Leonhard Euler's Magical Consonance Formula

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Consonance is a tricky thing. Some notes sound good together, others don't, and the difference between consonant and dissonant intervals has been a huge part of music theory for centuries, if not longer. Wouldn't it be great if we just had a formula that would tell us exactly which intervals actually worked together? Well, enter legendary mathematician Leonhard Euler, who took on the task of defining consonance with mathematical precision, so we could finally tell just how good any set of notes actually sound.
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SOURCES:
www.uni-miskolc.hu/~matsefi/Oc...
www.tonalsoft.com/monzo/euler/...
www.academia.edu/302747/Mathem...
Script: docs.google.com/document/d/1L...
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Also, thanks to Jareth Arnold and Jade Tan-Holmes for proofreading the script to make sure this all makes sense hopefully!

[@12tone]

Additional detail: I couldn't get my hands on an actual copy of Tentamen, but some sources I found claimed that Euler never actually wrote the complete formula down himself. He only looked at ratios that used primes up to 5. The actual function itself was a later modeling and generalization of his work. Either way, the principles are the same.

[@Ranzha_]

Hey, I picked up a copy of Tentamen from my university library if you want a look! Time to start translating Latin lmaoooo

[@markusmiekk-oja3717]

Paul Erlich's Harmonic Entropy might be an interesting next stop for this particular topic!

[@pkrbt6821]

Very interesting subject! One thing that always bothered me with the simple ratios explanation (and you also mentioned that at the end of the video) is that it doesn't explain why some pretty wild ratios sound nice to our ears, and even the "irrational ratios" from Equal Temperament. But how can we measure that? What is the formula for the actual dissonance our ears hear? Also, if we play two sine waves a tritone apart, is that really dissonant or do we need the harmonics to have dissonance? I think the top rated answer on this discussion really sheds a light into what such formula would be: music.stackexchange.com/questions/4439/is-there-a-way-to-measure-the-consonance-or-dissonance-of-a-chord

[@danielf3623]

I'm just waiting for the day one of you music nerd youtubers teams up with 3Blue1Brown to do one of these heavy math music theory videos.

[@Mau365PP]

That would be so cool !

[@filmNFX1]

Damn, I want that so bad now.

[@musicaljunkyart8756]

SUCH A GREAT IDEA!

[@sarius741]

Yes please!!! 3b1b is awsome and ur too so that comb has to be amazing

[@WadWizard]

*HEAVY MATH*

[@AdamNeely]

4:04 generalize nice. This is actually a really neat idea to "calculate" consonance, and it does seem to reflect our experience, which is always a good thing. I'm not sure if it makes sense to use the same "orders" for chords as intervals, though. Does it make sense to say that a 4:5:6 major triad is as soft as a minor 7th? Both share order 9. Maybe? Still cool to think about.

[@briankeegan8089]

At a glance this seems like another example of a useful explanatory idea that starts to break down if you carry it too far. That's an idea I connect to some critical thinking about models and theories: that we should think of them not as true, but as useful . . . so long as they have useful explanatory force, use them. As they don't, look for a different tools that have useful explanatory force at that point. For example, this formula generated a nice list of claims about the consonance of common intervals. But then it generates claims about much greater intervals that don't match our experience, like when we "round" 200:299 to 2 to 3, and so on. Another example is your video on the scale brightness spectrum, the one where you rank other scales as +1 or -2 from the major scale. It's really useful and explanatory for common scales and how some of the modes feel. . But then it starts to break down . . . like if I recall it correctly, an augmented scale scores a +2 . . . but our ears still hear the #2 as a b3, so . . .

[@12tone]

Yeah I didn't really get into it but I agree that it's better to view different numbers of notes as having their own order lists. Like, it's relevant to compare the orders of different triads, but comparing them to dyads or tetrads gives you some weird equivalences that don't really make a lot of intuitive sense.

[@Ranzha_]

@@12tone Recent math grad here. I'll be looking into generalizing Euler's gradus-suavitatis function for chords of all sizes using number theory, so as to avoid something as consonant as the 4:5:6 major triad having the same order as a 1:256 dyad eight octaves apart :) As you are a student of music, can you give a benchmark of more complex chords that sound equally sonorous? I was thinking of borrowing from negative harmony, having V7 and iv6 be of equal order.

[@Ranzha_]

Upon minimally closer inspection, negative harmonic reflection preserves intervals, so the chords should be of equal order if stacked oppositely (e.g. G-B-D-F is of the same order as D-F-Ab-C if intervals are created from overtones of G and undertones of C).

