I recommend "What Makes Music Sound Good?" by Prof. Dmitri Tymoczko: https://dmitri.mycpanel.princeton.edu/files/pdfs/MUS105hando...
There is a part near the beginning where he derives the diatonic scale from first principles. (There is a sequel as well, and a book if you prefer to read it in book form)
I skimmed through the whole article and only found some brief mention of Helmholtz, which no clear route from his research to the scale.
As a better derivation, I recommend this paper by William Sethares:
https://sethares.engr.wisc.edu/paperspdf/consonance.pdf
Or the informal summary version:
https://sethares.engr.wisc.edu/consemi.html
But note that Sethares's work assumes the Plomp and Levelt curve is correct, which is only approximately true. In practice there is also "higher order beating", which causes additional dissonance at ratios close to small integer ratios. This means the most accurate model of dissonance combines Sethares's approach with ratio complexity as in just intonation. This is backed by listening tests:
https://www.nature.com/articles/s41467-024-45812-z
The best informal overview and demonstration for this concept I know of is this Youtube video:
https://www.youtube.com/watch?v=iqxfE2nfGhY
Note also that there's a strong cultural/training component to dissonance perception. To minimize this you should ignore the preferences of musicians and only ask untrained listeners.
[1] I understand ancient scales were 3-limit instead of 5-limit scales, and that modern equal temperament has thirds which significantly deviate from 5:4 and 6:5, but once polyphony became a thing, 5-limit temperament made a lot more sense because they reduce beating.
Recommending The Jazz Theory Book to someone starting from scratch, especially someone self studying, is like recommending CLRS or TAOCP to someone who asks "I'm curious about learning the basics of algorithms and data structures, any good books for self study?" And of course that happens all the time too. Chapter 1 is "here's what intervals like major 3rd and perfect fifths are", Chapter 2 is "here's the major scale and all its diatonic modes and ii-V-I progressions", Chapter 3 is "here's how you might use the Phyrigian mode to play over a Sus flat-9 chord" (pg 48).
Recommending Schoenberg and Messiaen as steps to get a sufficient surface of topics covered for "music theory" is performance art in its own regard. Bravo!
I IV V chord progressions and major scales are basic enough I can explain them in a 400 word post from first principles (where no music theory exists except the assertion that the human ear finds the pitch ratios of 2/1, 3/2, 4/3, 5/4, and 6/5 pleasing to the ear, that the human ear wants at least three notes in a chord, and that the human ear wants at least three chords in a progression).
Those books get very complicated very quickly, and cover music that only a relatively few people actually listen to. It was pure gatekeeping, and I’m disappointed people pointing that out were voted down. Not a good moment for Hacker News. It would be like me recommending Lisp or Forth for one’s first programming language.
[1] https://www.abrsm.org/sites/default/files/2023-09/music-theo...
But yes, it takes an amazing amount of arrogance to look at a simple exploration of music theory from first principles and respond with "this isn't music theory"
Ah, well, that makes sense. I come from the "music theory is a quintessentially rigid, British-Schoolroom-coded subject that emphasizes rote memorization over actual understanding" school of thought. The musical equivalent to Americans requiring their children to recite the Pledge of Allegiance before they can even understand what the words mean.
The student seeking to build their understanding asks: why is the order of sharps FCGDAEB? Why is the mode with a sharp 4th note the Lydian mode?
Music theory responds: Because fuck you, that's why.
Same with many activities. Pedagogy and real-world use are different goals. Result is a confusing and horrible learning curve.
Some people are okay with being told "memorize this list of arbitrary factoids" and there's nothing wrong with that, I'm just not one of those people who can pretend that that's good pedagogy.
I don't think rote memorization is super useful for me as a player- I've heard a lot of analysis that sounds "accurate" but often it doesn't follow my own thoughts and understandings.
If folks are playing, that raises questions about "why this here" and it's nice to have vocabulary for to understand. But I see a lot of folks (there are several in this thread) who think that is a bit backwards- that somehow folks could start with an abstract rule set and learn much. If that's the view on what is happening then I agree that "teaching theory is impossibly hard".
But if learning to play comes before the theory about why we play what we play the situation is a lot easier.
Almost all the theory I have comes from applying the stuff from "theory" books to what I do.
I learned what a secondary dominant sounds like by learning I should stay in an A chord instead of an Am in the turn around in G major blues songs long before I got the concept of what that C# might do to the local tonicity of the tune from whatever theory book.
And I think that's a reasonable way to learn: start playing whatever, try to pick up something you can read for (I read double bass and clarinet pretty well) take up a lot of piano (weather you read it or not), play a bunch with musicians, and eventually the arbitrary facts about your practice sort themselves out.
Maybe their needs to be a “music theory for engineers” like we have “physics for poets.”
> Music theory responds: Because fuck you, that's why.
