Benedetti's Puzzle SOLVED (response to Adam Neely)

Benedetti's Puzzle is a music phrase that was posed in 1585 by Giovanni Benedetti to demonstrate that you cannot have mathematically perfect harmony without pitch drift. But modern technology allow to solve the puzzle by generating carefully constructed in-harmonic timbres.
_________________________________________________________________________
Download Samples:
drive.google.com/drive/folder...
_________________________________________________________________________
Follow:
Twitter: / new_tonality
Support:
Bandcamp: offcast.bandcamp.com/album/re...
Ko-fi: ko-fi.com/newtonality
BTC: bc1qzu3cscpaltcpqhe5pjrc3y64gux0ja6dtacvvw
Monero: 471K4TqU8nx3zVSjQVPgVUBco3r4NVWc8jhMCxuSJW72A8tFDDdfHKkgkPNUfRKebjZTJrpbkVAXZVA8A6tbADrbBRWnFpj
ETH / USDT: 0xb1596Ad3A60219906C699114E7DC5Fcc19ED9ec8
_________________________________________________________________________

[@peethedelund4089]

This is one of the most fascinating YouTUbe videos, I have ever seen. I've watched tons of stuff on the "tuning problem"; but never ever heard anyone suggesting to solve it by turning it on it's head like here. Absolutely brilliant!

[@stephenweigel]

This is a criminally undersubscribed channel

[@spacevspitch4028]

I've heard a lot about this kind of thing but I've never actually HEARD it like you presented here. I think it'd be amazing to digitally adjust the harmonic spectrum of an entire acoustic grand piano this way and see how a whole sonata sounds. Especially something like a late Scriabin sonata or something like that.

[@new_tonality]

Yeah that be really cool! If you are interested, in my latest video I expanded on this topic as for how it applies to microtonal music. There are sound examples using synthesized tones in "Challenging Simple Ratios" part. I play short piece from "Fur Elise" in stretched tuning and demonstrate how adjusting individual partials can reduce dissonance in such tuning system.

[@theskv21]

Look up the Hyperpiano if you’re interested in an application of the idea of adjusting a piano’s timbres to play music in other tunings

[@Henry-fv3bc]

There is clearly some sensitivity in the brain to the ratios of the partials themselves. E.g., if I form a sound from two sine waves an octave apart, and start to sharpen the upper partial, my brain eventually starts to notice a dissonance, even though there's no beating. This could be due to nonlinearities in the ear that create non-overlapping harmonics, but since harmonic ratios are so common in real-world audio, the brain is surely sensitive to harmonic ratios (that's how the brain works, it finds efficient ways to express patterns in the data). So, I suspect the brain's perception of dissonance is both due to ratios AND beating. But if you don't like the beating, then this clearly solves it. The 12-TET partials version still sounds different from the justly intonated major chord (which isn't necessarily a good or bad thing, but just noting that the brain does not perceive them as the same). It's clearly more consonant than a normal 12-TET chord. One disadvantage of making the synthesizer notes inharmonic is that they may not interact well with distortion (but if the distortion is built into the digital synthesizer itself, there's probably clever things you could do work around this, but it wouldn't work with analog distortion). This probably isn't an issue for most synth applications. Regardless, this is an awesome idea. All digital synthesizers should have this option. It doesn't make the synth any more out of tune with the rest of the music either, but probably makes it more "in tune" (since it's probably in 12 TET).

[@new_tonality]

100% agree on the point of perception of dissonancae being dependent on ratios as well as beating. Disortion is interesting, it works a bit differntly with inharmonic sounds. It still sounds like disortion, but because the inharmonic waveform is not periodic, disotrion creates a sence of chaotic rythm in addition to ritcher spectrum that is still inharmonic no matter how hard you push the gain. So there is a lot to experiment with on that front.

