Wow! This is the most concise, yet complete explanation of tuning systems, their limitations and practical implications I've ever seen. I've struggled with this topic since I began studying music nearly 25 years ago. I never quite understood it all and just lived with the idea that I probably didn't need to understand it to make enjoyable music. This, and your video on the origin of our 7-note major scale, really clarified a lot of things for me. Thank you so much!
Good work. This is perhaps the best short introduction to tuning and temperament I've seen so far. Subscribed. Towards the end, you point out that "you just can't win". I like to put it mathematically: no power of two is also a power of three or five. Even God can't change that, if She's logical. Being a tuning freak who mostly plays heptatonic scales, I play around a lot with tuning. Your description of 1/4 tone meantone was right on, but for me, the fact that C to D is 9/8 and D to E is 10/9 is a feature, not a bug. It feels harmonic to sing up a scale of gradually smaller intervals, as I do in the first five notes of my 1, 9/8, 5/4, 11/8, 3/2, 7/4, 15/8 scale. I can even sing the 11th harmonic pretty well in tune by just sort of sliding up that fifth. I haven't played in equal temperament for decades now. I'm perfecty happy to stay in restricted tonalities, I do other stuff to make my music interesting. Not that I have anything against 12TET, it's a wonderful comprimise that's indespensible for lots of music. But singing those thirds is a bitch. cheers from sunny Vienna, Scott
Nice and clear video! Only one little comment: at 15.08 you say: "the next value of a better approximation of the true interval ratio's is 30" But actually it is 31. 31 meantone fifths almost correspond with 18 octaves.
A lot of people say I'm consistently out of tune :) One time on the radio they played the lowest note on a pipe organ and the highest note several octaves above. They were out of tune with each other by quite a lot. Why is this ? Does it mean they didn't use equal temperament ?
Thank you for this approach to explaining the relationship between the nomenclature and the physics. I think mainly of the frequencies not the letters. Also if you spend time with an acoustic guitar, you can find the sympathetic lower frequencies relative to the "child" note. For example, if you play the harmonic on the high e 12 fret and then mute it, you will hear the A string 3rd fret C note harmonic ringing. My guitar teacher has never been able to explain this. How does the 5th A string, 3rd fret harmonic ring by itself? You can map almost the entire fretboard this way. This is were my love for music comes from. The structure is preordained. What are your thoughts?
I checked the ratio because I noticed that 202.7/200 isn't the same as 531.441/524.288 I did the math and the ratio comes from 3^12/2^19 (which is correct, you can derive this from the table you used to explain Pythagorean tuning. So the only error is the "=" sign at 6.04. And for me (I'm a math teacher) an "=" sign is important. Lef & right must be exactly the same.
Great vid! I would only comment that an instrument that has an infinite number of notes is literally any instrument besides a keyboard/fretted instrument (to an extent even guitarists can bend certain notes). All others have the capability to (and are trained to) make micro adjustments in real-time depending which note of the chord they're playing. For example, it's a rule of thumb to slightly raise the fifth/minor third and lower the major third. This is where ear-training is so important!
i like the duality if you want to be in tune and take advantage of frequencies positive relationships, either make something boring or have infinite notes, thats your two choices. i have an idea, you could attempt something in the middle. have we forgotten the concept of the training a single kick a thousand times vs a thousand kicks 1 time?
Thank you for the nice video. Referring to the table at 04:35 one can see that also the pentatonic circle C-G-D-A-E (denoted by 5-TET, a chinese scale) and the "Lydian" circle C-G-D-A-E-B-F# (7-TET) are worth considering with the out-of-tune error of 10.2 Hz and 2x6.8=13.6 Hz, respectively, when compared to our beloved western 12-TET scale with the error 2.7 Hz. The next local optimum is the impractical 53-TET scale/circle with the error at about ten times smaller that 12-TET.
I'm no expert so I have a question. In the Pythagorean tuning system, is the "octave" of the reference C (100 Hz) slightly higher because it actually corresponds to B sharp (202.7 Hz)? Because when we keep going with the fifths, we have C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B# and if we keep not using enharmonics, it will continue with F×, C× and so on. So every time we come to the next F, there will be one more sharp added to the group of the seven notes and we will eventually see triple and quadruple sharps etc. In this situation, it will not be a circle of fifths, but a *spiral* of fifths since none of the notes will reoccurr and it'll keep gettin sharper as we go up and flatter as we go down (C-F-Bb-Eb-Ab-Db-Gb-Cb-Fb-Bbb-Ebb etc.). Which looks mesmerising to me.
