The best explanation that I have found so far. I watched tens of videos. Many of them mentioned 3/2, but it was not clear where 3/4 come from and why there were only 7 notes. Everyone is focused on twelve. Your drawing is awesome and helps to understand it perfectly.
What an excellent video!! I've crawled around any number of learned explanations of tuning systems, becoming more confused as I went, and you managed to explain it clearly in just a few minutes. Well done!!
thank you so much for this demonstration, by far the best video on this, ive trawled through a number of resources still slightly puzzled but you've completely solved it for me!
I have been so confused with tuning but you have explained it so well. Thanks so much. Eagerly awaiting the next video re issues with Pythagorean tuning
This is a very nice demonstration, especially on the guitar. One can read many mathematical treatments, but seeing this demonstrated on a stringed instrument, just as the "Pythagoreans" might have, gives one much greater insight. Sometimes visualization provides just the right touch.
These videos are the best explanations to the tuning issues I have ever come across. Could you do a video about just intonation and/or other tuning systems? Explain the wolf intervals? Beating (interaction between close harmonics), etc?
I've been trying to find resources on Pythagorean tuning, and this was by far the clearest and easiest to follow video that I've found. Thanks for your generosity!!
Some confusion is introduced by measuring the non-played distance, i.e. the 32/21 does give 3/2 but _this_ 3/2 is not the 3/2 that is shown in the diagram. The 3/2 that is shown in the diagram is 64/43. This confusion is further amplified by showing the vibrating part of the string at the opposite end.
: The Reason why the fifth he found on guitar is not perfectly 1.5 is because, for well-tempered system, which divide an octave to equally 12 parts, is actually “out of tune” for every single note according to Pythagorean system. (That’s why perfect pitch today to me is fundamentally wrong!)
Hey, I really enjoyed this video! It would be so helpful if you posted a video on just intonation. Your explanations are very concise and easy to understand and this tuning system is definitely something I've been having trouble with. Let me know!!
My inquiry began in traditional chinese scales and have been going down the rabbit hole. I still have so many questions, I'm looking forward to more videos about the topic! 😊
I just started learning piano. At first I could not understand why F needs to start from C instead of the following the calculation of the others. Then I realised all the other keys involved 1/2 steps. So starting from B down to F does not have a half stop, but going down to bass F and then go up an octave does.
This was amazing! I ended up here after watching "The Concise History of Chinese Musical Temperament - Episode 1: Formation of Tones" and found out an Ancient Chinese emperor figured out a scale called Sanfen Sunyi that does "Divide by 3, subtract that, and add a third of the new". The author compared it to the Pythagorean tuning system, and since I was curious I ended up here. I'd love to hear your thoughts on Sanfen Sunyi vs Pythagorean tuning. I really enjoyed this video, thank you again.
Very interesting- it would be very valuable to me and other guitar players if you could discuss the possibilities and difficulties of tuning a guitar, bearing in mind that consonant, or pure intervals are the most desirable, and of course that they are limited. A player has the tempered scale to impede his progress, and also the fact that, due to the inclination of the string, 2 g notes on one string will be different pitches, two A notes on one string will also be different. The phenomenon of resonance and it's value to a tuner would also be very interesting to read- thanks
Hello. Please can you tell me why, at 11:57, you descend by a fifth rather than rising by a fifth? I appreciate that not doing this would produce an F sharp - which would sound wrong - but it makes the sequence a bit inconsistent - like the fundamental note is really F for the subdominant.
I have the same question (I think!). Why break away from the formula followed thus far? Why go up a fourth from C rather than up a fifth (and back an octave if necessary) from the last note? I notice that you posted this comment a year ago. Have you found an explanation since then? If so, I'd be very grateful if you could share it with me! :)
David I think you might need to add a video explaining how the frequency manipulation of going up/down intervals in the second half of the video is related inversely to the string lengths of the notes in the first part of the video. Also need to explain why constructing this scale using fifths produces notes that sound “consonant” together to create chords. I’m sure these are things you will eventually cover in future videos.
*I think you might need to add a video explaining how the frequency manipulation of going up/down intervals in the second half of the video is related inversely to the string lengths of the notes in the first part of the video.* All sound waves you produce with an instrument travel at the same speed: the speed of sound, which is approximately 343 m·Hz. Since the speed is a non-changing quantity, a constant, there must be some mathematical relationship describing why, and there is. The mathematical relationship is v = λ·f. v is the speed of the sound waves. f is the frequency of the sound waves, and λ is the length of the string producing the sound waves. Since v is a constant, λ and f are inversely proportional. *Also, need to explain why constructing this scale using fifths produces notes that sound "consonant" together to create chords.* Modern chords in Pythagorean tuning are not consonant. The perfect unison, the perfect octave, the perfect fifth, the perfect fourth, the major second, and the minor seventh, are the only consonant intervals in Pythagorean tuning. Notably, the Pythagorean thirds and sixths are more dissonant than their juster, Ptolemic counterparts. The reason Pythagorean tuning sounded well to the Pythagoreans is because they used quartal harmony (if any), not tertial harmany as we did in medieval times and later. The reason the perfect fifth is such a perfect consonance is because 3/2 is the simplest ratio of frequencies after 2, and 4/3 is merely its octave complement.
Thank you very much for this very detailed explanation. I unfortunately lose you when you explain the last calculation: how to arrive at 348.8 Hz from 496.7 Hz. Could you decompose it for me please? Thank you
Don't know much about Pythagorean tuning but if we take the ratio of natural 2nd at 10/9 instead of 9/8, it sounds more pleasant with the first 1st as well as the 4th if the 4th is at 4/3. But again it would be dissonant with rest other notes so the other notes have to be re adjusted. Interestingly if you go by this tuning, you can play all the notes from the Lydian scale at once and it won't sound dissonant.