[@Henry-fv3bc]

@@new_tonality I agree. When I said "interact well with distortion" I just meant that it would be a different sound from which people are used to, which is not necessarily a bad thing. On electric guitar we normally omit the 3rd in distorted chords since the 12TET major third is quite sharp and this is enhanced by the distortion. However, sometimes, the messy sound of a distorted 12TET major third is actually desirable. The difference with these 12TET-aligned partials is that even distorted single notes would sound messy - I'd be interested to see what that sounds like. Maybe a 12TET-aligned digital synth could somehow emulate "non-messy" distortion by generating additional intermod products at 12TET-aligned points in a way to mimic intermod products of distorted just intonated polyphony. That might be pretty hard to do efficiently though.

[@poproporpo]

This is awesome! Even if it cannot be achieved in real life with our voices (hence not "solving" the choir going out of tune problem), I found this solution to be extremely creative, and really gave me a new perspective. Большое спасибо!

[@derjushkinalubov]

It is an awesome combination of « physicists and lyricists » , mathematics and music

[@JLMoriart]

Very glad to see you making Sethares-inspired videos, more people need to dive into matching tuning and timbre!

[@lilac_hem]

your whole channel is extremely underrated and deserving of far more attention

[@eddyflux]

oh what a nice idea to not change the tuning but change the sound itself! never heard of that before, thanks for elaborating it so clearly

[@user-wr6ip7fk7f]

Hammond organ have the frequency of all overtones (3 & 5 ) equal to notes 12EDO. For example, 5th overtone is 400cent, not 386. It is the only popular instrument, to realy play chords in 12EDO with distortion. There is a model of Hammond, which has 7th and 9th overtones.

[@new_tonality]

Cool, always liked how it sounds distorted. Didn't realize it has to do with tuning of partials, thought it is amplifier or a cabinet. Now I realize it was something about it in the book I've mentioned. Thanks for info!

[@bacicinvatteneaca]

Doesn't distortion create harmonic partials? Wouldn't that mean that a single distorted note on guitar has no beatings but on played by a Hammond organ does? You know, since the distortion-generated harmonic partials should be out of tune compared to the inharmonic drawbar partials of the instrument itself.

[@user-wr6ip7fk7f]

@@bacicinvatteneaca Theoretically you are right. I have never played real hammond, only VST plugins. They (plugins) sound in compex chords clearer then, for example, distorted guitar (which i can play). Maybe this is because guitar strings have many higher order inharmonic partials (not only 3 & 5, but 9, 11 ,13 etc.) which are much higher then natural harmonics due to the strings inharmonicity. Also distortion not only create harmonic partials, it create summary and difference tones, which will be more in tune with 12EDO on hammond.

[@macronencer]

Sevish mentioned this in his recent live stream, and I had never realised it before. As soon as he said it, I stopped to think about the tonewheel mechanism and realised that I should have realised it long ago. Presumably the non-octave drawbars are getting their tones from the same wheels as the lower drawbars for higher notes. I used to play Hammond, but I never had a tonewheel model: mine had more modern circuitry - it was from the 1970s, the X5. I'm now wondering whether that had 12EDO partials or not... I want to find out! I know it used octave divider circuits with 12 semitone frequencies, each divided down to get octaves - but that doesn't necessarily make it clear to me how the drawbars functioned. Anyway, it's amazing that such an interesting thing could explain "that Hammond Sound" that people often speak of!

[@mirrorengine]

your videos are a treasure!!! thank you so much!!!

[@tango_doggy]

Very well made and demonstrated. Great video!

[@gabrielhenschen9665]

Let's make every interval pure through making every note impure! Surely this cannot be serious?

[@infinitonica2362]

Love the harmonic sweep demonstration, one of the best explanations of Just Intonation I've seen. The thing is.... ultimately we have to be "music makers" not just "tuning makers", the question lies... can these ideas be used to make truly inspiring new music? And also it should be noted that differences in harmonic content of partials can make things interesting.... if everything lines up too much, it can be dull sounding

[@JustinMasayda]

Excellent video. I had wondered about this but regarding just intonation, meaning, I wondered if modifying the partials could result in a just tuning in every key. But, I think that's probably mathematically impossible. It didn't occur to me that it could work in an equal temperament. It's a really clever solution. Great explanation!