[@KiteGiedraitis]

Remember, Euler's rating or scoring depends on the voicing of the chord or interval. 9:8 scores better widened to a 9th, 9:4. Which makes a lot of sense musically. 4:5:6 scores worse in 1st inversion 5:6:8, and better in 2nd inversion 3:4:5. Its best score, and IMO most consonant voicing, is 1:3:5, the voicing with all odd numbers. Its 2nd-best-scoring voicings are 2:3:5 and 1:2:3:5, among others. You can use this method to find the best voicing of any just intonation chord. See en.xen.wiki/w/Odd_limit#Proposed_Extensions For a good explanation of the 200:299 phenomenon, google "harmonic entropy". The 50:63 vs 4:5 issue is best explained as a cultural preference. Just like Euler's formula breaks down for really high numbers, it sort of breaks down for really wide intervals like 64/3 (a 4th plus 4 8ves). But then, in actual music there would never be a superwide dyad without other notes in there. IMO Euler's formula would be better if he didn't add 1 at the end. So that 1:1 is order 0, 1:2 is order 1, etc. Less work to calculate, also if you stack an interval, its score doubles. Like a major 9th is twice what a perfect 5th is.

[@rokcetmakesgarbage]

i don't understand 75% of what you say but boy do i feel smarter when i listen to it

[@musik350]

There's literally nothing Euler didn't also have his hands on in mathematics

[@musik350]

@@B3Band I hate this thought with all of my body

[@stanleybaluga4707]

except the math that didn't exist yet at his time

[@apuji7555]

@@stanleybaluga4707 Euler did create new areas/increased the popularity of them so he technically did have his hands on those

[@Kram1032]

I really wish Euler would have defined unison as the zeroth order. That "one added for good luck" is really annoying.

[@MrMisterkrazy]

Found the programmer!

[@Kram1032]

I mean I do program a little but I mostly just find starting at 0 really reasonable in a lot of contexts.

[@Kram1032]

Unison is literally playing the same note as itself...

[@MarsLos10]

I was thinking the exact same thing! The +1 is so ugly xD

[@phlimy]

Yeah that would have made much more sense... I kept getting mildly infuriated

[@kayleighlehrman9566]

I think that ratios like 200:299 are best analyzed after rounding to fewer significant digits. Its 1% away from a multiple of 2:3 which is very good

[@SelvesteDovregubben]

Love the fact that you included Euler's identity in this episode as well. I've seen you use it in other episodes, but this time it really fits.

[@pathagas]

Hey! So I’m working on coming up with my own mathematical model of consonance and dissonance as part of a research project. I’m gathering empirical data on seventh chords over the next two weeks at surrounding high schools. I found it was super interesting that Euler used LCM since that was also used by other people trying to find solutions. Look up the idea of *relative periodicity* for an explanation of why that is important.

[@Jetpack103]

Hello! It's been a little while, so maybe you've made some progress in your project? I'd love to hear about your findings, and read any works you've written on the topic.

[@gammon9281]

Couldnt you have posted this like 2 years earlier? So i had more stuff to write in my seminar paper... hahaha I love it when you find more stuff that combines music and math! keep it up!

[@CalloohCalley]

I'm loving this "mathiness". I went to basic music theory classes in grades 8-12, but we never really got into this side of music and I really like it, man! Keep up the good work.

[@panteleimonnielsen225]

Incredible, man, how do you make so many great vids? Keep it going!

[@SendyTheEndless]

This is really interesting, and a great feat by Euler, but the fact that the pure major 3rd sounds MORE dissonant to me than our crappy 50:63 3rd approximation in E.T. just goes to show that perhaps culture and mere exposure has more to do with it (like learning music as a language). Also, a major third sounds horrible in the bass range, and timbre plays an effect as well (hence why cultures with instruments with different harmonic partials generally seem to produce different scales and intervals to us). It's quite fun to try all of the intervals yourself by getting two synth oscillators and sweeping one very slowly up from the unison. Much like everyone experiences the rhythm to pitch threshold at a slightly different frequency, I think we all each have our own tastes for what sounds consonant (for example, a major 2nd to me always sounds really nice and almost resolved).

[@tompw3141]

That 4:5 major 3td sounded nastily flat to me... but maybe it was because I'd just heard the 50:63

[@Whatismusic123]

What you percieve as more dissonantnis really just yoy percieving it as out of tune

[@ChuckBurry]

This is great. It explains why 4 pt choral voicings sound so "soft" while a triad doesn't sound as "soft." Where it loses me is with the least common denominator of 4 or more pitches, you would get a crazy high order of softness. For instance 2:3:4:5:6:8:10:16:20:32 (A2, E3, A3, C#4, E4, A4, C#5, A5, A6) would have an LCM of 240, but it's essentially a major triad and sounds very consonant (when in tune). I just think it's so astounding that musicians like us train our ears to do math. When you hear an interval and are able to name the octave and pitch class, your ears are essentially hearing the equivalent of an auditory story problem on ratios. Keep up the awesome videos!

[@Gedom666]

Can heavy metal affect perception of dissonance? I listen to it all the time and most examples of dissonance just sound fine to me.

