josiah-theory

[josiah-theory]

Oops, Nearly 4 Months Went By

Yikes! Hello!

I guess I should summarize how the past 4 months have gone, music/theory-wise.

I began work once more on my music theory library, Harmonica, in Python, which hadn't been touched in some time. Then, I grew frustrated with Python and learned Rust, and began writing the library in Rust. That's how a lot of June and July went.

Then, over a month ago, I started messing with Pure Data again for the first time since 2020, after listening to some Autechre and watching some computer music talks and going, hey I want to create my own music tools from the ground up again. I grew frustrated with programming logic in Pure Data, so I decided to take a swing at SuperCollider again for the first time since 2017.

At first I was moreso focused on learning to use SuperCollider to, y'know, make sounds - learning the DSP tools and the client-server architecture, etc. Then, in recent weeks, I've been focusing more specifically on sclang, the programming language of the SuperCollider environment. Now, as of a week ago, I've been writing my library, Harmonica, as a SuperCollider extension.

My, quite the language-hopping this library has gone through. My rationale for writing it in SuperCollider is that I feel I may as well use it in an environment that's already specifically geared towards making music/sound experiments. After all, the nature of this project is just that: experimental. Also, I always suspected that my music theory explorations would be best suited towards algo music.

Also, sclang seems so far to be a genuinely competent programming language in its own right. I mean... it's also very... quirky (Quarky?) and I must say, coming from Rust, I am VERY underwhelmed by its error reporting. Overall though... so far so good. The gist of sclang is that it's very OOP, but also very functional and has first class functions. So actually, pretty similar to rust in some aspects!

I will say though, SuperCollider's reputation as a slow-burn of an environment with a steep learning curve is not unwarranted - good LORD it takes a lot of work to get going in this environment! Pure Data, as esoteric and opaque as it looks (those patches you see on YouTube with the million erratic lines connecting confusingly labeled boxes across the screen) is much more plug-n-play in comparison.

Alright, that's all I've got for you. Maybe some day I'll have some actual examples to share. Here's to another 4 months! Or, well, hopefully less this time.

[josiah-theory]

yes hello I haven't forgotten about this

I'm still working on Music Theory Things, I just like... have to package my ideas before I can share them here. For now I've been rambling to friends and discord servers. But yes... I will surely update this... Soon™

[josiah-theory]

This is how all music works

I might as well delete this blog now. That's it. Music has been solved.

[josiah-theory]

Come on Tumblr, add LaTeX support

It really irks me that any LaTeX I add only appears on my page when viewed on desktop. Ugh! My pretty formulas... 🥺

[josiah-theory]

How the hell do changes work?

Sorry, I don't have much of an answer. Only guesses.

There seem to be multiple different ways of getting around

1

Holy shit it made the number big. Anyways, uh, cadences. You know, like a G chord going into a C chord, or an F chord going into a C chord, to establish that feeling of "we are in C major."

Cadences seem like the most obvious scheme to work from, when trying to establish tonality. V -> I, IV -> I, ii -> V -> I, IV -> V -> I, V -> vi... these are the blueprints which can be injected with detours and be stringed together to form more sophisticated progressions.

The motivation behind their use seems to be "let's make it sound like we're in this key."

2

Pivot chords. Say we're in our previous example: C major. We've gone between the I and V a lil bit, and then we've reached our IV chord, F. Then we play Dm - still in C major, because Dm is the ii chord. But, wait - then comes a Bb chord. Huh? Then another C - are we still in C major? - but then we go back to our F chord - and it sounds resolved! In F major, not C major! We've used Dm as a pivot chord to modulate keys - from C to F.

This works because the chord D minor is present in the keys of C major and F major. In C major, it's the ii chord. In F major, it's the vi chord. the ii became a vi, and facilitated a smooth shift from one tonality to another. The pivot chord was our entry into the tonality of F major, and then the IV -> V -> I cadence, Bb -> C -> F, sealed the deal, communicating "yes, we are in F now."

3

Modal interchange. This is when you take a chord in a progression and temporarily borrow tones from another key to momentarily change the chord's "flavor". Let's assume you're playing F -> G -> C. Now, the chord Fm uses a tone that's outside of C major - the tone A flat. But hey, it sounds cool. So maybe we decide we want to make the F a minor chord at some point. We are free to choose this - I promise you won't go to jail.

Whenever we substitute a chord in a progression for a chord sharing the same root but different tones, that's basically what modal interchange is.

Now, you can combine modal interchange with pivot chords and cadences, and things can get really interesting! Now that' IV chord in C major can become basically whatever chord in whatever other tonality you want. Fm? FM7 augmented? F9? Fm7b5? [Insert corresponding keys these chords fit into here]? Have at it.