[@music-zv6je]

11:00 hearing this makes my heart happy and I don't know why or how that works, but it genuinely feels that way. What a sound.

[@nihilumaeternum6555]

Very interesting approach; neither consonance nor consistency is approximated, but timbre.

[@sagandalya108]

Now we just need artificial ears to appreciate this artificial harmonic series you have crafted :)

[@LatchezarDimitrov]

The just intonation isn't praticable because the half tones in each interval are differents! The perfect music equal temperament must have an universal half tone to avoid equally all just intervals!

[@MYGAS21]

Fascinating indeed.

[@gambe96]

One does not simply disregard the harmonic series

[@jasensargent2245]

so the only way this wouldn't be a true solution to the puzzle is that if we defined "perfect" music as that which uses sounds that coincide with the harmonic overtones we find in nature

[@Andrew524476]

Привет, thanks for the educative video, could you also make an overview of the Turkish music theory as to my understanding it tries to deal with a similar problem?

[@new_tonality]

Привет)) that is very interesting to do. I really don't know much about it at the moment

[@qondonyon]

this is AWESOME!!! SCREW NATURE!!! >:333

[@moriscengic]

So is there posibility to tune piano to this harmonic system and is it going to clash with other instruments like guitar?

[@new_tonality]

Piano and other acoustic instruments have harmonic spectrum (with slight deviations), that is just the physics of vibrating string. So for them perfect harmony can be achieved only in Just Intonation with all the downsides of Just Intonation. The only way to do what I did in the video but with real timbres (like piano) is first to sample piano to the computer and than take every individual sample and alter frequencies of its harmonics to suit 12 tone equal temperament (12-tet). Like I did with synthesized sound. That is how you will make piano sample library to sound perfectly in 12-tet tuning. It is really the question of building digital instruments rather than tuning the real ones. Will such library sound good with other libraries or with real instruments - I don't know. I don't think anyone tried that. I will, but with synthesized timbres for now.

[@tango_doggy]

@@new_tonality It could be possible to make an instrument with those approximate overtones although I'm not sure how

[@axgp_]

would you be able to upload and share the 12-tet spectrum sample you loaded into serum at 11:01 ?

[@new_tonality]

Here you go drive.google.com/drive/folders/1T0MxXGr7h941aw335M38cH285w_LBkEn?usp=share_link

[@axgp_]

@@new_tonality thank you!

[@moadot720]

5:57 *“PREFECT 4TH...”* perfect spelling 😂

[@Fire_Axus]

least squares meantone tuning would be much better for this puzzle.

[@ucanihl]

Well isn't this just another way of compromising? When we listen to two pure tones having frequencies f1 and f2 we also hear the non-existant fundamental tone gcd(f1,f2). So by changing the timbre to fit 12tet you are sweeping the problem under the rug.

[@new_tonality]

It is another way of compromising, that opens up interesting creative opportunities))

[@Spherey]

you look like a mix between vladimir putin and neil cicierega

[@alexandrusimo899]

holy shit bro

[@ldmdesign5610]

How can I make such 12tet compatible sounds?

[@new_tonality]

I need to make a video on that… I don’t know any software to do it so I built my own for that purpose. I have a repo on Github but installation and operation is not straight forward. I will update readme and post a link in several days.

[@ldmdesign5610]

Cool! I'm a #maxforlive developer so started thinking about it and making my own additive synth so far with independent partial frequency control

[@new_tonality]

​@@ldmdesign5610 here is the repo link. github.com/SevaDer14/xen-explorer

[@ElvisTranscriber2]

I've would like to know of a better way to explain that the different modes that produce the harmonics will use the energy put into the system so to feed an *infinite* number of partials you would need an infinite amount of energy as well as that flexibility of the string, place along the string wilhere it is struck will condition which harmonics are favoured. I cannot be possible that a strings vibrates in an infinite number of modes. It is possible to exit particular harmonics in a string the way you're doing 8t and in other ways. I'd like an actual demonstration of, let's say the 10000th harmonic 😜

[@adiaphoros6842]

The higher the mode, the lower the energy. On physical systems, the energies fall off fast enough such that their sum is still finite. With the nature of reality, yes, there should be a cut off point. Most audio editing software cuts off at the 20th partial, because any further partial would be so faint that they would be dominated by the inherent noise in the medium (i.e. air).

