way they are, as opposed to just read and rehearse rules or facts, to be able to learn them this is gold. Thank you. Now comes the question. Why did you stop and where did you go? You truly made some really good and useful videos. So if you ever decide to make some more I'm sure a lot of people would be happy.

What were the historical music theory books you read? I'd like to dig them up if I can. Thanks for the content!

The best video I’ve seen so far explaining this. Good job!

This is a very nice visual representation and also very nice in terms of the mathematical relationships. I really like having this additional information in my arsenal when explaining things to students. Yet, I do have to agree with many of the comments below. This is only clear, when one already "presupposes" that the student understands what a major and a minor chord is, or an interval of a major or minor, for that matter. As you were frustrated, wanting to know "why these notes", a student of yours, learning this as their first exposure to major scales may ask: "why a major chord, and, where the heck does it come from". It also only makes any sense when one, again, presupposes that the new learner has any clue as to which notes they want to "discover" in the major scale. A huge amount of the steps used in this explanation are built upon the assumption that the aim of the discovery is to "find" the notes of a major scale, and also on multiple (sometimes complex) principles in music. As a teaching technique, this does not clarify much for a new student off the bat. It does absolutely help someone who already knows music theory, to have a nice additional angle to understand relationships of notes from a mathematical stance, and also be aware of how the notes relate to each other. This is not, however, a plausible route for explaining how we came to have the actual 12 notes, or the actual 7 notes of either diatonic music or the major scale. Using the above process, one would not "discover" the notes of the scale unless you set out to "prove where they lie", based on your already existing understanding of all of them. Take as an example the 2nd. From an uniformed perspective, why would you use the 9th, and then lower it to the 2nd? That can only be done post-understanding the scale already. Personally, I am not entirely sure how we actually came to have the 7 steps of the major scale, as I stopped studying formal theory after completing G6 theory exam - I then continued learning as I performed in the industry along with other musicians. BUT, I am inclined to think that it grew out of the pentatonic scale more than it grew out of someone using the above method. The pentatonic scale has been around all over the planet, and for ages longer than the major/minor scales, and it does seem somewhat likely that a next experimental step from there could lead to a major scale. Over-and-above that, the pentatonic scale is something that appears to be very much "inbuilt" into most people, and has an easy execution by most humans, regardless of their musical training. Sure, this may be because of it's use over centuries, I agree, but, the ease with which it "falls on the ear" may play a role. As far as I understand it, also, the very major/minor/augmented/diminished intervals owe their description to how they appear in a major or minor scale (scales which you would need BEFORE you understand or describe the intervals), and then, the 4 main chord groups (major/minor/augmented/diminished) owe their existence and description to the combination of intervals (specifically a third and a fifth above 1). Thus, the explanation above, though insightful, does appear cart-before-horse.

Glad you're back!

Excellent. Learning guitar as a pianist and drummer has opened up theory in my mind, took 20 years. Golden ratio has entered the zeitgeist.

Video: Music theory is simple Video: Pythagoras listening to E=MC hammer

Mathematics in music is like the skeleton of our physical body. It's the soul of it that makes it beautiful.

Glad to see you back

Outstanding! I have been wondering about all of this for the last several years. Amature self taught musician. Thanks!

glad you're back!

Everyone learns a bit differently and this video is no more complicated than any other that talks about the "why" of music theory. If it didn't make sense to you, then you simply don't think the same way. I'm very much a neophyte, and I'm more in my element with math than music theory to begin with, but I learned plenty from this video. Take the tidbits you can use and they'll accumulate over time, just don't stop learning. And to anyone who feels they don't have the music theory chops to follow, I recommend doing some reading/watching about the basics of scales, degrees, and intervals. Just the 101 stuff, and you'll be following along in no time.

Music notation is a special purpose descriptive language. Mathematical notation is a general purpose descriptive language. Do not confuse one with the other.

