A study of musical scales (2017) | Hacker News Reader

A study of musical scales (2017)

ianring.com

59 pointsandai48 comments

A bit of a tangent, but what I always found confusing about music theory is which parts are arbitrary historical legacy and which are fundamental to human enjoyment of music. 12 notes in a scale? Their particular frequencies? That whole multiplying and dividing business? Why the letters are in the wrong order. Why have sharps and flats instead of just more letters? I always get stuck wondering how much of this is just a big ugly edifice and how much actually matters.

I can help with a couple of these. 12 notes is a kludge, because powers of 2^(1/12) well approximate simple ratios. These simple ratios (such as 3:2) sound good. I like this video for explanation: https://www.youtube.com/watch?v=1Hqm0dYKUx4

Also, the reason that the letters are in the wrong order is that we're playing the wrong scale. The monks who invented the scale usually sang in A-minor (so we use the first letter for that scale). Nowadays, we use the major scale, which has the same spacing between notes, but in a different order. It just so happens that C-major has the same notes as A-minor, so that's why we count from C. A great channel for this is Michael New, he explains everything very clearly: https://www.youtube.com/watch?v=UcviIQg_BlU

Music, much like language, has a lot of technical debt.

Thanks. Those are very enlightening. The fraction names are really nothing like what you'd expect them to mean! Now I realize why when someone was trying to explain a circle of fifths to me and how it was like a clock, it wasn't a 5-numbered clock!

The other thing to keep in mind is that in musical intervals, X and Y add up to X + Y - 1. For example, two thirds (say, C-E and E-G) make a fifth (C-G). That's because both endpoints get counted in the interval.

The timbres of almost all (pleasant) physical instruments, including the human voice, make octaves and fifths (1/2 and 1/3 frequency ratios) important (and pleasant) intervals (the clarinet is a rare exception where the twelfth is the important interval, but even then that's an octave plus a fifth - music intervals use 1-based indexing which is definitely historical legacy).

The circle of fifths and equal-temperament tuning is a kludge, and there are alternatives (and you can go back to baroque times and hear pieces written for different tunings), but no approach is perfect; fundamentally 2^x will never equal 3^y so you have to compromise somewhere.

7-note scales are not the only one possible - a pentatonic scale produces pleasant sounds, as do the various medieval modes - but you can't go much finer without things sounding unpleasant (and eventually with microtone tuning you reach a point where people will hear beats instead of distinct tones). A 7-note scale also fits conveniently with the circle of fifths to allow you to have a single instrument that can play in multiple keys.

Fundamentally you do want to be able to write runs up and down the scale, so a notation of 7 points with sharps/flats is a lot more convenient than a notation of 12 points where a run would have seemingly-arbitrary gaps and off-scale notes look the same as on-scale notes.

If you're actually interested, I'd highly recommend doing some practical musicmaking and/or conventional academic study rather than trying to theorise everything from first principles yourself. After all, many of these principles were refined over centuries of trial and error, while the fundamental physical underpinnings were only discovered very recently.

> with microtone tuning you reach a point where people will hear beats instead of distinct tones)

That's a fantastic notion. Music consists of rhythm melody and harmony. Harmony is created by relations of frequencies. But Rhythm itself IS a frequency, and there can be several overlapping rhythms in a piece of music. Melodic steps are also ratios between frequencies, but pieces of melody also repeat in a piece of music. So its all about frequencies and frequency of changing the frequencies. Something like that. The simple point is that rhythm is a frequency. A note is a frequency.

"Beats" is a specific term of art. It's not related to rhythm.

https://en.m.wikipedia.org/wiki/Beat_(acoustics)

It's a term of art describing the regular rhythmic peaks when discordant notes are played together.

True. Much of the above article's analysis of scales applies equally well to an analysis of rhythm, and the circle-bead diagrams for scales he uses are still the best pictograph method ive seen to get a visual impression of a beat over a single bar

-- bar a.k.a. scale

Any time you feel like you have too much money and spare time, you should get into modular synths - I think you might really dig it.

