mossheadmandy - Tumblr blog

mossheadmandy · 2 months

Text

Music Theory Notes (for science bitches) 1: chords & such

This is one of these series where I use my blog as a kind of study blog type thing. If you're knowledgeable about music theory, it will be very basic. But that's kind of the problem, I've really struggled to absorb those basics!

When I was a teenager I learned to play violin and played in orchestras. I could read music, and play decently enough, but I didn't really understand music. I just read what was on the page, and played the scales I had to play for exams.

Lately I've been trying to learn music again. This time my instruments are zhonghu, voice, and DAWs. At some point I might get my violin back too. But really, I'm a total beginner again, and this time I want to do it properly.

For a long time when I tried to learn about music I would get overwhelmed with terminology and jargon and conventions. I might watch videos on composition and they'd be interesting but a lot of it would just fly over my head, I'd just have to nod along because I had no idea what all the different types of chord and such were. I tried to learn from sites like musictheory.net, but I found it hard to figure out the logical structure to fit it all into.

I feel like I'm finally making a bit of headway, so it's time to take some notes. The idea here is not just to answer the what, but also to give some sense of why, a motivation. So in a sense this is a first attempt at writing the introduction to music theory I wish I'd had. This is going to assume you know a little bit about physics, but basically nothing about music.

What is music? From first principles.

This is impossible to answer in full generality, especially since as certain people would be quick to remind me, there's a whole corner of avant-garde composers who will cook up counterexamples to whatever claim you make. So let's narrow our focus: I'm talking about the 'most common' type of music in the society I inhabit, which is called 'tonal music'. (However some observations may be relevant to other types of music such as noise or purely rhythmic music.)

Music is generally an art form involving arranging sound waves in time into patterns (in the sense that illustration is about creating patterns on a 2D surface with light, animation is arranging illustrations in time, etc.).

Physically, sound is a pressure wave propagating through a medium, primarily air. As sound waves propagate, they will reflect off surfaces and go into superposition, and depending on the materials around, certain frequencies might be attenuated or amplified. So the way sound waves propagate in a space is very complicated!

But in general we've found we can pretty decently approximate the experience of listening to something using one or two 'audio tracks', which are played back at just one or two points. So for the sake of making headway, we will make an approximation: rather than worry about the entire sound field, we're going to talk about a one-dimensional function of time, namely the pressure at the idealised audio source. This is what gets displayed inside an audio editor. For example, here's me playing the zhonghu, recorded on a mic, as seen inside Audacity.

A wrinkle that is not relevant for this discussion: The idealised 'pressure wave' is a continuous real function of the reals (time to pressure). By contrast, computer audio is quantised in both the pressure level and time, and this is used to reconstruct a continuous pressure wave by convolution at playback time. (Just like a pixel is not a little square, an audio sample is not a constant pressure!) But I'm going to talk about real numbers until quantisation becomes relevant.

When the human eye receives light, the cone cells in the eye respond to the frequencies of EM radiation, creating just three different neural signals, but with incredibly high sensitivity to direction. By contrast, when the human ear receives sound, it is directed into an organ called the cochlea which is kind of like a cone rolled up into a spiral...

Inside this organ, the sound wave moves around the spiral, which has a fascinatingly complex structure that means different frequencies of wave will excite tiny hairs at different points along the tube. In effect, the cochlea performs a short-time Fourier transform of the incoming sound wave. Information about the direction of the incoming wave is given by the way it reflects off the shape of the ear, the difference between ears, and the movement of our head.

So! In contrast to light, where the brain receives a huge amount of information about directions of incoming light but only limited information of the frequency spectrum, with sound we receive a huge amount of information about the frequency spectrum but only quite limited information about its direction.

Music thus generally involves creating patterns with vibration frequencies in the sound wave. More than this, it's also generally about creating repeating patterns on a longer timescale, which is known as rhythm. This has something to do with the way neurons respond to signals but that's something I'm not well-versed in, and in any case it is heavily culturally mediated.

