The Mathematical Problem with Music, and How to Solve It

[@eladyaari]

i always wanted someone to explain this to me, so clearly and eloquently. great work. thanks Yuval

[@number42iscool]

pinned but no replies?

[@marcelobrunorodrigues7630]

Not only violinists, but all that play bowed string instruments.

[@RadicalCaveman]

@@number42iscool That was your way of saying, "First!"

[@number42iscool]

@@RadicalCaveman i guess!

[@mihailmilev9909]

@@RadicalCaveman haha lol

[@iamthegurr9554]

I'm a piano tuner. I've read and seen many explanations on this subject. This is the best and most intuitive description of tunings and why I've seen yet. The math concepts here have made the separate tunings including equal temperment so much more understandable. Thank you so much!

[@lawrencetaylor4101]

I had a knock at the door. It was a piano tuner. I told him that I hadn't called him. He said, your neighbors did.

[@ronniesimonet5922]

@@lawrencetaylor4101 😂

[@EdwinMcCravy1]

Do you tune temperament by electronic technology, or strictly by ear? How do you stretch treble octaves? Do you tune 4ths pure and let fifths "wow-wow"?

[@phillipshearman5597]

@@EdwinMcCravy1 By ear or with a professional tuner (not cheap one)? Either is OK but DO NOT USE EQUAL TEMPERAMENT! The "true" solution is NOT to temper the intervals and not to worry about overtones that do not affect the scale relationships. Listen to this tuning and never go back to ET again!!! This one with demonstrate just how much music is missing from our music by temper tuning the keyboard style instruments. His math is solid but does not related to acoustic instruments. As per the stretching of the octave: After both the 4ths and 5ths are pure; only the instrument can tell you how to set the octaves (both up and downwards. kzbin.info/www/bejne/bISZmoOiesqmmLs

[@dennisdougherty7538]

If you use equal temperament in piano preparation you should not call yourself a "tuner" but preferable a "tech". ET is out-of-tune and is the enemy of artistic musical expression.

[@quintonpierre]

Fun fact : In professional orchestras, musician change the tuning of each notes dynamically to get "just intonation" sounds, in particular they will slightly decrease a third and slightly increase a fifth.

[@colinbrown7947]

Not even just professionally either. My highschool tried (and often failed) but it was always a consideration.

[@Anonymous-df8it]

But what about comma pumps?

[@rogerfrood7377]

This seems like a very sweeping generalisation. I don't understand "musician change the tuning of each notes dynamically". Do you mean soloists, or ALL musicians in an orchestra? Changing ALL the notes?? Really???

[@quintonpierre]

@@rogerfrood7377 yes, they do change all the notes, there are some rules to do that, one reason is that the range of frequencies of all the instruments is incredibly spread, the fix becomes more important in that case.

[@colinbrown7947]

@@rogerfrood7377 yes :) they are all doing it. It's no generalization, you need to do it to sound the best

[@Musicrafter12]

Fun fact: violinists frequently use all three systems: pythagorean, just, and equal temperament. We tune the instrument's strings according to the perfect 3:2 pythagorean fifth and usually play single lines without accompaniment wholly in the pythagorean system, as the whole steps are wider and the half steps are narrower, giving melodic lines more direction. When playing chords, or playing with other instruments which also use pythagorean tuning (like other stringed instruments), we often will adjust certain notes to just intonation to avoid clashing. We try to avoid adjusting melodic notes this way, instead preferring to adjust only the harmony notes. When playing with equal temperament instruments like piano, if there are any long sustained notes where the intonation difference and resultant clash will be clearly noticeable, we occasionally adjust to equal temperament for just a moment to avoid this. Performance is an art of compromise!

[@richardervins]

Fun fact: Other instrumentalists do that too. Even rock musicians. I regularly detune my G string to make accompaniments using power chords sound better for instance.

[@LatchezarDimitrov]

You can't play the same notes in different places because of used system!!! This is fondamentaly WRONG to think that one note can be played differently staying still in tune! Big time to stop with math arguments from no musicians! The good tunning exists but nobody explane how to use it! Any questions?

[@srothbardt]

How about when playing cards after the concert?

[@juliafox52]

@@LatchezarDimitrov Yes. I have a question: do you play an instrument? If so, which one?

[@LatchezarDimitrov]

@@juliafox52 kzbin.info/www/bejne/h3qvnIiId66ggrs. Yes, 67 years...

[@tcheems]

Over the years, I've had several high school senior math students try to write their final math explorations on music intervals and tunings. This is exactly what I had in my mind that I wanted them to do. Sadly, they rarely came close. It's so much harder than it seems to make the math and musical terminology accessible to everyone. You do an absolutely fantastic job!

[@DeirdreMcNamara]

Just write the music! I'm going back to Pergolesi!

[@electric7487]

That, and the fact that modern music education rarely addresses any tuning other than 12 equal temperament with 440 Hz A. It also seems to treat the circle of fifths and enharmonically equivalent accidentals as necessities, when that is not the case at all.

[@EvidenceReasonsAcademy]

Outstanding. I was a music math/physics/electrical engineering student in school. This is the confluence of so much of my fields of interest. OUTSTANDING job sir. I salute you.

[@adityaagarwal636]

In Indian classical music, we have something called "Shruti", which are 22 in number. 12 of them are picked as the main notes. In some prices, some shrutis are used, giving it a different mood. A musician can pick the notes as per his compositions and needs.

[@rakeshvora8609]

Correct...u r right

[@manumusicmist]

I guess it means custom ratios for custom pieces. I'm an indian myself, but the whole raga and shruti stuff is too complicated to me. I can't even understand western scales.

[@doanahtari]

My teacher says that music is not an invention, music is a discovery. There is music in nature, in law of nature . We can understand it, see it, hear it with mathematics 🥰this is just melting my heart ridiculously🫶

[@altareggo]

Your teacher is smart!!!

[@gyrow1684]

Music is forbidden in Islam. I'm glad that I left that hateful religion which breeds blind ignorance about the world and the universe.

[@chetsenior7253]

I don’t use math to understand trees. I use love.

