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Music Theory: An Education from First Principles (lightnote.co)
955 points by akalin on Oct 25, 2016 | hide | past | web | favorite | 148 comments



A similar project: https://www.teoria.com/en/tutorials/

Focused more on practitioners. Includes exercises with MIDI/USB piano keyboards in the browser: https://www.teoria.com/en/exercises/

These resources also can be useful:

1. https://www.musictheory.net/lessons (exercises included)

2. http://www.earbeater.com/online-ear-training (ear training exercises)

3. https://www.iwasdoingallright.com/tools/ear_training/online/ (ear training)

4. https://trainer.thetamusic.com/

5. http://music.stackexchange.com/

N. Wikipedia is a good reference. E.g., https://en.wikipedia.org/wiki/Glossary_of_musical_terminolog...

If someone wants to try music, start with playing an instrument and then pick the theory. If in physics theory says how the world works, music theory is mostly about labeling things that sound good vs noise. And it's hard to get the words without playing first.


Great list. I would add hooktheory:

6. https://www.hooktheory.com/hookpad (interactive composition tool)

7. https://www.hooktheory.com/theorytab (searchable database of chord progressions in popular songs, playable in abovementioned tool)

They also sell an interactive music theory primer, which I've not read, but everything else there is free (and amazing).


One more resource for theory, ear training and rhythm -- also in its beginning stages -- https://utheory.com, which is a side project of mine.


I like this! I've always wanted to build a music theory textbook (like Laitz) where the examples could be played.

> When a song says that it is in the key of C Major, or D Minor, or A Harmonic, etc. this is simply telling you which of the 12 notes are used in this song.

Small nitpick, this is not accurate, C Major and A (natural) Minor have the same notes but different starting notes so they are different scales, and pieces written in them sound different from each other. It's one of those things that's slightly hard to explain if you don't sing/compose/play an instrument/read music but very obvious if you do.


The scales don't necessarily have the same notes. That only happens with equal temperament tuning (, which is it's great advantage).

https://en.wikipedia.org/wiki/Equal_temperament

If you optimized the tuning for harmony, the notes would be largely similar, but some of them would not line up that well. Playing one scale with you instrument tuned for the other wouldn't work that well.

https://arxiv.org/html/1202.4212v1/#sec_3_3_0

Read that whole article. Super interesting. It was posted on HN this summer.


Yes, I probably should have added "in equal temperament, ...". I would say that most of music theory assumes equal temperament because it's the simplest to deal with from a theory point of view, because it allows you to modulate to other keys with no restrictions, and because deviating from equal temperament is something that is more of the performer's choice than something the composer dictates. For instance when choirs hold some consonant chords (eg at the end of a cadence sometimes), we tune the notes in just intonation (ie away from equal temperament) to get the nice "ring", and barbershop quartets sing some intervals about one-third of a semitone differently from what equal temperament would be (https://en.wikipedia.org/wiki/Harmonic_seventh_chord), and leading tones are often sharpened quite a lot for melodic purposes; these adjustments are something that you do naturally on a continuous pitch instrument, and composers normally don't specify them.


I can explain it: if you drew a frequency graph of how often a piece of music in those scales used each note, the graphs would be different. The tonic (first note in the scale) would likely be the highest point in the frequency plot. The note seven semitones up would also likely be prominent.


Perhaps even more important, the tonic will be used as a place of repose, or resolution of tension. Most traditional tunes end in the tonic, and the majority begin in the tonic as well. In addition, the major and minor modes have been used to convey different emotions, that will probably be invoked at some level if you've been exposed to a reasonable amount of mainstream Western music.

I suspect that there are no first principles that explain these traditions (and many more), but they work well enough that they can be relied upon (for instance) to reinforce the emotional experience of a movie, sports broadcast, or religious ceremony.

Perhaps a psychological effect worth exploring is how we remember aspects of music, and how that memory shapes our experience of new and familiar music.


Counter-nit pick:

It is accurate that it tells you what notes are used. It's just not all it says.

Your move.


Counter-counter nit pick: it's still not accurate because a piece that stays in C Major all the way through can still use notes outside of the C Major scale (eg passing notes)

Also I would say "X is simply Y" means "all X does is Y", but I guess this isn't universal


Yeah, the "simply" part made me half agree with you.

We're not so different after all!


>Small nitpick, this is not accurate, C Major and A (natural) Minor have the same notes but different starting notes so they are different scales, and pieces written in them sound different from each other.

Only because they follow usual (classically derived) patterns about the chords played in each key (e.g. 1st, 4th, 5th etc, with are different in each key, and thus give different priority to each note of the scale).

One could play absolutely the same song in C major and A minor (same chords, not same relative chords) -- and it would sound 100% the same, the chords would just have different relationships with each key.


Yes, but a musician would (reasonably) say, for example, "this song is in CM, not am". I'm trying to say that keys are traditionally defined to be more than "what notes are used"


The chord relation 1st - 4th - 5th is derived from the natural overtone series every vibrating string or air column emits. Every human can deduce unconsciously the base key note of a song by hearing the song. This is a adaption to physical laws not a cultural convention.


>This is a adaption to physical laws not a cultural convention.

Finding it nice to listen is the cultural convention. Other cultures enjoy microtonal works, or completely different harmonies just fine.


Aren't they also complementary in some way, like complementary colors?

It's easy to switch between them (although that makes sense since they are the same notes...)


Yep that's true.