All right, so, this is the medium we have to play with. When we analyse an audio signal that represents music, we chop it up into small windows, and use a Fourier transform to find out the 'frequencies that are present in the signal'.

Most musical instruments are designed to make sounds that are combinations of certain frequencies at integer ratios. For example here is a plot of the [discrete] Fourier transform of a note played on the zhonghu:

The intensity of the signal is written in decibels, so it's actually a logarithmic scale despite looking linear. The frequency of the wave is written in Hertz, and plotted logarithmically as well. A pure sine wave would look like a thin vertical line; a slightly wider spike means it's a combination of a bunch of sine waves of very close frequencies.

The signal consists of one strong peak at 397Hz and nearby frequencies, and a series of peaks at (roughly) integer multiples of this frequency. In this case the second and third peaks are measured at 786Hz, and 1176Hz. Exact integer ratios would give us 794Hz and 1191Hz, but because the first peak is quite wide we'd expect there to be some error.

Some terminology: The first peak is called the fundamental, and the remaining peaks are known as overtones. The frequency of the fundamental is what defines this signal as a particular musical note, and the intensities of the overtone and widths of the peaks define the quality of the note - the thing that makes a flute and a violin playing the same fundamental frequency sound different when we listen to them. If you played two different notes at the same time, you'd get the spectrums of both notes added together - each note has its own fundamental and overtones.

OK, so far that's just basic audio analysis, nothing is specific to music. To go further we need to start imposing some kind of logical structure on the sound, defining relationships between the different notes.

The twelve-tone music system

There are many ways to do this, but in the West, one specific system has evolved as a kind of 'common language' that the vast majority of music is written in. As a language, it gives names to the notes, and defines a space of emotional connotations. We unconsciously learn this language as we go through the process of socialisation, just as we learn to interpret pictures, watch films, etc.

The system I'm about to outline is known as 12-tone equal temperament or "12TET". It was first cooked up in the 16th century almost simultaneously in China and Europe, but it truly became the standard tuning in the West around the 18th century, distilled from a hodgepodge of musical systems in use previously. In the 20th century, classical composers became rather bored of it and started experimenting with other systems of tonality. Nevertheless, it's the system used for the vast majority of popular music, film and game soundtracks, etc.

Other systems exist, just as complex. Western music tends to create scales of seven notes in an octave, but there are variants that use other amounts, like 6. And for example classical Indian music uses its own variant of a seven-note scale; there are also nuances within Western music such as 'just intonation' which we'll discuss in a bit; really, everything in music is really fucking complicated!

I'll be primarily discussing 12TET because 1. it's hard enough to understand just one system and this one is the most accessible; 2. this has a very nice mathematical structure which tickles my autismbrain. However, along the way we'll visit some variants, such as 'Pythagorean intervals'.

The goal is to try and not just say 'this is what the notation means' but explain why we might construct music this way. Since a lot of musical stuff is kept around for historical reasons, that will require some detours into history.

Octaves

So, what's the big idea here? Well, let's start with the idea of an octave. If you have two notes, let's call then M and N, and the frequency of N is twice the frequency of M... well, to the human ear, they sound very very closely related. In fact N is the first overtone of M - if you play M on almost any instrument, you're also hearing N.

Harmony, which we'll talk about in a minute, is the idea that two notes sound especially pleasant together - but this goes even further. So in many many music systems around the world, these two notes with frequency ratio of 2 are actually identified - they are in some sense 'the same note', and they're given the same name. This also means that further powers of 2, of e.g. 4, 8, 16, and so on, are also 'the same note'. We call the relationship between M and N an octave - we say if two notes are 'an octave apart', one has twice the frequency of the other.

For example, a note whose fundamental frequency is 261.626Hz is known as 'C' in the convention of 'concert pitch'. This implies an infinite series of other Cs, but since the human ear has a limited range of frequencies, in practice you have Cs from 8.176Hz up through 16744.036. These are given a series of numbers by convention, so 261.626Hz is called C4, often 'middle C'. 523.251Hz is C5, 1046.502Hz is C6, and so on. However, a lot of the time it doesn't matter which C you're talking about, so you just say 'C'.