[@unomikoshari2632]

I feel like it has a lot to do with context. Dissonance sounds super uneasy and "bad" in very harmonic, consonant contexts. In metal, where non harmonic riffs are bread and butter, its easier to accept the dissonance. For example, when i listen to other genres like pop that are super consonant, dissonance sounds gnarly (not in the good way), but listening to metal, any dissonance or "wrong" notes sound GNARLY, DUDE

[@JbfMusicGuitar]

I'm pretty sure it does; like if you listen to micro-tonal music, you'll start to get used to that was well.

[@mus-dos4763]

Yeah, same happens to me, tritones and minor seconds sound fine to me, but then jazz chords i found really dissonant and nasty soundind

[@briankeegan8089]

As far as I know, everyone answers your question "yes." rick beato has an interesting vid on tolerance for dissonance. Check it out. kzbin.info/www/bejne/l4TcdGOfg7CXasU

[@Aleph_Null_Audio]

So can Debussy and Schönberg.

[@JoonasD6]

MORE MATH! Thanks. This was amazing. :) To me it started to look like, if one doesn't want to work with the whole orders, simply comparing least common multiples could be a workable measure of consonance. The smaller the better, although with some approximations acceptable due to tuning systems and limitations in precision.

[@MarsLos10]

I find the minor second interval more dissonant than the tritone am I weird? Also this was one of the coolest videos of 12tone, musician and mathematician here haha

[@jodoinscott]

I also find this to be the case. Personally, I think that's because the m2 is found further along the harmonic spectrum compared to the tritone. It's also important to recognize that our brains do have a certain threshold for fusing slightly out-of-tune notes.

[@adamtaylor2142]

Excellent work! Thank you.

[@richardbloemenkamp8532]

Great to have a video on two subjects I like.

[@KrisCadwell]

Great video, very interesting. Not something I will be using in my music, but I love learning new weird stuff.

[@JbfMusicGuitar]

Really cool stuff! Sort of surprised to see the major second as high as that, also oddly happy that the minor 2nd isn't last.

[@musicaljunkyart8756]

Wonderfull video! Loved it!

[@mixuaquela123]

One observation i've found is that the context affects how the interval sounds too. Good example of this is minor 6th. Let's take for example notes C and Ab. In C+ chord (C, E, Ab) the minor 6th sounds a lot more dissonant, compared to Ab/C chord (C, Eb, Ab). Yes i know that ofc the third note affects the chord, but still the interval itself sounds different. You can test it by first playing C+ chord, then the interval. Next play Ab/C chord and then the interval. I at least noticed the interval sounds different because its role varies depending on the chord. Sry for bad english, tried do my best explaining :D

[@mixuaquela123]

another example is actually tritone. When you think tritone as scuffed fifth (role in diminished chord) it sounds evil. But when you think it as unresolved minor 6th (role in dominant 7th chord) it sounds funny. U can test it by playing tritone and then the other interval

[@jodoinscott]

@@mixuaquela123 I think this has to do with how the chord might fit into a harmonic series. If it does, then the harmonic grounding is clear. If not, it sounds dissonant.

[@mixuaquela123]

@@jodoinscott Yep, seems reasonable

[@abramthiessen8749]

Thank you for pronouncing Euler properly. Euler is (among other things) the shibboleth of math enthusiasts.

[@yaitz3313]

He didn't pronounce it correctly. He pronounced Euler's first name wrong.

[@abramthiessen8749]

@@yaitz3313 Euler is a Swedish name. It is pronounced similar to "Oiler".

[@yaitz3313]

@@abramthiessen8749 Yes, I said Euler's FIRST name. He pronounced Leonhard wrong. You're supposed to include the "ard". It's "Le-on-ard", not "Len-erd". German pronunciation, not English.

[@DanNobles]

This is so great. Thank you

[@aanon1342]

The harmonic entropy model can take small deviations in ratios like 200:299 into account. Might be worth a google.

[@Cloiss_]

Anon is this that thing 3b1b did a Video on?

[@aanon1342]

@@Cloiss_ I don't think so.

[@Anonymous-df8it]

Or you can just round to the best consonance within a 2 cent interval (just noticeable difference).

[@unspeakablevorn]

So once upon a time I decided to see if I could make a Highly Dissonant Interval. And, looking at intervals that were quite consonant I noticed that the nicest ones had really small fractional representations that were quite close to the actual ratio - the perfect fifth in equal temperament is close to 3:2 and then the next closest fraction is 747:295 which we really can't hear; the perfect fourth similarly is close to 4:3 and then 295:221; the major third is 5:4 then 29:23, etc. Anyone who knows some stuff about continued fractions knows where this is going: there is a ratio for which the closest fraction grows as slowly as possible: the golden ratio (1+sqrt(5))/2, which is 'close' to 3:2, 5:3, 8:5, 13:8, 21:13 and so on, climbing the Fibonacci sequence. It's about 33 cents above a minor sixth, if you want to hear how it sounds.