I get an annoying click at the end of every note on this page. Does anyone else get that? Otherwise, well done.


This is actually my side project. It's very much unfinished but if anyone has any feedback I'd love to hear it.


It's very western-focused, especially when you describe the chromatic scale as "modern music".

I would also lose the "from first principles" by-line. The pentatonic scale description was neat in that you talked about how frequencies that sound good together have certain clean ratios, but when you jumped to chromatic scale, there was nothing about how that was derived other than "filling in the blanks." I love the site in general, but the "from first principles" by-line implies a few things that are kinda missing. It's more accurate to call it a visual introduction to western music theory.

A while back this article popped up on HN: https://arxiv.org/html/1202.4212v2 That one dived more into the physics and math that derive different music systems. As I was reading it, I wanted some interactivity like this. So well done! There's a lot of detail in there that could give you ideas for additional visualizations. It was also criticized here on HN a bit for also failing short of it's "first principles" promise, so worth finding and reading that discussion too.

The other thought I have is make the notes keyboard-playable. A lot of the examples are around playing notes together, and shifting between them using a mouse is too slow.

Awesome site overall!


>It's very western-focused, especially when you describe the chromatic scale as "modern music".

99% of the music people write and listen to in the west is western-focused, and not even that adventurous at that, so this is a little moot point.


Sure, but even in western music the chromatic scale is considered to have been exhausted in the 20th century. Modern music has moved on to use other notes and sounds and in many cases is not even pitch-centric.


>Sure, but even in western music the chromatic scale is considered to have been exhausted in the 20th century.

Yeah, but by people like Schoenberg, Alban Berg, and the like, or the wilder jazz guys. Not exactly what most people enjoy with a cup of coffee.

>Modern music has moved on to use other notes and sounds and in many cases is not even pitch-centric.

I think that for e.g. electronic, dance, experimental music etc, the chromatic scale/classical harmony/etc. is still a good foundation -- the additional not pitch-centric focus is either in rhythm (which is orthogonal) or in sample/noise-based focus which usually just goes with "if it sounds good, it's ok" kind of approach.

So, not much micro-tonal or other approaches going on in practice.


It's out there if you open your eyes a bit. There is a TON of microtonality in blues guitar playing - they just don't use the $10 academic words for it.


Well, I'm from a country where lots of folk music is microtonal.

That said, I don't consider the blues guitar playing a major example of microtonic music, $10 words or not. There are some elements, but you can find much more impressive examples in African music for one (part of which could have been inspiration for early gospel/blues musicians).


Have any examples you could link to? I'm very curious to hear what that sounds like.


Just listen to any blues...there are string bends all over the place, and many of them aren't traditional western 12-tet intervals.


That's a bit like saying the alphabet has been exhausted, so we should come up with new letters.

There is a huge, potentially indefinite space of possibilities of combinations of melody, harmony, rhythm and timbre, even within the constraints of the chromatic scale


It's less from a be-PC point of view and more from a "what (in some sense arbitrary) decisions led to the western system of rules, and what are alternative ways of making rules?"

You know, understand that this isn't the only way of doing it, and see how the physics could be applied in other ways (and have been). Appreciate the full breadth of complexity out there.


While the name that you propose might be technically correct I would recommend sticking to the "from first principles" kind of name. This introduction was created for beginners (like me), and the name was attractive and was what I expected. Maybe a footnote with this information could be useful.


Here's the HN discussion on that one: https://news.ycombinator.com/item?id=12085844


Yeah, it would be nice if he mentioned the ratios the chromatic scale is based on, as well as the difference between Pythagorean tuning and equal temperament. Honestly, I thought even the pentatonic scale should have been explained.


I like the demos. The transition from ratios to equal temperament was kind of jarring. It might help to not jump straight from the pentatonic scale to a chromatic scale and then go back to the major scale later, but to introduce the pentatonic scale, and then describe major and minor chords as 4:5:6 and 10:12:15 ratios, and then show how the major scale can be constructed simply by trying to get as many usable major and minor triads out of the least number of notes (while emphasizing that the major scale isn't the only solution, and that other "modes" exist).

Once you've established that the major scale is useful and in no way mysterious, then you can point out that if you divide the octave up in 12 logarithmically equal parts, you end up with equal tempered notes that almost line up with the ratio-based major scale. (I once threw together a diagram to illustate this that you can find here if you scroll down a bit: http://jsnow.bootlegether.net/cbg/justintonation.html) From then on, you can treat equal temperament as an approximation that implies the ratios you described at the beginning.

(Side note: One idea I try to impress on anyone who demonstrates the slightest inkling to write about music theory is that I really wish there existed something like Christopher Alexander's "A Pattern Language", but for music. Music is full of lots of little "tricks" that you can find scattered about in books, but I haven't ever seen anyone try to systematically collect as many of those as they can in one place, with a clear dependency graph.)


> The transition from ratios to equal temperament was kind of jarring

I agree. I'd like the course to comment on the fact that the ratios used in the pentatone scale and the 0th, 2nd, 4th, 7th and 9th powers of the 12th root of 2 don't match exactly.


I think you explain things very well. I've tried several music theory courses and quit all of them partway through. They were too fast or too tedious. It's hard to strike that balance, but somehow you did.

One thing that you did that helped me get further is sticking to that rainbow row of notes, as you explained notes and chords. I think most of the lessons I've tried move quickly to piano keys, where the relationship among the notes is more cluttered, because of the black and white keys.

