The Fine Art of Combining Harmonics | Hacker News Reader

The Fine Art of Combining Harmonics

whatmusicreallyis.com

143 pointsflabber35 comments

I found the first 2/3 of this mostly acoustic-woo, but by the time it got to the actual 2D "HarmoniComb" I was quite excited by the concept of this as a playing surface.

However, then I decided to go visit another page at the same site, the one on "Tuning". It has a section on digital sample rates that is just so completely incorrect that it made me wonder about everything else I had read. The page linked above is specifically about "combining harmonics". However, the author doesn't appear to understand how the exact same concept (more specifically, how any waveform can be represented as the sum of a (potentially infinite) series of sinusoidal waves) makes their musings on digital audio totally wrong.

Yeah a lot of the explanations have a pseudo-scientific flavor to them with questionable mathematical hand waving.

Time Cube, but for sound!

I know next to nothing about music theory, but this felt like an abstruse statement of the bleeding obvious up until the section "What Tuning Really Is: Subsets of the HarmoniComb", which was interesting. I have no idea if it's accurate.

Incidentally, the author seems to use some strange 'İ' character (a capital I with a dot over it) a lot. I'm not sure why.

"İ" is from some Turkic languages: https://en.wikipedia.org/wiki/Dotted_and_dotless_I

My immediate thought on reading the first paragraph is, "this is like Time Cube if it actually made complete sense under the word salad." It's really, honestly rather obnoxiously, obtuse, but I get what they are getting at.

It feels like when academic ML papers use the abstract to try to boozle and astound the reader with tons of jargon, because we all know a paper wins more Academia Points (TM) the more they can make other smart people feel dumb.

Ah, Time Cube - instantly what I thought as well.

> Incidentally, the author seems to use some strange 'İ' character (a capital I with a dot over it) a lot. I'm not sure why.

This is because the author is actually a fractal within a 420-dimensional LSD hypercube, the 'İ' denotes membership of the set of infinity|UniTY<>Triome.

Also, you nailed in one, definitely timecube vibes going on here.

Harmonic intervals are interesting but they are also misunderstood. Humans are able to distinguish out of tune notes down to a value of about 1%. To get more accurate tuning we tend to listen for beating (a kind of amplitude modulation) instead. This means we can tolerate tuning systems other than just intonation. Another thing to consider - the missing fundamental phenomenon suggests that the ear/brain is actually doing something like autocorrelation. This makes more sense than the idea that we have a template for the harmonic series wired in our brains. Finally auto correlation works for chords too, not just intervals. Every chord has a fundamental period of repetition - shorter periods are widely ranked as more consonant. There are lots of grand music theories that fixate on the harmonic series. The maths is fun, but it can get in the way of more effective alternatives for organising sounds.

This is really cool, but -- as someone who's been researching just tuning for a while -- I think a lot of very important aspects of human psychology have been overlooked here.

Frankly, familiarity is a vastly more important aspect of music than any micro-tonal artist would like to admit. And while the overtone series is (in many many ways) the root of all music, not every overtone is equal.

For instance: 9/8 (a major second) is just (3 * 3)/(2 * 2 *2) and is generally more consonant (like dull, boring, unmoving: an octave or perfect fifth) than anything with a 5 prime number in it (eg 5/4 a major 3rd). AFAICT, the order of simplicity (dullness -> dissonance) on intervals is 2/1, 4/1, 3/2, 4/3, 3/1, 9/8, 16/9, 5/4, 8/5, 6/5... it's not exactly clear what the math is here.

Also, it's always really, really telling to me when someone shares a music theory without sharing a recording of that theory in action. The site I'm working on lets people use their keyboard to play with every scale, so you can verify that it's not just number play.

yep - I came in here to talk about "familiarity" as the bedrock of how we listen to tunes. In my estimation, music is a constant juggling act between familiarity and novelty, and bending too much on one side or the other trends towards boredom.

I would argue that some of the pure intonation music (i.e. Michael Harrison's Revelations) or alternate tuning (i.e. Lamonte Young's Well Tuned Piano) is striking initially because of how alien it sounds, but to my ear a lot of it doesn't feel like it does too much more than that both as a function of the tuning and the difficulty in building a moving composition in something so foreign.