[@adararbiv3990]

That was amazing, thx

[@conoroneill8067]

Hmmm... I'd be curious to see someone try and build a sense of 'consonance' based on the infinite fraction representation of a ratio number - that way, if it's just over, say, 3/2, you get 1 +1/(2+1/1000000) - then you could say that because of that extremely large number in the denominator it's functionally more consonant, and therefore provide something that's even more accurate than Euler's Method. Yeah, I geek out about this sometimes. Because mathematicians, like musicians, are very fun at parties.

[@edharbison995]

Rational numbers aren't infinite fractions. Irrational numbers are infinite. Also I am going to explore this with the idea of how irrational a number can be and if it affects this at all.

[@xenontesla122]

Ed Harbison What’s pretty weird is that the interval with a frequency ratio of the golden ratio (at least to me) sounds more consonant than a tritone, even though the golden ratio is “more” irrational than root 2.

[@edharbison995]

That's pretty interesting. I don't know why that is.

[@olipolygon]

@@edharbison995 I feel it just has to do with how our ears are trained to Western music. Nothing to do with the mathematics itself.

[@ZipplyZane]

I can't agree with putting a minor 3rd and major 2nd together. The order goes unison, octaves, p5,p4, M3,m6,M6,m3 m7,M2 M7,m2, tritone. Once past the major 3rd, wider versions of intervals are more consonant, and each in a class. This is why extensions sound more consonant, with, for example, the m9 being just ahead of the m2. If I were making a formula, I'd be trying to make that fit. Though the idea that consonant intervals sound less consonant the further apart they are is something I never noticed. I just thought distance made things more consonant, but I guess it just softens dissonance. Neat.

[@vOddy75]

Distance softens both consonance and dissonance

[@ZipplyZane]

True in general, but I did find something else interesting when I went to my keyboard to play around. An octave may sound more consonant than two, but every other interval besides the minor 6 sounded more consonant if I added an octave. Weirdly, the minor 13th sounded rather dissonant while the minor 6th sounded warm. All on equal temperament, of course. I expected this for the perfect fifth, since it's a simple 1:3 ratio and the second harmonic, but not the rest. And I definitely did not expect any form of a minor sixth to sound dissonant.

[@vOddy75]

​@@ZipplyZane​Maybe a third + an octave hides how the third is slightly out of tune in twelve tone equal temperament, thus making it sound more consonant? I have no hypothesis for the minor sixth being perceived as it is by you. I don't really hear the same things. 2:1, 3:2, and 4:3 are all equally consonant to me, and displacing any interval by octave mellows that interval's character, making it a bit duller, a bit less characteristic of itself.

[@Torthrodhel]

Those two intervals at the beginning of the video are also used at the post-intro beginning of the music for Sonic the Hedgehog's Eul Eaucean zone.

[@DanUsselman]

Nice Mandelbrot Set and Sierpinski Triangle :)

[@bokobo6295]

I’m doing an IB HL Math IA on Euler’s Gradus Function, does anyone have some good sources for understanding and using the formula from frequencies?

[@TheNightBender]

For those of you interested, I recently made a document about a different way of looking at dissonance: www.reddit.com/r/musictheory/comments/aho8yw/analyzing_the_function_of_notes_in_chords/ I explain in the comments a way to use it to organize intervals in terms of dissonance. The results are actually fairly similar to what Euler did

[@MrBillyspilly]

I love this video for understanding Euler's perspective...but I do have issues with it. Loss of "Octave-y Cohesion" at extremes on a piano may be more due to inharmonicity and tuning conventions in the modern age than how good a more distant octave really sounds. In your example, 6 octaves sounds more like a root note plus the 7th of the 6th octave than 6 octaves, and I think that is due to how your piano (or digital model if digital) was tuned. I tune pianos, and this discrepancy in tuning does not seem uncommon to me, but I also don't believe it is fair to consider it a best practice. In other words, I'm saying a deliberate choice is made with those higher notes sometimes to pull them down, and I prefer to widen them to align the notes better harmonics of lower notes. Some tuners don't believe that inharmonicity matters too much, but I have heard these flattened 6th and 7th octave notes enough to believe that inharmonicity is playing a role. There may also be a link to electronic tuning that Euler would not have been party to. I'm not sure. All I know is, when I tune 6 octaves? It doesn't sound at all like what you produced here, and the "offness" of your example is something I have run into before when somebody else tuned a piano before me. (Or a piano, perhaps, with tight high strings that has not been tuned in too long. Wrap your head around that one...) In short: "Loss of Octave-y cohesion" is probably due more to tuning conventions than mathematical laws of physics alone. My 6th octaves never sound like your example, and I do believe they sound consonant. All of this leaves me quite unhappy with Euler's concept of orders of softness arising from octaves. I couldn't disagree with that basic concept more. There is a fundamental "sameness" that I perceive that radiates out from the central octave. A fifth sounds similar in basic musical tension to a 12th, 19th, 26th, etc (Octaves of the 5th). I arrange with this framework, and the notion has never let me down. While I'm certain many musicians and musicologists will disagree with me, I don't believe "Octave-y cohesion" is a thing that dissolves that much as we add octaves. I just don't. I here a consistent sameness that breaches the with if we tune for it....and I see the same kind of "sameness" persist across all other basic intervals. I am a HUGE Leonhard Eular fan...but not in this effort of his. No question the man was a once-in-a-millenium kind of genius. His understanding of octaves, I believe, missed the mark.