Pythagorean tuning, as explained in the video (which I assume you did not watch, based on your comment), is a tuning system where all musical intervals allowed are constructed from only perfect octaves (ratio 2) and perfect fifths (ratio 3/2). The ratio 10/9 cannot be constructed from only perfect octaves and perfect fifths, so its corresponding interval does not exist in Pythagorean tuning. It does exist, however, in Ptolemic tuning.
@@angelmendez-rivera351 I knew that 10/9 2nd does not occur in Pythagorean tuning but did not know there exist another tuning that has that. Thanks for the info I'll look it up. But if you can experiment a bit, I would suggest do try mixing 10/9 4/3 and 5/3, they go well with each other. 10/9 and 3/2 are dissonant.
@@siddheshdeshpande7183 The interval between a perfect fifth (3/2) and a major second (10/9) has a ratio (3/2)/(10/9) = 27/20, and so the interval is an acute fourth. You obtain the same interval if you add a syntonic comma (81/80), which I call an acute unison, to a perfect fourth (4/3), since 4/3·81/80 = 27/20. The acute fourth is a dissonant interval indeed.
@@angelmendez-rivera351 I am relieved that you agree that 10/9 and 3/2 are dissonant, I came to know this through my teacher and of course through the experience of listening. Don't really understand the terms guess I have to do a bit more research, appreciate the mathematical explanation though. Cheers!
@@siddheshdeshpande7183 Ptolemic tuning is a system of just intonation where you only use three intervals to construct all the other tuning intervals. The three intervals you use are the perfect octave (ratio 2), the perfect fifth (ratio 3/2), and the major third (ratio 5/4). This system has some peculiar things you have to be very careful about, though. The interval between the perfect fifth and the perfect fourth (ratio 4/3) is the interval with ratio 9/8. However, the interval between the perfect fourth and the minor third (ratio 6/5) is the interval with ratio 10/9. Both of these intervals are seconds, and furthermore, it appears both of these should be called the major second. Yet they are different intervals, the difference between them is the acute unison (ratio 81/80), which while very small, is still audible, and can make certain intervals sound out of tune. How do we know which one is truly the major second? We think of a major second as being half of a major third. For example, in 12-EDO, a major second is equal to 2 steps of 100 cents, and the major third is equal to 4 steps of 100 cents. So which of 9/8 and 10/9 is half of a major third? It turns out, both of them, because 9/8·10/9 = 5/4. In fact, if you want to tune the C major diatonic scale using just intonation, then you need both 9/8 and 10/9 to do so. Since the major sixth is tuned to the ratio 5/3 and the major seventh is tuned to the ratio 15/8, we obtain a scale with degrees tuned to 1, 10/9 or 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2. The steps sizes, then, must be 10/9 or 9/8, 9/8 or 10/9 (respectively), 5/4, 16/15, 9/8, 10/9, 9/8, 16/15. There is no possible way to have an equivalent scale with only 2 step sizes. You can decompose every interval that is not a second into only seconds. For example, a major third is an interval of 9/8 plus an interval of 10/9, in any order. A perfect fourth is an interval 10/9 plus one of 16/15 plus one of 9/8, in some order. Why does this matter? Because it severely affects how you combine intervals together into chords, and how you stack them, and it also affects key changes. For example, a perfect fourth has 1 of 9/8, 1 of 10/9, and 1 of 16/15. If, instead, you play 2 of 9/8, and 1 of 16/15, you still have played two major seconds and one minor second, but instead, you now have an acute fourth of 27/20, rather than a perfect fourth of 4/3. It turns out that this also affects the chromatic scale. In the chromatic scale, the scale degrees are the perfect unison (1), the minor second (16/15 ?), the major second (10/9 or 9/8), the minor third (6/5), the major third (5/4), the perfect fourth (4/3), the tritone (25/18 or 45/32 or 64/45 or 36/25), the perfect fifth (3/2), the minor sixth (8/5), the major sixth (5/3), the minor seventh (16/9 or 9/5), the major seventh (15/8), the perfect octave (2). This results in chromatic steps 16/15, 25/24 or 135/108, 27/25 or 16/15 (respectively), 25/24, 16/15, 25/24 or 135/108 or 16/15 or 27/25, 27/25 or 16/15 or 135/108 or 25/24 (respectively), 16/15, 25/24, 16/15 or 27/25, 135/108 or 25/24 (respectively), 16/15. So there are 4 different chromatic steps sizes! The difference between the intervals of ratios 27/25 and 16/15 is 81/80, and the difference between the intervals of ratios 135/108 and 25/24 is also 81/80, the syntonic comma, or acute unison as I call it. Furthermore, the difference between the intervals of ratio 16/15 and 135/108 is this interval with ratio 2048/2025, called a diaschisma, which is approximately equal to a syntonic comma (the difference between the 2 is inaudible). So the smallest chromatic step is 25/24, the second smallest is that plus a syntonic comma, the second largest is the smallest plus approximately 2 syntonic commas, and the largest is the smallest plus approximately 3 syntonic commas. This is suspicious, is it not? The solution to this, quite frankly, is to not use the chromatic scale or diatonic scale we have been using. Instead, it appears as though the fundamental step size is, well, the syntonic comma itself, or at least some interval that is very close to it. There is a good reason to think this as well. We already have related the different chromatic steps to each other in terms of syntonic commas and diaschismas, and the diaschisma is indistinguishable, auditively speaking, from the syntonic comma. In fact, (16/15)/(25/24) = (16/5)/(25/8) = 128/125, and this is the ratio for the diminished second. The difference between a syntonic comma and a diminished second is a diaschisma, so the diminished second is indistinguishable from 2 syntonic comma steps. Furthermore, the difference between an augmented unison and 3 syntonic commas is equal to the intervallic ratio 1600000/1594323 = 2^9·5^5/3^13, whose interval is called an Amity comma, an interval much smaller than the syntinic comma that is barely inaudible. The syntonic comma and diaschisma differ by an inaudible interval called