Also the interactive little things, simple as they are, are very helpful. I know even these simple things took quite a bit of work to throw together. I think they are critical to how engaging your lessons are.


This is the first time I've understood diatonic chords, how a key called C Major could have a chord called D Minor.

(I am not much of a musician nor a serious student of music theory. If I were, I guess I would have eventually learned diatonic chords some other way. But it just goes to show you how good this course is, that a person with a passing interest in music theory was able to quickly learn even that.)


>> I've tried several music theory courses and quit all of them partway through

I feel like I'm a musician at heart. If I could delve into music with a purely exploratory and child-like fascination, I think I would enjoy it immensely.

But I can't fathom beginning. Every. single. intro. into. music., is a lecture forcing theory unto you as though "theory" is the very definition of "music". The consensus from experienced musical folks is to shove theory down your throat; you should not be allowed to even touch an instrument until you've drunk the "super serious music cool-aid"! Music is supposed to be one of the most intimate ways we express ourselves, and yet Western culture dictates that there is a singular method to begin exploring it, and it's far too serious and far removed from any kind of natural process.

Had my introduction to programming, starting at the age of 7, been the shoving of books down my throat about Object Oriented programming and how to write 20 different sorting algorithms, I never would have become a software developer. It took me 5 years of accomplishing nothing - and enjoying that exploratory process - just to figure out the basics. Another 5 years to begin digging into the core things that make a developer a developer. And then another 5 years (and counting, a decade later) actually learning how to tie it all together and use the acquired knowledge as a set of skills.

Music in the Western world is taught in such a way that they're trying to cram 10 years of knowledge - that you should be picking up along the way - as pre-requisite knowledge. It is taught as a job aiming to produce immediate ROI rather than skill/talent that accumulates over time.

Sample phrases from the OP that are spreading the all-too-serious cool-aid - the same things you hear from every music professional (italic emphasis mine).

>> A lot of music theory is about limiting which notes to use in your song to a small set that sound good together.

>> not all of them sound good together... Lets listen to some garbage

>> Because not all of the 12 notes sound good together, we must select a set of notes to use in a song.

>> start exploring which chords are safe to use in each Key

Why is it deemed necessary to instruct first-timers about "limiting" the notes we might use (some notes are not "safe" you know!), and telling me what does or does not "sound good". Music should have absolutely no "must"s attached to it. Let alone labelling certain combinations of notes as "garbage" - the use of that term is horrifically off-putting to someone who is looking to explore. This method of teaching completely destroys the discovery process. You tell me that "XYZ" chord is garbage, so I avoid it from day one. What if, left to my own devices, I were to discover that chord - and wind up loving it?

tldr; Teach me to play an instrument without teaching me anything about theory. Don't even teach me how to read sheet music. Just teach me how to play a few songs I enjoy. Once I can actually play with some confidence, then - and only then - bring in the information that ties together with what I have first learned hands-on. Basically, how any 3rd-world child with instruments carved out of wood would learn. No books. No sheet music. Just an instrument and a patient MENTOR, rather than a classroom teacher who's trying to make a classical prodigy out of me in record time.


I agree. If you want to learn to be a musician, you should start with practice instead of theory. The same goes for learning a language. It's better to learn a few phrases and sentences before the teacher sits you down and starts diagramming the nouns, verbs, prepositions, and so forth.

But just because this course exists doesn't mean you have to go through it first. Go and learn your instrument. Then come back in a couple years when you want to round out your understanding.


Very good post. My tip would be you should learn like Jimmy Hendrix did, and many others: turn on your radio/spotify/tidal/whatever and try to play along on any instrument you can find. Just finding the one note that sounds like the song's resolution will teach you what you need to know about the tonic, then try to catch bits of melody. 10 years of a bit of this every day and you're golden


Have you, by any chance, played Rocksmith? It seems like it might be an approach you'd enjoy.


Thanks for the idea, but sadly that would not fit me whatsoever. Guitar Hero, but with a real guitar - too independent to start.

I'd need the human element - a real person, in an intimate setting - to guide me through the initial pains. A mentor, to the extent that someone would be teaching me their trade - or perhaps even their (our?) ancestors' traditions. Someone who is going to have the humble attitude of: "You will learn this, and it is OKAY for you to be complete garbage for the first year. A music teacher would give up on you after 3 months, but I will not and we will get through the tough beginning together."

Yep, like a helpless dog who keeps peeing on the carpet and just can't figure out why the humans are so agitated - until it finally clicks and the problem permanently vanishes. ;)


I get where you're coming from. I'm that way with many things I'd like to learn. There's something that feels just right about learning from someone who already has the skills.

On the other hand, Rocksmith really is far more than guitar hero. It's instrument-learning gamified, with a somewhat quasi-human guide who's always talking you through what's going on, what you're learning, how to tune, etc. You get to tackle learning songs at your own pace, and there are quite a number of silly little games that--for me, having first picked up a guitar and messing with them off an on since I was 17--have really helped cement actual chords in my head ... without overloading me with theory and all that. I tend to make natural connections with things as I learn them, and Rocksmith has been a pretty fun way to engage with guitar & bass.

Anyway, the fact that it's all in a game form certainly can be an understandable turn-off. But it doesn't always feel like a game. It strikes a pretty good balance. Could be just a fun thing to do in between sessions with a human. :)


That is such a lot of bollocks. :-)

Go grab a guitar, twiddle about with the machine heads so the strings are randomly tuned, and start hitting the strings with a book.