[@doanahtari]

@@chetsenior7253 I love math🌻 Math is good to understand and explain a harmony. i think nature has a math, because has a harmony. Probably we love trees because of some mathematical reasons like in music. Because our emotions in our brains works with math. Even though we don't know.. Sorry for my bad English 💐 I love trees too. Actually i just love trees and music in this world ahahah😇

[@doanahtari]

We can't use love, to understand trees. We just love. We just love and then try to understand with math and science. So maybe we can understand why we love. Some people don't try to understand but just love and it's ok. Like in music like in everything 🌻

[@halmessec5876]

When I was 12 years old I tried to tune our piano. It took me weeks and I finally gave up. Thanks to your discussion I now understand why. Thank you so much. Hal

[@FormantMath]

You're welcome, always at your service ;-)

[@ChrisLee-yr7tz]

Great video. Just one suggestion which I think would be really useful is to show the wave interference when the ratios don't quite work. A visualisation of the 'messy' waveforms really helps explain a lot as to why we hear the dissonance and feel it as so jarring.

[@jimbrownza]

Agreed... if I remember first year physics at uni, you get the "beats frequency" when they're a little off... (or was that something else.... long time ago ;) )

[@BobYourell]

@@jimbrownza Yes, a beat frequency results from the *interference pattern*. Fun fact: listen to one tone only in each ear and they combine in your brain to produce the same beat frequency as they do in the air.

[@timharig]

The thing is, that the waveform is NOT AT ALL "messy" to look at. When you mix frequencies together you end up with the cross-Cartesian product of all of the sums and differences of the frequencies being mixed. So, if I mix two frequencies A and B together, I end up with four different frequencies: A, B, A+B, A-B (or B-A if B should be larger). In time domain graph of the waveform, you would simply see one frequency superimposed over another similar to images depicting an AM modulated waveform. The reason that two sounds close in frequency mixed together sound "messy" has to do with how humans perceive low frequency sound. Your brain has to have some mechanism to determine what sounds are tones and which sounds are individual reoccurring events. The cut-off happens somewhere below about 50 Hz. You can imagine a mechanical buzzer or bell (like a school bell) happening every second. You mind will hear the individual ticks. As you turn up the frequency you will hear the ticks/dings getting faster and faster together. At some point as the ticks/dings get faster, your mind will stop interpreting them as individual ticks/dings, they will start to run together, and you will start hearing them as a single tone -- albeit with the kind of undulating, buzzing overtones that are generated by a square wave. The reverse happens as you decrease from a high frequency passed 50 Hz until the frequency starts to sound like individual ticks/dings again. The same thing happens with when pure tones are mixed together. When the A and B frequencies are more than 50 Hz apart, you can hear the A-B/B-A component as it's own separate tone. As you move A and B closer together the A-B/B-A component crosses the threshold, stops sounding like it's own tone, and instead starts sounding like an undulation of the A and B frequencies. If you get A and B to within 1 Hz of each other, they will sound almost like a single tone together but they will drop in and out every second as the A-B/B-A wave rises and falls. This is the origin of the "jarring" "dissonance" that you hear. It is like listening to a radio station that is rapidly fading in and out with a constant hidden "beat." That is why the sum and difference frequencies are sometimes referred to as the "beat" frequencies.

[@ChrisLee-yr7tz]

@@timharig Hence why I put the word messy in ' ' because it's not really the right terminology, just a means to an end to make a point. Compared to mixing 2 sine waves which happen to have a 'nice' ratio (notice the ' ' for the nice) you do end up something rather more 'messy' whereby what i really mean is it's periodicity can be very very different to the underlying frequencies. Thanks for the totally unnecessary lecture though.

[@fredhughes4115]

@@ChrisLee-yr7tz No more unnecessary than your totally unnecessary suggestion. Unlike all of these other comments, which of course are completely necessary. Or, you could avoid your actually unnecessary saltiness, and note that your suggestion was a good one, and Tim's little lecture was in fact informative and fascinating.

[@1ps8sa]

I've been struggling to understand musical theory for the last week, and have read or watched scores of articles/books/videos on the subject. This video is by far the best!!! Bravo and thank you.

[@RyanK-100]

Thank you for not just talking about these differences, but mostly for actually doing the math with us and showing the results. I hare read several books on the subject over the decades. I had a "flavor" of what they meant, but now I see and hear the differences. Some things can't be described by words. You need to just do them. You are a great teacher as well as a theoretician.

[@mr88cet]

Amazingly clearly-presented! Thanks. This topic I’ve been into since 1977, and it’s gloriously intriguing! A couple minor “nits” need to be mentioned though: 11:50 - Equating a _Temperament_ with a _Tuning_ is a common mistake, but in fact not quite correct! _Temperaments a subset of Tunings_ ; all Temperaments are Tunings but not all Tunings are Temperaments! A temperament is a _scheme for adjusting pitches_ from their exact-integer-ratios. So, Pythagorean and Just Intonation are Tunings, but they are *not* Temperaments, because they use exact integer ratios. Equal-Temperaments, Meantone Temperaments, and Well-Temperaments *are* temperaments. They have deliberately and systematically adjusted their pitches away from exact whole-number ratios. 18:37 - Minor Historical nit: Meantone Temperaments were much more common in the mid-late Renaissance than in the Baroque, by which time Well-Temperaments began to take over (and persisted into the mid-late 1800s, BTW - longer than most people realize).

[@FormantMath]

Thanks. As I wrote under another comment here, I now understand that I should have been a bit more careful with my use of the words "tuning" and "temperament". Cheers!

[@ValkyRiver]

@@FormantMath This is the piece "misatetarta", written in 19-tone equal temperament: kzbin.info/www/bejne/gmndnISmacqappI

[@iamthegurr9554]

I didn't realize the differentiation either. Thank you too!

[@leif1075]

@@FormantMath WHO WROTE THE MSICin thus video especially that at the very beginning before you asked what is sound..I'd love to know...thanks for shaking.

[@FormantMath]

@@leif1075 The tune is “Snowfall Butterflies” by Asher Fulero. See the description. Cheers!

[@kendebusk2540]

I played the slide trombone so I could make any note I wanted. When I play the piano, I have no choices. This was a great explanation, thanks!

[@RobertoCorralTV]

Incredibly clear and rigorous. I love how you've rigorously combined math with musical fundamentals. I had seen several videos on this topic and I never quite understood it. Now yes. Thank you very much!!

[@tennisiswonderful]

You have done a GREAT job with this video!! I don't think very many people will ever understand how many years of music theory you combined into a half hour video. This is like the "Meru" of Music/Math videos.

