The Mathematical Problem with Music, and How to Solve It 

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

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.

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!

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!

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!

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🫶

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.

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.

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.

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

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).

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!!

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

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!

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

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.

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).

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.

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

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.

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 🍻

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!

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!

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.

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.

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

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!

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!

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!

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.

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!

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.

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!

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.

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!

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

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

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…

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 💛

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

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 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.

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!

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.

Awesome video! I was always interested in this, but never knew how complicated it can get. Thanks for making it!

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.

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.

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

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.

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.

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

I love these grandiose titles for youtube videos that are designed with one purpose: to make you click, if not watch.

This is exactly the explanation I've been seeking for many years.

Best explanation of musical tuning and temperament ever

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.

A video worthy of bookmark and rewatch again and again. Thank you.

Thanks for this great video. You explain these complex concepts in such a clear and effective way!

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. 😅

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....

The best exposition of this topic I have ever seen. Congratulations.

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.

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".

The best explanation I’ve seen on RU-vid. Well done!

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.

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).

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...

This is the best explanation of this subject I have ever seen. Well done and thank you

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.

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.

Such an interesting and well explained video! It's so much fun understanding this kind of stuff.

What a precious channel! Thank you for all that!

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.

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

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

I'm so relieved to finally understand this. Thanks so much

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.

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.

Amazing. Concise and informative! I enjoyed every second of your work, good sir.

Fantastic tutorial. Finally all this makes sense!! Regards from Baltimore.

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??

One of the best RU-vid videos I’ve seen in a long time.

GREAT - FIRST EXPLANATION IN SO MANY YEARS I UNDERSTOOD IT SO CLEARLY

"Stay tuned" 🤣

how'd u animate this?

Awesome material you produced here! Just awesome!

this is honestly one of my favorite youtube videos ever

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

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.

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

Great explanation! Clear, plausible and inspiring! Many thanks!

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.

Europeans: when saying a number after a decimal point, (for example, 1.25) we don't say "one point twenty five", we say "one point two five". Twenty implies there's a "tens column".

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

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

5:35 - 5:50 This is all true, but I do want to remark this is not universal, and to what extent this holds varies from culture to culture. Keep in mind that 99% of scientific studies that have been conducted on the subject matter have been conducted to study only the musical perception of Westerners: in particular, North Americans and West Europeans. There are no reliable grand scale studies out there studying phenomena of musical perception globally, they are all confined to studying certain Western musical traditions and perceptions. On the other hand, the few case studies that have been performed for non-Western traditions show that musical perception varies drastically from culture to culture. For example, in South Africa, there are instruments played with scales that do not have octave equivalence. So, do keep in mind that whenever something is presented as a psychological phenomenon in musical perception, we can be only certain it holds for Western musical traditions, not necessarily anywhere else. 11:19 - 11:30 This is often presented as a serious practical challenge, but in actuality, you are never required to design a piano that can play indefinitely many perfect fifths above or below the middle octave, since the number of octaves a piano can span is finite. Also, the video mentioned several individuals from the 1600s working on this problem, but technically, this is not historically accurate, since in the 1600s, the tuning system described, which is based on only perfect fifths, called Pythagorean tuning was not in use. Instead, they used a more complex tuning system, called Ptolemic tuning, in which intervals were generated by the perfect fifth, as well as the major third, and this is because in plainchant, tertial harmony eventually became preferred over quartal harmony. The major third is in the interval corresponding to the pitch ratio 5/4. Ptolemic tuning was the tuning that mathematicians, scientists, and music theorists from the 1600s were trying to fix, not Pythagorean tuning, which had long been replaced. The video alludes to this a bit later. In fact, even in Ancient Greece, Pythagorean tuning was the first, but not most widely used tuning system. Archytas popularized using septimal tuning, and Ptolemy popularized using major thirds. 11:37 - 11:46 We need to be more careful here. We need to distinguish between the concepts of "scale" and "tuning," which most people tend to conflate, and I think the phrasing used by the video here does not help. A scale is a scheme for dividing a diapason (usually chosen to be the octave in Western music theory, since we operate with octave-equivalence) into a number of steps or scale degrees, each step of some given size in relation to the other steps. A tuning system is a system that provides you with the possible frequency ratios that a scale degree from an already existing scale may be tuned to. Also, "temperament" and "tuning" are not synonymous. A temperament, by definition, is any tuning system that deviates from just intonation. 14:33 - 14:54 This is not quite accurate. Pythagorean tuning is an example of just intonation: specifically, it is a 3-prime limit intonation. The new tuning being described now is what I have been calling Ptolemic tuning, which in modern tuning theory, is called 5-prime limit intonation. Both are examples of just intonation, restricted to the lower harmonics. p-prime limit intonation is a tuning system in which we find all the harmonics of a frequency, and we only keep the ones that are prime numbers, and that are smaller than or equal to p. Pythagorean tuning is 3-prime limit, because we use the harmonics 2 and 3, the only prime numbers smaller than or equal to 3. Specifically, we use the harmonics and octave reduce them, so we have the intervals 2 and 3/2. Ptolemic tuning is 5-prime limit, because we use the harmonics 2, 3, and 5, the only prime numbers smaller than or equal to 5. Specifically, we use the octave-reduced harmonics, giving us the intervals 2, 3/2, and 5/4. All other intervals being generated are combinations of these three intervals, just as how, in Pythagorean tuning, all intervals being generated are combinations of 2 and 3/2. Septimal tuning is what we would call 7-prime limit intonation, yet another example of just intonation. Just intonation in general is just the tuning system by which all intervals are ratios of harmonics. This does not produce a scale, all it produces is the family of intervals you can tune the scale degrees of a scale to.