Absolutely no-one is stopping you from doing that (well, perhaps folk within earshot). Hell, there are guitarists, bass players, drummers and singers (whole bands!) who know pretty much nothing of music theory and still play stuff.

If you want to play, go ahead!


+1 on "Just grab a guitar".

Almost every rock/blues/metal guitarist worth naming started out by grabbing a guitar and fucking around on it until it started to make sense.


You'd probably enjoy the Suzuki method.


I majored in music and I like what I see so far. The interactivity is nice. Looking forward to seeing more. Pretty solid on the concepts that are covered but I could do with a review on part-writing and other intermediate/advanced topics :).


Thank you! Definitely. There's a lot more I'd like to add.


I haven't read it, yet, only did a quick search to see if I could fine the words "tonal center" somewhere. I think that concept is under-emphasized and resolves a bunch of common confusion about music. e.g. this post[1] that was on the front page a few weeks back in which the author did not understand what's the difference between C major and A minor.

[1]: https://eev.ee/blog/2016/09/15/music-theory-for-nerds/


Poking around on this, this strikes me as a good, simple introduction to the basic concepts, without burying the newcommer under a ton of jargon, which has been been a recurring frustration of mine.

Some suggestions:

I might include a bit on the diatonic scale between the sections of the pentatonic scale and the chromatic scale. Anyone familiar with Western music should recognize quickly, as a great deal of well-known music has been composed using it. It should also serve as a nice half-way point between the pentatonic and chromatic scales -- introducing more notes from which to compose melodies but all of them still mostly sound nice with each other.

I suspect the reason you covered the diminished chord in addition to the major and minor ones is because that's the chord that works on the 7th interval of a scale, as you cover in the later section. But you might want to cover the augmented and suspended chords as well (perhaps in a later section). I'm probably not the only one who, when attemting to finger a chord on a piano or guitar, missed the 3rd or the 5th, and discovered another triad that sounds interesting.


Nice work. I love that you included the pentatonic stuff first, actually. it's a great starting point for music. If I could change one thing, I would put the chromatic scale section more towards the end; I would teach it by starting like you did, but expanding pentatonic into diatonic, then harmony, and finally dealing with chromatic scale.


Everything clickable (the sounds) before http://www.lightnote.co/music-theory/pentatonic/ did not work for me (on Chrome).

Still really liked it. Await the chord progressions stuff.


Same problem here. No sound or interaction on Chrome.


Looks good, I think there is a great foundation. The Pentatonic Scale feels out of place; but that's probably because I feel that pentatonic is just a mutation of other modes. Chords are really just stacked intervals, it might make sense to move them immediately after the 'harmony' section? The interface is really nice, but I don't think it clearly communicates or reinforces the idea of 'tonic', which is fundamental to understanding music theory. Please keep working on this, there needs to be a good interactive explanation of music theory!

EDIT: You might benefit from filling out the intervals section. Clearly articulate each interval, which are classified as consonant vs dissonant, etc. That would get the user/student thinking about 5ths vs 7ths, etc.


Looks great! One question:

"This is known as an Octave and is not considered a new note. An Octave is the same note with a higher pitch."

Why is this so? I guess I'm totally tone deaf since I get no understanding of the sameness of notes separated by one octave by listening to them. In the same vain the good sound/bad sound example that opens the harmony section sound both good to me.


The notes that sound good together have a basis in the harmonic series, the 1st and 3rd are octaves of the fundamental, 2x an 3x the frequency. the 2nd harmonic and 4th harmonic make up the other 2 notes in a major chord. Periodic sounds that are not pure sine waves (like vibrating strings) have harmonics. Our ears are attuned to hear these harmonics and this natural phenomenon is the basis of all pitch, melody and harmony in music.

https://en.wikipedia.org/wiki/Harmonic_series_(music)


If you want the first-principles answer, it is that sounds in nature tend to occur as multiple overlapping tones one or more octaves apart, so our ears and brains have evolved to hear a sameness in them.

That is, when you pluck a guitar string that's tuned to 400hz, it also vibrates at 800hz, 1600hz, and so on. (Do an image search on "guitar string harmonics" if it's not obvious how that would work.) But we hear such overlapping tones as a single "note", whereas a 400hz tone overlapping with a 600hz tone sounds like a chord.


For the octave, if you heard them at the same time they would be a bit harder to distinguish than most other ratios, so it would be nice if the example let you do that.

And yeah, the "bad" example is not really very bad. It's a little bit tense, but there are much worse ratios they could have chosen.


Agreed. Saying they're the "same note" is an oversimplification. They're "equivalent" in certain contexts, which is an important distinction.

They aren't the same note: if someone plays A3 (fundamental is 220Hz) and then A4 (440Hz) you can easily tell them apart! In contrast, if they play 220Hz and then 221Hz an untrained listener will most likely hear them as the same note.

In some contexts a note and its octave can be thought of as equivalent: if you are playing a chord you can often replace A3 with A4 (and vice-versa) without changing the chord's feel and function. However, that isn't always true. For example, playing G#3 and A3 at the same time will sound more dissonant than playing G#3 and A4 (a minor 2nd is more dissonant than a minor 9th).


IIRC octaves are what's referred to as harmonic frequencies in physics/kinematics/etc. "Why is it the same note?" AFAIK it's because the higher ones are multiples of the base freq.

But TBH I know next to nothing about music so I could be off.