[@FormantMath]

Thank you so much. This one was especially nice to read, though "years of music theory" is clearly an exaggeration 😉

[@alysdexia]

@@FormantMath nice < niais < nescius := not-skilled you are: → well. I disliked this video for your saying that there are infinite tones on a string when there can only be finitely many bodies and configurations and much much fewer discerned pitches, not speaking up, and the clickbait title where you posed a solution but one that turns out old and not to solve the problem between harmony and transposition (which has no solution). I don’t know if it’s me, my Switch speaker, or the recording’s sample rate but I can’t hear the Pythagorean kord beats (which should be ~3 Hz?). little bit: pick one. I wrote this under two popular videos to revolutionize music theòry; in short, every song has a best key which is most important, so you don’t need kord notes to line up on the instrument nor to transpose as long as the representative or favorite pitches are covered: “I beg to differ: harmony isn’t everything: playing around with sine pitches on KZbin without the pesky harmonics I found my favorite pitches are 330 Hz E₄, 290 D₄, and 490 B₄ so that 10 Hz off doesn’t sound as good; there are also tristimulus loudness bumps which I found were off from the published plots, where mine are the bracketed: [4,800 D₈] 3,700 As₇, [875 A₅] 930 As₅, [2,500 Ds₇], [(1,700 Gs₆)] (1,900 As₆), [(3,300 Gs₇)], [10,600 E₉] 10,600 E₉, [(9,700 Ds₉)] (7,500 As₈); the familiar hearing modes and limits conspire to make the best key. The pitch discrimination suggests 36 steps/twofold is best. (I will not use “octave” as the 7-note scales are stupidly-lopsided and ordinals and fractions are equivocated.). Not only does that keep the important factors, as a square the staff can be simplified to three lines where the pitch is marked by a slope between the 6ths and 36ths. Length and loudness can be marked by dots on the four sides, again only in the small important factors 2, 3, 4, and maybe 6, 9; as length marks now are too fraction-heavy the base length should be on the beat. A piano with this scale gets six white keys and six black keys which can rock onto two frets.” Unless I’m deafish I forgot to say my investigation found hearing is quadristimulus not tristimulus.

[@BorisG13]

This is so well explained! And this comes from someone who almost failed math! Honestly one of the most useful videos I’ve watched on this platform.

[@randyg.7940]

Ok

[@jacobbass6437]

As a music major who used to do engineering, this is such an amazing intro to the niche rabbit hole of tuning theory. Avant guard composers of the 21st century often break the idea of 12 notes per octave (the term is 12EDO or 12 equal divisions of the octave) and pushing boundaries by writing music in tuning systems like 19EDO (19 because it has a ratio that is really close to 5/4 (important in music writing) making it pretty stable if used accordingly).

[@PragmaticAntithesis]

The 19EDO perfect 5th is actually quite a bit worse than the 12EDO 5th. The reason why 19EDO is used is because it's major 3rd is so good. 12EDO is like Pythagorean tuning, while 19EDO is like quarter comma meantone tuning.

[@jacobbass6437]

@@PragmaticAntithesis thanks, I remembered that part wrong!

[@PragmaticAntithesis]

@@jacobbass6437 You're welcome! 😊

[@ValkyRiver]

@@PragmaticAntithesis This is the piece "misatetarta", written in 19-tone equal temperament: kzbin.info/www/bejne/gmndnISmacqappI

[@Anonymous-df8it]

@@PragmaticAntithesis *minor third. *third-comma meantone

[@johngeverett]

I learned so much from this video - math, music theory, what is a 'howling fifth' (which I had heard of but never really understood), music history . . . A true plethora of knowledge, a multi-discipline smorgasbord!

[@awjaaa]

Wow. THANK you! I have this natural tendancy to struggle with things that I cannot understand the underlying reason for. You just completely opened music up for me, and I am already a musician. And, as an experimental musician w/ a tendancy towards the technical side... you just fed my imagination with enough ideas to try to play with for many YEARS to come! Liked/subscribed/bell'ed/commented, based solely on this one video. If the rest of your content is even only 1/5th (heh) as great as this, I will benefit.

[@ernestoherreralegorreta137]

I'm into Physics and Mathematics and couldn't play an instrument or sing a tune to save my life, but I've long been trying to get a good grasp of the maths behind music without any success...until today. I had concluded that most of these explanations relied crucially on some level of implicitly assumed knowledge coming from actual musical practice, but your exposition of the topic was superb in bootstrapping theory from scratch, so to speak, and kept me glued to the screen all the time. Thank you so very much!

[@sithdev8206]

This is genuinely one of the best explanations of musical temperaments I have ever seen. Amazing!

[@ericostling7410]

A great demonstration and presentation of the theory, thank you. For me, as a life-long musician who has studied this problem carefully, I see at least 3 key points must be added here in practice: 1. Using 'frequency' to discuss or understand pitch is a fairly modern concept unknown to earlier times, including Bach and Mozart.(!) It appears in the 19th century discovery and use of electronics and magnetism, a reference that we take as some 'given' now, a starting point others before that did not use at all. Ancient Greek musicians, Medieval organ and harpsichord builders, they used *very* different methods to find the right tuning for their instruments, much of which we do not know about any more. There is a lot there that is lost to us in the modern 'Hertz'-based era. 2. Similar, 'transposition' is fairly recent idea as well. It comes with the introduction of Harmony into Western music that is foreign to earlier times, during the pre-Renaissance, and seen in the development of keyboard manuals for pipe organs. There are no 'black keys' in existence before that, which is the basic tool that you clearly demonstrate we must use to 'transpose' anything to another 'key' in a music ensemble. 3. There is no evidence Pythagoras used the so-called 'Pythagorean' method of tuning attributed to him. More likely he used the Just Intonation approach which you show, especially given the perfect number ratios stored in it. It also reflects his teaching of these ratios for their deeper meaning and significance. You might know the famous legend with him involving the blacksmith hammers, which indicates he had a keen musical ear for hearing these ratios, as well as all their inner harmonics that you touch on. All without needing to use any kind of modern machinery to study them, just a monochord. Go figure...

[@jakegearhart]

29:34 The transposition of the melody here actually sounds great past the first chord. The final dominant 7th chord is more in-tune with the harmonic series, being constructed of 5/4, 3/2, and ~7/4. The preceding chord is a minor chord with a lowered minor third based on that 7/4 interval.

[@Tyrell_Corp2019]

Great video, thanks! I’ve always equated tuning systems to earths imperfect rotation and thus the need for leap-year. Calendars and tuning systems: The bane of humanity.

