Some mathematical principles of "tones" that resonate well with Navani's research

[Serack he/him]

Over the course of obtaining my undergraduate degree in electrical engineering, I came to really enjoy some of the higher level mathematics that describes what this post is about, but I'll spare you most of that with some handwaving, and maybe a few links for those interested in digging deeper.  For now please forgive me while I indulge in showing off a selfie of our department's club shirt that's about as math geeky as you can get.  This is actually the fundamental equation behind this post.  

  

Ok, that out of the way, Rhythm of War has a great deal of stuff in it that danced around real world stuff I learned mostly from a class called "Sound Engineering" that I found really exciting and wanted to post about.  Hopefully I can do a good job of communicating this that doesn't hand wave away too much of the underlying principles, while also not getting dragged down into boring details or assuming too much prior knowledge for most people that might also find this interesting.  I'm building this from a post I made here.  The post will be mostly about the RL principles.  I'll reserve a first reply where I might post speculation on how it might influence where the story is going.  

 

Sounds, particularly constant sounds like a note from an instrument can be analyzed according to their frequency content.  (That's what the equation in the image does) 

A "pure tone" will have only one frequency associated with it that is a perfect "sine wave" however, in real life, things generally aren't "pure" and actually, less "pure" tones are usually more desirable.  Here's a picture of a sin wave pulled off the internet.

Because of the nature of how a bow draws across a violin string (or a plate as in the scenes in The Stormlight Archives) the wave generated from them is more like a "saw tooth wave" like this:

Now what really matters, is that mathematically it has been shown that for non "pure" repeating waveforms like the saw tooth above, there are multiple contributing frequencies that add up to make them, and the primary one or "fundamental frequency" is the pure tone with the same frequency as the repeating waveform.  The next important characteristic is that all other frequency "components" are multiples of the "fundamental frequency."  So the note A at the middle of the piano has a fundamental frequency of 440Hz, and that note is also comprised of other frequency content at frequencies of n*440Hz.  That is, 440 (1x440), 880 (2x440), 1320 (3x440), 1760 (4x440), etc.  

This higher frequency content in a note is sometimes called the "harmonic frequencies" and whether two notes harmonize is dependent on if this higher harmonic content meshes together mathematically, or it can be dissonant and the higher harmonics can "beat" against each other (more on that beat later).

What makes an "octave" an "octave" is that it is the next note up or down on the scale that matches this pattern, so the "A" one octave up on the piano has the fundamental frequency of 880Hz, and all it's other harmonics are the same higher frequency harmonics as the note that corresponds to 440Hz.  This is the underlying principle for why properly tuned octaves are in perfect harmony with each other.  

The notes on a scale are placed such that choosing ones that harmonize will have other parts of their higher frequency content mesh up perfectly with each other.  (Edit: This is a gross simplification.  After reading my post for context, @Zincmind's post below has much more on how notes on a scale are placed where they are mathematically)

Back to Beats

The concept of "beats" relies on constructive and destructive interference.  Here's a 12 minute youtube video you can skip or watch for more on what that means, but here's the hand wavy version in a few sentences.  When two pure tones are almost but not quite exactly the same frequency, adding them together will have moments when they cancel out and become silent and alternating moments when they add together and become twice as loud in a rhythmic interval like a "beat."  How fast they beat depends on how close together the two frequencies are, with closer making for slower beats, and the beat speeds up as the frequencies get farther apart.  This 1.5 minute video, you shouldn't skip, because it demonstrates the phenomenon audibly, while displaying a representation of the waveform, and the experience really makes the concept click.

Dissonance

At higher frequencies, beating tones can be harsh and annoying, a property sometimes used by some alarms to really get your attention.  This is also the principle that underpins why dissonant notes are dissonant.  

So when you play two dissonant notes on a piano, on a superficial level, you just hear the two fundamental frequencies with no louder and softer pulses, but beneath that the higher frequency harmonics of the notes are of frequencies that are slightly off from one another and they "beat" against each other.  You can't necessarily hear this on a conscious level, but on a subconscious level you do, and it generates tension on this subconscious level.  This is why "chopsticks" is grating on your ears, and it is also abused in horror movies to generate tension with the music alone.  