[@StevieBoyesmusic]

Hammond B3 organ beat you to it!

[@matthijshebly]

Is this different from Hermode tuning? What are the actual frequencies of all the notes in your solution of Benedetti's puzzle?

[@new_tonality]

It is not Hermode tuning, because as I understand Hermode tuning is dynamic tuning that retunes notes during performance. What I used here is static tuning, it is regular 12 tone equal temperament, A = 440Hz and each half step is 12th root of 2 apart. All exact frequences can be found here pages.mtu.edu/~suits/notefreqs.html I heard about dynamic tunings but never came across Hermode tuning, thank you for introducing it!

[@matthijshebly]

@@new_tonality You wrote: > "…as I understand Hermode tuning is dynamic tuning that retunes notes during performance" Correct. > "What I used here is static tuning, it is regular 12 tone equal temperament, A = 440Hz and each half step is 12th root of 2 apart" Then I don't get what it is you're trying to achieve. Either you're trying to tell us about the virtues of 12-TET, or you're not. Either you're showing us how 12-TET fails, or not. Either you're telling us about an alternative tuning system you discovered, or you're not. > "All exact frequences can be found here pages.mtu.edu/~suits/notefreqs.html" I'm very well aware of how 12-TET works, thanks. What I don't get is how your video is presenting us anything new… I watched your video again; I don't get what it is that you're trying to tell us.

[@new_tonality]

@@matthijshebly What I am trying to say is that there is another way of tackling tuning problem, without sacrificing purity of harmony and without referring to dynamic tuning. The problem of tuning is fundamentally a problem of dissonance. We are taught that all musical sounds are harmonic, and inharmonic sounds are just noises that do not contribute to harmony. In that paradigm there is no other way of controlling dissonance but to play with pitches of notes. I believe that this “axiom” that we do not question, is misleading, because inharmonic sounds can be musical and do contribute to harmony, but they don’t lead to Just Intonation. In the same way as Just Intonation relates to harmonic spectrum, other tunings can be related to inharmonic spectrums. In this video I came up with inharmonic spectrum to which 12-tet is related. That allows me to play the puzzle, with both fixed notes and with pure harmony, I call it “solving” the puzzle. "Then I don't get what it is you're trying to achieve. Either you're trying to tell us about the virtues of 12-TET, or you're not. Either you're showing us how 12-TET fails, or not. Either you're telling us about an alternative tuning system you discovered, or you're not." That is all true ONLY for harmonic spectrum. What I tried to show is that harmonic spectrum is one of many. And that thinking does not apply to inharmonic spectrums. You can choose inharmonic spectrums that lead to any tuning be it 10-tet, 12-tet, 5-tet, 13-tet or non-equally tempered tuning. Maybe that approach can be interesting to microtonal or experimental musicians. In short, the problem of tuning is fundamentally a problem of dissonance, usually we control dissonance by playing with pitch, but we also can control dissonance by playing with spectrum. Hope that clarifies it.

[@matthijshebly]