…and when music students needed him most he vanished… Joking aside this is the type of stuff I needed to see first as a prep before picking up a guitar, someone who will actually show me what these terms mean and have a backstory as to its origins making it easier to remember, not just giving me some pneumonic device and calling it a day. Students already have the desire to learn, but this is a much more fundamental way of teaching it in the system we have. Unless we can all agree upon a new system for creating music that isn’t ancient we will have people who fell through the cracks like me. I’m luckily only in my mid twenties but I’ve been interested in music all my life, taking it the most seriously in High School and now College. Teach this to kids old enough to to understand it.

Straightforward algebra does indeed capture harmonics. It's only fractions, and you can algebraically solve for the rest.

It is actually more geometry than algebra.

Now that's pretty cool. From time to time I wondered how what we know now came to be and how in hell it is what it is..... amazing.... thank you

I finally realized why my parents gave me that silly spirograph game for Christmas.

I taught algebra to year 8. I taught it very well. The idea of teaching music theory to Year 8 is way more challenging I believe than algebra. The problem is partly that children are taught a lot of basic numeracy from year 1. I wish that the case was the same for basic solfège etc, but sadly it is not the case.

Almost. This confirms my past statements about graphic animations and teaching. That just because you can synchronize words with graphics does NOT mean that you are communicating or imparting knowledge.

Great vid! Going back and watching the old ones now. Subbed

Very well done. Fun explanation.

This is a great visualization, really appreciate the work you put in to this. I’m going to share this. Students of tuning, JI, and specifically those who have studied with or subscribe to the work of W.A. Mathieu would describe what you’re doing here as beginning to build out a 5 limit ‘lattice of tones’. The ratio tunings you cover in this video are the basic Major solfège tones of the Western Musical tradition and they’re also the most basic ‘Sargam’ tones sung by beginning students of Hindustani Classical music (North Indian). My understanding of Pythagorean tuning is that it is based on 3/1 ratio. Learning to recognize and sing pure Pythagorean tunings is challenging and rewarding, the Pythagorean 3rd which unlike the so-called ‘pure’ or ‘just intoned’ third (your third>5/4) is a ratio of 81/64. It is appreciably sharper than both the just third and the equal tempered third. In my experience when sung or played well those Pythagorean tunings have a quality of antiquity to them, it’s interesting to sing a ‘major third’ that feels like a relative of a pure fifth. If you’re not familiar with W.A. Mathieu’s book ‘Harmonic Experience’ I would highly recommend it. It begins by tuning the major scale as you’ve done here (as triadic overtones of the Root, 5th, and 4th) then goes on to deal comprehensively with tuning as it relates to Equal Temperament and makes the case that 12 Tone ET is a negotiation of on the one hand our innate partiality for pure ratio tunings, and on the other our desire for extended range and perceived motion within harmonic territory. There’s actually a lot of good information regarding tuning and tuning systems on Wiki but as far as I know there is no more comprehensive single work on tuning (in the modern era) than the Harmonic Experience. I studied with WAM but I’m not affiliated with the sale of this book, I just honestly think it’s the best book a person can have on the subject. www.amazon.com/Harmonic-Experience-Harmony-Natural-Expression/dp/0892815604

That final diagram minus the fretboard and numbers would make a great coloring book page. It's comforting to know that we had the scale long before the boring lesson in fractions.

Wonderful! Well done

This was amazing, thank you. It asnwered so many questions for me. I will likely print the final screen-shot and have it framed; its a wonderful visualization.

The intention of the video isn’t to teach music theory. It’s to explain, as the title implies, WHY western scales are formed the way they are. It’s very interesting to see how music has basis in math and nature.

Music theory is NOT supposed to be hard. It IS actually very simple when explained in a sequence of logical building block concepts from the foundation up. Although this information you have presented here is very interesting and logical from a mathematical standpoint I believe it to be very confusing to entry level musicians. I find that most musicians are trying to find practical information that can be applied directly to the music making process without too much extraneous thinking about abstract concepts. The math involved in music making is much more simple than you have presented here. It really never needs to go beyond addition, subtraction and division. Why would you use algebra to build major and minor scales when simple math will do the trick?

fantastic video & visualization, thank you

Gonna need some time for this to marinate. I’m not a guitar or string player, so I had to draw on my understanding of waves in physics to grok this.