[@josephcoon5809]

28:00 It’s best if you play the samples BEFORE presenting an explanation, or you run the risk of psychological priming. Learning is best done when the student makes the discovery for themselves rather than being told what to think. Other than that, this was a very excellent video. Thank you for your work. Cheers 🍻

[@DaveMiller2]

It would also be better for the listener to get all the explanations first, then play the two notes without any talking between them.

[@josephcoon5809]

@@DaveMiller2 That’s the opposite of what I said. 😂 Sample. Allow the student to think for themselves. Explain. If you give the explanation first, you condition the student to seek “experts” when discovering new things instead of thinking about it for themselves.

[@DaveMiller2]

@@josephcoon5809 Sorry. I see your point. I just meant that you should say "Here are the two tones, I won't say which is which". Then play them both without talking between them. I should have explained better.

[@josephcoon5809]

@@DaveMiller2 Ahh. Then I do agree!! 😂 Cheers 🍻

[@ashtar3876]

Honestly the second sounds worse because i'm used to equal temp

[@billysoy7383]

I was always waiting for a video that combined my favourite subjects of maths and music but never thought it would be done as spectacularly as this. Thank you!

[@rogerfrood7377]

First of all, CONGRATULATIONS! This is one of the best outlines of scales, intervals and temperaments I have seen on line. Some historical perspective, just enough to explain the new demands due to harmony singing, or modulation, but not going into excessive and misleading detours that you get so often in explanations of musical scales. (The author showing off how much he knows, even if it baffles and confuses the reader.) You do well to stick to a clear and lucid account, allied to well-organised graphics. I also like the fact that you point out clearly that equal temperament is a necessary compromise and not a perfect solution.

[@angelmendez-rivera351]

Well, no. Equal temperament is not a necessary compromise. As the video mentioned, the problem can be tackled by well-temperaments, such as the Well-Tempered Clavier, or by using a different number of notes on the scale, which the video also mentioned.

[@imetr8r]

Wonderful! The only discordance I could hear was in the transposed Bach with Just Temperament. I could hear no discordance in 12th root of 2 tuning. I once used Equal Temperament to program servo motors to play melodies for an industrial trade show. It worked so well that the trade show banned "loudness" from all future shows!

[@johnchessant3012]

Fun fact: The beep that's used to censor swear words is exactly 1,000 Hz, the pitch played at 1:33. Great video btw

[@Anonymous-df8it]

He swore there.

[@astro_cat030]

@@Anonymous-df8it lol

[@sator666666]

FINALLY YT recommended me what I've been looking for for years! Great explanation! Thank you.

[@adamholtz733]

Very nice start to your new channel Bunny ! Cool collaboration with all three of you !

[@popocatap]

Thank you so much for elucidating the relationship between music and maths! I started piano lessons at seven years of age. Much later, after overcoming that extremely frustrating period during school years, I discovered the pleasure of making freestyle music with a friend and the wonders of an electronic keyboard. As I progressed, my scientifically enquiring mind and interest in physics started seeking the logic behind tonality. Now, aged seventy, I've finally gained insight into this mystery called music. Thank you so much, Yuval, for your excellent didactic approach and relaxed presentation style with absolutely clear graphics!

[@FormantMath]

Thanks! It always warms my heart to read positive comments to this video, but this one especially touched me. Glad you liked the video.

[@erikgrundy]

I don't think I've seen such an elegant explanation of these concepts. I really enjoyed your explanation of the rationale of each tuning system, and their benefits and shortcomings. I often find people become too preferential, glossing over the problems of their favorite system to make it seem better. This was a lovely video to watch, thank you!

[@ai_is_a_great_place]

This is beyond beautifully done! I have always conceded that music and rhythm are two things I'll never be able to fully grasp intuitively, but watching this video was absolutely mesmerizing!

[@rainerausdemspring894]

Equal temperament was already proposed (and probably used) on the lute in the 16th century - among others by Vincenzo Galilei, father of Galileo Galilei. The first compositions through all 24 "keys" were written by Giacomo Gorzanis in 1567. Today many lute players (Renaissance lute) use 1/6 comma meantone tuning.

[@Hrabia_von_Wpiździeszturhau]

Oh, man! You made my day! I've been waiting for that explanation for the past 50 years, since when at music school, I said to my piano teacher that violin played along the piano constantly and always sounds out of tune to me, no matter who plays and on which particular instruments (to what the teacher replied that all was good and I must be tone-deaf or something 🤣). Well, now everything makes perfect sense. Apparently, I wasn't that deaf. Thanks a million!

[@glade_maker]

Thanks for the awesome presentation, very clear and detailed, will definitely save it to show students!

[@duchesspodcast]

Never thought math could give me as strong frisson as music, but you learn something new daily. Thanks for this exceptional video. I am not a mathematician and didn’t go high in math but am now doing sound design and composition, and this is absolutely fascinating and extremely relevant.

[@FormantMath]

Glad you enjoyed the video, and thanks for teaching me a new word - "frisson".

[@Mindlagoon]

This is fascinating. I'm an electronics engineer so I understand and work with harmonics and octaves all day long but have always wondered about musical tunings and why there are "missing" black notes. I shall now be able to write my masterwork with my discrete note theramin.

[@Jantsenpr777]

Awesome video! I love that it is purely informed by mathematics and entirely devoid of biases. And the "stay tuned" pun at the end was the proverbial frosting on an already delicious cake.

[@sanketshinde1142]

This is Gold. Had a discussion with a colleague about this, I’m an absolute beginner ans Never gave it a thought how complex the physics behind all of this is! Thanks for the explanation. I’ve watched the video 3 times already.

[@FormantMath]

Thanks, my pleasure!

[@dcamron46]

This the type of stuff you pick up over years of training and practicing in music, and/or by studying the science of it- and he just clearly laid it all out in 30 minutes…

[@neilhales4693]

So, I built a guitar about 55 years ago. I knew how long my fingerboard was and so I knew where I wanted to put the fret for the first octaves (which I know as the first harmonics). On the other hand I didn't have any idea if I was going to tune the open strings to the right frequency - though I would of course tune the strings to each other - so I guess I was going to transpose just about everything until I got that right! That more or less forces me to go with 12 geometrically equal intervals and it seemed so obvious that I simply got out the log tables - yes, that long ago, big shout out to John Napier here - and worked out where to cut my fingerboard for the frets. It seemed so obvious at the time, I never thought there could be any (sensible) other way, so that's very good to know now. Top advice here: measure twice and cut once. It's easier than starting again.