[@moth5799]

1:31 I've thought of an easier method for defining dissonance: How long it takes the sin waves of two tones to resolve. This takes into account pitch when measuring dissonance, which I think is quite helpful. For example: Say we have the interval of 4:7 you gave. Let's use A4 at 440hz as our starting note. We can use sin(440x) to represent this tone. Then the other note would be 440x7/4=770hz. So sin(770x). These two waves would resolve at roughly x = 0.05712 (assuming you're using radians, rather than degrees.) So that's a dissonance score of 0.05712 (lower is better.) Compare that then to 5:6, using A as our starting note again. The other note would be 440x6/5=528. So sin(440x) and sin(528x). These waves resolve at roughly x=0.0714. By this method a minor third is actually more dissonant than a seventh, disagreeing with Euler's formula. But as I said, the cool thing about this method, is that pitch is taken into account. For example a fifth between A0 and E1 takes longer to resolve and so gets a worse dissonance score than a minor second between B7 and C8, whilst an octave between A0 and A1 takes slightly less time to resolve than a second between B7 and C8, and so is less dissonant. This matches up with my hearing, even a fifth in that register sounds super dissonant, whereas a semitone super high up doesn't sound that dissonant. I'd assume that someone else has already came up with this method of measuring dissonance, but I haven't found any mention of it online. I do recall Adam Neely mentioning how long two tones took to resolve in one of his videos though. What do you think, does this match up with your hearing better than Euler's formula?

[@G1acia1]

What’s the equation you’re using here to find that “dissonance score” using radians?

[@user-vn7ce5ig1z]

Most people probably pick the second one because it's higher than the first, so it feels more cheerful. I'd be curious so see what people would choose had the order of the two chords been reversed (playing them backwards after doesn't work because the viewer has already been exposed to the original order).

[@brandonkillian4353]

Hey love ur vids ... i really enjoy your " understanding ______ " vids i think it would be cool to see u do a video like that on the song " devine intervention " by taking back sunday , was looking back on some music from my past and refound this song . The song has a interesying soumd that i would like to " understand "

[@xenontesla122]

I really want to see a plot of this function with frequency ratio as x and the output as y. I suspect that It will be really jagged and fractal-ey.

[@petarmilic9729]

Im a simple nerd. I see Euler i click like

[@sammiller9855]

For me, the level of consonance/dissonance of a dyad is also effective by the frequency range/register they are played in. For example, a minor 2nd dyad sounds more dissonant played in the lower bass register than the upper register to my ears.

[@jodoinscott]

That has to do with the way our ears hear - critical bands. Due to the structure of our ears, we have a harder time discerning tones in the lower register.

[@lawrencetaylor4101]

Uhh, I think I'll thank you for this. This was out of my pay grade, and I'll maybe leave this out of my music education. I'm just trying to get through Faber 1.

[@markmcknight9601]

Hey 12tone, have you gotten your Snark back in its box yet? That lead in to your advert for Brilliant was a bit mean spirited, but it still made me laugh. Well Done!!!

[@finnkenyon1289]

This is one of the aspects of music theory that I can conceptually understand, but never viscerally understand. I very much don't consider there to be a major difference in how pleasant two notes sound together. I generally would describe certain combinations as sharper or rounder. The supposedly consonant sounds tend to be rounder, while the dissonant ones sharper. I have not confirmed this really, but I believe it comes down how the notes interact. When notes people consider consonant interact, the tones made by the interference are closer to the notes you play, they sort of round out the sound if you will. supposedly dissonant combinations have a more erratic interference, that makes its presence known. It stands out from the notes that made it. There is a sharp contrast between the not made through interference and the ones you played. Western music does not usually or did not usually want that to happen, but some cultures do and I personally love it. For instance, Tuvan throat singing is kind of focused around this effect, the singer creates lower notes with their vocal cords and various resonance chambers in the body, and the interference creates a higher note that dominates the soundscape, creating what many would call an eerie sound. it is possible then to move around the notes the singer makes up and down together, to adjust them while leaving this interference tone constant, or to move the lower notes relative to each other to adjust the interference tone. But the entire idea rests upon this erratic and strong interference that western music refused to take advantage of. Also interesting to me, is that these sharper interference patterns are less notable in certain octaves. As you move up and down the scales, the interference seems to find spots where it settles in better despite the same ratio. It finds itself in a simpler relationship with the other notes and feels less sharp. I also feel like the opposite occurs with rounder tones, as you go up and down scales far enough, round tones fall out of relationships with their interference and become sharp. Anyway, just the weird idea of some kid who sings weirdly.