But the identification of "C" with 261.626Hz * 2^N is just a convention (known as 'concert pitch'). Nothing is stopping you tuning to any other frequency: to build up the rest of the structure you just need some note to start with, and the rest unfolds using ratios.

Harmony and intervals

Music is less about individual notes, and more about the relationship between notes - either notes played at the same time, or in succession.

Between any two notes we have something called an interval determined by the ratio of their fundamental frequencies. We've already seen one interval: the octave, which has ratio 2.

The next interval to bring up is the 'fifth'. There are a few different variants of this idea, but generally speaking if two notes have a ratio of 1.5, they sound really really nice together. Why is this called a 'fifth'? Historical reasons, there is no way to shake this terminology, we're stuck with it. Just bear with me here, it will become semi-clear in a minute.

In the same vein, other ratios of small integers tend to sound 'harmonious'. They're satisfying to hear together. Ratios of larger integers, by contrast, feel unsatisfying. But this creates an idea of 'tension' and 'resolution'. If you play two notes together that don't harmonise as nicely, you create a feeling of expectation and tension; when you you play some notes that harmonise really well, that 'resolves' the tension and creates a sense of relief.

Building a scale - just intonation

The exact 3:2 integer ratio used in two tuning systems called 'Pythagorean tuning' and 'just intonation'. Using these kinds of integer ratios, you can unfold out a whole series of other notes, and that's how the Europeans generally did things before 12TET came along. For example, in 'just intonation', you might start with some frequency, and then procede in the ratios 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, and at last 2 (the octave). These would be given a series of letters, creating a 'scale'.

What is a scale? A scale is something like the 'colour palette' for a piece of music. It's a set of notes you use. You might use notes from outside the scale but only very occasionally. Different scales are associated with different feelings - for example, the 'major scale' generally feels happy and triumphant, while a 'minor scale' tends to feel sad and forlorn. We'll talk a lot more about scales soon.

In the European musical tradition, a 'scale' consists of seven notes in each octave, so the notes are named by the first seven notes of the alphabet, i.e. A B C D E F G. A scale has a 'base note', and then you'd unfold the other frequencies using the ratios. An instrument such as a piano would be tuned to play a particular scale. The ratios above are one definition of a 'major scale', and starting with C as the base note, the resulting set of notes is called 'C Major'.

All these nice small-number ratios tend to sound really good together. But it becomes rather tricky if you want to play multiple scales on the same instrument. For example, say your piano is tuned in just intonation to C Major. This means, assuming you have a starting frequency we'll call C, you have the following notes available in a given octave:

C, D=(9/8)C, E=(5/4)C, F=(4/3)C, G=(3/2)C [the fifth!], A=(5/3)C, B=(15/8)C, and 2C [the start of the next octave].

Note: the interval we named the 'fifth' is the fifth note in this scale. It's actually the fifth note in the various minor scales too.

But now suppose you want to play with some different notes - let's say a scale we'll call 'A major', which has the same frequency ratios starting on the note we previously called A. Does our piano have the right keys to play this scale?

Well, the next note up from A would be (9/8)A, which would be (9/8)(5/3)C=(15/8)C - that's our B key, so far so good. Then (5/4)A=(5/4)(5/3)C=(25/12)C and... uh oh! We don't have a (25/12)C key, we have 2C, so if we start at A and go up two keys, we have a note that is slightly lower frequency than the one we're looking for.

What this means is that, depending on your tuning, you could only approximate the pretty integer ratios for any scale besides C major. (25/12) is pretty close to 2, so that might not seem so bad, but sometimes we'd land right in between two notes. We can approximate these notes by adding some more 'in between' piano keys. How should we work out what 'extra' keys to include? Well, there were multiple conventions, but we'll see there is some logic to it...

[You might ask, why are you spending so long on this historical system that is now considered obsolete? Well, intervals and their harmonious qualities are still really important in modern music, and it makes most sense to introduce them with the idea of 'small-integer ratios'.]