[@FormantMath]

Nice story! How did the guitar come out?

[@neilhales4693]

@@FormantMath As far as I could tell, OK. I never played it in a group with other instruments but I could tune all of the open strings to stopped strings and the harmonics all worked so I think that it was a job well done. Good enough for my ear - a well-tempered guitar for a not so well-tempered ear! Remember that I calculated the position of the frets to 4 decimal places and then cut the slots in the fingerboard with a junior hacksaw and a set-square. On reflection it's amazing that it worked at all. I've still got it though I rarely play nowadays and I always preferred the tone on my commercial/bought guitar - both acoustic, of course, so the choice of wood, dimensions, and strings makes quite a difference. The fact that it was strung with steel rather than nylon made it painful to play for very long. It cost me nearly nothing and for that it was super value. I never gave any thought to whether the positioning of the frets was tricky, I was much more concerned about bending the wood for the body.

[@gleedads]

That was great! The one thing I would have liked for "ear testing" would have been to hear the major chord back-to-back in more of the temperaments. You played it in Pythagorean and just temperaments, but I would have liked to compare it in more than those. But I'm sure you must have been struggling to keep the length of the video down to something reasonable. Great job!

[@Anonymous-df8it]

Pythagorean and just triads sound the same

[@Anonymous-df8it]

@@oliverfiedler8502 I think you're just imagining it

[@VoicesofMusic]

We never use equal temperament :) Small correction: extra keys on the keyboard are not originally baroque, they began in the early renaissance, even before music printing. Zarlino dates from the 16th century and tastini--extra frets--also.

[@rishabh.malviya]

This was brilliant! A friend and I spent an entire day and figured out that ratios are the important part. We then tried figuring an intonation system of our own. In any case, you went way beyond and I loved the audio pieces we could hear and compare! Really hope you win SOME2.

[@mihailmilev9909]

@Juan Ramon Silva Parra true, u should research, but once u have done the research aren't you now building on that thousands of years anyways?

[@seenonyt2210]

@Rishabh -- That's so cool! You and your friend are so ingenious 🙌

[@desirehowland7737]

I am glad you attempted that project. This is how it was actually solved and showed how equal temperament is certainly a "Crime against nature" as it was called when it was first experienced by the musicians and musical critics when presented 200 yrs ago. I took another 100 yrs of cramming the lies down our ears to get us to a finally stop complaining about it. "Tell a lie loud enough and long enough and people will believe it to be the truth" Adolf Hitler. Here is the best explanation presented to date: kzbin.info/www/bejne/bISZmoOiesqmmLs

[@AlonElishkov]

excellent breakdown, super clear and fun to follow. Good job!

[@FormantMath]

תודה רבה, משמח מאד לשמוע!

[@karhukivi]

Very good explanation - this should be shown in schools for music students starting music theory.

[@kengit2]

This is just amazing. Opened the flood gates and really gave me a path through many obstacles I have faced in my self study. My prayers have been answered. Thank you so, so much. Be blessed

[@JonSebastianF]

This is such a good video! The logical progression of your script is super easy to follow! :D

[@DJPastaYaY]

I am a musician looking to learn music theory and this helps a lot!

[@nitsanbh]

When I was in a boys Choir, when we sang A-Cappella, we often ended up in a slightly lower key than the one we started at. I think this has to do with thirds; If kid 1 sings C, and kid 2 joins with E, the E will probably be a little bit flatter than equal temp's E. If then kid 1 goes down to A, to create a Fifth with the ongoing E, his A will be flattish too, because the E is flattish. and so on.

[@nicolasclerc1986]

Exactly, that's a dilemma for a-cappella group (take Barbershop for example) to be able to both make chords "ring" properly, while staying in the base key of the song.

[@Rene-uz3eb]

30:56 just intonation is an interesting way of filling the scale. Whereas usually you fill with harmonics as multiples above the tonic, 4 and 6 in just intonation are borrowed from a virtual tonic three times lower than the actual tonic, so the tonic is actually the third harmonic (the fifth) of the virtual tonic. Very interesting and just as justified, as it only involves basic prime numbers like the forward harmonic scale.

[@SmithnWesson]

This was eye-opening for me. Thank you much. Yes, the modern solution. Take whatever note you are playing, and tweak it slightly, to get rid of any unwanted beat frequencies with any other notes that you might be playing. So in other words, any particular note, is not actually one frequency. Your B note will shift up or down slightly, as needed, to create a more harmonious tone. Now talk about cheating. There's no way to replicate that on a simple instrument like a guitar. It all has to be done with computers.

[@simpl51]

Not just eye-opening also ear, brain and soundbox in varying amounts 😵

[@mosabosa1]

This is the best and most intuitive description of tunings and why I've seen all may life, CONGRATULATIONS MY FRIEND...

[@robharwood3538]

This was fantastically well-explained! It made everything so clear. And I love the pun at the end: "Stay tuned!" 🤓 One thing that *could* have been explored a bit more (perhaps) is the reason why small integer ratios sound more harmonious, and how that's literally connected to constructive and destructive interference in physical waves. For example, a brief audio & visual representation of how 'beats' form when two notes slightly off-tune from each other are played. In any case, thank you so much for this video. I hope you keep making more videos, as your style and approach seem to work very well for this kind of exposition. Cheers!

[@Irondragon1945]

This guy

[@jumpingjflash]

The whole video was made just so he could use the pun at the end 😃😀

[@musicsimplyexplained]

I have watched many tutorials explaining this topic. This one is the best. Great job!

[@karelknightmare6712]

A wonderful video! Thank you. It also raises the question why are we still using square and cubic roots instead of ^1/2 and ^1/3. It also gives freedom to build new scales with any number of notes we want. 😅

[@coffeeisgood102]

It’s nice to know the “why” of what I play and the history behind it. Thanks for posting.

[@oleguernogues]

Been trying to find all this information put together in a single document for 2 years now, very greatful you 've done that in this video. Great job! This is the best "engineering-minded" description I've found of how we ended up to the equal temperament. The video is well structured, and presents all the key elements that lead to the the modern equal temperament tuning. The exposition speed is also adequate (it is, not too fast) so that all the "how's" and "why's" can be absorved and understood more or less in real time. Very informative to understand an essential component of music: harmony.