the schisma, with ratio (3^8·5)/2^15, and an Amity comma differs from 3 schismas by a ratio of 2^54·5^2/3^37, a ratio so tiny it is not worth talking about. As such, we have that the augmented unison is equal to 3 syntonic commas plus an Amity comma, which is approximately equal to 3 schismas. A Pythagorean comma (3^12/2^19) is equal to a syntonic comma plus a schisma. Thus, the augmented unison is almost exactly equal to the augmented unison, and the difference is the inaudible comma of ratio 2^54·5^2/3^37. The difference between the Pythagorean comma and the syntonic comma is inaudible, so we can actually almost treat them as the same interval, and treat them as the step size of the scale we must work with. I said "almost," because there is one detail that still needs to be taken care of. 12 perfect fifths approximate 7 octaves, with the difference being exactly one Pythagorean comma. This is actually the definition of the Pythagorean comma. On the other hand, 53 perfect fifths approximate 31 octaves even better, with the difference being exactly one Mercator comma (3^53/2^84), and 53 Pythagorean commas minus 12 Mercator commas are exactly equal to 1 octave. Therefore, 53 syntonic commas plus the error of 53 schismas minus 12 Mercator commas is exactly equal to 1 octave. The problem is that 53 schismas minus 12 Mercator commas is quite a large comma, and minus 12 Mercator commas by themselves are also a large negative comma. So we need to find the correct number of schismas from which to subtract 12 Mercator commas to minimize the error. That number is 22. This means the smallest error possible is with 22 schismas minus 12 Mercator commas, which means we have a 53-step scale, where 31 of the steps are Pythagorean commas, and 22 of the steps are syntonic commas. This gives a scale that is almost identical to 53-EDO. This is would be the correct scale for dealing with just intonation, and this is the scale where you can make most sense of chords and harmony with just intonation.
256 is not a perfect number. By definition, a perfect number n is a natural number n such that n = σ(1, n) - n, where σ(1, n) the sum of the positive divisors of n. 256 = 2^8, and thus, does not satisfy this equation, since the set of positive divisors is {1, 2, 2^2, 2^3, 2^4, 2^5, 2^6, 2^7, 2^8}, so σ(1, 256) = 1 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 = (2^9 - 1)/(2 - 1) = 2^9 - 1, so σ(1, 256) - 256 = 2^9 - 1 - 2^8 = 2^8 - 1, which is not equal to 2^8, though it is as close as it possibly could be. So 256 is called a quasi-perfect number. That being said, this is all irrelevant. The numerical part of 256 Hz is 256 only when the unit of choice is Hz. However, if the unit of choice is 1/hr or anything else, the numerical part will change accordingly. There is nothing special about this frequency in the context of music theory. It is true however, that the ratio from A to C in Pythagorean tuning is 27/16, and 256 Hz·27/16 = 432 Hz. The problem is that it is very unlikely Pythagoras would have cared about the exact frequency of what we call A, and it is very unlikely he would have gone out of his way to choose its frequency to be 432 Hz. Finally, I end this by saying that I understand that there is a nonzero probability that this comment is a joke. Nonetheless, I know of way too many people that seriously believe in the nonsense that A = 432 Hz in Pythagorean tuning is some sort of objectively superior tuning to all others, and it is something that needs addressing.
Great video! I'm trying to get a handle on the music of the spheres. The last move in the video, from B to F but starting at C, has me baffled, including why you start at C instead of B. C is 261 Hz, then you descend a fifth which I'm told is 384 Hz, then you go up an octave, which I'm told is 440 Hz. I get 317 Hz but the teacher got 348 Hz. Can you help?
I wonder if there is a reason why when showing the guitar neck that the octave is the shortest when I would think it would be the longest in the illustration?
Hi Thomas. I'm not quite sure what you're asking. The basic principle is to go up in fifths (i.e. increase the frequency by 3/2 each time) but halve the frequency if it takes you into the octave above.
@@discovermaths Hello! Thanks for the answer! What I'm confused about is, at 7:26, the image shows if you split the string at 2/3 of its length you get the fifth, so I'm imagining you would have to multiply 261.6 by 2/3 instead of 3/2 to find the fifth.
@@thomasvale5545 Ah, I see. The reason they're different is that one is a length division and the other a frequency multiplication.The fifth sounds if 2/3 the original length of the string is allowed to vibrate, producing a frequency of one over 2/3rds, or 3/2 the frequency of the open string. In the same way the octave sounds if the string is stopped at the halfway point, producing a frequency of one over a half, or twice, that of the open string.
@@discovermaths AAAAAH I get it now! Somehow I missed the youtube notification and tought I was left unanswered! Sorry for coming across as careless! Thank you for your patience. :)
It's genius, thanks! That's what i was looking for. But i want to ask: the greeks at the time did not have our modern arabic positional system of numbers, all these ratios were found on the monochord by a geometrical method of line segments and their commensurability. So if i'll use monochord to divide its string with only 2/3 and 3/4 ratios would i get perfect 7 notes harmonic sound?
The music theory of the Pythagoreans was a philosophical exercise that had little to do with ancient Greek music in practice. I recommend reading John Chalmer's Divisions of the Tetrachord (should be available for free online)
Nicely done but it would be very helpful if you would summarize all your tuning videos in some appropriate order for us to study. Also, since a guitar is (mostly) tuned in 4ths, why is the Pythagorean tuning so important? Thanks!
Excellent idea, Dan. In fact, I've gone a step further and supplied links to all of my music theory videos in the comments below each music video. (I'll have to give your point about guitar tuning some thought - perhaps in another video!)
Guitars did not exist during the time of Pythagoreans. Also, the perfect fourth is the octave complement of the perfect fifth. You _cannot_ have perfect fourths without perfect fifths.