I have probably seen almost every listed video on this topic and this is by far the best explanation. Clear, concise and not afraid to get into the math. You've answered questions I've had for decades. Much gratitude to you Sir 🙏🏽
Great explanation of the various systems and, importantly, what motivated them and their limitations. Really clearly presented. I did notice one small problem when you introduced equal tempered tuning at around 14:00: you say and write that the nth root of 2 can also be written as 1/2^n, which is not true. I think you meant to write 2^(1/n). Similarly, 1/2^12 is definitely not 1.059… it is 1/4096 which is much less than 1.
Thanks for these great videos! Quick question as I'm trying to understand the harmonic series and just intonation: I see the list of just intonation ratios (3:2 for a perfect 5th, 5:4 for a major 3rd, etc.), but I'm curious which overtone in the series corresponds to each of these ratios. Do you have a list or know where I can find one that would give this info? Some of them are clear to me. For example, I see that the 2:1 ratio of an octave first appears with the first overtone, and the 3:2 ratio occurs with the next overtone. But what about the 4:3 ratio of the perfect 4th? Which overtone corresponds to that? Thanks again for all the great learning here!
the fourth appears as the interval *between* the third harmonic (the p5) and the 4th harmonic (the octave). Then the major third between 4 and 5, then the minor third between 5 and 6. The sixths are inversions of the thirds (and the fourth is also an inversion of the 5th).
So I could theoretically sound more in tune if I retune my guitar for each key that I play in? So if I play a song in C major then tune from the 8th fret of the low E string. If I’m playing in E then tune from open e string?
It won't work. You may just keep it as it is. The guitar is a very imperfect instrument to tune well. Firstly, because it's strummed by hand, so the (A) differences in your touch against the strings, (B) differences in pressing against the frets, (C) bending of the strings, (D) bending of the neck, etc. already create too many inconsistencies. To minimise some imperfections, you may try using the pick (plectrum), or use always the fresh set of strings, but the problem is not alleviated, it just becomes a slightly different problem. But we play the guitar just because it is very imperfect; more perfect instruments (like the modern piano or the electric piano) are boring. We humans enjoy more in "finding the desired frequency" through series of small corrections (which add flavour to playing) than playing it straight away.
@@zvonimirtosic6171 I understand all of that, but my question wasn’t about making a guitar perfectly in tune or trying to improve the sound of the guitar, it’s if theoretically the guitar would be more in tune for the specific key, which it would be.
@@Zach-ls1if Some guitars lend tendencies towards certain scales and can accentuate moods. Say, the Spanish guitars with gut / nylon strings, produces certain overtones and harmonics (or lack of them) that work well in minor keys. The electric guitar, it may play the same notes, but it will never sound as mournful; the electric guitar may weave through all minor keys, and it sounds like rock'n' roll; it all sounds like major C to D.
@@zvonimirtosic6171 definitely tendencies, but there’s a lot of grey area obviously. Like what Mexican Sierreñas do with 12 string guitars, they get a completely different sound out of it than any other guitar, and if most people hear it without seeing it they can’t figure out how the sound is made. It’s pretty interesting. Listen to Miguel y Miguel “el cisne “ or Ariel Camacho “el karma” and take a listen. They do something with the guitars that most “experts” and 12 stringers say can’t be done
Typo at 14:00 - forumla should say 2^(1/n) NOT 1/2^n
Apologies!
Wow! This is the most concise, yet complete explanation of tuning systems, their limitations and practical implications I've ever seen. I've struggled with this topic since I began studying music nearly 25 years ago. I never quite understood it all and just lived with the idea that I probably didn't need to understand it to make enjoyable music. This, and your video on the origin of our 7-note major scale, really clarified a lot of things for me. Thank you so much!
i really hate pythagoras
Good work. This is perhaps the best short introduction to tuning and temperament I've seen so far. Subscribed.
Towards the end, you point out that "you just can't win". I like to put it mathematically: no power of two is also a power of three or five. Even God can't change that, if She's logical.