[@aronsarmasi2368]

I feel like I've learned more about music just now than in all 15 years of playing music and learning about music. That was super cool! Now I want to go out and write a program to synthesize music...

[@floretion]

I think that's the most I've ever learned about music in 30 minutes.

[@aonelau5167]

I realized this problem back in college, and luckily enough to find the (partial) answer for myself. Since then I've tried to explain this to many people, unfortunately with only a few success, even to fellow musicians. I even used Wave Equation I found in my Engineering Mathematics textbook to explain why harmonic series look like the way it is....haha Now I know I can direct them to your video. It's concise, well-organized, much better than I ever could have. Thank you.

[@markslowhand4214]

Great video explaining very clearly the history and the mathematical backgroung of approaches in musical tuning, their pros and conts ans their imperfectnes. May be it's worth to also discuss the inharmonicity of string instruments because physical strings are not able to create exact 2nd, 3rd, 4th, ... harmonics. They do not create their harmonics perfectly like it's mathematical iedal model which is beeing handele by progressively increasing the higher and decreasing the lower notes along a characteristic curve when a piano tuner tunes a piano. This curves are compromises and they are different from one piano to another.

[@FormantMath]

Thanks. The list of videos I want to make includes one about inharmonicity, stretched tuning, etc. Don't know, though, how long will it take me to get there...

[@xunge]

As a guitar beginner, I strummed to play the harmonic above the 19th fret (B on the E string) to find it's just the (nearly) 1/3 length of the string, and B is the fifth note of E major. I proceeded to check the 1/2 length (the 12th fret) to find the same pitch name for 1/1 length, and guessed human brain just use the logarithm, whose base number is 2, to make the cycle. So the interval of half note is just geometrically dividing into 12 pieces, and what makes the fifth note so special is that 2^(19/12) ≈ 3. I've been looking for a music-theory book under advanced mathematics to get a further scope about harmony theory, which plays an important role in constructing chord on guitar. Thanks for affording the links below and your Manim programming is so fabulous for me to check the history of what I've known before.

[@yinq5384]

Great video! Good luck in the SoME2!

[@SteelBenderMan1]

Hello Yuval, I’m a musician and play numerous instruments. I found this video wonderfully informative. For the first time I understand different tunings. Thank you

[@mlseg5143]

great video! incredibly interesting. The pure notes are a bit loud and harsh on the ears though, maybe dampen or soften them a bit?

[@FormantMath]

Thanks! Glad you liked the video. Is there a way in youtube to change the sound level of a specific section, after upload?

[@winna101ify]

Learned a lot about how instruments are tuned and why some are tuned differently from others. Very well explained.

[@jimdruckenmiller4612]

I ALSO want to thank, Julien Basch... for 'almost' providing valuable feedback. As it stands now this video sits at the perfect balance between music and math. Had Julien's contribution been anything other than what it was. I'd be afraid to see in which direction the balance would have shifted,... and I might have been forced to dust off my old college text books in order to account for my deficiencies, and inability to follow along.

[@AlexChristaki]

I think this could actually the best video on KZbin - Perfectly put together, whoever you are, I hope you are proud.

[@FormantMath]

Thanks, though this is a massive exaggeration... Yes, I am proud, and glad you liked it. It's easy to see who I am - just look at the video, right before the acknowledgements at the end, or at the channel's "about" page. Cheers!

[@PowerhouseCell]

Beautiful video! I can't believe I just found your channel - as a video creator myself, I understand how much time this must have taken. Liked and subscribed 💛

[@flowerpt]

Excellent presentation. I knew the concepts before but never realized Bach used temperament roughness and smoothness for resolution jn addition to pitch. Neat.

[@timothywalsh866]

Best explanation of musical tuning and temperament ever

[@twloong5135]

As a non musician who has been reading around this topic for years, I just want to say Bravo! . . . encore please!

[@hadassahsoddsandends]

This was soooooo much fun! Because of my tinnitis, I couldn't always hear the differences, but some were more obvious than others. I am wondering if this is why it is so hard to learn a 12 tone scale by using each piano note, as opposed to "aiming" for the thirds, fifths, and octaves. It may also explain why, in choir, I have trouble when I try to treat a 1/2 tone drop from 6 the same as a drop from 1 to 7. It might be technically the same, but harmonically speaking isn't quite the same?? I usually have to adjust it some to "blend in".

[@phillipbradshaw]

This is fascinating! The best way to express the variety of modes available from different tunings are Harp and violin duets.

[@FadkinsDiet]

However, any instrument with a soundboard/sounding box, if multiple tones are played, will automatically improve the ratios, due to the principle of resonance. The same key on the piano can provide slightly different pitches, depending on what other notes are being played (of course this varies based on the manufacture of the piano).

[@franciscomagalhaes7457]

Just that little picture at 0:28 taught me something I've been trying to understand for years. Thank you.

[@angelmendez-rivera351]

