Pythagorean vs Just Intonation | A Paradox of Playing in Tune 

It's all wrong! Only historical facts... All the notes must to have constant place like the A440Hz. Only the unisson must be perfect. All others intervals we must avoid to be in just intonation. Yes, the octave also! 12ET is the best way, but only without a just octave! We need a true reference how to stay in tune, an universal temperament! Nobody today ne propose different mesures for time or distance, stop please to lead people in wrong way!

pythagorean is a type of just intonation btw

A very nice short introduction to this rather complex issue. I would just add, that the conflict between Pythagorean and Just Intonation is not really a paradox; it's a simple consequence of the math. No power of two is also a power of three: thus no "circle" of perfect fifths can close on an octave. Also, no power of three is also a power of five, making it impossible to combine perfect fifths and just thirds in a circle. Cheers from a tuning and temperament freak in Vienna, Scott

A solution that i have never heard is to have more then one tuning for certain notes and as you move from a note in one context the pitch could change to match another context. For example lets say you play Dm7 and then CM9 now both chords share the C note and the D note. So we have three choices. Either the chords will not be just tuned or the C note will change it's tuning or the D note will change it's tuning. Let's say you are playing C major key so it would make sense to keep the C note the same tune so then when play CM9 you would play 9/8 for the D note but when you are playing Dm7 you would play 10/9. So as the D note from the Dm7 goes to the CM9 it would change 20 cents. Has anyone tried such a solution? And does it sound good? If it sounds good then there is no paradox at all. Well at least to those people who think it sounds good.

The piano is not tuned to an exact equal temperament. It would sound out of tune with itself. As notes go lower, they are not tuned exactly to the note an octave above. Instead, the slightly sharp octave overtone is tuned to that note. The result is that piano notes get flatter and flatter as you go down the keyboard. This is called “stretch” tuning. Wasn’t it Casals who told his students not to play in tune to a piano? Choirs singing a cappella get flatter and flatter over the course of a piece if they sing in tune vertically. You can blame math for that one, too. BTW, I heartily agree with her book recommendation. I read it a few years back and found it entertaining as well as informative.

I don't think this video is true for music as a whole because whether or not to use Pythagorean tuning for the melody depends on the music and taste of the individual. For example it could be the reason why we prefer Pythagorean tuning for the melody is because we are used to the Equal temperament tuning which is much closer to Pythagorean tuning than to just intonation. But when it comes ti harmony the pleasent sounds of just intonation are too strong for cultural influence to have an effect. Also by focusing on leading tones we may be skewing our view. It might say sound like a stronger resolution to have a sharper fifth but there may be other benefits in having it a bit flatter.

I have a question, if I play a major 3rd with C and the open E string, would that 3rd be in just intonation or in equal temp? Because the same 3rd played an octave lower (the one of the Bach Andante), you say it must be a pure major 3rd to be harmonically in tune, but wouldn't that E be a little lower compared to the open E string? For example if I have to play a scale of 3rds (C major), which of the two 3rds would be most appropriate? Sorry if I’m confused

I so wish you had played musical examples. You provide a good explanation, but it's abstract.

How about 53TET ? Nearly perfect Pythagorean fifth (31/53TET = 701.89 < 701.96 = JI 3:2) and fourth (JI 4:3 = 498.04 < 498.11 = 22/53TET ) and an excellent meantone third (17/53TET = 384.9 < 386.3 = JI 5:4). Pythagoran Db and C# and Meantone C# and Db (4/53 and 5/53).

Bravo, Inna!! 👏🏻👏🏻 Very clear explanation of the two systems! I can’t wait to see their application to some repertoire in the next videos 🎶🎶

So they make clickbait books too nowadays? Cool.

ugh my head is spinning. so basically I have to play out of tune to play in tune, in equal temperament and in tune is out of tune in Just intonation and vice versa depending on the tuning?? ughhhhh what about playing so that it sounds in tune, whenever I play anything? and make adjustments(cello) sharp or flat as needed?

What system should I use when tunning open strings?

5-limit JI is not out of tune melodically. That's a very curious statement. A 9/8 whole step is just as in tune as a minor 10/9 whole step, and a 16/15 half-step is just as in tune as a 256/243 half-step. Maybe Western musicians and audiences aren't as accustomed to the sound of the 5-limit JI melodic intervals (10/9, 16/15) as much as they are to the 3-limit JI/Pythagorean ones (9/8, 256/243), which are better approximated by 12-TET, but this in no way makes these intervals "horizontally out of tune". Being in tune is, after all, a function of musical intention, and there is no established intention of emulating exclusively Pythagorean intervals in Western music as far as I am aware. In addition, the example of Arabic classical music goes to show that even the very wide 88/81 (3/4 of a whole step in 24-TET) interval can be successfully conceived of and used as a leading tone.