Our Sensei has returned!

Hi Mark, thanks for creating this video! I saw in a comment from another video you made that you wrote a book. What's the name of your book? I want to buy a copy. Regards, Kevin

I like this take - I don’t think I’d heard anyone talk about building a major chord on the 1, 4, and 5 before to derive the other diatonic nots of the scale. Cool!

Well done, I have been pondering that question for some time...

In a lot of towns, villages and countries around the world are things called schools, with things called books in them.

Loved the video! Great job 🎉

Whoa. That was excellent. Thank you for making this.

What a tangle. Our twelve note scale is taken out of the 15 (16-1) pitches of the sixteenth harmonic and the thirty-second harmonic. Split a string in half, and you have the second harmonic, into four and you get the fourth harmonic. Split it again and.again and.again, and you get 8th, 16th and 32nd harmonics. Between each such halving and the next one, there are more and more harmonics, creating more and more of the scale, and by the time you have Split the string in 16 and up to 32, you have all the true harmonic notes, which now has to be tempered to 12 notes, with close to equal pitch increases, in order that the notes can fit together in a twelve key system and a quint circle. The notes left out from the 15 are between the major third, and the fifth, and between the fifth and the harmonic seventh, which also is adjusted. The fifth is close to true between the 12 and 15 note scales, and so are the pitches outside the third and outside the harmonic seventh. The notes replaced are fewer by 3 and their pitches averaged to suit the pitch distances of the more true pitces, which are also averaged , ie moved minimally in order to make equal all pitch distances. This is the physics of it. However a piano tuner has a slight difference in his tempered tuning for, I think, the key of F to sound cleaner.

This was incredible. Very intresting. What program did u use for the ratios? It would be useful to have the ratios be different colors to more easily see the relationships like superimposing the circle wheel over the circle of fifths. Look into cymatics and synesthesia

This is so cool!!!!!! I’m kinda lost, so I’m going to watch it a few more times so it sinks in, but thanks for making this video!!!!! Looking at music theory/structure from the way it was made is really interesting and makes more sense than “these are the notes, because reasons. It just is, don’t try to understand why, just try to memorize it”

I know music theory and leant about the harmonic series, but this was a very interesting to listen too. I think music theory classes would be much more interesting for people if teachers took more time to explain the mathematical aspect of it.

Amazing video!

what books do you recommend reading in order to become an expert at music theory?… starting from the bottom…😱

I am not sure the word "scale" is because of weight-measuring scales. In Italian the "scala musicale" can be translated as "ladder" because there are intervals like a ladder.

Fantastic video!

Interesting as can be! I have to say, if modern practical music theory was truly algebra, I would never get it, thankfully, it is intuitive and easily understandable because I can hear it, while math makes my mind go fuzzy and dark. By practical music theory I mean being able to construct scales, spell any chord, understand chord progressions, know what bass notes work best, master the circle of 5ths, be able to transpose on the fly, etc. If music theory was taught like this video, I would be sick to my stomach, but still, this is amazing and probably useful for the right person. I'll watch it a few times, who knows, I might play better or be able to record more professionally if I can grok this. Thanks, this is great.

I'm a bit confused actually! Not about the Pythagorean ratios of the notes and notes, but by what happens when you construct scales. "Major" and "minor" are recent inventions (Renaissance) - hold overs from the modes. I wish you had constructed the rest of the "chromatic" notes. I'd like to see how that works.

finally! Welcome back

Thank you for sharing this. As an engineer and guitar player for 20 years I enjoyed it. It's a great way to view the roots of how scales came to be and I love how simple the derivation was.

