Thanks for making this video. I've searched on this theme after starting to learn basic music theory. Pythagoras was really cool guy for his time, I'm always getting all excited, when I think about ancient greek minds.
I liked your previous video that led to this one, but the ratios presented at 1:11 are related to 5-limit tuning, not Pythagorean. It should be E = 81/64; A = 27/16; and B = 243/128 (as you had correctly labeled in your prior video).
multiply the internal angle of a regular pentagon (108 degrees) by the pythagorean comma ratio 531441/524288 you get the 109.473 very close to the central angle of regular tetrahedron (109.471 degrees)
"Two thirds along its length" is not the same as "divide it in the ratio 3:2", that would be "three fifths or two fifths along its length, depending which end you started at".
Multiplying the frequency by 3/2 is the same as playing the string 2/3 along its length. These are physically the same thing, this is literally a law of physics.
@@angelmendez-rivera351 "Two thirds along its length" is not the same as "divide it in the ratio 3:2", that would be "three fifths or two fifths along its length, depending which end you started at".
This really bugged me too :( Creates a lot of confusion for me why this apparently erudite gentleman keeps referring to the idea of dividing a string into two thirds and a third as dividing it in a 2:3 ratio. The latter is 5 parts, the former is (as it should be) 3. Or am I missing something?
Very nicely explained except for the fact that we don't have white notes and black notes on a keyboard like the one you used: Do we? We have white KEYS and black KEYS. Notes are all black.
This comes of being a guitar player! Hopefully, most people will understand what I mean. I have to say, though, that musical terminology is confusing, especially when you factor in the differences between US and UK usage. Keys, notes, tones, etc., can all mean different things depending on context.
05:30 With equal temperament, should we expect C2 to be exactly double C1? The diagram lists C1 as 32.71 hertz and double that would be 65.42 but C2 is listed as 65.41. Double C2 would be 130.84 but C3 is listed as 130.08... Why isn't the octave exactly doubling?
At the very beginning, you show a diagram of a monochord with a movable bridge in the 2/3 position then say that it produces a perfect fifth. Then you say that a perfect fifth sounds extremely consonant with the open string - - how do you know that if you have only one string that can make only one tone at a time? Also, this diagram seems to show a monochord that can produce a musical tone on each side of the movable bridge (neither side is dampened), so which side produces a perfect fifth and what does the other side produce?
1:06 - 1:14 Well, no, this is not accurate. Pythagorean tuning is a form of just intonation: specifically, 3-prime limit just intonation. This means that intervals are formed from by taking ratios of the harmonic series, such that the integers in the ratios only have prime factors 2 and 3. An entirely different matter is the discovery that all such ratios can be generated solely by the perfect octave, of ratio 2, and the perfect fifth, of ratio 3/2. The Pythagoreans did think 3/2 was a special ratio, which is why the prime numbers 2 and 3 were chosen. Also, the diagram on screen is inaccurate. 5/4 is not a Pythagorean interval, and neither are 5/3 nor 15/8. There are no integers n, m such that 2^n·(3/2)^m = 5/4, thanks to the fundamental theorem of arithmetic. 