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
Tuning is made way more complicated than it really is. What is called just intonation in this video is not really just, musically speaking. When put to the test, it is revealed to be totally incoherent. This 5 limit tuning which people call as just intonation is relatively recent. Pythagorean tuning was the standard for at least 2000 years. And yes, it works well for chords, melodies, double stops. Its major thirds are not too high, they are ideal, exactly at 81/64 ratio. What mpst do not understand is that the 5/4 is not a pitch class, it is not a major third. It is an overtone, one that is out of tune if you tune a fundamental pitch in your scale or chord to it. 5 limit rational tuning is melodically completely out of tune for mny reasons. Whereas Pythagorean Just Intonation with 17 notes available per tonic chord, this is perfectly in tune both melodically and harmonically at the same time. The only thing is, you must listen holistically, youmust listen horizontally to melodies (even within the voices of chords), AND you also must listen vertically to each instance of harmony at the same times each moment as you listen. Then it beomes perfectly clear. Unfortunately when we take up an interest in tuning and intonation, we tend to examine intervals in a vacuum, that it, not in context of real music with real timbres. And while the 5/4 sounds buttery and smooth as an interval out of context, in context of real music with real instrument timbres, it is woefully flat sounding, and the constantly varying step sizes, notes having multiple frequencies, no limit to the amount of notes in a key, commas appearing, and drifting of the tonic, the buttery smooth buzz of the 5/4 in no way makes up for the avalanche of other problems it causes. 81/64 major thirds are ideal for melody and harmony. Look on my channel, I have many many examples.They are not too wide as so many love to chant on repeat. It is important to have understanding when it comes to music, and so long as you are messing around with making the 5/4 the major third, your understanding is going to be limited at best, when it otherwise would not be. When musicians talk of keys like C major or E minor, it is very important for musicians to know what a "key" even is, and most dictionaries get it wrong. A key is 7 notes derived from an unbroken chain of six 3/2 fifths (or a close approximation of the 3/2, but ideally 3/2). That is it. So a major third is defined as the pitch class obtained from going up 4 perfect fifths. All musical pitches can be expressed as whole number ratios by some power of 3 in relation to some power of 2. A major third is 3 to the fourth power over 2 to the sixth power. One you understand what a key is. They it is important to know what chromatic notes are available per each tonic triad. Most people think there are 12 chromatic pitches, this is wrong. There are 17 chromatic pitches per tonic chord. There are 7 naturals, 5 flats, and 5 sharps in C major. And these 17 notes modulate up or down the chain of fifths when the tonic triad changes from major to paralell minor (or any of the paralell modes), or when the tonic note itself changes. So for example, and you will see this in real sheet music today, in C major, you can have Gb through to A# in a chain of fifths. So the scale from lowest pitch to highest would be: C, Db, C#, D, Eb, D#, E, F, Gb, F#, G, Ab, G#, A, Bb, A#, B, (C) Like I said this scale modulates with the key. The above 17 note vicinity scale is not just for C major, but also for A minor, E Phrygian, D Dorian, etc., all the relative modes. But for major, it is from the b5 though to the #6 in the chain of fifths. When you look at any of the renowned composers, especially the older ones, but not exclusively, they were careful to spell notes properly, because while they did not play in tune, they understood that the notes were informing of the underworkings of music and how tuning it supposed to be originally. You won't find an Cx in C major, you won't find an Fb in C major, but D and E respectively. But you will find both C# and Db in C major without pushing the ear to hear that the tonic chord has changed.
By default, most violinists tune to pure 5ths (i.e. Pythagorean). However in a string quartet it's common to very slightly brighten up G. And in baroque ensembles it might be completely different tuning (e.g. tuning each string individually to the harpsichord).
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).
41edo is also a viable alternative Pythagorean tuning system. Its fifth is at the interval 24\41, about 702.44 cents, only 0.5 cents sharper than pure 3:2 (although, 53edo does better than 41 in its fifth). Its fourth is at the interval 17\41, about 497.56 cents, 0.5 cents flatter than the pure 4:3 fourth. Its 5:4 approximation is slightly worse than 53edo's at 380.49 cents instead of the ideal 386.31 cents. Although 53edo does better than 41 in the 5-limit, 41 does better than 53 in the 7-limit and higher, so this could be an alternative tuning system to both Pythagorean tuning and 53edo tuning (it's also fewer notes).