[@emilyliedtke7059]

It should be noted that this function is extremely important in pure math as well! It's a key piece in elementary number theory, and even leads into cryptography!

[@GretaZewe]

Have you done a video on the ideas in Rameau's treatise on harmony?? Could you link it if you have?

[@12tone]

Not yet, but it's on my list!

[@GretaZewe]

@@12tone cool! Thanks for the reply!

[@casim8842]

Is there a slower more detailed version of this presentation available anywhere, even if just for Patreons? I'm working on a post-doc paper (for NIME2020) on colour-consonance-emotion at the moment. And this seems well-researched and practice-based. But waaaaay too fast.

[@Bigandrewm]

Speaking of trying to figure out a consistent model of dissonance, I'm guessing that a video on harmonic entropy is forthcoming at some point.

[@fryingwiththeantidote2486]

yeah get with 3Blue1Brown to do a video on the music theory of ben johnston.

[@kiro9291]

I would pay to see that

[@Bigandrewm]

Technically, Ben extended Harry Partch's ideas, so it might be a good idea along these lines to start there. Partch, of course, did not conceive in a vacuum either.

[@fryingwiththeantidote2486]

Andrew Meronek extended is a very soft word for what johnston did. And i believe 12 tone went over partchs 43 tone scale, which is a pretty insignificant part of the system but he has ventured into this territory somewhat previously.

[@theharry801]

ive thought about dissonance and consonance quite a bit and found that all dissonant intervals are linked to the tritone. Thoughts?

[@jodoinscott]

I would argue that m2 is the most dissonant. I don't see how it is related to the tritone.

[@denglish5275]

I stand by the fact that Euler is the most important person in history. His far reaching and broad areas of research have impleme tation in the base structure to so many things today.

[@rca88]

Darwin solved a problem as hard as any math problem, maybe as hard as all math problems combined. Outside of astronomy (Galileo, Copernicus), it's pretty hard, at least for an amateur like me, to think of math problems that conflict with deeply embedded world views and religion.

[@erics3317]

I'm assuming that you wouldn't want to compare the softness ratings of groups of notes of different numbers. So dyads can be compared to each other, but you shouldn't compare dyads to trichords for example, otherwise you would be equating the major triad with a minor 7th which doesn't make much sense intuitively. It occurs to me that you could use this to get a softness score for each of the Forte sets and rank them all within each cardinality based on softness. That might be an interesting project.

[@jameskennedy7093]

Did the voice over change for this? It sounds deeper.

[@lukebs1212]

The 4:7 ration occurs at the start of steely dans josie

[@SendyTheEndless]

4:7 is awesome

[@heavynov]

If you calculations are true, I must set out to modify the formula, as I do like the approach, but the fourth is nowhere near as consonant (in a two voice framework) as the major third or sixth and the major second isn't nearly as consonant as the minor third.

[@Mau365PP]

Damn.. Euler was everywhere ! *e^(π*i)+1=0*

[@user-hy6cp6xp9f]

Literally he has like 20 formulas named after him. There is a joke that every formula has to be named after the second person that discovered it because Euler probably discovered it first.

[@dsws2]

All sorts of thirds -- not only major and minor just intervals, but also neutral thirds -- are rather consonant. I think you need something like Plomp-Levelt consonance. (I make no claim to be any kind of expert. I've merely read parts of Tuning Timbre Spectrum Scale by William Sethares.)

[@mrlabon123]

Awesome

[@kedrak90]

1/64 sounds softer to me than 1/20. Or has this something to do with me goofing around on a keyboard playing that interval and the adding in the next Cs inbetween those.

[@hcesarcastro]

At first this function reminded me when I studied groups and Euler's totient function. Also, the on-line encyclopedia of integer sequences has a record Euler's gradus ("suavitatis gradus", or degrees of softness) function: oeis.org/A275314.

[@LesterBrunt]

This is super intetesting but what exactly is the point of this? What is the purpose of ordering intervals?

[@An_Amazing_Login5036]

The list for intervals gets fuzzy regarding the tritone and fourth. A major third is nearly always consonant (i don’t think i’ve heard a dissonant one at least) and a fourth is sometimes dissonant, especially in very consonant music. Why is the fourth ”softer” than the third, and could the system be rectified? Also, consider the tritone and the minor second. Would you call the tritone more dissonant than the minor second? I would not say so. It looks like when you are calculating the ”softness”, the octave distance to ”get home” (in the first octave) should count. Maybe i’m wrong.

[@vOddy75]

I don't understand this idea of the perfect fourth being dissonant. I've never heard it, I've only heard it as consonant.