Close. Each shift of an octave doubles the frequency, so octaves are specifically harmonics that are powers of 2. Other harmonic ratios make other intervals (including intervals that aren't in our scale, when you get to bigger prime numbers).

Most instruments produce harmonic multiples of the base frequency. (Percussion instruments don't.) So if you play a note at 110 Hz, there are also pitches at 220 Hz, 330 Hz, 440 Hz, etc. contained inside it. If you play a note an octave up, at 220 Hz, it contains pitches at 220 Hz, 440 Hz, 660 Hz... which is similar enough that it sounds like "the same note".

(This doesn't explain why 330, 660, 990, ... sounds like a different note, though. Maybe it doesn't to an untrained ear.)


I haven't tested this, but my intuition is that any musical sound with harmonic (as opposed to inharmonic) partials will sound like a single note, if the first fundamental is clearly the loudest, and the harmonics follow some kind of natural distribution in terms of loudness and temporal decay.

Two notes played in an octave would be quite distinct since you have a clear first fundamental for both notes, and the loudness/decay of the harmonics is slightly different between both notes


As someone just getting into music theory the interactivity is extremely nice. The visualizations and sound just click for me.


UI: The webpage is not usable on my iPad. The menu and main content divs overlap. I thought you should know.


Thanks. Yeah, the experience on mobile devices too is admittedly pretty poor right now. I wasn't ready yet for the traffic but I'll get this fixed.


It's not optimized for mobile but I went through it on an iPhone 6s and thought the experience was pretty dang good.


Hmm, odd; works on my machine. iPad Air running iOS 10. Haven't spent a lot of time on the site, and I see the occasional alignment glitch, but a long way from "not usable".


I've been wanting to do something like this a while now, and I think you've done a great job! I really appreciate how you used the sine waves to demonstrate consonance, and the pentatonic section is a great intro into the world of 'just-intonation'. Cheers!


Keep up the work. If only a few people get their start into understanding the physics of music through this, it will be worth the effort. Plus I think this kind of education will be a lot more prevalent in the future. Congrats & thanks for the huge effort!)


It would have been so much easier and fun if the notes were played on hoover not on click. Or even using the keys. And you could activate a specific section by clicking it, or parking the mouse in a certain spot.


Well done! Exactly the way I would want to and could actually learn this.


Please keep going. I've been dabbling with instruments for almost 30 years, and this is the first time theory started to click with me. Look forward to seeing more. Thanks!


Out of curiosity, what's your day job?


I think this is fantastic. Awesome work. Looking forward to seeing the missing chapters!


This is just lovely. Thanks!


This is excellent. The graphics and sounds are nicely done.

One thing that I don't see in it, but that I find fascinating, is that in western music each half step represents a ratio of the twelfth-root of two, in terms of frequency. That way 12 half steps (an octave) will double the frequency.

Certain notes within this are close, but not exactly, on a "simple ratio". It's just coincidental that it works out pretty good. (although you could make it work out just as good with something other than a 12 step scale....a 19-step scale has been used: https://en.wikipedia.org/wiki/19_equal_temperament )

Anyway, I think that would fit in well with what you've done so far, but obviously, explained in the nice simple graphics that you seem very good at.

I also must say I love the way you use color, I have a music project of my own (that I'm hoping to debut very soon) that also uses color in a very similar way. Did you know that Isaac Newton fixated on 7 colors (ROYGBIV) because he thought there was a connection between the diatonic scale and colors? That's why indigo seems to have been promoted from some obscure color to one of the "basic" colors of the rainbow. (I prefer BOYGBPP, red-orange-yellow-green-blue-purple-pink)


> Certain notes within this are close, but not exactly, on a "simple ratio". It's just coincidental that it works out pretty good.

It isn't coincidental. Equal temperament was a fork of just intonation.


Not sure what you are saying. It is chance how close they are. It works out fairly well in 12 tone scales, worse in most others. Is "coincidence" the wrong word? Maybe. But it didn't have to work out that way, luck or whatever you want to call it was involved.


Just intonation is based on intervals that are ratios of small whole numbers.

The 12-tone equal temperament scale is historically a development of just intonation (a "fork" in software terms).

So it's not a coincidence that 12-tone equal temperament intervals are close to ratios of small whole numbers.


Nice. I've been thinking a lot about writing a interactive blog series of something similar (music fundamentals for CS people). It'll probably be a while before I start writing that though.

I think the biggest thing that's missing at the moment is a section on rhythm/time. I'm sure you have plans for that. Looking forward to seeing finished course.


I wish this was even more "from first principles." I wish the "harmony" section would point out that the "simple ratios" they initially show both have powers of two in their denominators, and thus are just octave adjustments of the harmonic series. I wish the "chords" section would derive the major triad as the fundamental frequency combined with the first two (non-octave) frequencies in the harmonic series octave-adjusted down to be close to the fundamental.


Do you have any simple resources (similar to the OP link) to help learn a bit more of the math behind musical ratios?


I don't have any good web links, but a couple of books that I've found useful are The Science of Musical Sound by John R. Pierce and Harmonic Experience by W. Mathieu. (The former is more focused on sound and scientific theory, whereas the latter is more focused on understanding music in a way that would be useful to a musician or composer.)

On the Sensation of Tone by Helmholtz is also worth checking out if you want to understand where a lot of the modern understanding of just intonation comes from.


Sadly, I really don't. I've picked it up over a long time, and I still probably have plenty of misconceptions and less-than-rigorous understandings.