24:43 - 24:55 A "tiny bit higher" is an overstatement. 12 perfect fifths exceed 7 octaves by exactly one Pythagorean comma, ratio 3^12/2^19. The Pythagorean comma is actually almost equal to the syntonic comma, but is actually a tiny bit larger than the syntonic comma, by exactly one schisma (I leave it as homework for the reader to find out the ratio for the schisma). 25:07 - 25:12 To be clear, equal temperament is not restricted to only 12 divisions. There are equal temperaments with a different number of divisions of the octave. An equal temperament is a well-temperament that is also a meantone temperament. If an equal temperament divides the octave into n steps, then the resulting system is called n-EDO. So, the system in the video is called 12-EDO. EDO stands for "equal divisions of the octave." 12-EDO is equivalent to a meantone temperament where the perfect fifth is tempered by about 1/11 of the syntonic comma, or 1/12 of the Pythagorean comma. 25:52 - 26:02 We need to stop using percentages, since again, this is not very illuminating. 2^(7/12) is flat from 3/2 by exactly 1/12 of a Pythagorean comma. 26:23 - 26:31 "Perfect" transposition is an exaggeration. Technically, all keys are the same key, and the only difference lies in the root frequency, which only really matters when it comes to singing, or to instruments that can only play on a specific frequency range. There is no key color. So, in fact, the meaning of transposition here is trivialized, and it amounts to nothing more than setting the concert pitch. What it does prevent is comma pump, and it allows for atonal music to thrive, whereas well-temperaments are not suited for that. Polytonal music also benefits greatly from equal temperaments. Also, equal temperaments are easier to tune in practice, it requires less effort on person in charge of tuning. 26:31 - 26:50 Again, it merits clarifying that well-temperaments had already succeeded in doing this in the late 1700s and early 1800s. 27:12 - 27:19 The major third in 12-EDO is tuned to 2^(1/3). Three stacks of a just major third exceed an octave by 128/125, a diminished second, so 128/125 = (5/4)^3/2. Therefore, 5/4 exceeds 2^(1/3), the equal tempered major third, by exactly 1/3 of the diminished second, which is almost exactly equal to 2/3 of the syntonic comma. 27:41 - 27:56 Yes, these are the well-temperaments that I thought you were not going to mention. Meantone temperaments were abandoned for well-temperaments, and unlike meantone temperaments, well-temperaments actually fully solved the problems that were present previously, all while inventing the concept of key color. Well-temperaments were used for quite some time before equal temperaments began becoming popular in the second half of the 19th century. 30:33 In Turkish music, Byzantine music, and even in Chinese music, you encounter 53-EDO and 31-EDO, both of which are approximated by just intonation much better than 12-EDO. This music is xenharmonic, rather than microtonal. Other divisions of the octave exist in other parts of the world. We can encounter 3-EDO, 5-EDO, 7-EDO, 10-ED0, 11-EDO, 15-EDO, 17-EDO, 19-EDO, 22-EDO, 27-EDO, 41-EDO, 48-EDO, and so on. Not all systems used around the world are tuned to equal divisions of the octave, either, and as I alluded to earlier, many use completely different notions of scale altogether, some not even including octave equivalence. For instance, some scales use tritave equivalence instead, based on the third harmonic, rather than the second. Some scales also change from octave to octave, as is the case in many Native American traditions, or in some African traditions.

[@Getnill]

I don't think it's necessary to compare these differences with proportionalities to other commas, though the mention of what a comma is might have been useful. For a video like this it may be more appropriate to mention JND and the relation of sizes to that. Also an equal temperament is not required to be meantone.

[@angelmendez-rivera351]

@@Getnill *I don't think it's necessary to compare these differences with proportionalities to other commas,...* I never said it is necessary, but is definitely far more illuminating to do so for viewers of this video, than it is to work using percentages and what not, which are actually meaningless without further context. The mathematical relationship between the different types of tuning systems cannot be understood without having a discussion of how the different tuning systems deal with the various. That is at the crux of the mathematics of tuning theory. You would think a video that is about the maths of tuning theory would therefore take its time to explain that. *For a video like this it may be more appropriate to mention JND and the relation of sizes to that.* Nowhere in the video is the concept of JND mentioned. Also, as I stated, none of that actually tells you anything as to why temperaments work. *Also an equal temperament is not required to be meantone.* Yes, they are required to be meantone. It is mathematically impossible for them not to be. Equal temperaments being meantone is as much of a mathematical theorem as "all squares are rectangles" is. EDIT: alternatively, if you can find one example of an equal temperament that is not a meantone temperament with respect to any commas at all, then I will concede.

[@Getnill]

@@angelmendez-rivera351 I brought up JND because the purpose of his listening examples were to exemplify how perceptible these differences were to hear which is relevant to JND. I agree commas would have been a useful tool to talk about the math of these situations but by themselves (in number form) they do not communicate anything about hearing their size. As for equal temperaments being meantone, an equal temperament just means that an interval (usually an octave) was cut into equal parts and there have been mappings of target pitches (usually JI) assigned to each note. For example the optimal mapping of 53tet is not meantone because the mapped interval of a 3/2 stacked 4 times does not modularly equal the mapping of the 5/4

[@angelmendez-rivera351]

@@Getnill *As for equal temperaments being meantone, an equal temperament just means that an interval (usually an octave) was cut into equal parts and there have been mappings of target pitches (usually JI) assigned to each note.* The mapping of target pitches assigned to each note is not a necessary characteristic of equal temperaments. It works if you are working with 12-EDO, but in xenharmonic tuning systems, there are no particular mappings to choose from. So, your statement is inaccurate. *For example, the optimal mapping of 53-EDO is not meantone because the mapped interval of 3/2 stacked 4 times does not modularly equal the mapping of 5/4.* This is completely irrelevant, as it has nothing to do with the definition of meantone temperaments. If you are focusing specifically on quarter comma meantone, then yes, a tempered perfect fifth should be octave-reduced to a tempered major third. But there are other kinds of meantone temperaments for which this stacking requirement is false. For example, third comma meantone temperament has been used in the past and was somewhat popular, but it does not satisfy the requirement you stated. As I stated in my comments somewhere, 12-EDO is a type of meantone temperament, a 1/11-comma meantone temperament. All equal temperaments form some type of meantone temperament, because every equal temperament tempers the perfect fifth by some fraction of some comma, and that fraction is almost always expressed in terms of the syntonic comma, since every comma can be expressed in terms of other commas, the syntonic comma is no exception. If an equal temperament is equal to m-EDO, and the perfect fifth is tempered to n steps in the scale, then the perfect fifth has pitch ratio 2^(n/m). So, it the difference between the just intoned and the tempered perfect fifth is equal to 3/(2·2^(n/m)), which is equal to 1/m of the comma with pitch ratio 3^m/2^(n + m), which can be rewritten in terms of the syntonic comma and some schismas. This works for all m. The point is that tempering the perfect fifth by some fraction of commas means the other fractions get redistributed throughout the scale. Because of that, you can prove there exists some number of fifths that, when stacked together, will give you some justly intonated third. That number need not be 4, and the fifths need not be ascending.

[@Getnill]

@@angelmendez-rivera351 a xenharmonic tuning system without mappings is not a temperament. A temperament requires mappings. This is often why people these days distinguish between the terminology of tet and edo because tet claims there are tempered intervals which implies that there is an untempered version of the tuning, i.e. the JI mapping you chose. 1/3 comma meantone still satisfies the mapping I explained as do all temperaments that are meantone. 1/3comma or any comma meantone is so called because it is the amount you reduce the tuning fifth by (relative to 3/2), which does not effect the mapping at all, just the exact tuning. Your fifth is still your fifth and you major third is still 4 fifths minus two octaves. A different mapping means a different set of temperaments such as schismatic temperament, magic temperament, miracle temperament, etc..

[@vanbistrow7145]

Excellent! Very clear and systematic explanations with great graphics! Thank you!