This is completely misleading. On my acoustic, nut to fret 12 (octave) is 324mm, and nut to fret 7 (fifth) is 217mm. Et voila 324/217 = 1.493 but that is just a coincidence. Do the same for fret 4 (third) 134mm, and you get 324/134 =2.42, not 5/4, so what's going on? The relevant string lengths are fret to bridge, and nut to bridge. For the fifth that is 648/(648-217) = 648/431 = 1.503. For the third, its 648/(648-134) = 648/514 = 1.2607. Pitch or frequency is the inverse of the relevant string length, and nothing at all to do with the dead zone above the fret.
In Pythagorean tuning, a major third is tuned to 81/64, not 5/4. The ratio 5/4 does not exist in Pythagorean tuning, because there is no combination of perfect octaves and perfect fifths that can produce an interval with this ratio. This is because there are no integer exponents m, n such that 2^m·(3/2)^n = 5/4.
Your equal-tempered instruments are the wrong tools for this lesson, and the C major scale is not historically appropriate, but your mathematical descriptions are very thorough. It is a shame that no one listening to this video will hear any Pythagorean intervals.
The 12 divisions of the octave came about through the diatonic scales. The diatonic scale has several octave species, involving the raising or lowering of certain scale degrees, and some of these are treated as enharmonic to one another due to only being different by small intervals called commas. When you account for this enharmonicity, you obtain 12 divisions.
This is somewhat misleading. Pythagorean tuning, by itself, has nothing to do with how the various notes in the diatonic scale we know of are tuned. There are all sorts of different scales that are compatible with Pythagorean tuning. How the Greeks used Pythagorean tuning has no relationship to how they arrived at the 7-step diatonic scales. Nonetheless, it is true that the Pythagoreans knew that the difference between 12 perfect fifths and 7 octaves is the Pythagorean comma, a very small interval.
Fun fact: Italian opera composer Giuseppe Verdi used Pythagorean tuning based on the fundamental frequency of the Schumann resonance (≈8 hz) to justify his use for middle A set to 432 hz.
"Maths" is the abbreviation for "mathematics" in the UK and much of the English-speaking world outside the US. Similarly, we say "stats" instead of "stat" for statistics. The US edition of my recent book "Weird Maths" is titled "Weird Math" - so you can take your pick!
@@discovermaths To me its sounds like saying " musics" as in " Mozart's musics" or " the musics of the masters". Just sounds very odd to me. A grammatically misplaced plural.
@@johnvenable5638 It is not a misplaced plural. Mathematics are, quite literally, a plurality of discipline, just like quantum mechanics and physics, for example, are. You seem to not understand these subjects nearly as well as you think you do.
@@angelmendez-rivera351 would it be ok to use physic, mechanic, math, music or this would be considered odd? in other languages like danish, norwegian, swedish, german, french, finnish, italian, portugese and others all is singular.
@@rupertthurner8311 *would it be ok to use physic, mechanic, math, music or this would be considered odd?* Mechanic, math, and music are all regularly used words in the English language. The only odd one out is physic, which even then, is not incorrect if used in the appropriate context. *...in other languages like danish, norwegian, swedish, german, french, finnish, italian, portugese and others all is singular.* What?
It so refreshing to watch a video on youtube without someone asking the viewer to like and subscribe. Thanks for bringing us back to organic media.
So true! I hadn't realised what felt so uncommonly pleasant about this video till I read your comment.
And now i understand at 58 years old. Never got the gist of this back in my younger days. Thank you 😎
Thanks for watching, Barry!
@@discovermaths My pleasure. Yes you teach well so we can understand. Excellent
Most informative video on the subject I have seen so far. Absolutely fascinating. Thank you
The best explanation that I have found so far. I watched tens of videos. Many of them mentioned 3/2, but it was not clear where 3/4 come from and why there were only 7 notes. Everyone is focused on twelve. Your drawing is awesome and helps to understand it perfectly.
What an excellent video!! I've crawled around any number of learned explanations of tuning systems, becoming more confused as I went, and you managed to explain it clearly in just a few minutes. Well done!!
thank you so much for this demonstration, by far the best video on this, ive trawled through a number of resources still slightly puzzled but you've completely solved it for me!
Writing a paper on the different tunning systems for my physics degree, you sir have just saved my bacon!
For a physics degree? What a fascinating topic. Glad it was helpful!
You need to study Just Intonation and compare it to Pythagorean tuning.
@@TravisTellsTruths there's a lot to study.. like scientific pitch tuning system on a 12 TET temperament too..
Hopefully you spelled it correctly on your paper
Best explanation I found, Thank you so so much sir from the land of mountains, Nepal
This is brilliant. Thanks a million for this.
Thank you!
I have been so confused with tuning but you have explained it so well. Thanks so much. Eagerly awaiting the next video re issues with Pythagorean tuning
Thanks, Jane. I'm planning to upload the video on the Pythagorean comma tomorrow.
This is a very nice demonstration, especially on the guitar. One can read many mathematical treatments, but seeing this demonstrated on a stringed instrument, just as the "Pythagoreans" might have, gives one much greater insight. Sometimes visualization provides just the right touch.
Thanks for your explanation of this concept. Writing out the calculation and historical perspective help a lot.
I loved watching this video. A really clear mathematical and practical explanation. I have learnt a lot from this.
Thank you!
Are you THE Andrew Barker, author of "The Science of Harmonics in Classical Greece?" If so I'm a big fan!
@@Hazuls Sorry. It's not me.
These videos are the best explanations to the tuning issues I have ever come across. Could you do a video about just intonation and/or other tuning systems? Explain the wolf intervals? Beating (interaction between close harmonics), etc?
Many thanks, QF. Yes, these are topics I'd been planning to cover.