Edited March 12, 2022 by Serack

[Serack he/him]

Reserved

[Serack he/him]

When I posted the above, I wrote it thinking I would make some applications of those RL principle to Navani's research, but never really got around to doing that in a fleshed out manner.  However, today I did some thinking about how this applies to the "3rd heightening" which allows for a sort of perfect pitch, but for vision.  

I cringe a little when Lightsong describes some red lips as “three shades shy of the seventh harmonic.” Particularly since Red is the lowest frequency of the primary colors, and higher “harmonics” are usually defined as higher frequency.

I mean, you could get colors to “harmonize” if you treat the math /very/ carefully, but how we perceive colors is more related to the peak absorbance spectra of our eye’s 3(ish) cone receptors, not how their wavelengths “harmonize” mathematically. Of course Bent Weeks hand waved all of this in his Lightbringer series, but he used the term “resonance points” rather than harmonics.

So I just went down the optical harmonics rabbit hole a bit.

Borrowing the term "octave" from sound/music to mean a doubling of the wave frequency, Human hearing with a range/spectrum of about 20Hz to 20,000Hz (for those with superb hearing) is 10 octaves of harmonics that can be perceived.

However, unlike our hearing, our vision does not have a wide range or "spectrum" but is limited to between a deep red of about 740nm to a violet of about 380nm, or about one "octave" of harmonics. (even outside of typical color blindness, not all eyes can quite perceive all of this spectrum. I can't see all the way to 380nm).

Even though light sources are quantum in nature (involve photon particles of specific energies/wavelengths) they can have these photons combine to emit light in higher harmonic frequencies. The commonly used infrared Nd:YAG laser with a wavelength of 1064nm is frequently "doubled" to emit green light at 532 nm, or higher harmonics at 355, 266 and 213 nm. If you have ever bought a green laser pointer, it was probably a "doubled YAG." Which demonstrates that yes indeed, light sources can have emission that has harmonic content like sound saves do.*

Even though a green "doubled YAG" laser pointer is a perfect harmonic of a standard Nd:YAG infrared laser, the human eye can't see that far down the infrared spectrum. Speaking of light in terms of harmonics is kinda irrelevant for the range of human perception, as the photons of of higher or lower harmonics of light sources in the visible spectrum can't be detected by the eye to perceive their "harmonic/dissonant" constructive or destructive interference.

(I originally wrote this in a Reddit response, but I find the forums to be a better medium for long form/long term theorizing so I'm also placing it here)

Edit to add:  The eye can't come even come close to seeing it, but when I was in college, I built an instrument that detected "beating" of 2 close together peaks of emission from a doubled YAG laser, exactly like the sound waves beating together in the video in the OP above.  It took hours of measurements and calculation to show that that might be what was happening in my slipshod michelson interferometer though.  

*edit to add:  This is sort of comparable to how a bugle can get different notes from a fixed length of tubing.  The bugle tubing has a fundamental harmonic frequency that has it's wavelength (Frequency and wavelengths are inversely proportional to each other and are practically interchangeable in these discussions) defined by the length of the tube.  The bell shape at the end helps the player to emphasize the resonance of higher frequency harmonics and get those notes out of it instead.  If I remember my schooling right though, those notes have to be higher frequency harmonics of the primary.  Just like the higher frequency photons you can get out of a Nd:YAG laser by tuning it to the higher harmonics.  

Quoted from a post I make below that really tries but doesn't quite manage to make a theory about the different shardlight frequencies using the above information:

Quote

Check this out.  Earphones can't reproduce true base sounds because their resonator/speaker isn't physically big enough to produce sound waves that are that long.  However, they do produce all the harmonics in such a way that your brain/complex perception mechanisms go, "the only way these harmonics would be here is if the fundamental harmonic were there too" and fills in that gap so that you actually "hear" the long wavelength fundamental harmonic that isn't actually physically present.  

So if you have a set of frequencies of a certain sequence that fits the harmonics of a lower frequency just right, you can extrapolate what that frequency is.  And since we have one data point for the "7th harmonic" as red, which a quick google search defines as from 620nm to 750nm, we could make some guesses/ranges for all the harmonics.  