The thing I don't quite understand is why many seem to argue that major chords aren't inherently more "consonant" than minor chords, and that it's just psychological/cultural, despite the mathematical consonance that seems to be inherent of major triads.


"Consonant" vs "dissonant" is subjective: it's a matter of perception, taste. The way you test the hypothesis "This chord is more consonant/more pleasant than that other chord" is by asking a lot of people. Just like "steak tastes better medium-rare than well-done". Physics & chemistry can inform the discussion but it cannot prove or refute the hypothesis.


Shameless plug: I wrote a book a couple years ago that people in this thread may find interesting: "Music for Geeks and Nerds". It's a short book that uses Python to teach music concepts:

http://pedrokroger.net/mfgan/


I found this to be a way better resource than the "Music theory for nerds" article floating around a while ago.


There's not a smidgen of principle in the jump from harmonic ratios to tempered tuning and standard scales and chords.

True first-principles music theory must (A) focus primarily on psychology over physics (B) not tell people that complex ratios sound bad but simply help people notice that they are different from simple ratios (C) actually go through the full logic of how the tempered system is derived from chains of harmonic ratios adjusted to temper out commas. The easiest approach to the latter is to simply teach diatonic scales as harmonic ratios and not introduce temperament at all until much later.

Far far far from perfect but having the direction that this attempt is missing: http://www.tallkite.com/AlternativeTunings.html

Anyway, I'd write the ultimate thing if I ever found the time. There's at least some good elements to this attempt so far. It really needs to be licensed CC-BY-SA though so that people can adapt and contribute and improve to get it to where it's really good.


>Anyway, I'd write the ultimate thing if I ever found the time.

Sure you would...

Here's the thing, it's not that your criticism isn't valid, because I actually agree with it, but what's the point of finishing your remarks with "So anyway, this is why you're wrong and I'm right. Also, I would actually write the ultimate thing myself except I don't have the time."? Yeah, and I would write the ultimate language and compiler, if only I had the time, and I would invent fusion and cure cancer, if only I found the time. I mean, what's the point of remarks like those?


I meant more specifically: "I know how hard it is to get this sort aof thing optimally done. I've been considering doing this sort of thing for a long time, but it's a massive amount of work" programming languages and compilers. Which is to say, I acknowledge how hard it is to do this right. So, it's a statement of understnading along with a clarification that I'm someone with the knowledge to do this if only I had time, which I cannot say about


Don't forget: There is no ultimate best way to teach. People absorb knowledge very differently. I learnt that the hard way. Some people need to see examples first and play with it; other people (that's me) need to see the structure and principles first.

The "must" in your claims seems a bit too absolute / dogmatic to me.


The best way to learn anything is to study multiple sources, multiple approaches to the topic.

The "must" in my comment is precisely about what is required to achieve the "from first principles" premise. To learn a subject, "from first principles" is not necessarily the right way to go, but there are things that are or are not from first principles.


I hear ya (pun not intended), but I think the current approach would work, more or less, for people who went through physics school or something. Perhaps I'm biased by my background (physics, and reasonably familiar with music theory at amateur level).

I do agree that making it more intuitive would make it more useful for, e.g., kids in middle school.


I wasn't talking about intuitive. I'm saying that to follow the first-principles premise, certain things are needed.


I'm kind of late to the game in this thread, but my thought about theory is that it should start with physiology and technology. At each level we're reminded that we study the aspects of music that people have already invented, and that we may overlook a lot of important things, such as rhythm.

Physiology: Some of this may be speculative, but it seems likely that "harmonious" intervals, that have a superposition of harmonics, have a physiological effect.

Technology: The 12 tone scale could be described as a technology for tuning an instrument with harmonious intervals. Temperament is a technology for solving the problems of tuning primarily keyboard instruments.

Naturally, math is involved in understanding these things, as with many areas of science and technology.

I would talk about a handful of widely used instruments, such as keyboards, strings, winds, guitars, and drums.

Then you can begin to talk about scales, chords, melody, form, and so forth.


This is absolutely fantastic. I have seen many musicists trying to explaing this with not much success, mainly because they dont understand the physics behind.

You explanation makes perfect sense and it is so clear and so well that once you read it you can remember it forever.

Thanks!


Well done! It summarizes a lot of theory it took me months to puzzle out on my own.

One nit: At the top of each section there is a section title, at the bottom of each section, there is a "Next section" title, a description, and a next section button, and on the side there is a list of sections. Some of the titles are inconsistent from list to top of current and from bottom of current to top of next. It's a little confusing right now, and there are only a few sections; when there are more, it will be far more confusing. I don't have a suggestion as to how fix it, just pointing out the confusing inconsistency.


Great idea! Just FYI I'm noticing a fair bit of static when playing the various tones. It's mostly at the beginning, which makes me think it's just due to the discontinuity at the beginning (maybe start the volume at zero and quickly increase?) But I'm getting blips of static in the middle of most tones as well, so there must be something else going on. Tried both Chrome and Firefox. I suppose this could be an artifact from my onboard sound card or something like that, but I haven't noticed anything similar elsewhere.


This is nice, very nice. But it is quite disappointing that at least for everybody reading HN the whole "Music Theory" topic seems to be just a few concepts that can be learned in 1 or 2 hours.


I know right? I left some lengthy comments in the more interesting stuff in earlier threads...upvotes but no responses. Thanks but I'd actually like to have this discussion.


Are you a spambot?