The semitone

We've seen if we build the 'major scale' using a bunch of 'nice' ratios, we have trouble playing other scales. The gap above may look rather haphazard and arbitrary, but hold on, we're working in exponential space here - shouldn't we be using a logarithmic scale? If I switch to a logarithmic x-axis, we suddenly get a rather appealing pattern...

All the gaps between successive notes are about the same size, except for the gap between E and F, and B and C, which are about half that size. If you try to work that out exactly, you run into the problems we saw above, where C to D is 9/8 or 1.125, but D to E is 10/9 or 1.11111... Even so, you can imagine how people who were playing around with sounds might notice, damn, these are nice even steps we have here. Though you might also notice places where, in this scheme, it's not completely even - for example G to A (ratio 10/9) is noticeably smaller than A to B (ratio 9/8).

We've obliquely approached the idea of dividing the octave up into 12 steps, where each step is about the size of the gap between E and F or B and C. We call each of these steps a 'semitone'. Two semitones make a 'whole tone'. We might fill in all the missing semitones in our scale here using whole-number ratios, which gives you the black keys on the piano. There are multiple schemes for doing this, and the ratios tend to get a bit uglier. In the system we've outlined so far, a 'semitone' is not a fixed ratio, even though it's always somewhere around 1.06.

The set of 12 semitones is called the 'chromatic scale'. It is something like the 'colour space' for Western music. When you compose a piece, you select some subset of the 12 semitones as your 'palette' - the 'scale of' a piece of music.

But we still have a problem here, which is the unevenness of the gaps we discussed above. This could be considered a feature, not a bug, since each scale would have its own 'character' - it's defined by a slightly different set of ratios. But it does add a lot of complication when moving between scales.

So let's say we take all this irregularity as a bug, and try to fix it. The solution is 'equal temperament', which is the idea that the semitone should always be the exact same ratio, allowing the instrument to play any scale you please without difficulty.

Posed like this, it's easy to work out what that ratio should be: if you want 12 equal steps to be an octave, each step must be the 12th root of 2. Which is an irrational number that is about 1.05946...

At this point you say, wait, Bryn, didn't you just start this all off by saying that the human ear likes to hear nice simple integer ratios of frequencies? And now you're telling me that we should actually use an irrational number, which can't be represented by any integer ratio? What gives? But it turns out the human ear isn't quite that picky. If you have a ratio of 7 semitones, or a ratio of 2^(7/12)=1.4983..., that's close enough to 1.5 to feel almost as good. And this brings a lot of huge advantages: you can easily move ('transpose') between different scales of the same type, and trust that all the relevant ratios will be the same.

Equal temperament was the eventual standard, but there was a gradual process of approaching it called stuff like 'well-tempered' or 'good temperament'. One of the major steps along the way was Bach's collection 'the well-tempered klavier', showing how a keyboard instrument with a suitable tuning could play music in every single established scale. Here's one of those pieces:

youtube

Although we're using these irrational numbers, inside the scale are certain intervals that are considered to have certain meanings - some that are 'consonant' and some that are 'dissonant'. We've already mentioned the 'fifth', which is the 'most consonant' ratio. The fifth consists of 7 semitones and it's roughly a 1.5 ratio in equal temperament. Its close cousin is the 'fourth', which consists of 5 semitones. Because it's so nice, the fifth is kind of 'neutral' - it's just there but it doesn't mean a lot on its own.

For the other important intervals we've got to introduce different types of scale.

The scale zoo

So, up above we introduced the 'major' scale. In semitones, the major scale is intervals of 2, 2, 1, 2, 2, 2, 1. This is also called a 'mode', specifically the 'Ionian mode'. There are seven different 'modes', representing different permutations of these intervals, which all have funky Greek names.

The major scale generally connotes "upbeat, happy, triumphant". There are 12 different major scales, taking the 12 different notes of the chromatic scale as the starting point for each one.