All equal temperaments are bad, especially when you could tune purely. 53 is among the best of the smaller number edos, but ONLY if you use it correctly, which basically no one does. People usually take 53 edo and make Fb the major third and make the E the diminished fourth above C, the exact opposite of what is just. The most important thing anyone can know about tuning is that Pythagorean just intonation (that is, with correct note spellings, acknowledging the 17 note chromatic vicinity encompassing each key in the chain of fifths) is True Intonation, is idea just intonation, in melodies AND chords. Check my channel for examples. And the thing is, if you are already having 53 notes per octave, you are better off with pure Pythagorean imo, having 53 notes per octave, it will sound better, albeit quite subtly. And at the two ends of that chain of 53 fifths, is what is called a Mercator Comma. If D is in the exact center of your chain of 53 fifths, then on the flat end you will have Ebbbb and on the sharp end you will have Cxx. Ebbbb is less than 4 cents flat of a perfect 3/2 above Cxx. Ebbbb is enharmonic to Gxx, they are almost the same note. Like wise Cxx is enharmonic by the mercator comma nearly a perfect fifth below Ebbbb. Cxx is enharmonic, almost the same pitch as Abbbb. So if for some reason you wished to modulate really far, you could "jump" or modulate over that mercator comma and no one would hear that you are now on the other end of the chain of fifths.
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.
Yes, actually your solution is exactly what I discuss in part 2 of this, in rep examples. When I mentioned the word "paradox" in my title it was more along the lines of "there is no one single way to tune a note" and "there is no such thing as perfect intonation"
The solution has been under your nose all along. Play EVERYTHING in Pythagorean. 5 limit tuning is incoherent and is constantly creating puzzles that need solutions that are filled with compromise. It is an incoherent conception of intonation when it comes to music when you really put it to the test. Pythagorean Just Intonation, or what I more aptly call True Intonaitoin, sounds perfectly in tune in chords, melodies, and double stops. It is what is displayed in sheet music and what is ideal in tuning. This 5 limit thing is a more recent trend of the last few hundred years and is the reason why all music is out of tune. There are examples of this tuning in tertian harmony on my channel. Make sure when dealing with intonation to listen holistically. That is both melodically, each voice moving from one note to another processing that over time, AND at the same time listen to the sounding harmony every instance. If you can allow your brain to do this, and it must be trained to NOT do this, you will hear Pythagorean sounds perfectly in tune every time. Sadly we who become interetsed in intonation usually unknowingly condition ourselves to only listen vertically to harmony in an instance and in no way track the melodies that make up chords and process that information over time. We are stuck in the moment, but music is primarliy something which unfolds over time. That is the melodic intonation awareness that must be acknowledgeed even within chords. Pythagorean, or True Intonaiton, handles melody and harmony perfectly, 5 limit creates confusion and puzzles of compromise and commas.
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?
It is the only just intonation, btw. The others are rational intonations, but not just in the ultimate sense. Just in a physics sense, but not a musical sense. Pythagorean just intonation should be used in all melodies and chords and double stops and tuned according to the score (unless there is a misspelling) as the note sits in the chain of 3/2 ratio fifths.
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
You are correct! I actually talk about this exact problem in Bach Andante in part 2 of this topic, with repertoire examples (should be linked in description box somewhere). It's not possible to play this Bach opening "100% in tune" and different players interpret what sounds best, where there is more than one compromise that's possible - this is where intonation actually becomes subjective!
@@esteban100 Always go with Pythagorean Just Intonation, not 5 limit tuning (which this video calls just intonation), and also not equal temperament. If you tune Pythagorean it will sound best, every time, and knowing which note to play and how it is tune will be much much simpler An E is an E is an E. The note C is always the same frequency in its given octave. Fb is Fb. B# is always B#. Each note name always has the same frequency in its given octave. Everything is then clear, not just in your mind, but it will reflect perfectly what is on the page, and will sound perfectly in tune to the audience. Check out some of the examples on my channel. True Intonation is Pythagorean Just Intonation.