I'm good on music theory right up to time signatures. It seems like black magic to me, and often it is taught by just asking you to hear the beats and I just don't hear them.


Can you do metre? This is about as clear an example of 3/4 time (dactyls, in poetry) as you can get, I think:

RAINdrops on | ROSes and | WHISkers on | KITtens [beat] | BRIGHT copper | KETtles and | WARM woolen | MITtens [beat]

Fast syncopated 4/4 (trochees) is Johnny B. Goode:

(deep) | DOWN in LOUsi|ANa CLOSE to|NEW or [BEAT beat] |LEANS [beat BEAT] way | BACK up IN the | WOODS aMONG the | EVer [BEAT beat] | GREENS [beat BEAT] there | STOOD...

Okay, maybe not the best example. Other common times are just variations on those two.


I want to say if you draw out a heat map with some dimensions, it'd be more clear.


Love this, it's the way i would want to learn.


With Safari entirely unable to handle this page, I really feel like I'm missing out on the modern web with it for the first time.


Interesting, but too bad it's not much more than the basics. I struggle to find a good explanation of how harmony works, i.e. what they mean by the terms "resolution" or how chords are made, in short: how a musical piece is built.

My piano teacher's refusal to explain these to me is one of the reasons I lost interest in the instrument


This is the perfect minimalist introduction to music theory. A similarly good explanation is in Daniel Levitin's "This is Your Brain on Music". It's first few chapters explain music theory to beginners in a really elegant way.

I previously thought music was composed without any rules.


Inspired by this, I decided to write a JavaScript keyboard that only plays notes in the key you're in. https://nickretallack.github.io/theme/


> The Major Chord is the most common chord. Whenever you're asked to play a chord without specifying what type, then it's a Major chord.

Certainly not in modern pop music. By far the most common chord is power chord.


Power chords aren't actually chords, since they're only two notes.


That's kind of like saying a bathroom isn't a room :)

But they're also not major chords, they're fifths - E5, A5, etc, although I personally often add the octave to make it 3 notes.


This is great! I found the more robust examples really useful, like the ones that show notes and triads in a key. I'd love to have something like this in the form of a VST or something usable in Ableton.


If you are more into mechanical practice of notes reading I had created http://notationtraining.com


So why do notes sound good together when the frequency ratios are simple? Needs a section on the inner ear, stereocilia, and how overtones affect them.


This is still an open question after multiple millennia. Some posit that it is innate and have primate studies to back it up. Others posit that it is a culturally learned phenomenon and have social studies to back it up.


Does anyone know of a music theory class that works with GNU Lilypond format files, as discussed on lilypondblog.org ? Thanks in advance !


this is great but I there was recently an article that made the rounds on HN pointing out that what sounds 'Nice' to people not exposed to western music is very different from what sounds 'Nice' to westerners. This is a great site and people can learn a ton but its very western centric and it might be worth pointing that out early on.


Is there a resource for other styles? I'd love to learn about them.


I disagree with the equivocation of "sound good" with consonant notes and "sound bad" with dissonant notes.



Tension & resolution a much better framework to understand. But not all music takes advantage.


Does it again in "Too many notes" section when reffering to some nice dissonant chords as "Lets listen to some garbage."


This is fantastic! Looking forward to the next one.

Can anyone suggest good resources in this vein that apply these basic principles to guitar?


Love it. Thanks for sharing. Wish I had this when I had some music theory classes back in college.


Thanks! Really enjoyed the tutorial


FYI sound does not play in Firefox


Does in mine.


I'm having issues in Chrome. (Does quick speaker check), yeah my sound is good, but not working on the site right now.


Software engineer here. Finding my creativity is drying up.. Whats the fastest/easiest way to learn a musical instrument at the highest possible level?


Former music teacher here (and present software developer).

When I was teaching bass lessons, I'd often have friends, usually also in their mid-20s ask me, "Could I still start learning to play an instrument?"

My answer was, "Yes, but you probably won't be any good. Not because you're too old to learn, but because if you had the right inclination to be good, you'd have already found your way to an instrument." (This is totally different from learning additional instruments later in life.)

Learning to play an instrument creatively (as is your stated goal) is a long process of learning to build up intuition both about the structure of music and the way your instrument works, both mechanically and harmonically. Most of that happens in your head, not in your hands. There are methods that may be more or less efficient, but they all require years of effort.

That said, playing an instrument is fun. Learning to play a few songs on the guitar or piano is a totally attainable goal for a few months of effort. Really, something equivalent to that (including, e.g. middle school band class) is where most people get started. That remains fun even if you don't get to the point that you're using the instrument as a creative outlet.


Yes, but you probably won't be any good.

I started piano when I was 23/24 with the goal of wanting to play Jazz / improvise and quickly realized that the biggest challenge is lacking the technical skills to translate the "mental orchestra" onto the keys.

Especially before bed my mind wanders and produces these amazing Blues/Jazz riffs or Moonlight Sonata-esque ideas, but my talent is far behind my taste when it comes to either trying to play or even transcribing all that I "hear"... at least without significant manual effort.

So yeah, maybe that just confirms what you're saying, but what I realized around year 1 was that "improvisation" is really just a musician's bag of technical skills/tricks combined together in their own unique ways... And ultimately the more technical capability you have, the more "improv" things will appear.


After a few years, if you practice quite a bit, you get to where if you can hear it in your head, you can more or less automatically translate that to finger movements. A seasoned player doesn't really have to think much about the mechanics. You think a melody, and then it comes out.