I've been trying to find resources on Pythagorean tuning, and this was by far the clearest and easiest to follow video that I've found. Thanks for your generosity!!
Thank you so much! the diagram you drew near the end helped me so much!!
This is very informative and very well explained. I can’t thank you enough. Liked n subscribed.
ABSOLUTELY WONDERFUL MISTER! I LOVE MATHS, no I LIVE MATHS I Love your thumbnails
Thank you for this video! It helped us a lot!
Some confusion is introduced by measuring the non-played distance, i.e. the 32/21 does give 3/2 but _this_ 3/2 is not the 3/2 that is shown in the diagram. The 3/2 that is shown in the diagram is 64/43. This confusion is further amplified by showing the vibrating part of the string at the opposite end.
Just checked comments to see if somebody already pointed to this. Thak you.
this was so concise! other videos i've watched have made it seem much more complicated. thank you.
Thanks, Mike. I'm glad it was helpful.
Really good! stayed with this and got it right until the end!
Clear, great explanation. Thank you.
Thank you so much!!!! The best explanation I’ve seen!!!
I'm very happy it was useful, Liliana - thank you!
This helped a lot. I’ve been searching. Thank you
That's good to know - thanks for watching.
Same here. Finally some clarity!
Thank you for the clear explanation
Very well explained. Thank you.
: The Reason why the fifth he found on guitar is not perfectly 1.5 is because, for well-tempered system, which divide an octave to equally 12 parts, is actually “out of tune” for every single note according to Pythagorean system. (That’s why perfect pitch today to me is fundamentally wrong!)
Great explanation!
Thanks that diagram was a great help 👍
Hey, I really enjoyed this video! It would be so helpful if you posted a video on just intonation. Your explanations are very concise and easy to understand and this tuning system is definitely something I've been having trouble with. Let me know!!
Yes, I do plan to do one on just intonation in the near future to complement the ones on Pythagorean tuning and equal temperament,
Thanks for the video ❤️❤️❤️
thank you this is great!
Thank you!
My inquiry began in traditional chinese scales and have been going down the rabbit hole. I still have so many questions, I'm looking forward to more videos about the topic! 😊
I just started learning piano. At first I could not understand why F needs to start from C instead of the following the calculation of the others. Then I realised all the other keys involved 1/2 steps. So starting from B down to F does not have a half stop, but going down to bass F and then go up an octave does.
Thank you!
This was amazing! I ended up here after watching "The Concise History of Chinese Musical Temperament - Episode 1: Formation of Tones" and found out an Ancient Chinese emperor figured out a scale called Sanfen Sunyi that does "Divide by 3, subtract that, and add a third of the new". The author compared it to the Pythagorean tuning system, and since I was curious I ended up here. I'd love to hear your thoughts on Sanfen Sunyi vs Pythagorean tuning.
I really enjoyed this video, thank you again.
Very interesting- it would be very valuable to me and other guitar players if you could discuss the possibilities and difficulties of tuning a guitar, bearing in mind that consonant, or pure intervals are the most desirable, and of course that they are limited. A player has the tempered scale to impede his progress, and also the fact that, due to the inclination of the string, 2 g notes on one string will be different pitches, two A notes on one string will also be different. The phenomenon of resonance and it's value to a tuner would also be very interesting to read- thanks
Hello. Please can you tell me why, at 11:57, you descend by a fifth rather than rising by a fifth? I appreciate that not doing this would produce an F sharp - which would sound wrong - but it makes the sequence a bit inconsistent - like the fundamental note is really F for the subdominant.
I have the same question (I think!). Why break away from the formula followed thus far? Why go up a fourth from C rather than up a fifth (and back an octave if necessary) from the last note?
I notice that you posted this comment a year ago. Have you found an explanation since then? If so, I'd be very grateful if you could share it with me! :)
David I think you might need to add a video explaining how the frequency manipulation of going up/down intervals in the second half of the video is related inversely to the string lengths of the notes in the first part of the video. Also need to explain why constructing this scale using fifths produces notes that sound “consonant” together to create chords. I’m sure these are things you will eventually cover in future videos.
Perfect pitch is out of tune!
*I think you might need to add a video explaining how the frequency manipulation of going up/down intervals in the second half of the video is related inversely to the string lengths of the notes in the first part of the video.*
All sound waves you produce with an instrument travel at the same speed: the speed of sound, which is approximately 343 m·Hz. Since the speed is a non-changing quantity, a constant, there must be some mathematical relationship describing why, and there is. The mathematical relationship is v = λ·f. v is the speed of the sound waves. f is the frequency of the sound waves, and λ is the length of the string producing the sound waves. Since v is a constant, λ and f are inversely proportional.
*Also, need to explain why constructing this scale using fifths produces notes that sound "consonant" together to create chords.*
Modern chords in Pythagorean tuning are not consonant. The perfect unison, the perfect octave, the perfect fifth, the perfect fourth, the major second, and the minor seventh, are the only consonant intervals in Pythagorean tuning. Notably, the Pythagorean thirds and sixths are more dissonant than their juster, Ptolemic counterparts. The reason Pythagorean tuning sounded well to the Pythagoreans is because they used quartal harmony (if any), not tertial harmany as we did in medieval times and later. The reason the perfect fifth is such a perfect consonance is because 3/2 is the simplest ratio of frequencies after 2, and 4/3 is merely its octave complement.
Great video
All I know is that Pythagorean Tuning gives keys their own characteristics like Dm being depressing.
Any well-temperament that is not an equal temperament does that.
Nice one
Thank you very much for this very detailed explanation. I unfortunately lose you when you explain the last calculation: how to arrive at 348.8 Hz from 496.7 Hz. Could you decompose it for me please? Thank you
Don't know much about Pythagorean tuning but if we take the ratio of natural 2nd at 10/9 instead of 9/8, it sounds more pleasant with the first 1st as well as the 4th if the 4th is at 4/3. But again it would be dissonant with rest other notes so the other notes have to be re adjusted. Interestingly if you go by this tuning, you can play all the notes from the Lydian scale at once and it won't sound dissonant.