Amazing visualization, inspirational! Might I ask the question: what happens if you add the minor thirds? I would love to see a full possible vitualization😃

This is perfectly explained. I knew the 12 root of 2 relation between notes from making a digital top octave gen in an FPGA( and an ancient Radio Shack databook). So This is perfect for the way I think and play. Regular Music theory always sounded like alchemy to me: Augmented fith blah blah, harmonic minor blah. Harmonics are nth multiples danbangit !!, I just play by ear. Might be a hanidcap, but works for me. Thank you!!! My problem is I get so into the jam, that I forget the changes :-(

So wonderfully explained. I hope to see more! Subscribed! Cheers, Johnny English

Great video! Thanks!

This is sooooooo helpful and informative. Really great stuff here

Your video definitely has my attention in helping me understand something? My question is what you're actually explaining is this actually what they call harmonic ratios?

Thank you for this.

I feel like you really get it and this is probably a great video for people who are memorising music theory and wondering why they have to memorize certain things, but as a curious non-musician who quite likes music I'm kinda lost with the technical jargon; I don't intuitively know the note names so I'm missing the right map to grok this and have it click in my head. I feel like there's an intuitive understanding to do with the differences between vibrations and what and why they feel like they do, with evolutionary psychological reasons that transcend humanity, and great ways to illustrate it with the right artist involved, and music theory should be put on top of that rather than the other way round. I could only contribute in the stupid questions department though 😂

Geeze! Tough crowd - I loved your video and look forward to exploring your others - presumably getting into equal temperament and the other tonalities it crowded out.

I am very interested in hearing more about the history of how and why musical notes are chosen and who "invented" scales. Why does Egypt have a different popular scale or why and who decided to tune a guitar to standard? Love this info. Thanks!

I loved 8th grade algebra. I've now helped my own kids through 8th grade algebra. Now, maybe I can understand music

This may not be exactly what you have in mind, but, two good math eye openers are the Bossa Nova and Chopin Fantaisie-Impromptu Op. 66 in C# Minor rhythmic structures.

Pythagoras was known for figuring out how right-angle triangles work, not so much for equilateral triangles. Here's a humorous account of Pythagoras and his work from a talented musician/mathematician: kzfaq.info/get/bejne/jpd1aqxoxNiXc6s.html

Thanks for sharing this, glad your posting again! this sparks a question that maybe you could help with. why isn't P4 separated like we separate P5? if separating P5 gives us the major and minor 3rds, wouldn't separating P4 give us a major and minor ...Blue note"? When i look at this, it seems P5 and P4 are to the octave what the major and minor 3rd are to P5 .. and at that rate you could subdivide any division. would be curious to know the logic behind why we stop where we do.. like why not take the Root, P5, Maj3rd, and the subdivision of Maj3 to create the chord on which the scale is built..? hope that makes sense. Thanks again

Its so similar to a chaos fractal. Every division holds the entire system in itself smaller and smaller or bigger and bigger.

I could follow up to a while, then it got all confusing. I'll be back for this again and again, thanks for putting it out. Maybe using different colours would help as later on it gets very confusing.

amazing video

The artistic application of that "8th grade algebra* is far beyond the average 8th grader.

This is trippy to watch, very cool

Chromatic scale comes from picking a pitch and ascending in fifths until you arrive back at the same note as the starting pitch. Line those fifths up in scale order and you get the chromatic scale. Chromatic scale is questioned and messed with frequently by modern composers and musicians.

It looks like we have nice ratios like (1:2, 3:2 etc) in our fretboard, but it is not how "Equal temperament" works (it is another system). If I remember it correctly, 12 notes in an octave in Equal temperament does not have perfect ratios (in another word, there are a lot of decimals).

You are right if music theory only concerns about proportion, length and frequency . However, because music is much more about aesthetics, that means music theory is always subjective and flawed in some ways. With just-tone intonation, or Pythagoras intonation, you get intervals in proportion to each others, but then the distance between each pitch is uneven. Equal temperament is a compromise between getting pure harmony and even steps between two pitches, so that you can modulate from one key to another easily.

great info brother, although it can be pretty confusing, even to those who are not new to this approach of natural harmony in western music. I was thinking it would be great to have a cursor moving over the graphic to see what you are talking about each time. My 2 cents. Thanks again