[@nomukun1138]

Another reason octaves are psychologically perceived as the same is that men's and women's voices are on average about an octave different, so if a man and a woman sing the same melody together it's extremely likely they'll end up an octave apart. The high parts of the melody will sound high in both voices, and the middle notes will be in the relaxed center range of both voices. The experience of trying to sing the same melody with someone else and both being comfortable an octave apart is something most group singers have had.

[@harrisonbergeron9119]

Excellent assessment

[@DjVortex-w]

One would think that in the era of digital music they could devise some kind of "dynamic temperament", where the tuning of each note changes to be optimal for the current key.

[@cmyk8964]

That is what orchestras do but with a brain instead of a computer

[@bobaldo2339]

@@cmyk8964 I believe that is essentially what happens in certain genres of folk music, especially when only one or 2 keys dominate the music. .

[@jonathanosborn4800]

With digital instruments, i like to change tuning as the harmony changes, just have them saved and trigger the next patch as the harmony changes, or at least as the tonal centers changes.

[@noelwalterso2]

A notable fact about equal temperament is that the tritone is equal to the square root of two.

[@Zaphod313]

Yes! While the perfect fifth is the arithmetic mean of the octave, the 12-TET tritone is its geometric mean.

[@pocojoyo]

Maybe Ive watched too many videos on the subject but this may be the definitive video on the subject.

[@MarcoComercibjt2]

Not many DAWs support temperaments. I have developed a VST plugin and released for free on the internet, that does support them (it's programmable and can do many more things. It's like Sforzando, but more flexible). I have published it in a musician board, that AFAIK allows the download also to not members. If interested, I can give the link.

[@campbellsmith9849]

yes please!

[@Zawiedek]

I love this topic so much! Many think it is complicated, mysterious, arcane knowledge, but it is totally logical and - as you show here - mathematically precise.

[@heraclitusofephesus8749]

Well, it looks like you already had a commenter who nitpicked this video to death. Sorry -- this in general was a great intro to the topic, and many of the places where you glossed over some details, I assumed you were doing so to make this approachable (as that's kind of the point of SoME). Two basic nomenclature things I would point out, though, because they are somewhat non-standard and will make some viewers wince. At one point you say you will use the terms "tuning" and "intonation" and temperament" interchangeably. But these terms actually refer to distinct things -- things you're actually trying to distinguish a bit in this video! While the term "just intonation" today sometimes gets associated with the specific scale you mention (a scale that is mostly associated with a recommendation by Zarlino, a 16th century music theorist -- it's not really a medieval thing), that's not what the term "just intonation" means to tuning theorists in general. "Just intonation" is merely a term for ANY tuning system where all of the notes are tuned to exact mathematical integer ratios. A "just interval" is an interval tuned to an exact whole-number ratio. Thus, your "Pythagorean tuning" is also an example of another kind of "just intonation." And there are literally hundreds of other possible scale tunings that are "just." Meanwhile, a "temperament" is a scale that has some TEMPERED intervals. To "temper" an interval is to adjust it slightly away from the whole-number ratio. So, "meantone temperament" is called that because a note isn't tuned to 9/8 or to 10/9 (both exact ratios), but instead "tempered" to be some sort of approximate interval that doesn't correspond to a whole-number ratio. "Just intonation" is thus the opposite of "temperament" in a way. The former refers to scalar systems all tuned to precise whole-number ratios, while the latter refers to tuning systems with at least some notes deliberately tuned away from whole-number ratios. In your case, you discuss situations which have irrational ratios (with square roots, etc.), but historically these irrational ratios were considered harder to locate precisely compared to whole-number ratios (which could be located just with simply geometric division, as on a monochord). I know these terms are getting diluted and used in less precise ways these days, even sometimes in academic literature. But to some of us who are familiar with these terms and their exceptionally long history, calling your "just intonation" scale a "temperament" sounds way off. Just systems can be called "tuning systems" or "tunings," but a "temperament" should contain at least one "tempered" interval/note. Otherwise, thank you for a nice introduction to the content with some great visualizations and audio examples.

[@FormantMath]

Thank you for your comments. Yes, I now understand that I should have been a bit more careful with my wording. Next time it will be better 😉

[@christopherhorstmann5733]

That's a lot of work you put into this video. Very impressive. Thank you for this! Very well explained.

[@CasualGraph]

29:24 "But the just intonation version which is played on a piano tuned to the original key sounds absolutely horrific..." no? it's a little more wobbly but it's still got a nice warmth to it

[@FormantMath]

Well, that's a matter of personal taste. If you found it pleasant, enjoy!

[@eriktempelman2097]

Pianos, like all steel stringed instruments, actually do NOT have perfect multiples in their overtones. This gives that tell-tale sound. It is due to the high stiffness of the material; nylon strings have it much, much less. Read "The Acoustic Foundations of Music" by Backus.

[@FormantMath]

@@eriktempelman2097 That's right. You can read in the video description how I tried to minimize string inharmonicity in the sound samples

[@electric7487]

30:32 A lot of these alternative EDO's are simply standardisations of historical tunings. 1/3-comma meantone = 19 EDO 1/4-comma meantone = 31 EDO 1/6-comma meantone = 55 EDO Pythagorean tuning = 53 EDO

[@johnvonachen1672]

When multiple singers sing without accompaniment they listen while singing and produce better harmony, if the singers don’t have any problems with pitch, especially if singing without vibrato. If you could make computer controlled instruments that “listen” while playing, that would be the solution, not perfect, but better than the good enough we all use now.

[@chrislynch1235]

A book I was reading recently described the best key for a particular piece of music. I was thinking, “what difference does it make? “, But then I remembered: different keys sounded different, back then. Next thing you know, I’m listening to exactly what I wanted to hear:something played in a horrific key. Well done!

[@alexsvetlev1646]

"Stay tuned"... I see what you did there!

[@Mas1337]

Back at college we had subject "Math outside of Math" and each student needed to write a seminar and give presentation to other students. If only this video exited then to motivate me for for topic. Great video combining my two passions.

[@fatalnyl655]

how'd u animate this?

[@FormantMath]

I am too embarrassed to answer this question honestly...😳

[@beltanewalk8797]

@@FormantMath LOL, brilliant answer. I'm going to guess you did it it with Powerpoint.

[@FormantMath]

@@beltanewalk8797 I take the fifth 😉

[@Alexander_Sannikov]

this is exactly the video I was searching for a long time. there's many sources that contain some information on the exact physics of how temperaments work, but this is the most substantial video on the subject.