Being a tuning freak who mostly plays heptatonic scales, I play around a lot with tuning. Your description of 1/4 tone meantone was right on, but for me, the fact that C to D is 9/8 and D to E is 10/9 is a feature, not a bug. It feels harmonic to sing up a scale of gradually smaller intervals, as I do in the first five notes of my 1, 9/8, 5/4, 11/8, 3/2, 7/4, 15/8 scale. I can even sing the 11th harmonic pretty well in tune by just sort of sliding up that fifth.
I haven't played in equal temperament for decades now. I'm perfecty happy to stay in restricted tonalities, I do other stuff to make my music interesting. Not that I have anything against 12TET, it's a wonderful comprimise that's indespensible for lots of music. But singing those thirds is a bitch.
cheers from sunny Vienna, Scott
"The first formal tuning system" was created in India long before Pythagoras.
Nice and clear video! Only one little comment: at 15.08 you say: "the next value of a better approximation of the true interval ratio's is 30" But actually it is 31. 31 meantone fifths almost correspond with 18 octaves.
A lot of people say I'm consistently out of tune :) One time on the radio they played the lowest note on a pipe organ and the highest note several octaves above. They were out of tune with each other by quite a lot. Why is this ? Does it mean they didn't use equal temperament ?
Thank you for this approach to explaining the relationship between the nomenclature and the physics. I think mainly of the frequencies not the letters. Also if you spend time with an acoustic guitar, you can find the sympathetic lower frequencies relative to the "child" note. For example, if you play the harmonic on the high e 12 fret and then mute it, you will hear the A string 3rd fret C note harmonic ringing. My guitar teacher has never been able to explain this. How does the 5th A string, 3rd fret harmonic ring by itself? You can map almost the entire fretboard this way. This is were my love for music comes from. The structure is preordained. What are your thoughts?
I checked the ratio because I noticed that 202.7/200 isn't the same as 531.441/524.288
I did the math and the ratio comes from 3^12/2^19 (which is correct, you can derive this from the table you used to explain Pythagorean tuning. So the only error is the "=" sign at 6.04.
And for me (I'm a math teacher) an "=" sign is important. Lef & right must be exactly the same.
Great vid! I would only comment that an instrument that has an infinite number of notes is literally any instrument besides a keyboard/fretted instrument (to an extent even guitarists can bend certain notes). All others have the capability to (and are trained to) make micro adjustments in real-time depending which note of the chord they're playing. For example, it's a rule of thumb to slightly raise the fifth/minor third and lower the major third. This is where ear-training is so important!
Very well explained. Something explained badly decades ago now clearly done. Thank you.
i like the duality if you want to be in tune and take advantage of frequencies positive relationships, either make something boring or have infinite notes, thats your two choices. i have an idea, you could attempt something in the middle. have we forgotten the concept of the training a single kick a thousand times vs a thousand kicks 1 time?
Thank you for the nice video. Referring to the table at 04:35 one can see that also the pentatonic circle C-G-D-A-E (denoted by 5-TET, a chinese scale) and the "Lydian" circle C-G-D-A-E-B-F# (7-TET) are worth considering with the out-of-tune error of 10.2 Hz and 2x6.8=13.6 Hz, respectively, when compared to our beloved western 12-TET scale with the error 2.7 Hz. The next local optimum is the impractical 53-TET scale/circle with the error at about ten times smaller that 12-TET.
I'm no expert so I have a question. In the Pythagorean tuning system, is the "octave" of the reference C (100 Hz) slightly higher because it actually corresponds to B sharp (202.7 Hz)? Because when we keep going with the fifths, we have C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B# and if we keep not using enharmonics, it will continue with F×, C× and so on. So every time we come to the next F, there will be one more sharp added to the group of the seven notes and we will eventually see triple and quadruple sharps etc. In this situation, it will not be a circle of fifths, but a *spiral* of fifths since none of the notes will reoccurr and it'll keep gettin sharper as we go up and flatter as we go down (C-F-Bb-Eb-Ab-Db-Gb-Cb-Fb-Bbb-Ebb etc.). Which looks mesmerising to me.
456th like
Very interesting video, Thank you. I watch the first part, I am going to come back and watch bit by bit. Cheeers!