The standard tunings in Western Music are well-worn, but they can give a rich vocabulary for dissonance and consonance and have the psychological familiarity to build up emotional abstractions that some of the more adventurous scales do not.

My theory is that music is something of a "matching puzzle."

- using only these tones

- and only the intervals between them

- I will construct an elaborate yet consistent path

- which will stray into a place that's hard to resolve from

- then impress you with how I "solve" it

--------

So when playing with just intoned, I've applied this theory and it works. I've shown it to friends and they like it... they can't even tell it's just intuned.

My favorite so far is using 7/6 and 9/7 as my major and minor thirds. If you multiply them together you get 3/2. Just like you do with 6/5 and 5/4. Pretty cool eh?

As long as the first note and the last are familiar, people seem to be okay with it.

>There is of course a questionable degree of dissonance that can be tolerated, but the most obvious fact is that the brain can well be conditioned to accept and even enjoy these dissonances, as long as they are part of the only music it has ever experienced.

I disagree. I have listened to a fair amount of music in just intonation, and I am capable of hearing the tuning flaws of 12 equal temperament, but I am still capable of enjoying dissonance, and of appreciating the artistic flexibility that those tuning flaws give you.

This article presents a very limited view of music. Just intonation does have artistic merit (check out the works of Harry Partch), but it's not the only valid form of music. The human brain doesn't count frequencies, and integer ratios are not fundamental to perception of consonance. They just happen to work well with a common class of timbres. But for other timbres, e.g. tuned percussion, other tuning systems can sound more consonant. You can even hear this in Western classical music: pianos are tuned with non-integer octaves to compensate for the inharmonicity of their strings. See: https://en.wikipedia.org/wiki/Stretched_tuning

William Sethares published a more general model of consonance perception that handles all these cases:

https://sethares.engr.wisc.edu/paperspdf/consonance.pdf

I previously posted about this on HN with some more background and references:

https://news.ycombinator.com/item?id=23638307

That's right. If you spend too much time eating sweet chocolate you might start enjoying something more savory, too.

I am learning to play Irish trad on Irish flute and I have learned some time ago that dissonance is a real tool that can be used to create certain kind of moods, especially in slow airs.

I accidentally abandoned many of my preferences (likes/dislikes) and automated judgments. I took this newfound perspective into music, embodying the idea of "there are no wrong notes." I find deep joy in sitting at a 120+ year old pump organ, holding a mishmash of dissonant notes, including some out of tune ones, and wiggling one of the keys to learn to spot it in the mashup. It gives me a chance to witness the internalized feelings conditioned from classical musical training and listening to a ton of western music throughout my life, as well as to watch those feelings fade away into the joy.

We can learnjoy anything, which is why choosing to do anything without joy doesn't make sense to me anymore.

> I accidentally abandoned...

It sounds like you made a conscious and intentional decision. How does one drop their preconceived judgements by accident?

Psychedelics, perhaps?

I was aiming to abandon judgments of things and had no idea it meant how my body reacts to things would change, that I'd effectively disable disgust in myself, that country music would become rockin (and that I'd enjoy all other forms of music, food, smells, drinks, and people...children went from obnoxious to adorable in weeks).

The intention was to try something very broad that would likely cover moral judgments. I did make a conscious decision, but the side effects were for sure happy accidents.

Sethares and Tymoczko are the go-tos on this.

But there's a lot more to creating interesting musical structures than creating chords out of mathematical structures that are based on the harmonic series.

The more time I spend trying to understand music, the less I understand it.

That's what makes it so endlessly fascinating.

I've read quite a bit on this, but not Sethares, do you have a recommendation for starting on his work? thanks!

I'm not the parent, but I think they are referring to William Sethares' book "Tuning, Timbre, Spectrum, Scale": https://www.springer.com/gp/book/9781852337971

It's a fascinating take on how those concepts relate to each other. You can find some of the papers that were synthesized into that book on his web page: https://sethares.engr.wisc.edu/pubs.html

I like to think that in an alternate world, somebody wrote:

> There is of course a questionable degree of consonance that can be tolerated, but the most obvious fact is that the brain can well be conditioned to accept and even enjoy these consonances, as long as they are part of the only music it has ever experienced.

> as long as they are part of the only music it has ever experienced.