I got pretty excited about this thinking there might be a subset of harmonics of this hypothetical set of harmonics that is 16 within the visible range giving us hypothetical wavelengths for all 16 shards.  Unfortunately my math gives us a best case  of only 9 harmonics in the "visible" spectrum if you fudge a few things.  

So if we crank some math to find "hidden symmetries" using the information that the "7th harmonic" is red, that doesn't give us enough "harmonics" in the visible spectrum to represent all 16 shards.  

 

Edited March 17, 2022 by Serack

[AquaRegia he/him]

Just wanted you to know I'm reading and enjoying your posts here.  Nothing much to add, other than that I'm looking forward to seeing how you connect the RL stuff to Navani's research!

[Serack he/him]

16 hours ago, AquaRegia said:

Just wanted you to know I'm reading and enjoying your posts here.  Nothing much to add, other than that I'm looking forward to seeing how you connect the RL stuff to Navani's research!

If you like this, be sure to check the posts collected in the link in my signature.  

Brandon has explicitly called cosmere science "fake science" and has also discussed that the mechanism of "Intent" is pretty much supposed to allow for handwavy stuff to happen in cosmere physics that wouldn't be possible in real physics, such as slightly changing the shape of a plate to be sure that the tone it plays perfectly cancels out and negates an intended pure tone.  

Reality works way differently.  On a resonance plate like what Navani was playing with, if you put sand on it, the places where the sand collects are WHERE the tone is negated by itself due to reflections off the edges of the plate, and the clear spots are where the tone is doubled.  These are called anti-nodes and nodes respectively, and the sand collects where the plate is relatively still, and is bounced away from the parts vibrating from the sound waves.  In Cosmere physics, stuff (investiture) is attracted to the tone itself, rather than the being corralled by the nodes and anti-nodes of the resonance chamber.  Via handwavy Intent, a separate "tone" can attract the anthesis of the "stuff" attracted by the first tone, and the 2 together can cancel each other out and go boom.  

So as much as I want to try to project forward on where the cosmere physics are going, I'm limited in how I can guess what cosmere principles Brandon is going to hand wave into diverging from my limited understanding of RL physics going forward.  I may try anyways.

Which isn't a dig at him.  RL physics can be exactly the same way, in that until we look really close at the fringe cases, we don't know where the rules are going, and when we see it it sometimes looks really handwavy.  Einstein called quantum entanglement "Spooky action at a distance."

Edited October 6, 2021 by Serack

[Zincmind]

On a related topic, Navani's research aligns really well with some cool music theory principles about tones and how "tuning" an interval actually works. 

Like Serack described, sound waves "beat" against each other because of their relative frequencies. Sometimes, this interaction sounds pleasing, or consonant, and sometimes it does not. In general, the more consonant intervals correspond to simple, whole number ratios between the frequencies of the two notes. For example, an octave is 1:2, and a perfect fifth is 2:3.

Another way to describe the frequency of a note is to subdivide an octave into 12 equal pieces. Each subdivision is called a half-step or semitone . A perfect fifth corresponds to a difference of 7 semitones between the two notes being played, while an octave is separated by all 12 semitones. Using these measurements, if we start at a note and move up in pitch by a perfect fifth 12 times, we should arrive at the same pitch / frequency as if we had started at the same place and moved up by an octave 7 times (7*12 = 12*7). 

However, going back to our definition of an interval as "simple ratios", we now run into a problem. Namely, if we take the ratio of 12 perfect fifths from our starting note, it is not exactly the ratio of 7 octaves ((3/2)^12 does not equal 2^7). It is close, but not exactly the same. This difference is called a "Pythagorean comma", and means that our two definitions of notes (frequency ratios and a 12 tone system) do not agree, meaning we must choose one or the other. In music theory, the ratio system is called just intonation, and splitting an octave into 12 equal pieces is called pure intonation. Each system has its own advantages and disadvantages.

The major benefit of pure intonation is that is always the same. Each note will always have the same frequency, regardless of what key the piece is in, or what other notes are being played with it. This means individual notes can be tuned exactly, and used in any piece (useful so that you don't have to retune an entire piano in between songs that are in different keys). However, to a careful ear, pure intervals often sound slightly off. 