I have probably seen almost every listed video on this topic and this is by far the best explanation. Clear, concise and not afraid to get into the math. You've answered questions I've had for decades. Much gratitude to you Sir 🙏🏽
12 fifts dosent doubble the octave, but 12 3√2 does (cube root of 2). I just dont know how to apply this to a guitar ... Can you help me plz ?
Great explanation of the various systems and, importantly, what motivated them and their limitations. Really clearly presented.
I did notice one small problem when you introduced equal tempered tuning at around 14:00: you say and write that the nth root of 2 can also be written as 1/2^n, which is not true. I think you meant to write 2^(1/n). Similarly, 1/2^12 is definitely not 1.059… it is 1/4096 which is much less than 1.
Ah yes! Typo on my part! Sorry.
Thank you for vid. You saved me from having to go to my musical instruments class today! :D
17:55 Doesn’t a string instrument technically have an infinite number of notes?
Hi, amazing video! Just wanted to know how u calculated the frequencies from the Pythagorian tuning system to the interval ratio?
When you need to feel integrated in the explanation process...7:35 🤭🤣
Great break down. Its like the math guys say about Pi and 3. Close enough for our purposes! Greetings from New Mexico!
Sounds more like something that a physics guy would say
Thanks for these great videos!
Quick question as I'm trying to understand the harmonic series and just intonation: I see the list of just intonation ratios (3:2 for a perfect 5th, 5:4 for a major 3rd, etc.), but I'm curious which overtone in the series corresponds to each of these ratios. Do you have a list or know where I can find one that would give this info?
Some of them are clear to me. For example, I see that the 2:1 ratio of an octave first appears with the first overtone, and the 3:2 ratio occurs with the next overtone. But what about the 4:3 ratio of the perfect 4th? Which overtone corresponds to that?
Thanks again for all the great learning here!
the fourth appears as the interval *between* the third harmonic (the p5) and the 4th harmonic (the octave). Then the major third between 4 and 5, then the minor third between 5 and 6. The sixths are inversions of the thirds (and the fourth is also an inversion of the 5th).
So you can't tune a piano OR piano a tuna?!?!?
So I could theoretically sound more in tune if I retune my guitar for each key that I play in? So if I play a song in C major then tune from the 8th fret of the low E string. If I’m playing in E then tune from open e string?
It won't work. You may just keep it as it is. The guitar is a very imperfect instrument to tune well. Firstly, because it's strummed by hand, so the (A) differences in your touch against the strings, (B) differences in pressing against the frets, (C) bending of the strings, (D) bending of the neck, etc. already create too many inconsistencies. To minimise some imperfections, you may try using the pick (plectrum), or use always the fresh set of strings, but the problem is not alleviated, it just becomes a slightly different problem. But we play the guitar just because it is very imperfect; more perfect instruments (like the modern piano or the electric piano) are boring. We humans enjoy more in "finding the desired frequency" through series of small corrections (which add flavour to playing) than playing it straight away.
@@zvonimirtosic6171 I understand all of that, but my question wasn’t about making a guitar perfectly in tune or trying to improve the sound of the guitar, it’s if theoretically the guitar would be more in tune for the specific key, which it would be.
@@Zach-ls1if Some guitars lend tendencies towards certain scales and can accentuate moods. Say, the Spanish guitars with gut / nylon strings, produces certain overtones and harmonics (or lack of them) that work well in minor keys. The electric guitar, it may play the same notes, but it will never sound as mournful; the electric guitar may weave through all minor keys, and it sounds like rock'n' roll; it all sounds like major C to D.
@@zvonimirtosic6171 definitely tendencies, but there’s a lot of grey area obviously. Like what Mexican Sierreñas do with 12 string guitars, they get a completely different sound out of it than any other guitar, and if most people hear it without seeing it they can’t figure out how the sound is made. It’s pretty interesting. Listen to Miguel y Miguel “el cisne “ or Ariel Camacho “el karma” and take a listen. They do something with the guitars that most “experts” and 12 stringers say can’t be done
@@zvonimirtosic6171 thank you for the conversation btw
34edo, 41edo and 53edo ftw
This series kicks ass!
𝐩𝓻Ỗ𝓂Ø𝓈Ｍ 🤷