So noise music is only enjoyable if listened to from birth in isolation?

I based my bachelor's graduation project on Setahres' work.

The one thing I wish we could find is the drum version of this. As in instead of starting from a string (which leads to harmonic series), start from a circular membrane (which leads to a very complicated mathematical result based on bessel functions ircc).

+1 for Harry Parch. Here's a good introduction for new people [1]. I'd add in Lou Harrison [2] and Tristan Murail [3]

Lou Harrison was heavily influenced by Gamelan music, and Tristan Murail is one of the founders of spectralism, which uses spectrograms of individual instruments to build distinct harmonies by having orchestral instruments play the harmonic series as shown on the spectrogram. They stretch this idea even further by adding in "harmonics/overtones" that are not shown on the spectrogram to create dissonance.

[1] https://youtu.be/kCuYcS_Lcro?t=372

[2] https://www.youtube.com/watch?v=f7IwH01YFAI

[3] https://www.youtube.com/watch?v=q6WXzOIsBuQ

And Gerard Grisey.

From the 'Tuning' page:

> In the case of the human voice, these overtones will all be harmonics, because they display relationships based on perfect integers. On non-biologic instruments (which display slight deviations from perfect numbers due to mechanical attributes like stiffness), the whole matrix will get stretched or compressed according to the deviation (or “inharmonicity”) factor

I'm not sure I understand an instrument made mostly of meat and cartilage and maybe a little phlegm can be more 'perfect' than a mechanical one.

This is incredibly dismissive of any harmony that is based on equal temperament, or anything that isn’t the harmonic series. I don’t think that’s justified - a lot of jazz harmony only sounds good in equal temperament. For instance, minor 6 chords are going to sound terrible.

Huh? To me Dm6 in the just intonation A minor scale sounds fine.

Beautiful visuals. Ever since the pythagoreans there is a mystical fascination with harmonic ratios and harmonic perception but after reading Philip Ball's "music instinct" I am not sure its justified. I don't know if there is something more updated since 2010 (for non-brain neurologists)

Music producer Andrew Huang does a fantastic job demonstrating the harmonic series concepts in a video from last year: https://www.youtube.com/watch?v=Wx_kugSemfY

I have been thinking very much this for over a decade now. Down to even using the times and the division signs the same way (or + and - given they are more easily typed).

An interesting thing is that 2 is very special musically, but not mathematically, because it makes all notes musically the same (establishes pitch classes).

A very funky thing (kinda annoying really) is as follows:

Given a note, any note, as a tonal center or starting point (I call it O for origin) you 'make' a new note from this one by multiplying by 3 (O+3) [note 1]. Similarly we can make O+5.

O+5 is in traditional western music closest to the major third

AND O+3 corresponds to a 12-tet perfect fifth (flat 2 cents).

In short +3 is the fifth and +5 is the third...weirdly annoying.

note 1: Here's were 2 (which defines octaves) comes into play because it allows to say that O+3 defines a pitch-class no matter the octave.

Please, try playing your harmonic ideas with real sounds.

Here's some things to try:

- How bad or good do two synthesized violins sound together at 800 and 900 hertz? Think about them as overtones: Does adding a tone at 100 hertz resolve the sound? Think of them as undertones: Does adding a new tone at 7200 hertz resolve that sound?

- Does the instrument change the harmony? What happens to harmony when the instrument is a clarinet? What happens to harmony when the instrument is a sine wave? What happens to harmony when the instrument is a tuned tympani?

- Try to play together the overtone series, for example, 1000 hz, 2000 hz, 3000 hz, ... How does that sound?

- Try to play together the undertone series, for example, 2520 hz, 1260 hz, 840 hz, 630 hz, 504 hz, ... How does that sound?

Holy cow! I just came up with this theory and started proving it like... 2 months ago. Crazy how these things line up sometimes.

One thing not included in here: given tests on human response rates to sound, it is very likely that grouping frequencies together based off a harmonic series is done "before the brain." (ie, that bird is this set of frequencies, that tree in the wind is this other set)

I'm sorry, but this is a pile of mumbly-bumbly.

If you want the goods on this stuff from someone who actually understands it, read Kyle Gunn's "The Arithmetic of Listening".

Reply on news.ycombinator.com