@@new_tonality > "What I am trying to say is that there is another way of tackling tuning problem, without sacrificing purity of harmony and without referring to dynamic tuning." Neither your video, nor this comment, clarifies it, I'm afraid… > "The problem of tuning is fundamentally a problem of dissonance." Incorrect. Dissonance is an integral part of music, and it has been since, well, centuries. E.g., in mediaeval contrapuntal church music, dissonance was a part of the tension and release-cycle, and that persists until today in almost all forms of music. The dominant seventh chord is only the dominant seventh chord because of its inherent dissonance and therefore tension. No, the issue with tuning is of a mathematical nature; it is the fundamental problem of rational vs irrational numbers: * All harmonics of a note have BY DEFINITION a rational relationship to one another: the octave to the root is 2:1, the perfect fifth (just intonated) to the root is 3:2, the major third (just intonated) is 5:4, etc. === whereas === * Any subdivision of the octave into two or more equal intervals will ALWAYS result in irrational relationships: 2^(1/n) is ALWAYS an irrational number, for any n>1. It is this quirk of mathematical reality that makes tuning impossible, when the aim is to tune both just as well as equally tempered. We can either tune justly and not modulate, or modulate and tune equally tempered. We can't have it both ways. (Hermode is interesting because it continuously shifts a just tuning to well-tempered tonal centres, but it cannot exist in acoustic instruments). > "We are taught that all musical sounds are harmonic, …" …No, I wasn't taught this. > "and inharmonic sounds are just noises that do not contribute to harmony." …No, I wasn't taught this. Dissonance is part of music. > "In that paradigm there is no other way of controlling dissonance but to play with pitches of notes." There's loads we can do, but it will ALWAYS, by definition, be a compromise, because we can't escape the disconnect between rational and irrational numbers. > "I believe that this “axiom” that we do not question, is misleading, because inharmonic sounds can be musical" Define "inharmonic"… > "and do contribute to harmony, but they don’t lead to Just Intonation." I have the feeling you're confusing dissonance and irrational interval relationships… > "In the same way as Just Intonation relates to harmonic spectrum, …" Just intonation relates to having rational intervals between pitches. > "…other tunings can be related to inharmonic spectrums." You just made up the term "inharmonic spectrum". The word "harmonic" in "harmonic spectrum" is not a value judgment, it merely refers to having rational ratios between pitches. > "In this video I came up with inharmonic spectrum to which 12-tet is related." I can't tell. Your video doesn't make it clear. > "That allows me to play the puzzle, with both fixed notes and with pure harmony, I call it “solving” the puzzle." What's still unclear is how you have "solved" the puzzle. I asked you to provide the exact pitches of your solution of Benedetti's puzzle. Like, exact Hz values. Until you do, I won't be able to understand your solution. > "What I tried to show is that harmonic spectrum is one of many." …No… The "harmonic spectrum" refers to harmonics that have, by definition, rational relationships to one another. > "And that thinking does not apply to inharmonic spectrums." There's no such thing as an "inharmonic spectrum". You can't just go about making words up to prove a point. That's not how science works. > "You can choose inharmonic spectrums that lead to any tuning be it 10-tet, 12-tet, 5-tet, 13-tet or non-equally tempered tuning." Yes, but they will ALWAYS suffer from the dilemma that the universe has confronted us with: that irrational numbers cannot be rational at the same time. > "Maybe that approach can be interesting to microtonal or experimental musicians." Sure, different tuning systems are interesting, but they can't solve Benedetti's puzzle. > "In short, the problem of tuning is fundamentally a problem of dissonance, …" No, not at all. The problem of tuning is ONLY the problem of rational vs irrational numbers. Dissonance exists in both equal temperament as well as in any just tuning. > "…usually we control dissonance by playing with pitch, …" We control dissonance by allowing (or not allowing) the tension of it to be followed by release. > "…but we also can control dissonance by playing with spectrum." There's nothing to "play" with. Pitch relationships are either rational or irrational. > "Hope that clarifies it." I'm afraid not.

[@new_tonality]