Just intervals (tuned using frequency ratios) usually sound slightly better, but are more context dependent, since they are defined as a ratio with respect to some other frequency. The choice of the frequency to define all other notes relative to corresponds to choosing a key for a piece of music, and will very slightly offset the positions of notes from where they would be in pure intonation. 

OK, now that the context is out of the way, onto the connection to Navani's research. She ends up finding tones that correspond to Honor and Odium. At one point, she tries to sing Honor's tone at the same time as Raboniel sings Odium's, but something is wrong at first. Though they are each "pure" tones on their own, they are just slightly off when put together. This corresponds exactly to how individual notes can be tuned perfectly with pure intonation, but when they are put together in an interval, they sound just a little bit wrong! It is only after making a slight change to both frequencies that they truly combine, to create the Rhythm of War. This is exactly what just intonation does; it moves each note slightly from where it would normally be on its own, but as a result makes the interval formed by both notes sound better! 

Anyways, I have no idea if this was intentional, but there are certainly some neat parallels between the way Sanderson wrote the magic system and real world math and music theory. 

[Serack he/him]

@Zincmind, thanks for your addition to the discussion!  These are things that were taught in my Sound Engineering class over a decade ago that I no longer know well enough to have explained nearly as well as you did.  

I do remember being told that a good acapella barbershop quartet style ensemble will typically sound much more harmonious than something played on instruments with standard tuning because the singers are trained to adjust to perfect harmony with each other whereas the instruments are stuck in their strict tunings.  

edit to add:  Also, if I understood/remember correctly, in a pure intonation tuning, the dissonances that can come from it not being perfect makes for dissonances that can sometimes be the goal of picking certain note combinations that you can't get if tuned with just intonation.  But I'm not confident of that claim.  

Edited March 12, 2022 by Serack

[+Oltux72 he/him]

On 5.10.2021 at 7:42 PM, Serack said:

I cringe a little when Lightsong describes some red lips as “three shades shy of the seventh harmonic.” Particularly since Red is the lowest frequency of the primary colors, and higher “harmonics” are usually defined as higher frequency.

I mean, you could get colors to “harmonize” if you treat the math /very/ carefully, but how we perceive colors is more related to the peak absorbance spectra of our eye’s 3(ish) cone receptors, not how their wavelengths “harmonize” mathematically. Of course Bent Weeks hand waved all of this in his Lightbringer series, but he used the term “resonance points” rather than harmonics.

So I just went down the optical harmonics rabbit hole a bit.

Borrowing the term "octave" from sound/music to mean a doubling of the wave frequency, Human hearing with a range/spectrum of about 20Hz to 20,000Hz (for those with superb hearing) is 10 octaves of harmonics that can be perceived.

Do you need to borrow that term with this definition? The doubling of frequencies of lasers fundamentally comes from a linear relationship between energy and frequency in the physics of light, doesn't it?

What I am trying to get at is that light from Stormlight is not ordinary light. If you had a relationship like E = k * f ^2, the octave would be an increase of frequency by a factor of 2^(1/2), wouldn't it?

 

[Serack he/him]

@Oltux72 I am not sure what relationship that equation is supposed to represent (or even what all the variables represent other than frequency)

The Wikipedia article I linked in the post you quoted probably does a better job of explaining it than I can, but laser frequency doubling fundamentally comes from tuning to a harmonic of the laser’s fundamental frequency.  Harmonic frequencies are always an integer multiple of the fundamental frequency, the second always being 2*fundamental.

And harmonics are the term used in the book.  Speaking of then in terms of octaves just just serves as a ruler that allows us to compare the range of “harmonics” we can see compared to the range of harmonics we can hear.  
 

The conclusion that the visual frequency range is too small for the term “harmonics” to be a useful term is independent of the term octave 

[Brgst13]

On a related note, if we divide the 10 octave hearing range up and assume equal distribution of pure tones, each Shard's tone will be 8 semitones from the one beneath it, which corresponds to a minor sixth.  Even more interesting, a minor sixth interval has a wavelength ratio of 1.6:1.

[cometaryorbit]

The Warbreaker color recognition, color harmonics, etc. are kind of odd in several ways, not just the limited wavelength range of visible light, but also regular objects like lips seen by reflected light aren't going to be a single wavelength the way a laser or LED is.