4

Hey, where'd my huge numbers go? I guess you only get three.

Anyways, the next one I wanted to mention was voice leading. This is another way of deciding which chord comes next - you can shift around the notes in the chord you're currently playing.

This technique has a lot of overlap with modal interchange - after all, when you substitute a minor chord for a major chord, you're just shifting the third (and seventh) tones down a semitone. But where modal interchange works with respect to the root note and its position in the key, manipulations of voice leading aren't really focused on key/tonality so much as they are focused on how the voices in a chord move around.

There's a kind of music theory called Neo-Riemannian theory which covers these kinds of manipulations. There are fancy names given to these different manipulations - they're called "transforms" in this theory - but I'm not going to go over them, because I can't be bothered to google it right now.

But basically, you can do things like shift from minor to major, or major to minor - C to Cm, or Cm to C - or perhaps shift the C note down to yield B E G, the notes of Em - or perhaps shift the G up to G# to yield C E G#, the notes of C augmented - or a personal favorite, shifting the first and fifth up a semitone to yield C# E G#, the notes of C#m - there are many possibilities.

With manipulations of voice leading, you're not focusing so much on tonality as you are the way the chromatic movements feel in their own right. But just like modal interchange, this can be combined with the other techniques to establish a feeling of tonality.

5

Oh cool, my big number privileges have been restored!

There is actually no 5th thing I wanted to say, that's really all that's on my mind right now.

All of these concepts (and more than what was listed here) may be used to get around harmonically. I like thinking of harmony in terms of a big complicated city, or maybe like constellations in the night sky. You learn to get around, you learn the routes. Tonality concerns orbiting around a certain star, but wouldn't it be boring to orbit the same ball of gas forever?

I'll stop before my metaphors become even more tedious.

[josiah-theory]

But wait, does a key HAVE to be perceived as EITHER C major OR C major with chromatic tones? Is there no inbetween?

Wow, great question, dear reader! No, it doesn't have to be one or the other. This is an important aspect of how I feel tonality is perceived. Rather than thinking of each note as being in a binary of "perceived" vs "not perceived", you can instead think of them as all being weighted - anywhere from 0% to 100% perceived. This way of thinking in terms of 0% to 100% is important to how I think of harmonic perception.

Again, I feel that when it comes to models of perception and expectation, these things are best modeled in nuanced, statistical ways.

[josiah-theory]

Building a model for scale perception

I have this idea in my head of a tool that takes a MIDI file, and tells you how the perceived scale changes over time in the music.

It's a bizarre idea for many reasons. For one, MIDI is an abstraction of real music. MIDI is music as much as sheet music is music - both are just abstractions, on the same level as a recipe, a set of instructions.

Let's say you have a MIDI note with pitch = C3, duration = 2 beats, and velocity = 100. What does this sound like?

Well, maybe it's just a straightforward sine wave.

Or maybe it's a saw wave.

But neither of these even have release - sound that endures after the MIDI note ends. Take the following synth sound for example:

Or this one, which not only has an entirely different timbre, but is in an entirely different register, despite being the same MIDI pitch:

The same piece of MIDI information can describe entirely different sounds. That's because MIDI is an abstraction of sound.

These abstractions are my primary subject of interest. When working with them, when trying to analyze them, I have to accept that models of perception can only go so far.

What these abstractions forego entirely is one of the most crucial features of sound and how it's perceived: timbre. When you ignore timbre, you ignore the very thing that makes sound... well, sound.

Does this mean my study is ruined? Is it pointless to study these musical qualities without considering timbre? I must admit it is, to some degree. Take the score of the most sombre, serious piece of music, and perform it on kazoos, and see just how sombre that piece comes out. Timbre is incredibly important.

...But, timbre isn't everything. While we can't expect a model of perception to be complete without accounting for timbre, I feel pretty darn confident claiming that these abstractions - the information communicated by MIDI data, and abstractions thereof - at least factor in to how a piece of music is perceived. And therefore, I believe that so long as you're willing to account for the rough incompleteness of such a model, there is at least SOME utility in its consideration.

And, more importantly, it's just a curious thing to me, trying to analyze how scales are perceived, just using MIDI data.

Scales, keys, tonalities, chords, whatever - these things are all themselves abstractions anyways. So I'm willing to accept that I'm working with at least one foot in la-la-land. I have no illusions about the major scale being ~fundamental~ or ~natural~, or that there's some natural, physical pretense to a V chord "wanting to resolve" down to a I chord. It's ultimately a matter of cultural programming. And there are immense variations in these associations between person to person.