That doesn't mean that composing becomes automatic -- often once you cross that chasm, there's still frustration between having an abstract musical idea, and translating it into actual notes. Also, by that time, the way you listen to music has changed pretty drastically, and you also realize that a lot of your own ideas are mundane in comparison to the things you can half-way understand in music that you listen to. That's where most of the work actually happens -- in becoming better at thinking about musical ideas.

If you're able to sing the ideas that you've got, I'd recommend recording yourself singing them, and then spending the time learning to play those ideas. That at least lets you work on the gap that you mentioned between thought and mechanics in a concrete way.

For most instrumentalists though, myself included, you pretty quickly get to where you can play things more easily than you can sing them. If you ask a trumpeter to sight-sing a piece of music, you'll usually see them miming the hand movements because they can imagine how it will sound when they play something, and work backwards from there to singing.


I've just started playing piano and I've been really inspired by pianowithjonny.com. It's an excellent web site, well worth the $20 or so per month for monthly membership so you get access to all his courses (put in your email address and get a discount code).

I just started his Blues Bible course[1]. Maybe that could help speed you up. It's tons of riffs to help get you improvising. But yeah, he says riffs are basically just a bag of tricks you can dip into.

[1] https://pianowithjonny.com/products/bible-of-blues-riffs-vol...


> Especially before bed my mind wanders and produces these amazing Blues/Jazz riffs or Moonlight Sonata-esque ideas, but my talent is far behind my taste when it comes to either trying to play or even transcribing all that I "hear"... at least without significant manual effort.

Double down on ear-training, it'll take you a long way toward being able to being able to automatically translate the imaginary sounds into real play.


Some thoughts from someone who knows music at a pretty high level and can play multiple instruments well. Hopefully these will help you avoid the very common scenario of buying an instrument, picking up a beginners book, playing for a month and then becoming bored of it and never touching it again:

1) Think about the music you enjoy listening to the most and what instruments are most prominent in it. That's the instrument you should choose. Like getting good at anything else, practicing music is hard and can be chore at times. You won't always have the inspiration to play. The best way to consistently practice and improve is to be learning song that you really want to be able to play.

2) Playing with others is absolutely essential. You will improve more jamming with others for an hour than you will in ten hours of practice on your own. You do need some amount of skill to do this, but not an awful lot. Once you have a little bit of confidence built up, look for others at a similar skill level to meet once a week or so and play together. You don't need to form a band or anything, but you just might find that you want to.

3) Music theory is nice to know, but not essential at first. It is very useful when writing music of your own and improvising with others, but there's a bunch to nail down first before not knowing it will hurt you at all. Learn basic notes and chords, then focus on picking songs you want to know and learn to play them well. While learning songs, think about what music theory you have learned and how it works within the song.

4) The truth is that becoming "good" will take a decade of dedicated work. But if it is something you truly enjoy then after a year or so you will be able to get to the point where it is both fun and very rewarding. So really do focus on your enjoyment of it. You don't have a parent requiring you to practice for an hour a day anymore, and you won't find immediate benefit from it at the start. Anything you can do to make it an enjoyable routine is very very important. I won't recommend specific books or methods because you fall into the same problem as telling someone how to learn programming. There are so many different options, but what's important is to find what you like the best.


Get a good teacher and take weekly lessons for the first few months. Make sure your instrument is up to snuff. Make a habit of practicing at least 30 minutes a day of technically hard stuff. Most importantly, find an instrument (and music genre) that you love.


Follow sporkologist's advice and also read "The Talent Code" by Daniel Coyle. It will give insight into why some approaches to practice produce far better results than others.


Thanks for mentioning that book, I hadn't seen it before but it formalizes a lot of things that have helped me in the past. Also they mention the Finland educational system. Yeah I'll have to get a copy of this one.


> Most importantly, find an instrument (and music genre) that you love.

Absolutely.

There's no point picking up the guitar if you don't listen to any guitar music.


This was the correct answer.


I don't really have an answer to your question but here is an observation I've made over the years: A lot of people want to be able to play, and want to get there as quickly as possible. But the people who succeed are the ones who enjoy the journey. In other words: Don't view practice as a hump you need to get over. Enjoy it.

Having said that... As with learning all things... It's definitely more enjoyable when you progress quickly.


I heard this anecdote a few years ago.

Someone met a world-class concert musician after a show and said "That was incredible. I'd give my whole life to play like you," and she replied "Well, I did."


As someone who loved playing music I would suggest you avoid brass or woodwind instruments. It is maddening to spend years and years practicing and not even be able to play all of the notes (individually) on a 'standard' piece of music.

The ways of producing good sound and good tone are still dark arts with wildly conflicting opinions on them. To me it seems the good players already had a sense of what to do right and then built on that with tons of practice. You might end up going down a dead end with your practice and no one can really set you right because they 'just got it'.


I am feeling the same as you! I've slowly been going through the course Syntorial to learn about Synths and general music theory. Not sure if it's the best for what you want but I have found it pretty interesting. Would be curious to hear others thoughts.

http://www.syntorial.com/


Synthesia is crazy good. You can learn to play 10-15 seconds of quite a complex piece in a few hours with zero music background.

A good private tutor if your other option.


Shameless plug, I can help: www.magicinstruments.com if you're in SF, stop by for a demo. My email is andrew at the website mentioned above.


Ctrl-- in firefox 45 to see Subscribe button. Apart this, thanks :)




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