Precisely what I was thinking of, how you can start with a low frequency oscillator and turn it up, until it becomes a note. Rhythm turns into note. Multiple rhythms turn into harmony.

> 12 notes in a scale? Their particular frequencies? That whole multiplying and dividing business? Why the letters are in the wrong order. Why have sharps and flats instead of just more letters?

The number of notes and the ratios between their frequencies are fundamental. The rest are historical accidents.

A good place to start is David Benson's 'Music: a Mathematical Offering'. Available free online here: https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-mus....

This is a pointer to a more intuitive understanding.

Playing music, and especially improvising, is more much hung on the intervals—those ratios—rather than concerns about fixed notes. Give or take, emotional colouring of tones and harmonics is another.

The number of notes is also historical tradition, e.g. pentatonic vs heptatonic.

Sure, but it's based on something real (2^17 ~ 3^12) rather than being a pure historical accident.

It all starts to make sense once you start playing an instrument. Personally I think of music theory as a convenience. If I know music theory, then that makes it easier for me to play with other people who also know music theory, and a lot of musicians know music theory because it's a convention. Of course we can always throw these conventions away and do something completely new, but the problem with that is I'm not talented enough to be able to break away and make something cool like Jazz or microtones.

I really just want to have fun playing music with my friends, and 12 tones is enough to be able to do that.

A bunch of different scales is used in different cultures and traditions around the world.

The choice of a scale is an interesting combination of universal math and tradition. Scales are built from intervals which a are basic ratios - 1/2, 2/3, 3/4 etc., but the choice of how many and which steps to include differs. But you cant just make up arbitrary scales, they still have to be based on intervals.

Western music traditionally uses a 7-note scale. The 12-note scale is really a clever hack, because it allows some instruments to play a whole range of different 7-note scales by using different subsets if the 12-note scale. But the music is still based on a 7-note scale, which is indicated by the letters and music notation.

Your question assume the historical and culture specific parts of music are "ugly" and only the universal and fundamental properties "matter". I fundamentally disagree - much of our enjoyment of music is due the cultural context and tradition.

All of it matters, as a lot of what you find enjoyable about music is learned. Today's music might sound dissonant to someone a few hundred ago. Eg. a particular interval the _tritone_, also known as the devil's tone was considered particularly unpleasant before, but is now ubiquitous. Even today you might find music with 12+ tones not enjoyable (try listening to Indian or Middle Eastern scales). To understand what the physical fundamentals are, think of a taught piece of string that produces a note when you pluck it. The length of it is arbitrary (frequency), what you call this note is arbitrary (letters, sharps, flats etc.). But if you start cutting the string and compare the note it makes to the original, that _ratio_ of strings (frequencies) is a fundamental property of sound and the basis of all music. The simpler the ratio, the more 'pleasant' it will sound when played together, eg. 1-2 is the octave, 2-3 is the fifth. As you can cut the string any number of times, the number of these ratios (the number of notes in a scale) is unlimited, but western music settled on 12 because that number has some nice properties.

*Note - 12 tone equal temperament, that allows for the kind of transformations in the article does not exactly match these fundamental ratios. If we were using pure ratios, the distance between any 2 notes in the scale would differ slightly. Meaning you would have to retune a piano if you wanted to play in a different scale.

Other people have good answers to this question in this thread, but here's my attempt at the beginnings of a programmers introduction to music theory:

https://github.com/abetusk/scratch/blob/release/notes/Music-...

voting749227383 has a good response but to elaborate on the 12 note choice more, there's some justification of why 12 as opposed to some other number [0]. 12 notes gives a good compromise of the ability to combine notes in the scale while still not being too large to be unwieldy. In other words, some large ratio of note pairs have frequency ratios with small numerators and denominators when chosen to be from 12 notes as opposed to 11, 13 etc.

I hadn't heard about the C vs. A issue, so it's interesting to learn that it's from the monks. I assume the sharp and flat notes are because they're trying to differentiate, as much as possible, the discordant neighboring notes and maybe trying to promote some base mode/scale (like C-major).