Pythagorean tuning, as explained in the video (which I assume you did not watch, based on your comment), is a tuning system where all musical intervals allowed are constructed from only perfect octaves (ratio 2) and perfect fifths (ratio 3/2). The ratio 10/9 cannot be constructed from only perfect octaves and perfect fifths, so its corresponding interval does not exist in Pythagorean tuning. It does exist, however, in Ptolemic tuning.
@@angelmendez-rivera351 I knew that 10/9 2nd does not occur in Pythagorean tuning but did not know there exist another tuning that has that. Thanks for the info I'll look it up. But if you can experiment a bit, I would suggest do try mixing 10/9 4/3 and 5/3, they go well with each other. 10/9 and 3/2 are dissonant.
@@siddheshdeshpande7183 The interval between a perfect fifth (3/2) and a major second (10/9) has a ratio (3/2)/(10/9) = 27/20, and so the interval is an acute fourth. You obtain the same interval if you add a syntonic comma (81/80), which I call an acute unison, to a perfect fourth (4/3), since 4/3·81/80 = 27/20. The acute fourth is a dissonant interval indeed.
@@angelmendez-rivera351 I am relieved that you agree that 10/9 and 3/2 are dissonant, I came to know this through my teacher and of course through the experience of listening. Don't really understand the terms guess I have to do a bit more research, appreciate the mathematical explanation though. Cheers!
@@siddheshdeshpande7183 Ptolemic tuning is a system of just intonation where you only use three intervals to construct all the other tuning intervals. The three intervals you use are the perfect octave (ratio 2), the perfect fifth (ratio 3/2), and the major third (ratio 5/4). This system has some peculiar things you have to be very careful about, though. The interval between the perfect fifth and the perfect fourth (ratio 4/3) is the interval with ratio 9/8. However, the interval between the perfect fourth and the minor third (ratio 6/5) is the interval with ratio 10/9. Both of these intervals are seconds, and furthermore, it appears both of these should be called the major second. Yet they are different intervals, the difference between them is the acute unison (ratio 81/80), which while very small, is still audible, and can make certain intervals sound out of tune. How do we know which one is truly the major second? We think of a major second as being half of a major third. For example, in 12-EDO, a major second is equal to 2 steps of 100 cents, and the major third is equal to 4 steps of 100 cents. So which of 9/8 and 10/9 is half of a major third? It turns out, both of them, because 9/8·10/9 = 5/4. In fact, if you want to tune the C major diatonic scale using just intonation, then you need both 9/8 and 10/9 to do so. Since the major sixth is tuned to the ratio 5/3 and the major seventh is tuned to the ratio 15/8, we obtain a scale with degrees tuned to 1, 10/9 or 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2. The steps sizes, then, must be 10/9 or 9/8, 9/8 or 10/9 (respectively), 5/4, 16/15, 9/8, 10/9, 9/8, 16/15. There is no possible way to have an equivalent scale with only 2 step sizes. You can decompose every interval that is not a second into only seconds. For example, a major third is an interval of 9/8 plus an interval of 10/9, in any order. A perfect fourth is an interval 10/9 plus one of 16/15 plus one of 9/8, in some order. Why does this matter? Because it severely affects how you combine intervals together into chords, and how you stack them, and it also affects key changes. For example, a perfect fourth has 1 of 9/8, 1 of 10/9, and 1 of 16/15. If, instead, you play 2 of 9/8, and 1 of 16/15, you still have played two major seconds and one minor second, but instead, you now have an acute fourth of 27/20, rather than a perfect fourth of 4/3.
It turns out that this also affects the chromatic scale. In the chromatic scale, the scale degrees are the perfect unison (1), the minor second (16/15 ?), the major second (10/9 or 9/8), the minor third (6/5), the major third (5/4), the perfect fourth (4/3), the tritone (25/18 or 45/32 or 64/45 or 36/25), the perfect fifth (3/2), the minor sixth (8/5), the major sixth (5/3), the minor seventh (16/9 or 9/5), the major seventh (15/8), the perfect octave (2). This results in chromatic steps 16/15, 25/24 or 135/108, 27/25 or 16/15 (respectively), 25/24, 16/15, 25/24 or 135/108 or 16/15 or 27/25, 27/25 or 16/15 or 135/108 or 25/24 (respectively), 16/15, 25/24, 16/15 or 27/25, 135/108 or 25/24 (respectively), 16/15. So there are 4 different chromatic steps sizes! The difference between the intervals of ratios 27/25 and 16/15 is 81/80, and the difference between the intervals of ratios 135/108 and 25/24 is also 81/80, the syntonic comma, or acute unison as I call it. Furthermore, the difference between the intervals of ratio 16/15 and 135/108 is this interval with ratio 2048/2025, called a diaschisma, which is approximately equal to a syntonic comma (the difference between the 2 is inaudible). So the smallest chromatic step is 25/24, the second smallest is that plus a syntonic comma, the second largest is the smallest plus approximately 2 syntonic commas, and the largest is the smallest plus approximately 3 syntonic commas. This is suspicious, is it not?