1:14 - 1:20 This is called the circle of fifths, but again, this is inaccurate. The circle of fifths is not a defining feature of Pythagorean tuning, and in fact, it is not unique to Pythagorean tuning: there are many xenharmonic tuning systems where the circle of fifths is still applicable. Also, what was said in the video is not the full story. While equal tempermants do allow you to play in all keys, well-temperanents also do this, and equal temperament did not originate with the intent of replacing just intonation, which had already been replaced by well-temperaments. Also, this issue with the Pythagorean comma is not even a genuine problem for Pythagorean tuning, at least not if you decide to be more careful about it. The video claims that the Pythagorean comma is about one quarter of a 12-equal-tempered semitone, which suggests we should divide the octave into 48 intervals, rather than 12, and that there should be no issue with treating the Pythagorean comma as a semitone in this context. However, it would be more accurate to say that 9 Pythagorean commas approximate a Pythagorean major second, only exceeding it by the ratio 3^106/2^168 = (3^53/2^84)^2. Even more specifically, 2 major seconds are equal to a Pythagorean major third, tuned to the ratio 81/64, and the distance between this and a perfect fourth is 256/243, which is a Pythagorean minor second. The distance between a Pythagorean major second and a Pythagorean minor second is a Pythagorean augmented unison, tuned to the ratio 3^7/2^11. Why is this important? 5 Pythagorean commas exceed a Pythagorean augmented unison by only 3^53/2^84 a ratio whose interval is known as Mercator's comma. This is an interval so small, it is actually inaudible, unnoticeable, unlike the Pythagorean comma. Notice how 9 Pythagorean commas exceed a Pythagorean major second by 2 of those commas, and 53 perfect fifths exceed 31 octaves by exactly Mercator's comma. This suggests that we should use the circle of fifths to construct a chromatic scale of not 12 divisions of the octave, but 53 instead. Furthermore, notice that 53 Pythagorean commas exceed the octave by exactly 12 of Mercator's commas. This is a rather large amount, so this seems like a terrible idea, but this can be fixed. How can it be fixed? The secret to this is actually hidden in one of the mistakes made earlier in the video. The video claimed 5/4 is a Pythagorean interval, even though it is not. Nonetheless, if we allow this interval, to form something called Ptolemic tuning, which is a modification of Pythagorean tuning, also knows as 5-limit, then you can use the interval 5/4 and all the intervals you can generate with it, in conjunction with the Pythagorean intervals. The Pythagorean major third, tuned to 81/64, exceeds a just major third, tuned to 5/4, by a small interval called a syntonic comma, of ratio 81/80. This is important, because the syntonic comma is a tiny bit smaller than the Pythagorean comma, but is still almost exactly the same size as the Pythagorean comma. The difference between the two is called a schisma, and it is less than 2 cents, so it is an inaudible difference. As such, a Pythagorean comma is a syntonic comma and a schisma. If instead of using 53 Pythagorean commas, we use 31 Pythagorean commas and 22 syntonic commas, then the difference between these and the octave is not 12 Mercator commas, but 12 Mercator commas minus 22 schismas. This new comma is an extremely tiny comma that measures less than 0.4 cents! As such, this circle of fifths is indistinguishable from an unbroken circle of fifths. Since the syntonic comma and the Pythagorean comma are indistinguishable, the scale produced here sounds indistinguishable from 53-equal temperament. This means you can play in all keys, you can form harmony, and all the important intervals are perfectly in tune. You can even play polytonal or atonal music with this. This type of music has been rarely used by American composers, but has been used widely in Chinese music, Ottoman music, and Turkish music.
I am full of admiration at such scholarly expertise . I have tuned guitars with wiggly frets in Kirnberger III, Young, and Werkmeister. But guitars hardly ever play in difficult keys so I may try Pythagoras as well . These frets are just bronze wire ,glued on with Evostik .All very reliable and durable . But a soldering iron will release them easily . One guitar was accurately tuned to Equal Temperament and the frets were nowhere near in line .That plays well in all keys . All notes were checked with a Korg Orchestral Tuner.
The Well Tempered Clavier, by Bach, gave us equal temperament. But does anyone really believe it took that long to figure out? Nobody is even looking at this video though, they are too distracted.