[0] Measures of Consonances in a Goodness-of-fit Model for Equal-tempered Scales by Aline Honingh

I spent lots of time during the quarantine on learning music (theory). My consensus is: 99% of what is considered "music theory" is purely interpretative cruft. 1% of it is based on physics.

The only reason why we (westerners) learn the same cruft is because we have been agreeing that this cruft is the least confusing one we have come up with (so far).

I think @votingNNN answered you well. The sharps and flats are because the majority of music stays within the scale -- particularly for the singing monks not in front of a piano. So they hear and consider the scale and notes falling outside of that sound wrong i.e. they sound sharp or flat

Sharp and flat are only needed when moving outside or between scales, or using "fixed do", like modern western sheet music does with sharps and flats. Here is a discussion of the benefits

https://music.stackexchange.com/questions/41764/solfege-fixe...

For the record i prefer and use a numbered tin kay method like the article (but starting from zero rather than one), because it lets me abstractly consider what im playing, in much the same way as the above article

Interestingly the underground Chinese church uses a method called tin kay for teaching worship piano. For instance all major chords are represented as 047

And if a beginner counts their piano keys, just with those three numbers, they can now play all major chords. Add 037 and they can now play all minor chords

This vastly beats memorizing each key combo for each scale

This method came from asking God for a revelation on how to teach piano, and they attribute it to Him

It is also possible that they read just about anything on mathematical music theory, where representing chords as sets of integers modulo twelve is standard.

If you feel like the other commenters clarified anything for you, I'd like to throw a new bit of confusion back into the mix ;) https://strandbergguitars.com/eu/boden-true-temperament/ this guitar actually sounds more "in tune" than a standard.

https://www.youtube.com/watch?v=-penQWPHJzI

Nice guitars and certainly nice strings. Fun, and in tune

To my ears, I'm hearing a normal equal temperment tuning, with the normal slight disharmonies over the chords...

Maybe somebody better than me can give another opinion?

Edit: I was looking at the website, rather than your linked video. Yeah, that guitar with wobbley frets definitely has slightly better tuning (imho). For guitar (and equal temperment) generally, some chords will always sound flatter than others, so a concern is presumably that guitar sounds worse as you move away from Cmaj/Emaj songs??

But in any case, fun

Oddly enough, I just finished watching this video which explains all the scales succinctly. I was struck by all the permutations that are possible.

Simple method to organize ALL MUSICAL SCALES of harmonies: https://www.youtube.com/watch?v=Vq2xt2D3e3E

But note that only one octave is covered in the examples. There are musical styles (e.g. Jazz) where a scale extends over more than one octave, e.g. you have to extend the system to e.g. 24 or 36 places.

Thanks ... this clarifies so many

We get a post like this every couple of weeks, it seems. Given the audience here, I understand the impulse - it's very fun to see the structure, and it really helps you understand the framework upon which a lot of western music is built. But beware that this is not a way to actually understand what is interesting or compelling about "music", because you'll notice that the first step in all these articles is to reduce "music" to something you can analyze like it's computer code. This was tried in the early 20th century, with Schönberg etc. You get things like "12 tone music". It was tried in the mid 20th century onwards, when you had all kinds of flexibility to use crazy mathematical constructions with electronics and computers - you get things like Karlheinz Stockhausen. That is, things which don't sound any more interesting that any other arbitrary way of generating material with which to compose. These articles are fun, but don't think that they're about "music" any more than an article about the chemistry of a common pigment is an article about "painting".

Representing music in all of its relevant dimensions (scale, time, polyphony, melodic patterns, dynamics, etc.) is a very interesting and still not optimally solved problem. It is e.g. relevant for machine learning based systems for musical composition or improvisation; with transformers there was a big progress made compared to nearly thirty years of stagnation. And this is not about "exotic" compositions associated with the composers you mentioned, but about any kind of popular or classic music. The "necklace" representation of scales (also used in the referenced site) is an interesting one since it allows to e.g. represent modulation by rotation and/or mirroring. You rarely meet it in traditional music education. Such representations lead to useful insights about music if one wants to deal with it algorithmically. And there are many people here (including me) who do.