Best video on youtube about intonation. Thank you, also for the book recommendation.

Everything you explain is so amazing! I wish I understood anything you say! :D

So awesome! Great coverage of a fundamental concept of tuning but also important to understanding microtonal as well as non-western music! You rule.

How is it that the octaves aren't in tune? Is that in relation to equal fitting tuning?

That chart at 0:52 deserves its own video!!!

I have a question, when playing thirds in violin solo repertoire, which voice is the one that remains in the pytagorian system?, example, when i practice 3rds i always keep the lower note in pytagorian and the higher note is the one that i move aroun to tune the third, but i dont know if i am doing it wrong, should i keep the higher note in pytagorian and move around de lower one?

Thanks Inna. Very helpful to understand this a bit better.

Please stop talking about it and demonstrate what it sounds like. Sounds not speech.

Streaming your channel now!!!

I would think that a string quartet that adds a piano would necessarily have to play in Equal Temperament and not Pythagorean or Just Intonation. As you put it it seems like it only has to do some adjustments. Would those adjustments entail playing in just intonation in some passages and Pythagorean in others and on Equal Temperament in others ? If I made some sense, would that mean that all 3 tunings can coexist in the same piece ?

Quick sub

A Pythagorean major third is 81/64. A just major third is 5/4, or 80/64. The third scale tone is Ground Zero for tuning, because it can either be a Pythagorean note in a melodic line, or a just part of a sustained tonic triad.

Great video!! Thank you for comparing these two systems. Really interesting comparison!

So beautiful and wonderful talent

Brava. This is great information on this subject.

THIS SHOULD HAVE A MILLION VIEWS

This is such a great topic! Very interesting and useful!

Inna, very good discussion.

Great treatment to this fascinating topic!!!

I m so glad i found you

Great video, informative and very well done!!!

This is such a great video with amazing insights!

Another AmaZing video!

Thanks Inna! That was a great explanation!

I bought that book. Thanks

BRAVISSSSIMOOOOO, I LOVE IT!!!

So interesting again!

super great vid

This needs more views!!

Wonderful video!

Thank for sharing,,,, 😍

The concept is sound.

Bravissima!

I know no one else will tell you this, but it is not a paradox. Pythagorean tuning is always right. All the notes on the page are Pythagorean notes and should be tuned as such. A chord of C, E and G should have the E 81/64 and G 3/2 above the C. E should never be 5/4 ratio EVEN IN CHORDS, even in double stops. The 5/4 is a note of timbral distinction, NOT a note of the scale, not a note of pitch class, it is an overtone of timbral distinction. Does the 81/64 major third beat, yes, more or less depending on the timbres of the instruments sounding, but it is always in tune. Even it's beats are whole number ratios to the fundamental. The 81/64 gives a difference tone of the 17th harmonic a few octaves down. This type of beating is totally different to the irrational nonharmonic beating of equal temperament. 5 Limit just intonation is out of tune melodically, while giving a nice buzz harmonically, BUT both the vertical listening and horizontal listening must be used to determine tuning, and while simultaneously taking in both directions, Pythagorean is the true intonation in all instances. When you think about it, it is the only thing that makes sense. Our entire music notation system and conception is Pythagorean tuning. There are no other accidentals than sharps, flats, naturals, and their compounds. Pythagorean tuning gives you every note with a name. 5 limit tuning gives you detuned versions of all those same notes. This confusion and lie about 5 limit tuning being the standard for harmony is what has lead to equal temperament and the destruction of all harmony. 5 limit tuning has a comma in the diatonic scale, and the two sizes of whole tones makes the music out of tune, unless you condition yourself only to listen to harmony and not melody. If you listen in both dimensions, it is clear 5 limit is out of tune. If you want to play in tune you must use true intonation, what the world calls Pythagorean tuning (even though Pythagoras did not discover this tuning). I just call it what it is True Intonation. Contrary to popular teachings of today, an orchestra can end on a major chord tuned with true intonation, and it will sound completely at rest and in tune, and MUCH BETTER, than if they used 5 limit. The going back and forth between 5 limit and 3 limit in classical music creates a disjointed spirit in the music. When I say Pythagorean or true intonation, I am not talking about 12 tone Pythagorean, but Extended Pythagorean tuning, even with 53 notes per octave, from Ebbbb to Cxx on a chain of fifths.

This is good. But if you haven't already, watch some other music instruction channels on what they do to make their videos more watchable for repeated subscribers. You talk really fast and almost look uncomfortable.