This video is totally wrong. Pythagorean Just Intonation is what is just intonation and is always the right tuning, even for chords. 5 limit tuning is founded on a relatively new and backwards way of conceiving of harmony that has resulted in basically all music being out of tune in the world. 5 limit harmony, when put to the test is totally incoherent as a musical system of pitches, and is melodically badly out of tune. And using both Pythagorean and 5 limit tuning alternating is confusion because the note C is C. There are no pluses or minuses on the page, no up and down commas, all that is a recent invention. Even meantone tuning, invented and popularized by Jesuitical Roman Catholics, is based on a perversion. It just declares by fiat that the major third should be a 5/4 ratio, and then perverts and detunes the 3/2 perfect fifth to make that which is by nature a just major third, the 81/64 ratio, into the 5/4. The 5/4 is NOT a major third. All of these people are lying to you,, knowingly or unknowingly. The just major third is Pythagorean, it is 81/64.
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 ?
Correct - depending on the context of the notes in relation to harmony, the tuning systems can all live within the same piece. For example, an F# as a passing tone in a melodic G major scale would likely be Pythagorean. But the same F# would be in Just Intonation for a held D major chord. Any part that doubles one of the hands in the piano would have to opt for the equal temperament (or whatever the piano is tuned to).
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?
The exact opposite, you have to play in tune, but being in tune depends on the context. The notes of the 12-T system change based on what context you play in. Basically just use your ears to tune,
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.
It should always be the 81/64 Pythagorean for the major third. The 5/4 is not a major third, but simply the fifths harmonic. Check out my channel for examples of Pythagorean chords in real musical context. True Intonation is what the world calls Pythagorean Just Intonation.
@@RememberGodHolyBible I stand by what I said. If you're listening to a sustained tonic triad, such as at the end of a cadence, a 5/4 just third scale tone will fit the tonic and dominant tones, with fewer heterodyne beats than an 81/64 Pythagorean third scale tone. Singers and non-fretted string players make fine adjustments in their tuning depending on whether, as noted, they're tuning to a melodic line or to a sustained chord. If you happen to prefer 81/64 at all times, happy listening. Some people prefer equal temperament; to each his own.
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.
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.
Since the syntonic comma of 81/80 is tempered out in 12edo, most musicians (even those who have to check their intonation) think in terms of one whole tone and one half-step and not as two separate whole tones and half-steps separated by 81/80.
@@YoVariable It's true that 12-TET splits the difference. However, it doesn't do so evenly, like meantone. The 12-TET whole tone is negligibly lower than the Pythagorean one, while the half-step is indeed somewhere in the middle, but still a tiny bit closer to 256/243. This talk about JI being melodically out of tune just goes to show how the acoustic identity of the individual notes in the scale is tied to the Pythagorean system.
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.
Yes, it definitely depends on the situation! And different cultures are accustomed to hearing intonation differently. This video is a basic overview/introduction of the two types of tuning system and the general definition, if that makes sense.
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!
No, no its not. Its not sound or right at all. Pythagorean just intonation is what is desirable in every instance. 5 limit tuning, what she calls just intonation here is based on a wrong conception of harmony and creates all sorts of confusion, and is the reason why all our music is out of tune these days. Without people seeking after 5 limit and rejecting Pythagorean thirds, we would never have been sold on 12 tet. You can have perfect just intonation (Pythagorean) on acoustic keyboard instruments, AND modulate indefinitely. The issue is that people forsook that which was right because they became enchanted, seeking after this carrot, the 5/4 as the major third, which they can never reach, because by its very nature it causes problem after problem. It is outside the key yet people treat it like it is a diatonic note, and it is so close to the real diatonic note, that the ear hears it as that, even though it is not, and the confusion it has wrought is too much to really fathom. Because of this incoherence which cannot be solved in any truly satisfying way, it then opens the door to "anything goes" tuning or ever increasing amounts of temperaments to try or x-limit tunings to try. All which keep you on the merry-go-round, and for what , a tuning system that sounds bad and is also uneccessarily complicated. A key is 7 notes derived from an unbroken chain of six 3/2 fifths. Not the confusion described in 5 limit. There are infinte notes in the diatonic scale in 5 limit. It is confusion. The only way off and to land in something good, intonatioin wise, is to realize that Pythagorean tuning, 17 notes per octave per tonic triad derived from an unbroken chain of fifths, that is just intonation, and sounds by far the best in chords, melodies, and double stops, AND it stands up to scrutiny and testing in various pieces of varying complexity, styles, and timbres.
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.