@@matthijshebly Ok. I think I get what you talking about and I think the misunderstanding lies in harmonic vs inharmonic spectrum. > "You just made up the term "inharmonic spectrum". The word "harmonic" in "harmonic spectrum" is not a value judgment, it merely refers to having rational ratios between pitches." Firstly and most importantly I did not made up term inharmonic spectrum. It is widely used scientific term that means that spectrum is not harmonic)) That is that partials in that spectrum are not whole number multiples of fundamental frequency. So if fundamental frequency is 100Hz then in harmonic spectrum partials (harmonics) will be 200Hz, 300Hz, 400Hz etc (harmonic series). But if their frequencies are for example 100Hz, 235Hz, 314.15Hz, 1265.8475Hz then such spectrum is called inharmonic. There is related term inharmonicity which is the degree to which partials depart from harmonic series. It can be small like in spectrum of a piano (though still large enough for piano tuners to stretch octaves to acheive greater consonance in octaves) or quite big like in church bells. I am not able to tell you the exact frequencies of partials that I used for spectrum with which I claim I solved the puzzle, as I don't have accsess to my PC right now. But it was inarmonic you can see it at 9:56. Inharmonic spectrums can have irrational ratios between frequencies of partials and therefore irrational ratios between frequenceis of notes. Rational ratios appear ONLY in case of harmonic spectrum. I wanted to explain it in more detail in further videos as it is not trivial topic and I had to write software to do it. Next video I will be making about spectrum of real instruments, inharmonicity and Gamelan music to show how in practice spectrum of instrument impact tuning. And then I will go to theoretical videos about nature of dissonance (or I better get used to call it percieved or instateneous dissonance) with references to scientific literature. If you dont want to wait, as I dont make videos that often you can read "Tuning, Timbre, Spectrum, Scale" the book I referenced in this video, it cover almost everything on that topic. >"I have the feeling you're confusing dissonance and irrational interval relationships…" I am not confusing them. When I am talking about dissonance I mean precieived dissonance (other authors may call it instateneous, tonal or simultaneous dissoance/consonance). It the the perceptual phenomenon that correspond to roughness or pleasantness of simultaneosly sounding pitches. There are many theories of what mechanisms contribute to that. The main ones are cultural, periodicity and spectral interference (the one I touch upon in this video) There is great review this and other theories "Instantaneous consonance in the perception and composition of Western music" (pre-print is avalible here psyarxiv.com/6jsug/).

[@DUNCZI]

Benedetti's theorem is obvious, just because there is no "one" separated unic soloistic isolated sound. One sound is an OUTPUT of many of little part-resonances =harmonics. So, one sound IN THIS case aren't equal with one number. Even numbers, like imaginative numbers or, matrix number are exists, and those behave spectrum like. Basicali one number is not one sound/ note. The note, as a conclusion of the proportion is quite good model. But just a model. One note (sound) is "extend" enlarges itself, behave in a ZONE, hence the musical note is spectral-like. Utopia, musical sound has no just ONE singular space. No pipoint like. This axioma exclude the matematicaly exact possibility of the "celestial etheric" harmony. BUT. Matematically non-precious proprotions, constellations, harmonies, chord could represent the original beauty. Compare and see Penrose or Maurice Esher pictures on illusions. Not phisically exists (Esher's impossible spaces) BUT we generated them. So, beauty is a paradoxon AND THE WHOLE WORLD based on many paradoxons!! The illusion are exist as illusions. Refer to the basis, point to the possibility show the creativity, and something impossible step out (even a little bit only), in the "real world". But what it is real? -so, it need a new discuss and definition... IF illusiions are behaving as reality, reality could be an more arranged illusion? ART is manipulation, magical act throughout illusions....

[@Theeduckie]

I feel like if you had actually solved it, you’d have more than 3 (now 4) comments and more than 9 likes and more than 200 views.

[@new_tonality]

I think it is better to judge things like that by substance of the argument and not by popularity metrics

[@Theeduckie]

Objective Harmony but you didn’t. The changes are because when we sing, we are making the ratios unconsciously based on what is currently being sung.

[@new_tonality]

@@Theeduckie Absolutely! Voice without any processing is always harmonic, but first you can come up with algorithm that re-tunes harmonics to 12-tet compatible frequencies and second you can use prepared (12-tet compatible) samples for guitar, piano and cello. Modern music is digital and all instruments are heavily processed. So my point is that there is TECHNOLOGICAL SOLUTION to the problem, just another way of processing sound. And it works, as I've shown it.

[@Theeduckie]

Objective Harmony yes. Adam also pointed out. It’s an unsolvable puzzle if it’s “natural” production of music.

[@Theeduckie]

Objective Harmony also I decided to subscribe out of frustration lol