The perfect pitch analogy suggests that Perfect Color Recognition means being able to recognize light wavelengths exactly... but reflected light is going to be a mix of wavelengths, and things are still described as a single specific color.

[Brgst13]

Perhaps perfect color recognition means the ability to discern the exact wavelength of the reflected light.  This is similar to perfect pitch identifying notes in a chord.

[Serack he/him]

54 minutes ago, Brgst13 said:

Perhaps perfect color recognition means the ability to discern the exact wavelength of the reflected light.  This is similar to perfect pitch identifying notes in a chord.

this seems to miss @cometaryorbit's point that reflected light is almost never monochromatic.  The "exact wavelength" is actually a spread out spectrum of wavelengths with a peak at a particular frequency.  In optics, we call this frequency coherence, with the tighter the spread, the more it is said to be coherent.  Even most LED's are spread out compared to a laser or an emission line from an excited gas.  

I can accept that Stormlight is pure monochromatic light source, and even that the Tears of Edgli are a source of dyes that in the right conditions will provide monochromatic reflections allowing for the perfect color shadings characteristic to the court of the gods in Warbreaker.  But paint them on someone's lip, and that coherence is probably degraded significantly.  

 

Edit:  See this post for a better treatment on light coherence (both frequency and spacial), particularly how it can be achieved using low tech like what's available in the stories.

 

Edit2:  Cool, I just noticed that @cometaryorbit was participating in that discussion too!

Edited March 16, 2022 by Serack

[AquaRegia he/him]

I can empirically confirm that @cometaryorbit and @Serack are correct about reflected light being polychromatic.  I work in the dyehouse lab of a fabric mill; one of our jobs is to make sure the color of dyed fabrics matches a standard. 

Color perception is a ridiculously confusing combination of three individually complicated factors: the light source, always polychromatic (we use at least 4 different "standard" light sources, none of which is the same as natural daylight or direct sunlight); the infinitely complex absorption curve of the object; and the way our eyes and brains work (three different pigments, each with its own unique absorption curve, sending signals to our squishy brains to be coded somehow into one perceived "color").

For one specific example of the nightmare that is color matching, check out this article on metamerism: https://en.wikipedia.org/wiki/Metamerism_(color)

[Serack he/him]

the You are Not So Smart (YANSS) podcast has a fantastic episode about how some researchers replicated the visual/social phenomenon that was "the dress" using croc's and socks.  It has a lot of interesting information about what @AquaRegia describes as "sending signals to our squishy brains to be coded somehow into perceived 'color'"

I highly recommend YANSS in its entirety (I went through their entire back catalogue over the course of 2 years while mowing the lawn), but that one is relevant to this particular discussion

Edited March 17, 2022 by Serack

[cometaryorbit]

6 hours ago, Serack said:

this seems to miss @cometaryorbit's point that reflected light is almost never monochromatic.

Yeah, exactly.

Plants, for example, are green because the chlorophyll absorbs red and blue and reflects green - so what we see is the green light. But it's not a true monochromatic green like a green laser (and in fact, it's not *purely* green wavelengths, just predominantly).

And it gets even worse ... our eyes have 3 cone types, but just as colors that look radically different to people with 3 functioning cone types can look identical to someone with red-green colorblindness (2 functioning cone types)  there can be colors that are indistinguishable to human eyes but actually made up of very different wavelengths.

So I am not sure what Perfect Color Recognition actually means.

If it's a true perfect pitch analogy (applying to "pure" monochromatic light), it wouldn't do much of anything -- certainly not for natural reflected colors like lips.

But if it's a total recognition of all wavelengths (so blue+yellow=green would look completely different from a pure green) then art, clothing, etc. for regular humans would often be utterly bizarre to the Returned, and we don't see that. They see the same colors in more detail, not totally alien colors.

[Brgst13]

8 hours ago, Serack said:

this seems to miss @cometaryorbit's point that reflected light is almost never monochromatic.  The "exact wavelength" is actually a spread out spectrum of wavelengths with a peak at a particular frequency.  In optics, we call this frequency coherence, with the tighter the spread, the more it is said to be coherent.  Even most LED's are spread out compared to a laser or an emission line from an excited gas.  