While Pythagorean tuning has its strengths, especially in melodic contexts and with certain interval purity, the claim that it universally provides “true intonation” in all situations overlooks some key aspects: 5-Limit Just Intonation and Harmonic Consonance: In harmonic contexts, especially with triads and chords, 5-limit just intonation’s 5/4 major third is often preferred because it more closely aligns with the harmonic series and provides a purer consonance compared to the 81/64 major third in Pythagorean tuning. The 5/4 ratio gives a smoother harmonic blend, which is why it has been a cornerstone of Western harmony for centuries. Dismissing it as "out of tune" ignores its actual harmonic effectiveness. Beating and Dissonance in Pythagorean Thirds: While the 81/64 third produces a specific type of beating, it’s inaccurate to suggest this is inherently different or “better” than the beating in other tuning systems. Pythagorean thirds can create noticeable dissonance in harmonic contexts because they diverge from the harmonic series, particularly in music emphasizing smooth consonance. Practicality of Extended Pythagorean Tuning: The suggestion of using a 53-tone octave based on a chain of fifths may work in theory but is not practical for most musicians. Additionally, using a large microtonal scale doesn't inherently align with Pythagorean tuning principles used in traditional Western music, which typically adheres to a 12-tone system. The complexity added by microtonal options does not resolve the challenges of consonance in harmonic contexts. Historical Context of Tuning Systems: The assertion that 5-limit tuning led to the development of equal temperament is inaccurate. Equal temperament was developed to balance key modulation needs, enabling musicians to move between keys with minimal tuning inconsistencies. This allowed harmonic and melodic flexibility beyond what strictly Pythagorean or 5-limit systems could offer. In summary, each tuning system has its ideal uses, and claiming that Pythagorean tuning is “true intonation” in all situations oversimplifies the nuances of harmony and the different purposes of various tuning systems.
@@jenshoffmann2210 Man, the music speaks for itself. All your theory is one thing, but the music on my channel and in other places as well testifies to the contrary. Pythagorean thirds and intervals in general are not just usable in chords, they are ideal. This is not the case with 5 limit intervals, they sound okay in isolation but woefully flat when tested in various pieces of music and the listener is listening both for melodic and harmonic purity simultaneously. Western music is not based off of 12 notes per octave, regardless of what everyone else says repeatedly, believe your own eyes, not what parrots tell you. If you ar in C major, or any key, really there will be 17 possible notes available. This is what you will see when you study scores. In C major these are C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B. These are all the notes in the vicinity of C major in all western music. You won't find an Ebb in C major. You won't find B# or E# or Cb or Fb in C major, Those notes exist in other keys, but not C major. Every key has a chromatic vicinity of 17 notes per octave. The 53 note scale I mentioned is a larger scale that the 17 notes can modulate within. Not so much for all the notes to be used as one big chromatic scale at once. Now because keyboards and guitars in 12 tet have taken over, people just wrongly assumed that western music is based on 12 notes, even though when they read sheet music there are more than 12 notes per key per octave. Because of the keyboard and edication based on the keyboard deemed as definitive, people's understanding has become very dark. The reason why 5 limit is responsible to a large degree for 12 tet is because: people seeking after the 5/4, and being told lies repeatedly about the 81/64, and not being able to acheivethe 5/4 in many keys for modulation, they sought temperaments, they temperaments usually detuned the more fundamental 3/2 interval to make that which is by nature an 81/64 into something resembling or exactly, a 5/4, by a chain of detuned fifths. But because these meantone temperaments led wolf intervals and still limited modulation, and the fifths were dull from being detuned, people sought well temperaments, all with the wrong presupposition that the 81/64 is not ideal. Then finally because of the Jesuit Order, a crucial part of Babylon the Great, the Mother of Harlots and Abominations of the Earth, 12 tet was brought over from China where it was rejected, and then it was heavily promoted in the west. And because the major thirds are closer to the 5/4 and the fifths are better than meantone AND you can modulate in a circle with only 12 notes, people were hooked, bewitched by it. But if people never fell for the lie of the 5/4 major third, keyboards would have been developed for more than 12 notes peroctave that can play perfectly in tune. There is not just one layout that can play Pythagorean perfectly in tune on acoustic instruments, there are many, and most of them are better ergonomically than the keyboard we have today, for scales, chords, transposition, and they can modulate indefinitely. One does not even need to have all 53 notes available on the keyboard at once. You can have 24 