Next is the minor scale, which tends to feel more sad or mysterious. Actually there are a few different minor scales. The 'natural minor' goes 2, 1, 2, 2, 1, 2, 2. You might notice this is a cyclic permutation of the major scale! So in fact a natural minor scale is the same set of notes as a major scale. What makes it different?

Well, remember when we talked about tension and resolution? It's about how the notes are organised. Our starting note is the 'root' note of the scale, usually established early on in the piece of music - quite often the very first note of the piece. The way you move around that root note determines whether the piece 'feels' major or minor. So every major scale has a companion natural minor scale, and vice versa. The set of notes in a piece is enough to narrow it down to one minor and one major, but you have to look closer to figure out which one is most relevant.

The 'harmonic minor' is almost the same, but it raises the second-last note (the 7th) a semitone. So its semitone intervals are 2, 1, 2, 2, 1, 3, 1.

The 'melodic minor' raises both the 6th and 7th by one semitone, (edit: but usually only on the way up). So its semitone intervals are 2, 1, 2, 2, 2, 2, 1. (edit: When you come back down you tend to use the natural minor.)

If you talk about a 'minor scale' unqualified, you mean the natural minor. It's also the 'Aeolian mode' in that system of funky Greek names I mentioned earlier.

So that leads to a set of 24 scales, a major and minor scale for every semitone. These are the most common scale types that almost all Western tonal music is written in.

But we ain't done. Because remember I said there were all those other "modes"? These are actually just cyclic permutations of the major scale. There's a really nerdy Youtube channel called '8-bit music theory' that has a bunch of videos analysing them in the context of videogame music which I'm going to watch at some point now I finally have enough background to understand wtf he's talking about.

youtube

And of top of that you have all sorts of other variants that come from shifting a note up or down a semitone.

The cast of intervals

OK, so we've established the idea of scales. Now let's talk intervals. As you might guess from the 'fifth', the intervals are named after their position in the scale.

Let me repeat the two most common scale modes, in terms of number of semitones relative to the root note:

position: 1, 2, 3, 4, 5, 6, 7, 8 major: 0, 2, 4, 5, 7, 9, 11, 12 minor: 0, 2, 3, 5, 7, 8, 10, 12

So you can see the fourth and fifth are the same in both. But there's a difference in three places: the third, the sixth, and the seventh. In each case, the minor is down a semitone from the major.

The interval names are... not quite as simple as 'place in the scale', but that's mostly how it works. e.g. the 'major third' is four semitones and the 'minor third' is three.

The fourth and fifth, which are dual to each other (meaning going up a fifth takes you to the same note as going down a fourth, and vice versa) are called 'perfect'. The note right in between them, an interval of 6 semitones, is called the 'tritone'.

(You can also refer to these intervals as 'augmented' or 'diminished' versions of adjacent intervals. Just in case there wasn't enough terminology in the air. See the table for the names of every interval.)

So, with these names, what's the significance of each one? The thirds, sixths and sevenths are important, because they tell us whether we're in minor or major land when we're building chords. (More on that soon.)

The fifth and the octave are super consonant, as we've said. But the notes that are close to them, like the seventh, the second and even more so the tritone, are quite dissonant - they're near to a nice thing and ironically that leads to awkward ratios which feel uncomfy to our ears. So generally speaking, you use them to build tension and anticipation and set up for a resolution later. (Or don't, and deliberately leave them hanging.)

Of course all of these positions in the scale also have funky Latin names that describe their function.

There's a lot more complicated nuances that make the meaning of a particular interval very contextual, and I certainly couldn't claim to really understand in much depth, but that's basically what I understand about intervals so far.

Our goofy-ass musical notation system

So if semitones are the building block of everything, naturally the musical notation system we use in the modern 12TET era spaces everything out neatly in terms of semitones, right?

Right...?

