The Mathematical Problem with Music, and How to Solve It
There is a serious mathematical problem with the tuning of musical instruments. A problem that even Galileo, Newton, and Euler tried to solve. This video is about this problem and about some of the ways to tackle it. It starts from the basic physics of sound, proves mathematically why some musical instruments can never be perfectly in tune, and then introduces the main solutions that were proposed to solve this problem, along with their upsides and downsides: Pythagorean tuning, Just intonation, the Meantone temperament, and finally - the equal temperament, which is the tuning system almost everybody uses today in the West.
=== Further Resources ===
▶ To learn more about the connections between music and mathematics, I highly recommend the book “Music - A Mathematical Offering” by David Benson. This terrific book is a treasure trove of information, extremely well written, and its thorough discussion of temperaments is just one of the many topics it covers. The book can be downloaded legally and for free from here: homepages.abdn.ac.uk/d.j.bens...
▶ Sevish is a master of electronic microtonal music. His compositions, despite their ominous genre, sound light and fun. Check him out. sevish.com
▶ Paul Davids explaining how and why John Frusciante (of the Red Hot Chili Peppers) “mistuned” his guitar in the song “Scar Tissue”. • Why John Frusciante is...
▶ A Madrigal by Nicola Vicentino (1555), played on a 24-tone harpsichord tuned in meantone temperament, by Johannes Keller. • Nicola Vicentino: Musi...
▶ A concise introduction to Arabic music. Pay attention especially to the Albayati, Alsaba, Alsard, and Ahuzaam maqams, with their intense microtonality. • Western Music VS Arabi...
▶ The Lumatone Isomorphic Keyboard is a cool interface to microtonal music. • Introducing: LUMATONE ...
=== Thanks to ===
▶ Yehezkel Raz, the Ableton wizard, for transforming me from a complete Ableton noob to a good-enough user in less than two hours. yehezkelraz.com
▶ Dina Lurie, a dear neighbor and a great violinist (in the Irish fiddle tradition!), for contributing two notes and one double glissando. / dina.lurie
▶ Alon Schab, a musicologist, and also a friend, bandmate, and academic colleague, for advice on some musical and historical issues. haifa.academia.edu/AlonSchab
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▶ The intro and outro music is “Snowfall Butterflies” by Asher Fulero (via KZhead Audio Library). asherfulero.com
▶ The flute and Qanun sound samples are from FreeSound. freesound.org
▶ Photo of the Antegnati Organ in Santa Barbara, Mantua (1565), courtesy of the organist Simon Lloyd. simon-lloyd.com
=== Contents ===
00:00 - Intro
00:44 - What is sound?
02:42 - Melodies
04:48 - Intervals
07:00 - Choosing frequencies
11:56 - Pythagorean Tuning
14:33 - Just Intonation
18:36 - Meantone Temperament
24:34 - Equal Temperament
29:50 - Other temperaments
31:09 - Outro
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The sound samples were prepared with Ableton Live 11. To make them piano-like but still as accurate as possible, I used the physical-modelling Pianoteq plugin, with the unison width set to 0, octave stretching ratio set to 1, string length set to its maximum value (to minimize string inharmonicity), hammer noise set to 0.5, pedal noise set to 0, and the velocity-to-dynamics curve considerably lowered (ending at mezzo-piano).
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Created by Yuval Nov for the 2022 "Summer of Math Exposition" (SoME2) competition, hosted by the one and only 3Blue1Brown (Grant Sanderson).
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#math #music #temperament #tuning #Pythagorean #meantone #3b1b #SoME2
i always wanted someone to explain this to me, so clearly and eloquently. great work. thanks Yuval
pinned but no replies?
Not only violinists, but all that play bowed string instruments.
@@number42iscool That was your way of saying, "First!"
@@NuisanceMan i guess!
@@NuisanceMan haha lol
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.
Not even just professionally either. My highschool tried (and often failed) but it was always a consideration.
But what about comma pumps?
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???
@@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.
@@rogerfrood7377 yes :) they are all doing it. It's no generalization, you need to do it to sound the best
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🫶
Your teacher is smart!!!
Music is forbidden in Islam. I'm glad that I left that hateful religion which breeds blind ignorance about the world and the universe.
I don’t use math to understand trees. I use love.