That mathematical synthesis at the end is exactly what I wanted to finish wrapping my head around the problems with this tuning. It's always bothered me how music can't be "perfect", how we allow for certain deviation of the "pure" intervals, in service of coherence. But after getting the problem with this temperament as well as others, I do believe 12-tone equal temperament, at least for the uses that it's given within a mostly Western framework, is really the best way to go. Thank you for this series
Assolutamente no. Dal punto di vista filosofico infatti, adottare a tutti i costi una "manipolazione arbitraria" dei suoni conduce a un risultato sbagliato, contraddittorio quanto non degenerativo per la società. E per cui è e rimane una scelta sbagliata (anche se serve a giustificare inopportunamente il prurito, i vezzi, le farse e l'idealismo degli artistoidi, finti artisti...). L'errore infatti è e rimane sempre un errore. E lo rimane ancorché gli si dia una veste e una giustificazione matematica "apparentemente rassicurante per la maggioranza". Infatti, anche i sofismi rimangono sofismi, nonostante impieghino motivazioni "matematematicamente rassicuranti". LJC
Music absolutely can be perfect. Ottoman music uses just intonation. Turkish music does as well, and even some forms of Chinese music do. The Pythagorean comma is not actually the huge problem it is presented to be. There are many workarounds. Also, this video is historically inaccurate. Equal temperaments were not invented to solve the problem described in the video. That problem was solved by meantone temperaments, and later, by well-temperaments. Equal temperaments came later, in the 20th century, and we started using them for reasons that have absolutely nothing to do with Pythagorean tuning. Also, the video's description of Pythagorean tuning is not accurate, because 5/4 is not a ratio that exists in Pythagorean tuning.
This is very informative and interesting, and underrated. Thank you for this knowledge, it is making me a better musician
Thanks, Pedro. I'm glad it's helping.
Thanks for making this video. I've searched on this theme after starting to learn basic music theory. Pythagoras was really cool guy for his time, I'm always getting all excited, when I think about ancient greek minds.
Fascinating!! I wish I had found your page sooner. Good work.
I liked your previous video that led to this one, but the ratios presented at 1:11 are related to 5-limit tuning, not Pythagorean. It should be E = 81/64; A = 27/16; and B = 243/128 (as you had correctly labeled in your prior video).
multiply the internal angle of a regular pentagon (108 degrees) by the pythagorean comma ratio 531441/524288 you get the 109.473 very close to the central angle of regular tetrahedron (109.471 degrees)
This is so interesting, thank you.
"Two thirds along its length" is not the same as "divide it in the ratio 3:2", that would be "three fifths or two fifths along its length, depending which end you started at".
Multiplying the frequency by 3/2 is the same as playing the string 2/3 along its length. These are physically the same thing, this is literally a law of physics.
@@angelmendez-rivera351 "Two thirds along its length" is not the same as "divide it in the ratio 3:2", that would be "three fifths or two fifths along its length, depending which end you started at".
You are right. Two thirds is correct for the case.
This really bugged me too :( Creates a lot of confusion for me why this apparently erudite gentleman keeps referring to the idea of dividing a string into two thirds and a third as dividing it in a 2:3 ratio. The latter is 5 parts, the former is (as it should be) 3. Or am I missing something?
@@ashharijaywardena Most people don't think these things through, or notice a problem.
All good, but I think the ratio becomes 1/3 to 2/3 or 1:2 not 2:3
Exactly!
@discovermaths have you heard of the “howard comma”…?
Very nicely explained except for the fact that we don't have white notes and black notes on a keyboard like the one you used: Do we? We have white KEYS and black KEYS. Notes are all black.
This comes of being a guitar player! Hopefully, most people will understand what I mean. I have to say, though, that musical terminology is confusing, especially when you factor in the differences between US and UK usage. Keys, notes, tones, etc., can all mean different things depending on context.
05:30
With equal temperament, should we expect C2 to be exactly double C1?
The diagram lists C1 as 32.71 hertz and double that would be 65.42 but C2 is listed as 65.41.
Double C2 would be 130.84 but C3 is listed as 130.08... Why isn't the octave exactly doubling?
Yes, it should be exactly double. It's because I've rounded to 2 decimal places.
@@discovermaths ok thanks... I believe there is a typo going from C2 (65.41) to C3. C3 should be 130.82 but it is listed as 130.08.