I completely agree that it's very interesting. But it's a Glass Bead Game you're playing. A fun game, a game which can make you a better composer, even but one should not be fooled into thinking that this is really essentially to how the music you love was created. One of the few things I think I know about music is that good way to make it is to take whatever system you have and subvert it a little bit.

I'm a musician and composer myself (even used to work as a professional musician a couple of years). There are totally different approaches to composing music, from totally intuitive/non-intellectual to totally formal/rule-based. I would say that today the majority of composers compose music intuitively and directly on the instrument, without being aware of the exact processes involved. Formalizations in music are therefore mostly academic, arising from analysis of the compositions of others, as an attempt to explain how these compositions work. For algorithmic composition, such formalizations are necessary, but so far not sufficient to create a high-quality composition (just as one could not become a good musician just by studying music theory). Nevertheless, it is necessary to study such formalizations if one wants to advance the field of algorithmic composition. However, I consider most "classical" approaches to formalizing music obsolete for this purpose.

Yes - this is more like stamp collecting than composition.

It can be interesting for generating unfamiliar colours, but that's all. It's certainly not a deep key to harmonic theory.

And it ignores the fact that scales don't necessarily repeat at the octave, scale steps don't have to be equal, and there can be any number of steps in a scale.

Cmon this is hacker news not 'art is ineffable news'.

Yes the point of things like this is to help understand the framework that underlies (edit: most western) music. There's actually a lot of structure to western music that is obfuscated by the completely bonkers terminology. I find articles like this very intersting and useful, and there's no way you'll catch me listeneing to stockhausen.

For example the explanation of modes vs scales in this article is the clearest I've ever seen.

I thing the point of the GP is to remind us that this is not 'the framework that underlies music', it is 'a framework that is used to structure much European/American-inspired music'.

That is, there are cultures that formalize or structure their music differently, and there are properties of music that are outside of these models. There are even many popular or niche songs that work against some aspects of this structure.

And all this is not about the ineffability of art, it is simply about not confusing the map for the territory. Many people who wax poetic about the mathematical properties of music, and start deep theories about the human brain and our enjoyment of music from there, tend to forget that the mathematics are describing a particular formal system that some choose to use when making music; they are not mathematical properties of anything that a human being would recognize as pleasant music.

Many people ... tend to forget ...

I dont think thats true, especially on HN. We're perfectly capable of being interested in the mathy bits without losing sight of the arty bits

e.g. the OP who wrote that very analytical piece about every possible scale composes music like this https://ianring.bandcamp.com

Yes this is so true. It is a sign of our general alienation from nature. We virtualize everything and think the simplified, abstract, intellectual picture is the reality. Nature is much more than a mental model of it. https://publicdelivery.org/magritte-not-a-pipe/

I don't understand the complaint here. The author writes music. He made this analysis out of curiosity and because he loves music. Isn't this perfect HN content?

So far, these posts are fluff, but this does not mean that just because humanity did not succeed in mathematically understanding harmony in the early twentieth century that it must be impossible.

> You can't play a song in C minor, because that would require three flats. So instead you play the song with A as the root (A B C D E F G). That scale is a mode of the major scale, called the Aeolian Mode.

This paragraph confused me until I realized it's supposed to say "That scale is a mode of the MINOR scale".

Other than that I thoroughly enjoyed this. Never thought of a scale as a bitmask before.

Aeolian mode is not a mode of the minor scale. It is that mode of the major scale we commonly refer to as the minor scale. So if you call it a mode, its a mode of the major scale. Or you could call it a minor scale itself. But not a mode of the minor scale

Got it. You’re right thanks for the explanation.

The article is correct here. The C major scale (C D E F G A B) has 7 modes, and A minor (A B C D E F G) is one of them.

oh I see, I think you might be right. Thanks for the correction

TIL "nominal" as a class of number (in addition to "cardinal" and "ordinal"). How interesting. In some way like a code point in a character set.

Thats the best explanation of scales vs modes that I've seen

That computer speech narration on the video sucked the soul all out of TFA and I closed the tab.

Reply on news.ycombinator.com