@@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😇
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 🌻
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!
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.
@@lawrencetaylor4101 😂
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"?
@@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. kzhead.info/sun/abdvmbSkfJeplZE/bejne.html
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.
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!
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.
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?
How about when playing cards after the concert?
@@LatchezarDimitrov Yes. I have a question: do you play an instrument? If so, which one?
@@juliafox52 kzhead.info/sun/hK2Fm7mKeXujf5E/bejne.html. Yes, 67 years...
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!
Just write the music! I'm going back to Pergolesi!
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.
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
He swore there.
@@Anonymous-df8it lol
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.
Correct...u r right
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.
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.
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 ;) )
@@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.
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.
@@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.
@@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.
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.
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
You're welcome, always at your service ;-)
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.
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).
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.
@@PragmaticAntithesis thanks, I remembered that part wrong!
@@jacobbass6437 You're welcome! 😊
@@PragmaticAntithesis This is the piece "misatetarta", written in 19-tone equal temperament: kzhead.info/sun/f5yzm7Woa5edo2g/bejne.html
@@PragmaticAntithesis *minor third. *third-comma meantone
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).
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!
@@FormantMath This is the piece "misatetarta", written in 19-tone equal temperament: kzhead.info/sun/f5yzm7Woa5edo2g/bejne.html
I didn't realize the differentiation either. Thank you too!
@@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.
@@leif1075 The tune is “Snowfall Butterflies” by Asher Fulero. See the description. Cheers!
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!
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!!
I think that's the most I've ever learned about music in 30 minutes.
This is genuinely one of the best explanations of musical temperaments I have ever seen. Amazing!
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!
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!
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 🍻
It would also be better for the listener to get all the explanations first, then play the two notes without any talking between them.
@@davemiller6055 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.
@@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.
@@davemiller6055 Ahh. Then I do agree!! 😂 Cheers 🍻
Honestly the second sounds worse because i'm used to equal temp
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.
Thank you so much. This one was especially nice to read, though "years of music theory" is clearly an exaggeration 😉
@@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 KZhead 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.
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!
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.
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!
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.
Ok
Here's my take. There is no reason to make it back to 440Hz in this equation since the Pythagorean temperament method keeps shifting the reference point. Each time you go back down an octave, you have lost your relativity. So the equation is bogus due to this continual shifting relativity since we are dealing with ratios, not amounts. Musical notes are relative, not absolute. It's just like the vanishing point in 3d vision. So all we have to do is divide the octave by the 12th root of 2 until we double the frequency. The intervals are thereby perfectly relatively spaced, and all transpositioning remains perfect. Also, how to tune without aids? By detecting and reducing beat frequencies. Every two notes return beat harmonics. This type of tuning always lands on the equal temperament.
I already heard most of the information in this video at some point, but having it all condensed here into such a clear and well-illustrated way was great. Thanks for this video!
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…
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!
Thanks! It always warms my heart to read positive comments to this video, but this one especially touched me. Glad you liked the video.
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.
Awesome video! I was always interested in this, but never knew how complicated it can get. Thanks for making it!
This is such a good video! The logical progression of your script is super easy to follow! :D
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!
This is the best and most intuitive description of tunings and why I've seen all may life, CONGRATULATIONS MY FRIEND...
I have watched many tutorials explaining this topic. This one is the best. Great job!
Very good explanation - this should be shown in schools for music students starting music theory.
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.
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.
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.
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.
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.
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 💛
Learned a lot about how instruments are tuned and why some are tuned differently from others. Very well explained.
It’s nice to know the “why” of what I play and the history behind it. Thanks for posting.
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.
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.
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!
This is exactly the explanation I've been seeking for many years.
A video worthy of bookmark and rewatch again and again. Thank you.
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!
Pythagorean and just triads sound the same
@@oliverfiedler8502 I think you're just imagining it
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.
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.
Thanks for this great video. You explain these complex concepts in such a clear and effective way!
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.
Best explanation of musical tuning and temperament ever
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!
High quality work! Congratulations and keep it coming. When I was younger I had to deduct myself all the mathematical logic behind tuning, in the absence of relevant material that could help me. What an effort it was and it could have been so much simpler if this video was available at the time....
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.
Glad you enjoyed the video, and thanks for teaching me a new word - "frisson".
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.