I can accept that Stormlight is pure monochromatic light source, and even that the Tears of Edgli are a source of dyes that in the right conditions will provide monochromatic reflections allowing for the perfect color shadings characteristic to the court of the gods in Warbreaker.  But paint them on someone's lip, and that coherence is probably degraded significantly.  

 

Edit:  See this post for a better treatment on light coherence (both frequency and spacial), particularly how it can be achieved using low tech like what's available in the stories.

 

Edit2:  Cool, I just noticed that @cometaryorbit was participating in that discussion too!

I am well aware of the multiple wavelengths of reflected light.  I dropped an s in my initial reply which may have led to some confusion.  I was picturing something like an LED array where different diodes emit different colors but when they illuminate an object the light appears to be one coherent color.  My vision was that perfect color recognition is like looking into the LED and seeing all the individual diodes lit up.  Obviously reflected light would have many more "diodes".

 

As for harmonics, I think we are all overlooking the fact that there is an Investiture component as well.  Electrons can have a wavelength but so could the theoretical Investiture particle.  It is quite possible that this particle may be tied to color in some way but may have a much broader spectrum in visible light.  This would allow a harmonic to be perceived by a highly invested being but not violate basic color physics.

[Serack he/him]

14 hours ago, Brgst13 said:

As for harmonics, I think we are all overlooking the fact that there is an Investiture component as well.  Electrons can have a wavelength but so could the theoretical Investiture particle.  It is quite possible that this particle may be tied to color in some way but may have a much broader spectrum in visible light.  This would allow a harmonic to be perceived by a highly invested being but not violate basic color physics.

I concede that investiture/fake physics makes the physiology of rods and cones and the vast complexity of processing their information into an actual perceived color something hand waved into the realm of magically special for those with enough breath.

The same for the dyes made from the Tears of Edgli which are explicitly invested by Endowment Per WoB (referenced in the linked coppermind article).

Neither of which overcome the fact that human vision is so narrow a portion of the electromagnetic spectrum that it barely encompasses a single harmonic of its lowest perceived frequency.  A red "7th harmonic" would put the reference fundamental frequency's wavelength somewhere between 4.3 and 5.2 μm which would only be perceivable by humans as heat.  

Edited March 17, 2022 by Serack

[Serack he/him]

Hold up...

In another physics/science heavy post @G2F4E6E7E8 said:

Quote

These hidden symmetries would be the symmetries of the "investiture field". Some stormwarden might, just out of curiosity, decide to figure out which other equations satisfy these hidden symmetries and suddenly discover the patterns governing other shards. This same type of story has played out a lot in the history of real-world math.

Check this out.  Earphones can't reproduce true base sounds because their resonator/speaker isn't physically big enough to produce sound waves that are that long.  However, they do produce all the harmonics in such a way that your brain/complex perception mechanisms go, "the only way these harmonics would be here is if the fundamental harmonic were there too" and fills in that gap so that you actually "hear" the long wavelength fundamental harmonic that isn't actually physically present.  

So if you have a set of frequencies of a certain sequence that fits the harmonics of a lower frequency just right, you can extrapolate what that frequency is.  And since we have one data point for the "7th harmonic" as red, which a quick google search defines as from 620nm to 750nm, we could make some guesses/ranges for all the harmonics.  

I got pretty excited about this thinking there might be a subset of harmonics of this hypothetical set of harmonics that is 16 within the visible range giving us hypothetical wavelengths for all 16 shards.  Unfortunately my math gives us a best case  of only 9 harmonics in the "visible" spectrum if you fudge a few things.  

So if we crank some math to find "hidden symmetries" using the information that the "7th harmonic" is red, that doesn't give us enough "harmonics" in the visible spectrum to represent all 16 shards.  

[Germybacon]

Hey Serack!  We had a conversation about the Pythagrium concept of Musica Universalis (also known as Harmony of the Spheres) and how it relates to your write up and the Cosmere as a whole. 

You rock!

[Frustration]

36 minutes ago, Germybacon said:

Hey Serack!  We had a conversation about the Pythagrium concept of Musica Universalis (also known as Harmony of the Spheres) and how it relates to your write up and the Cosmere as a whole. 

You rock!

Welcome to the shard.

This topic has been inactive for three years, the shard discourages posting on topics that old.