@@Better_Call_Raul it should be listed this part of 130 rather as .81 and not .08 to be exact to hear a true frequency 1 octave below Middle c
At the very beginning, you show a diagram of a monochord with a movable bridge in the 2/3 position then say that it produces a perfect fifth. Then you say that a perfect fifth sounds extremely consonant with the open string - - how do you know that if you have only one string that can make only one tone at a time?
Also, this diagram seems to show a monochord that can produce a musical tone on each side of the movable bridge (neither side is dampened), so which side produces a perfect fifth and what does the other side produce?
Interesting
1:06 - 1:14 Well, no, this is not accurate. Pythagorean tuning is a form of just intonation: specifically, 3-prime limit just intonation. This means that intervals are formed from by taking ratios of the harmonic series, such that the integers in the ratios only have prime factors 2 and 3. An entirely different matter is the discovery that all such ratios can be generated solely by the perfect octave, of ratio 2, and the perfect fifth, of ratio 3/2. The Pythagoreans did think 3/2 was a special ratio, which is why the prime numbers 2 and 3 were chosen. Also, the diagram on screen is inaccurate. 5/4 is not a Pythagorean interval, and neither are 5/3 nor 15/8. There are no integers n, m such that 2^n·(3/2)^m = 5/4, thanks to the fundamental theorem of arithmetic.
1:14 - 1:20 This is called the circle of fifths, but again, this is inaccurate. The circle of fifths is not a defining feature of Pythagorean tuning, and in fact, it is not unique to Pythagorean tuning: there are many xenharmonic tuning systems where the circle of fifths is still applicable.
Also, what was said in the video is not the full story. While equal tempermants do allow you to play in all keys, well-temperanents also do this, and equal temperament did not originate with the intent of replacing just intonation, which had already been replaced by well-temperaments. Also, this issue with the Pythagorean comma is not even a genuine problem for Pythagorean tuning, at least not if you decide to be more careful about it. The video claims that the Pythagorean comma is about one quarter of a 12-equal-tempered semitone, which suggests we should divide the octave into 48 intervals, rather than 12, and that there should be no issue with treating the Pythagorean comma as a semitone in this context. However, it would be more accurate to say that 9 Pythagorean commas approximate a Pythagorean major second, only exceeding it by the ratio 3^106/2^168 = (3^53/2^84)^2. Even more specifically, 2 major seconds are equal to a Pythagorean major third, tuned to the ratio 81/64, and the distance between this and a perfect fourth is 256/243, which is a Pythagorean minor second. The distance between a Pythagorean major second and a Pythagorean minor second is a Pythagorean augmented unison, tuned to the ratio 3^7/2^11. Why is this important? 5 Pythagorean commas exceed a Pythagorean augmented unison by only 3^53/2^84 a ratio whose interval is known as Mercator's comma. This is an interval so small, it is actually inaudible, unnoticeable, unlike the Pythagorean comma. Notice how 9 Pythagorean commas exceed a Pythagorean major second by 2 of those commas, and 53 perfect fifths exceed 31 octaves by exactly Mercator's comma. This suggests that we should use the circle of fifths to construct a chromatic scale of not 12 divisions of the octave, but 53 instead. Furthermore, notice that 53 Pythagorean commas exceed the octave by exactly 12 of Mercator's commas. This is a rather large amount, so this seems like a terrible idea, but this can be fixed. How can it be fixed? The secret to this is actually hidden in one of the mistakes made earlier in the video. The video claimed 5/4 is a Pythagorean interval, even though it is not. Nonetheless, if we allow this interval, to form something called Ptolemic tuning, which is a modification of Pythagorean tuning, also knows as 5-limit, then you can use the interval 5/4 and all the intervals you can generate with it, in conjunction with the Pythagorean intervals. The Pythagorean major third, tuned to 81/64, exceeds a just major third, tuned to 5/4, by a small interval called a syntonic comma, of ratio 81/80. This is important, because the syntonic comma is a tiny bit smaller than the Pythagorean comma, but is still almost exactly the same size as the Pythagorean comma. The difference between the two is called a schisma, and it is less than 2 cents, so it is an inaudible difference. As such, a Pythagorean comma is a syntonic comma and a schisma. If instead of using 53 Pythagorean commas, we use 31 Pythagorean commas and 22 syntonic commas, then the difference between these and the octave is not 12 Mercator commas, but 12 Mercator commas minus 22 schismas. This new comma is an extremely tiny comma that measures less than 0.4 cents! As such, this circle of fifths is indistinguishable from an unbroken circle of fifths. Since the syntonic comma and the Pythagorean comma are indistinguishable, the scale produced here sounds indistinguishable from 53-equal temperament. This means you can play in all keys, you can form harmony, and all the important intervals are perfectly in tune. You can even play polytonal or atonal music with this. This type of music has been rarely used by American composers, but has been used widely in Chinese music, Ottoman music, and Turkish music.