The solution to this, quite frankly, is to not use the chromatic scale or diatonic scale we have been using. Instead, it appears as though the fundamental step size is, well, the syntonic comma itself, or at least some interval that is very close to it. There is a good reason to think this as well. We already have related the different chromatic steps to each other in terms of syntonic commas and diaschismas, and the diaschisma is indistinguishable, auditively speaking, from the syntonic comma. In fact, (16/15)/(25/24) = (16/5)/(25/8) = 128/125, and this is the ratio for the diminished second. The difference between a syntonic comma and a diminished second is a diaschisma, so the diminished second is indistinguishable from 2 syntonic comma steps. Furthermore, the difference between an augmented unison and 3 syntonic commas is equal to the intervallic ratio 1600000/1594323 = 2^9·5^5/3^13, whose interval is called an Amity comma, an interval much smaller than the syntinic comma that is barely inaudible. The syntonic comma and diaschisma differ by an inaudible interval called the schisma, with ratio (3^8·5)/2^15, and an Amity comma differs from 3 schismas by a ratio of 2^54·5^2/3^37, a ratio so tiny it is not worth talking about. As such, we have that the augmented unison is equal to 3 syntonic commas plus an Amity comma, which is approximately equal to 3 schismas. A Pythagorean comma (3^12/2^19) is equal to a syntonic comma plus a schisma. Thus, the augmented unison is almost exactly equal to the augmented unison, and the difference is the inaudible comma of ratio 2^54·5^2/3^37. The difference between the Pythagorean comma and the syntonic comma is inaudible, so we can actually almost treat them as the same interval, and treat them as the step size of the scale we must work with. I said "almost," because there is one detail that still needs to be taken care of. 12 perfect fifths approximate 7 octaves, with the difference being exactly one Pythagorean comma. This is actually the definition of the Pythagorean comma. On the other hand, 53 perfect fifths approximate 31 octaves even better, with the difference being exactly one Mercator comma (3^53/2^84), and 53 Pythagorean commas minus 12 Mercator commas are exactly equal to 1 octave. Therefore, 53 syntonic commas plus the error of 53 schismas minus 12 Mercator commas is exactly equal to 1 octave. The problem is that 53 schismas minus 12 Mercator commas is quite a large comma, and minus 12 Mercator commas by themselves are also a large negative comma. So we need to find the correct number of schismas from which to subtract 12 Mercator commas to minimize the error. That number is 22. This means the smallest error possible is with 22 schismas minus 12 Mercator commas, which means we have a 53-step scale, where 31 of the steps are Pythagorean commas, and 22 of the steps are syntonic commas. This gives a scale that is almost identical to 53-EDO. This is would be the correct scale for dealing with just intonation, and this is the scale where you can make most sense of chords and harmony with just intonation.
Could you explained scientific pitch tuning system on the 12 TET temperament intonation?
In the spirit of Pythagoras you simply MUST tune C to 256 Hz, a perfect number!!! This will give you a 432 Hz for A.
256 is not a perfect number. By definition, a perfect number n is a natural number n such that n = σ(1, n) - n, where σ(1, n) the sum of the positive divisors of n. 256 = 2^8, and thus, does not satisfy this equation, since the set of positive divisors is {1, 2, 2^2, 2^3, 2^4, 2^5, 2^6, 2^7, 2^8}, so σ(1, 256) = 1 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 = (2^9 - 1)/(2 - 1) = 2^9 - 1, so σ(1, 256) - 256 = 2^9 - 1 - 2^8 = 2^8 - 1, which is not equal to 2^8, though it is as close as it possibly could be. So 256 is called a quasi-perfect number.
That being said, this is all irrelevant. The numerical part of 256 Hz is 256 only when the unit of choice is Hz. However, if the unit of choice is 1/hr or anything else, the numerical part will change accordingly. There is nothing special about this frequency in the context of music theory. It is true however, that the ratio from A to C in Pythagorean tuning is 27/16, and 256 Hz·27/16 = 432 Hz. The problem is that it is very unlikely Pythagoras would have cared about the exact frequency of what we call A, and it is very unlikely he would have gone out of his way to choose its frequency to be 432 Hz.
Finally, I end this by saying that I understand that there is a nonzero probability that this comment is a joke. Nonetheless, I know of way too many people that seriously believe in the nonsense that A = 432 Hz in Pythagorean tuning is some sort of objectively superior tuning to all others, and it is something that needs addressing.
The sacred number in music comes from the ratios, not the frequency.
I don't think Pythagoras would care about the actual pitches measured in hz.
Great video! I'm trying to get a handle on the music of the spheres. The last move in the video, from B to F but starting at C, has me baffled, including why you start at C instead of B. C is 261 Hz, then you descend a fifth which I'm told is 384 Hz, then you go up an octave, which I'm told is 440 Hz. I get 317 Hz but the teacher got 348 Hz. Can you help?
i have the same question!!
Can you please elaborate more on the "odd ball" that you mentioned and why that is? I'm referring to what comes next after 496.7
Wonderful. Thank you. What key are the gulls singing in?
Off to start a band called dissonant. Thanks
I can’t find your video on just intonation. Did you post that video? Thank you! Your videos are so helpful!
Hi Ronda. That's on my "to do" list some time in the next few months. Thanks for watching and I'm glad they're proving useful.
I wonder if there is a reason why when showing the guitar neck that the octave is the shortest when I would think it would be the longest in the illustration?
How Pythagorus get 261.6 for C in the first place?
He didn't. The Pythagoreans would have just used relative pitch. That figure is based on our modern tuning system.
Pythagorean tuning by key signature might ring.
I can’t find the video on the problems with Pythagorean tuning!
I can’t find the video on “just intonation!”
i don't think its been created, even I was looking for it
Hi, I just have one question. How come that on the Pythagorean scale you have to multiple by 3/2 instead of 2/3?
i still dont grasp why you multiplied 261.6 by the complement of 2/3 which, as you showed at 7:26, is the ratio where the fifth is found
Hi Thomas. I'm not quite sure what you're asking. The basic principle is to go up in fifths (i.e. increase the frequency by 3/2 each time) but halve the frequency if it takes you into the octave above.
@@discovermaths Hello! Thanks for the answer! What I'm confused about is, at 7:26, the image shows if you split the string at 2/3 of its length you get the fifth, so I'm imagining you would have to multiply 261.6 by 2/3 instead of 3/2 to find the fifth.