[@An_Amazing_Login5036]

@@vOddy75 For example, play a perfect fourth, then a major third. The fourth resolves to the third, marking it is slightly dissonant (at least more dissonant than the third). Or take someone else's word for it. Vincent Persichetti writes in his book "20th Century Harmony _Creative Aspects And Practise_" the following: (s. 15)" The perfect fourth sounds consonant in dissonant surroundings and dissonant in consonant surroundings."

[@vOddy75]

@@An_Amazing_Login5036 I've heard it said, but I haven't heard it. I won't take any one's word for it. When I hear the fourth, I don't feel that it needs to resolve. It isn't innately pulled anywhere in particular. I feel pulling toward tonal centres, but I don't feel it here. I think that hearing the fourth as wanting to resolve to the major third is just being used to triad based music. Innately, the fourth (4:3) is a simpler ratio than the major third (5:4). Why would a simpler ratio want to resolve to a more complex one? And being pulled toward a pitch is generally a melodic phenomenon anyway, not a harmonic one (like the half tone away from the tonal centre is pulled toward the tonal centre).

[@joshvictor110]

@Hagel Adding to your point of triad dominated music: perhaps a fourth just doesn't form pleasant intervals with other notes, as shown in, say, a sus4 chord. Overtones might contribute to this as well. Just a quick thought, don't mind me.

[@oldmatedave1]

@@vOddy75 One possible answer would be that the perfect fourth (from the root) isn't found until really high in the harmonic series - So if you take the harmonic series on C: C C G C E G Bb C D E F So the major 3rd (E) is a much earlier and stronger overtone than the 4th (F). Although of course there is an implied fourth between the G and C. So yeah 4ths are weird. Consonant in some scenarios and dissonant in others. In baroque music the 4th is often used (as a suspension) as a mild tension/resolution to the 3rd. In modal jazz tonic chords are often built out of stacked 4ths.

[@ayanbiswas2943]

Is there a measure of consonance better than that of Euler?

[@eti313]

When you hear a note, you also hear the frequencies of any harmonics that might be there.

[@stephenhall3515]

A 'solution' in search of a question. Any actual music Euler might have produced would have sounded similar to that of others of his time. That is part of the nature of art.

[@AbhiBass96]

How did you know that minor 7th was order 9 and stuff? Wish you showed us how you did. At least a few. Thanks. Also, for a major triad You said it was 60 and then we would have to find the numbers with exponents that make up 60, right? I did LCM of 60 2 - 60 2 - 30 3 - 15 5 - 5 - 1 ^That is how I was taught = 2*2*3*5 2^2 3^1 5^1 2-1=1*2=2 3-1=2*1=2 5-1=4*1=4 8+1 = 9 and order of 9? Let know if I'm right. @12tone

[@12tone]

That looks correct, yes!

[@AbhiBass96]

@@12tone I couldn't get minor 7th As there are two ratios 16:9 If we stack two forths 4*4 / 3*3 = 16/9 Or The fifth and minor third 3*6 / 2*5 = 18/10 = 9/5 So, I took the latter 3^2 and 5^1 3-1=2*2=4 5-1=4*1=4 4+4=8+1=9. The order of 9.

[@Mikasks]

I finally understand those stuff, its a bit too musical for me so i cant understand it when i saw the formula the first time.

[@anirudhsilai5790]

What about irrational ratios? Would 1:sqrt(2) be an exact tritone?

[@bendowson3124]

Technically, 1:sqrt(2) would have order infinity according to Euler's formula. The biggest limitation of Euler's formula is it doesn't consider intervals that are very close approximations of other, more consonant intervals. Since 1:sqrt(2) roughly approximates a 5:7 ratio (which has order 10), its order of softness should only be a little greater than 10.

[@bendowson3124]

Oops, the 5:7 ratio has order 11, not 10.

[@T3sl4]

Noteworthy that, in Euler's time, tuning (tempering) was harmonic, as has been discussed in prior videos I think. Modern equal tempering is a big fat infinity to this formula -- all non-octave ratios are /irrational!/

[@lucianodebenedictis6014]

I don't know if this has been answered, but I always ask myself why the intro click is in 5/4 starting on the second beat

[@rca88]

I hear 4/4 starting on the one.

[@lucianodebenedictis6014]

@@rca88 it's four low pitched clicks followed by an high pitched one

[@bonecanoe86]

I love the mathy side of music!

[@rolanddeschain6617]

it's weird the b2 and the #7 are in a different order

[@AlexKnauth]

I don't think that's weird. Sure if you moved the #7 down an octave they should be the same, but that separation of them being almost an octave apart actually helps them sound less dissonant to me. Like the Maj7 chords at the start of Bohemian Rhapsody don't sound very dissonant to me, but if you voiced it differently with the 7 down next to the root, it would probably not sound as nice. The same is true for the start of the chorus in Space Oddity, I think

[@edslushie570]

Cool... but I can’t comprehend putting the major 2nd on the same order as the minor 3rd. Going up a minor 3rd (+a fifth)makes a minor chord, which is just as consonant as a major chord, but going up a major second (+a fifth) makes a sus chord. Wait... does a major second sound good WITHOUT a fifth? Like a major seventh in reverse?