In your opinion what is the most harmonically perfect tuning system?
@@GravytyMusic 53-EDO
@@angelmendez-rivera351 thank you very much!
I am full of admiration at such scholarly expertise . I have tuned guitars with wiggly frets in Kirnberger III, Young, and Werkmeister. But guitars hardly ever play in difficult keys so I may try Pythagoras as well . These frets are just bronze wire ,glued on with Evostik .All very reliable and durable . But a soldering iron will release them easily .
One guitar was accurately tuned to Equal Temperament and the frets were nowhere near in line .That plays well in all keys .
All notes were checked with a Korg Orchestral Tuner.
The Well Tempered Clavier, by Bach, gave us equal temperament. But does anyone really believe it took that long to figure out? Nobody is even looking at this video though, they are too distracted.
Wrong . Get to the back of the class .
Hello sar I'm Indian 🇮🇳🇮🇳🇮🇳
Welcome to Discover Maths!
@@discovermaths THÂÑK YŒU SÃR 🙏🙏🙏
After 53 fifths you do get really close though.
Yes. 53-EDO is indistinguishable from Ptolemic just intonation. This video is bordering on misinformation.
Trigo function sir. 😐😐
This Wednesday.
That mathematical synthesis at the end is exactly what I wanted to finish wrapping my head around the problems with this tuning. It's always bothered me how music can't be "perfect", how we allow for certain deviation of the "pure" intervals, in service of coherence. But after getting the problem with this temperament as well as others, I do believe 12-tone equal temperament, at least for the uses that it's given within a mostly Western framework, is really the best way to go. Thank you for this series
Assolutamente no.
Dal punto di vista filosofico infatti, adottare a tutti i costi una "manipolazione arbitraria" dei suoni conduce a un risultato sbagliato, contraddittorio quanto non degenerativo per la società. E per cui è e rimane una scelta sbagliata (anche se serve a giustificare inopportunamente il prurito, i vezzi, le farse e l'idealismo degli artistoidi, finti artisti...).
L'errore infatti è e rimane sempre un errore.
E lo rimane ancorché gli si dia una veste e una giustificazione matematica "apparentemente rassicurante per la maggioranza".
Infatti, anche i sofismi rimangono sofismi, nonostante impieghino motivazioni "matematematicamente rassicuranti".
LJC
Music absolutely can be perfect. Ottoman music uses just intonation. Turkish music does as well, and even some forms of Chinese music do. The Pythagorean comma is not actually the huge problem it is presented to be. There are many workarounds. Also, this video is historically inaccurate. Equal temperaments were not invented to solve the problem described in the video. That problem was solved by meantone temperaments, and later, by well-temperaments. Equal temperaments came later, in the 20th century, and we started using them for reasons that have absolutely nothing to do with Pythagorean tuning. Also, the video's description of Pythagorean tuning is not accurate, because 5/4 is not a ratio that exists in Pythagorean tuning.