Oh boy... I'm really going off on a tangent. Yeah, get used to that, I guess. I do that a lot. Anyways, what was the point of all this? Basically what I'm saying is that it's honestly kind of dumb to build a model of harmonic perception using only MIDI data, but it might be worth experimenting with anyways.

Maybe it's confusing what I even mean by a "model of harmonic perception". Well, what I mean by that is writing computer code that can take MIDI data, and say something like, "oh yeah, this is the ii -> V7 -> I progression in D major. Then it modulates to G major, before resolving back in D major." Or something like that, iunno.

Honestly, this is a bit of a fool's errand. But hey, it's something I'm curious about.

Given a measure of music, one way to analyze what scale it is in is to just collect all the notes present, and then report the scale they form. For example, if all the notes present form the collection C, D, E, F, G, A, B, then you clearly have a measure that's in C major.

But what about cases where you have chromatic grace notes? Short, subtle notes between other notes, which lie outside of C major. The aforementioned algorithm would report a much more chromatic scale - even though a listener would most likely still PERCEIVE it as C major.

This is where I'm at right now. I need to figure out how to weigh different notes based on different parameters - some candidates are:

The frequency at which they occur

Their velocities (how loud they are)

Their duration (how short/long they are)

Another parameter which I feel factors heavily into perception - my perception, at least - is how low a note is. I find that the lower a pitch, the more it informs the perceived scale. This is NOT a rule, but a general rule of thumb I've learned in harmonic perception is that bass notes inform perceived tonality more than notes in higher registers. Again - NOT a rule! Just an observation that's generally held up for my own ears over the years.

Frankly, there's so much more that factors into the perceived tonality of a section of music. A big factor is the context that the section sits in. I'm talking temporal context - you can analyze a few bars of music in isolation, but its perceived tonality can ENTIRELY change once placed within the larger piece of music. This gestalt quality of harmony profoundly complicates matters.

There is a lot to this. I'm going to end this entry here. Uhm... in closing, how musical harmony is perceived is a strange, elusive sort of thing, and it varies from person to person. Honestly, I feel a statistical/probabilistic model of sorts would be the most realistic. But whatever. It's curious stuff!

[josiah-theory]

LaTeX

Oh yeah, I wanted to test LaTeX real quick, too. It should display on my page.

\[a\cdot (b+c)=a\cdot b + a\cdot c\]

[josiah-theory]

What I Am Doing Lately

I've been focusing on answering that classic question: "Why does the music I enjoy sound good to me?"

Of course, being me, this music theoretical question pertains to harmony in specific. Why do some chord progressions/key changes sound better/more "logical" than others?

To answer this question, I enjoy transcribing my favorite music into MIDI, and then taking a look at the chords/scales/etc, figuring out ways to describe the quality of the changes.

Here's an example transcription of a bit of "Aigrette" by Hatfield and the North:

Oh boy, I forgot you can upload audio on here. Well, there it is, in all its MIDI glory - although, the melody is the only accurate part of this transcription. I took liberties with the texture of the accompaniment, and the bass part is straight up oversimplified... but it communicates the harmonic information quite sufficiently.

Anyways, using this, I can look at what chords feed into what other chords, how the melody traces over it all, the interplay between the parts... (though it'd be more meaningful to do a more 1:1 transcription...)

The problem, however, is that this becomes tedious to do on pen and paper.

I'm a fan of programming in python, however, and have been working on a music library of sorts for the past year or so - I call it "Harmonica", because that's fun. I may go more into detail about that project in another post. For now, I'll say it represents an accumulation of attempts to automate analytic/theoretical explorations which previously took place on pen and paper.

So yeah, that's the gist. Lately, I've been

Oh, sorry, accidentally pressed ctrl + v. Uhm, anyways, as I was saying: Lately, I've been turning my focus back to developing these music theory tools in python. That's good fun.

Look, I'm meandering. I'm not used to blogging. There are plenty of things I'd like to write about, though. Can I drop some bullet points?

Reflections on over-engineering & over-generalizing

How I view musical harmony combinatorially

Various ideas on musical analysis & generation via MIDI

Aw yeah!

Alright, that's all I got for this post. swag

[josiah-theory]

This is an image test. Look, combinatorial necklaces. Wow!

[josiah-theory]

I decided to make a blog.

It's been a long while since I've been on Tumblr. Almost a decade, I reckon.

I decided to make this because I get interested in things sometimes, and enjoy rambling about them to my friends. Usually it's math or music related.

I don't want to spend money on my own website, so Tumblr will suffice.