@@thomasvale5545 Ah, I see. The reason they're different is that one is a length division and the other a frequency multiplication.The fifth sounds if 2/3 the original length of the string is allowed to vibrate, producing a frequency of one over 2/3rds, or 3/2 the frequency of the open string. In the same way the octave sounds if the string is stopped at the halfway point, producing a frequency of one over a half, or twice, that of the open string.
@@discovermaths AAAAAH I get it now! Somehow I missed the youtube notification and tought I was left unanswered! Sorry for coming across as careless! Thank you for your patience. :)
Where is the video for Just Intonation? I can't find it.
I don't think I ever got around to making it. Thanks for reminding me - I'll add it to the list.
It's genius, thanks! That's what i was looking for. But i want to ask: the greeks at the time did not have our modern arabic positional system of numbers, all these ratios were found on the monochord by a geometrical method of line segments and their commensurability. So if i'll use monochord to divide its string with only 2/3 and 3/4 ratios would i get perfect 7 notes harmonic sound?
No. The way you described their tuning system is not how they actually did. Their tuning system was what they called the Greater Perfect System.
The music theory of the Pythagoreans was a philosophical exercise that had little to do with ancient Greek music in practice. I recommend reading John Chalmer's Divisions of the Tetrachord (should be available for free online)
why constructing the last one is different?
Nicely done but it would be very helpful if you would summarize all your tuning videos in some appropriate order for us to study. Also, since a guitar is (mostly) tuned in 4ths, why is the Pythagorean tuning so important? Thanks!
Excellent idea, Dan. In fact, I've gone a step further and supplied links to all of my music theory videos in the comments below each music video. (I'll have to give your point about guitar tuning some thought - perhaps in another video!)
Guitars did not exist during the time of Pythagoreans. Also, the perfect fourth is the octave complement of the perfect fifth. You _cannot_ have perfect fourths without perfect fifths.
You might enjoy this -- John Sase: th-cam.com/video/TC-Dwq-qr-Y/w-d-xo.html
This is completely misleading. On my acoustic, nut to fret 12 (octave) is 324mm, and nut to fret 7 (fifth) is 217mm. Et voila 324/217 = 1.493 but that is just a coincidence. Do the same for fret 4 (third) 134mm, and you get 324/134 =2.42, not 5/4, so what's going on?
The relevant string lengths are fret to bridge, and nut to bridge. For the fifth that is 648/(648-217) = 648/431 = 1.503. For the third, its 648/(648-134) = 648/514 = 1.2607. Pitch or frequency is the inverse of the relevant string length, and nothing at all to do with the dead zone above the fret.
In Pythagorean tuning, a major third is tuned to 81/64, not 5/4. The ratio 5/4 does not exist in Pythagorean tuning, because there is no combination of perfect octaves and perfect fifths that can produce an interval with this ratio. This is because there are no integer exponents m, n such that 2^m·(3/2)^n = 5/4.
So, no, it is not misleading.
Your equal-tempered instruments are the wrong tools for this lesson, and the C major scale is not historically appropriate, but your mathematical descriptions are very thorough. It is a shame that no one listening to this video will hear any Pythagorean intervals.
Great Video, thanks very much Pythagoreans didn't have the square root of two, they had a hard time with ratios.
Why 12 notes in an octave. Why not 16 notes in an octave ?
The 12 divisions of the octave came about through the diatonic scales. The diatonic scale has several octave species, involving the raising or lowering of certain scale degrees, and some of these are treated as enharmonic to one another due to only being different by small intervals called commas. When you account for this enharmonicity, you obtain 12 divisions.
Lord ... are you really not going to play the Pythagorean scale after all that ....
It was the P system, would it limit music, could a trio even be put together, without have to go back to the drawing board
Why is 1/3 a fifth? Why isn't it a third?
We call this proportion a fifth because it is the fifth scale degree of our diatonic scale.
This is somewhat misleading. Pythagorean tuning, by itself, has nothing to do with how the various notes in the diatonic scale we know of are tuned. There are all sorts of different scales that are compatible with Pythagorean tuning. How the Greeks used Pythagorean tuning has no relationship to how they arrived at the 7-step diatonic scales. Nonetheless, it is true that the Pythagoreans knew that the difference between 12 perfect fifths and 7 octaves is the Pythagorean comma, a very small interval.
Fun fact: Italian opera composer Giuseppe Verdi used Pythagorean tuning based on the fundamental frequency of the Schumann resonance (≈8 hz) to justify his use for middle A set to 432 hz.
4th note can't be 1/4 point.
When did "maths" become a word? Sounds illiterate. Just me?
"Maths" is the abbreviation for "mathematics" in the UK and much of the English-speaking world outside the US. Similarly, we say "stats" instead of "stat" for statistics. The US edition of my recent book "Weird Maths" is titled "Weird Math" - so you can take your pick!
@@discovermaths
To me its sounds like saying " musics" as in " Mozart's musics" or " the musics of the masters". Just sounds very odd to me. A grammatically misplaced plural.
@@johnvenable5638 It is not a misplaced plural. Mathematics are, quite literally, a plurality of discipline, just like quantum mechanics and physics, for example, are. You seem to not understand these subjects nearly as well as you think you do.
@@angelmendez-rivera351 would it be ok to use physic, mechanic, math, music or this would be considered odd? in other languages like danish, norwegian, swedish, german, french, finnish, italian, portugese and others all is singular.
@@rupertthurner8311 *would it be ok to use physic, mechanic, math, music or this would be considered odd?*
Mechanic, math, and music are all regularly used words in the English language. The only odd one out is physic, which even then, is not incorrect if used in the appropriate context.
*...in other languages like danish, norwegian, swedish, german, french, finnish, italian, portugese and others all is singular.*
What?