Lmao no. Actually sheet music is written so that each row of the stave (or staff, the five lines you write notes on) represents a note of the C major scale. All the notes that aren't on the C major scale are represented with special symbols, namely ♯ (read 'sharp') which means 'go up a semitone', and ♭ (read 'flat') which means 'go down a semitone'. That means the same note can be notated in two different ways: A♯ and B♭ are the same note.

The above image shows the chromatic scale, notated in two different ways. Every step is exactly one semitone.

Since a given scale might end up using one of these 'in between' notes that has to be marked sharp or flat, and you don't want to do that for every single time that note appears. Luckily, it turns out that each major/minor scale pair ends up defining a unique set of notes to be adjusted up or down a semitone, called the 'key signature'. So you can write the key signature at the beginning of the piece, and it lasts until you change key signature. For example, the key of 'A♭ major' ends up having four sharps:

There is a formula you can use to work out the set of sharps or flats to write for a given key. (That's about the point I checked out on musictheory.net.)

There is some advantage to this system, which is that it very clearly tells you when the composer intends to shift into a different scale, and it saves space since with the usual scales there are no wasted lines. But it's also annoyingly arbitrary. You just have to remember that B to C is only a semitone, and the same for E to F.

What are those weird squiggly symbols? Those are 'clefs'. Each one assigns notes to specific lines. The first one 𝄞 is the 'treble clef', the second one 𝄢 is the 'bass clef'. Well, actually these are the 'G-clef' and the 'F-clef', and where they go on the stave determines note assignment, but thankfully this has been standardised and you will only ever see them in one place. The treble clef declares the lines to be E G B D F and the bass clef G B D F A.

There is also a rarer 'C-clef' which looks like 𝄡. This is usually used as the 'Alto clef' which means F A C E G.

This notation system seems needlessly convoluted, but we're rather stuck with it, because most of the music has been written in it already. It's not uncommon for people to come up with alternative notations, though, such as 'tabs' for a stringed instrument which indicate which position should be played on each string. Nowadays on computers, a lot of DAWs will instead use a 'piano roll' presentation which is organised by semitone.

And then there's chords.

Chords! And arpeggios!

A chord is when you play 3 or more notes at the same time.

Simple enough right? But if you wanna talk about it, you gotta have a way to give them names. And that's where things get fucking nuts.

But the basic chord type is a 'triad', consisting of three notes, separated by certain intervals. There are two standard types, which you basically assemble by taking every other note of a scale. In terms of semitones, these are:

Major triad: 0 - 4 - 7 Minor triad: 0 - 3 - 7

Then there's a bunch of variations, for example:

Augmented: 0 - 4 - 8 Diminished: 0 - 3 - 6 Suspended: 0 - 2 - 7 (sus2) or 0 - 5 - 7 (sus4) Dominant seventh: 0 - 4 - 7 - 10 Power: 0 - 7

There is a notation scheme for chords in pop, jazz, rock, etc., which starts with a root note and then adds a bunch of superscripts to tell you about any special features of a chord. So 'C' means the C Major triad (namely C,E,G) and 'Cm' or 'c' means the C Minor triad (namely C,E♭,G).

In musical composition, you usually tend to surround the melody (single voice) with a 'chord progression' that both harmonises and creates a sense of 'movement' from one chord to another. Some instruments like guitar and piano are really good at playing chords. On instruments that can't play chords, they can still play 'arpeggios', which is what happens if you take a chord and unroll it into a sequence of notes. Or you play in an ensemble and harmonise with the other players to create a chord together. Awww.

Given a scale, you can construct a series of seven triad chords, starting from each note of the scale. These are generally given scale-specific Roman numerals corresponding to the position in the scale, and they're used to analyse the progression of chords in a song. I pretty much learned about this today while writing this post, so I can't tell you much more than that.

Right now, that's about as far as I've gotten with chords. On a violin, you can play just two strings at the same time after all - I never had much need to learn about them so it remains a huge hole in my understanding of music. I can't recognise chords by ear at all. So I gotta learn more about them.

As much as I wrote this for my own benefit... if you found this post interesting, let me know. I might write more if people find this style of presentation appealing. ^^'

240 notes · View notes