[@alainpbat3903]

Isn't the piano used in this video technically 3 frequency?

[@GarryLarryBarry]

Woah that's a lot of math.

[@Aleph_Null_Audio]

I agree with Euler's results except in one instance: I minor ninth sounds way more dissonant to me than a tritone. Maybe that's because I'm used to hearing tritones used in the context of a dominant seven to tonic motion, whereas minor ninths are much more rare in functional harmony. Something to ponder...

[@12tone]

Yeah, cultural context definitely factors into this as well. For instance, the perfect 4th is often heard as wanting to resolve to the major 3rd, even though the latter is by any reasonable estimation the more dissonant interval. We're just so used to triad-based harmony that our expectations override the underlying mathematical structure.

[@Aleph_Null_Audio]

@@12tone - Just had another thought: maybe "softness" has more to do with instability than with dissonance in the "unpleasant to listen to" sense. The tritone is definitely more unstable than the minor ninth in that it does seem to want to resolve somewhere. The minor ninth just sits there like an ice pick in the ear.

[@zeke7209]

​@@Aleph_Null_Audio The tritone is somewhat close to the 5:7 ratio, which is order 11. Also a minor *ninth* (15: 32) is an octave higher than the minor second (15: 16). The minor ninth would have order 12, not 11. This could possibly explain why the minor ninth may sound more dissonant than the tritone.

[@bendowson3124]

Interestingly, according to this formula, sus2 and sus4 chords are more consonant than major and minor triads.

[@Pablo360able]

Okay, but there's still one thing I'm confused about. When I hear a note, am I hearing a sound wave with a specific frequency?

[@lukesaunders4776]

Maths. Making the world more transparent for over 2000 years

[@consequenceable]

wow

[@drezzylol]

Perfect fourth doesn't sound so consonant in the context of a major chord. It can, though, if you know what you're doing. Everything is relative in the music theory. Tritone, for example, will be your sharp four in the major chord. Which is brighter sounding than the regular major and arguably more pleasing to the ear.

[@deakenwylie3819]

256 likes, 2 dislikes... Great, I can't rate this video EITHER WAY. >:C

[@user-om9jl5jv5y]

Look at it like this: at the time, difference between likes and dislikes was 254. And EXACTLY ONE LIKE would bring it up to 255! So, like, you had to do it.

[@n7275]

Is there any peer-reviewed publications phenomenological and psychoacoustic aspects of this formula.

[@jodoinscott]

I don't think there is any peer-reviewed phenomenological publication! Maybe you mean something else.

[@enricopersia4290]

Dud, i thought to have left math nightmares to high school periodo :D joking, great content as always

[@stevenkaminski1317]

I love watching these but then get more confused before I started. Not the videos fault, I didnt realise music was so intense

[@beatrixwickson8477]

Two things I don't follow: why are 1:3 & 1:4 ratios lumped together and what falls into Order 6?

[@zeke7209]

If my calculations are correct, the intervals in Order 6 are 2:5, 3:8, 2:9, 1: 10, 1: 18, 1: 24 and 1: 32, all larger than an octave.

[@beatrixwickson8477]

@@zeke7209 thank you kindly!

[@brockobama257]

MORE MATH

[@natea5225]

Oiler?

[@bumpty9830]

A little (okay, a lot) mathier still, and with an answer to the "200/299" problem with the Euler Gradus function: kzbin.info/www/bejne/maq6Zq1jgpessNk

[@guitarandhow4264]

Is your intro in 5/4?

[@rillloudmother]

i will always pick the tritone

[@johnt.mickevich2772]

Shhh! I've been trying to tell people music theory ISN'T math!

[@cbfedge5593]

Did anyone else think of muse with the first two sounds?

[@thedabblingwarlock]

Huh, this is pretty interesting. Who knew you'd find sigma in music theory. :)

[@jodoinscott]

Fast Fourier Transformations (FFT) is described using sigma and are used to calculate spectrograms which are used in digital tuners and electroacoustic music. :)

[@marco-xe9je]

0:49 AN INTEGR- oh wait...

[@pepxand4194]

Can you please breakdown "To live is to die" from Metallica??

[@hotdogskid]

I find it really interesting that to me a tritone doesnt sound that dissonant. It might be a “repetition legitimizes” situation, but things like m7b5 chords and the locrian mode dont sound as jarring to me as other people. Repeat it enough and a tritone will start sounding major.

[@emilpysenisoncrack420]

6:51 When you said "on top", that E sounded so autotuned

[@ssalamander2134]

To me, the 4:7 sounds wrong. Not disonant just like flat or sharp

[@feridunabi7723]

a lot of fun: draws a ball in a cup