Those are questions that music theory will eventually answer. However knowing the answer to them is not very useful for anything music and so ends up being uninteresting even though when you ask it seems interesting. I can derive the order of sharps - or at least I think I can (I know my circle of fifths) - but realistically the isn't useful for anything and so I won't bother.
Which is to say music education is mostly focused on making good music, and generally keeps building on that. Theory should be learned as it is useful, and I think works best when every point you learn results in the lightbulb "so that is why I do X all the time"
D major: D E F♯ G A B C♯
D minor: D E F G A B♭ C
Db major: D♭ E♭ F G♭ A♭ B♭ C
Db minor: D♭ E♭ F♭ G♭ A♭ B♭♭ C♭
My point was not to showcase my confusion, rather to reinforce the fact that music theory texts are just generally low quality, and music theory as a field seems to be okay with that.
It's the culmination of years spent chasing an explanation of where all this stuff came from.
Highly informed by Tymoczko, who was mentioned in a sibling reply, and a bunch of other thinkers on the topic.
Ableton's teachings are far more interesting in my opinion, even if extremely more limited.
I get that maybe it's just the way it is, perhaps the underlying logic is so complex that it would be counterproductive to present it up front. I'm not saying that the theory should adapt to my pet peeves. But I'm very prone to learn things by understanding first principles and building from there. Having to memorize scales where suddenly you need to skip keys or use black keys without much apparent logic is a real deterrent for me considering it's something I learn as a hobby.
This is also the reason why I like algebra, calculus, etc. but dislike statistics... in algebra or calculus you start from first principles and work up from there understanding everything. In statistics (at least the way I was taught) you are shown the formula for the probability density of the Gaussian distribution and need to accept it as an act of faith.
Psychoacoustics is interesting, but it's not something that really helps you become a better musician for at least a couple of _decades_ of your journey. For the vast majority of students, it's a complete waste of time.
It's extremely unlikely that you learnt algebra or calculus from first principles, either. You started from the other end, learning the tools and skills to put it to good use, and then dove deeper with those tools already under your belt. Musicianship is no different.
I'm the same but never had an issue with music theory, because I came to it backwards from just playing. I learned to play by having fun with the instrument, and then the theory I learned afterwards felt like just being given names for the things that had become obvious through playing. What I'm saying is it will click if you let the music be your first principle.
I've been playing music for 20 years, learned and forgotten plenty of music theory. The only way I actually "grok" it, is by re-deriving the whys, but then that slows my learning to a crawl and I begin to forget the earlier lessons.
The memorization doesn't work well for me. I can read music, I can derive scales and chords, but the memorized knowledge of "this chord is technically the same as this one so we'll start calling things other things but it all means the same thing" gets me totally lost.
These sort of traditional music courses are designed around understanding western classical music. There are some scientific reasons for the way things are but they weren't developed by scientists. This sort of music theory will 100% help build an intuition into western, mostly classical but not exclusively classical, music.
If you want to know why the major scale is the way it is, go take math classes, a few physics courses will likely help as well, probably by the time you're in graduate school you'll understand it and can explain it better than 99% of musician. It'll help you in no way in creating music.
Essentially, what these complaints boil down to would be something similar to someone trying to learn to drive by first learning how an entire car works. It might be interesting to some, but it isn't the point.
In the linked textbook the real actual theory only starts in chapter 33, "Set Theory". (The exposition of the actual theory is pretty bad, though - zero explanation of what consonance/dissonance is, where the 12 notes come from and the reason why our particular scales exist.)
- 1.3 grand staff: "Note that middle C is always clearly notated in either the upper or lower staff and never floats between the two staves." It doesn't tell that the choice of which staff the notes belong to indicates which hand should play them.
- 1.4 accidentals "... are characters that can be placed before notes to raise or lower them." Not to reuse staff locations to distinguish the notes one needs to use.
- 1.5 Enharmonic notes: "When two notes sound the same but are written as two different notes on the staff, the written notes are said to be enharmonic." This is totally wrong, not merely maddening, but before discussing scales any decent explanation is impossible.
- 3.3 Scale Degree Names: in the middle of chapter 3 (minor scales) because there's no f** definition of what a scale is.
- 6.1 Introduction to Triads: "A triad consists of three notes stacked in consecutive thirds. A triad is also called a chord as well as a harmony. (Harmony also refers to chord progressions.)"
It's sad because many other places benefit from the concise and elegant treatment, for example:
- 5.5 Augmented and Diminished Intervals: "Augmented intervals are one half step larger than perfect or major intervals and diminished intervals are one half step smaller than perfect or minor intervals."
Example: memorize the major scale as a series of whole and half steps but we won’t provide you with any context or motivation for why this is so.
Similar: in a major scale all notes are played and none are skipped or repeated. Except if your scale goes D–E–F#–G, in which case you skip F. The only way for this to be correct is apparently we consider F and F# to be the same note for purposes of scales even though this directly contradicts a previous section.
I was expecting a “21st century” music theory reference to have an emphasis on building intuition. Music theorists are apparently allergic to good pedagogy.