What an outstanding presentation! Clearly and concisely explained, with elegant scripting and visuals. Thank you.
Excellent! Very clear and systematic explanations with great graphics! Thank you!
Great video! Good luck in the SoME2!
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. 😅
That's a lot of work you put into this video. Very impressive. Thank you for this! Very well explained.
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.
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.
@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?
@Rishabh -- That's so cool! You and your friend are so ingenious 🙌
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: kzhead.info/sun/abdvmbSkfJeplZE/bejne.html
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".
Amazing. Concise and informative! I enjoyed every second of your work, good sir.
thank you for this video, I was searching someone that can explain this in a simple way and you done it in a very clear and understandable way.
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.
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!
This guy
The whole video was made just so he could use the pun at the end 😃😀
The best explanation I’ve seen on KZhead. Well done!
Such an interesting and well explained video! It's so much fun understanding this kind of stuff.
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.
Not just eye-opening also ear, brain and soundbox in varying amounts 😵
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.
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...
This was extremely well explained and presented!
FINALLY YT recommended me what I've been looking for for years! Great explanation! Thank you.
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).
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.
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 😉
This video was VERY well assembled and presented, I really enjoyed the exp between the Oct & 5th intervals and their key roles in how our note divisions are constructed, that meant a lot to me.
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.
Nice story! How did the guitar come out?
@@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.
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.
That is what orchestras do but with a brain instead of a computer
@@cmyk8964 I believe that is essentially what happens in certain genres of folk music, especially when only one or 2 keys dominate the music. .
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.
I love these grandiose titles for youtube videos that are designed with one purpose: to make you click, if not watch.
That was excellent, and thanks for the helpful links. I just downloaded Benson's book. I'll be thinking of more math now when I tune my guitar!
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.
great video! incredibly interesting. The pure notes are a bit loud and harsh on the ears though, maybe dampen or soften them a bit?
Thanks! Glad you liked the video. Is there a way in youtube to change the sound level of a specific section, after upload?
I am not musical educated, but was always interested in how the tones "belong" together! A very precise explanation in this video, although difficult to grab at one time.
At last, an explanation I can understand! Wonderfully done. Thank you. 🏆
"Stay tuned" 🤣
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.
yes please!
The best exposition of this topic I have ever seen. Congratulations.
As a non musician who has been reading around this topic for years, I just want to say Bravo! . . . encore please!
The things that amaze me are, first, that we maintain the horrific notational overhead of key signatures after moving to a tuning theory with perfect transposition; and, second, that with electronic music and (argh) autotune available, technologies which allow the use of arbitrary precise intervals, we have become _more_ insistent on 12EDO. We truly hate ourselves. :) (Of course I speak of general and popular musical culture, not tuning wonks, who do, thankfully, also exist.)
And how is the problem solved??
this is honestly one of my favorite youtube videos ever
A wonderful, masterfull explanation of something I've been benefitting from all this time and never even knew. Thank you! And kudos to to the comments section as well...seldom do I find myself enjoying such civil sophistication and elaboration as what I found here.
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
Well, that's a matter of personal taste. If you found it pleasant, enjoy!
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.
@@eriktempelman2097 That's right. You can read in the video description how I tried to minimize string inharmonicity in the sound samples
how'd u animate this?
I am too embarrassed to answer this question honestly...😳
@@FormantMath LOL, brilliant answer. I'm going to guess you did it it with Powerpoint.
@@beltanewalk8797 I take the fifth 😉
What a precious channel! Thank you for all that!
I’ve been thinking about how an optimal MOS scale would have a perfect major third, but I didn’t realize that was meantone! Because I’ve been thinking about it, I can say an interesting fact. When you stack 31 fifths like this, you end up at a note almost exactly the same as your starting note. It’s only 6 cents flat! If you make each fifth just slightly sharper (like 0.2 cents sharper), you got another equal tempered scale, 31-tet! The next scale that’s closer to this perfect MOS (with a bias towards being lower bc it gives us a better minor third) is 81-tet, which is a lot more notes for very little improvement.
And to avoid the 'wolf fifth' you can flatten the octave.
@@Anonymous-df8it XD noooo
This is the piece "misatetarta", written in 19-tone equal temperament: kzhead.info/sun/f5yzm7Woa5edo2g/bejne.html