or 17 or 12, and then with foot pedals modulate the keys (via a fretting mechanism is the string based keyboards). This can be done on piano instruments, organs, harpsichords, and certainly with electric instruments. But because people fell for the lie (as I did for 9 years), none of that was ever developed and even now people are still drinking the kool-aid of the 5/4 major third and still thinkning the 81/64 does not sound good or ideal in chords. The problem is that people who are in to intonation, understandably examine intervals outside of musical context. And because they like the sound outside the context, in a vacuum, they think that extends over into real musical context. But music is not made in an instance of time. It is made over time. It is not one moment, it is information of sound processed over time. And in the "over time" part 5 limit is found to be incoherent and impractical both, audibly and conceptually. There are 7 notes in True Intontion in the diatonic scale, 17 with chromatic notes. How many notes are there in the diatonic scale in 5 limit? Think about it. There is no defined answer. Its not 7, not 9, not 12, there is no limit to the amount of notes in the diatonic scale, and there is a comma in the diatonic scale. And then think about the diatonic modes, it is chaos. Dorian has two different tonics and the scale at minimum straddles two different syntonic comma scales. There is no consistency of pitch or harmony. And even if you find a way to make it work conceptually, it always finds situations that require compromise to keep the pitch ffrom drifting, or to keep notes somewhat more melodically related to the tonic. Putting it to the test, 5 limit doesn't work. Sure with very simple music you could pass it off as acceptable. But when you really start working with it, it both does not work and is not practical and sounds bad, when one is listening holistically (vertically and horizontally simultaneuosly. I too was VERY resistent to this idea I am telling you now. I invested 9 years of my life into 5 limit. But when I stopped long enough to put these things to the test, I was astounded and also glad at the findings. Just intonation, True Intonaiton, is possible and practical for acoustic instruments and keyboards. But it must start with proper education, both in terms of the 17 notes per octave per tonic chord, and how that all works, and with understanding the right things around intonation, and getting people who are already into intonation to retrain their ears to hear as they once did before all the 5 limit conditioning was imparted.
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.
![BLACK BAG - Official Trailer [HD] - Only in Theaters March 14](http://i.ytimg.com/vi/Du0Xp8WX_7I/mqdefault.jpg)
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
Perfectly explained! Thank you! =)
True!
@@LatchezarDimitrov Hey, Latchezar, how are you doing? I remember chatting with you about intonation a while back. Nice to hear from you.
While not exact, it does get absurdly close after 53 fifths, making extended Pythagorean functionally equivalent to 53 edo.
@@Dayanto Yes, 53EDO is quite an interesting tuning. But it's a lot of notes, so it's a bit unwieldy for acoustic instruments.
That chart at 0:52 deserves its own video!!!
Haha! It's a rabbit hole of a chart!
Bravo, Inna!! 👏🏻👏🏻 Very clear explanation of the two systems! I can’t wait to see their application to some repertoire in the next videos 🎶🎶
Thanks so much, Laura!❤️
So awesome! Great coverage of a fundamental concept of tuning but also important to understanding microtonal as well as non-western music! You rule.
Thanks for watching :)
Tuning is made way more complicated than it really is. What is called just intonation in this video is not really just, musically speaking. When put to the test, it is revealed to be totally incoherent. This 5 limit tuning which people call as just intonation is relatively recent. Pythagorean tuning was the standard for at least 2000 years. And yes, it works well for chords, melodies, double stops. Its major thirds are not too high, they are ideal, exactly at 81/64 ratio.
What mpst do not understand is that the 5/4 is not a pitch class, it is not a major third. It is an overtone, one that is out of tune if you tune a fundamental pitch in your scale or chord to it. 5 limit rational tuning is melodically completely out of tune for mny reasons. Whereas Pythagorean Just Intonation with 17 notes available per tonic chord, this is perfectly in tune both melodically and harmonically at the same time. The only thing is, you must listen holistically, youmust listen horizontally to melodies (even within the voices of chords), AND you also must listen vertically to each instance of harmony at the same times each moment as you listen. Then it beomes perfectly clear. Unfortunately when we take up an interest in tuning and intonation, we tend to examine intervals in a vacuum, that it, not in context of real music with real timbres. And while the 5/4 sounds buttery and smooth as an interval out of context, in context of real music with real instrument timbres, it is woefully flat sounding, and the constantly varying step sizes, notes having multiple frequencies, no limit to the amount of notes in a key, commas appearing, and drifting of the tonic, the buttery smooth buzz of the 5/4 in no way makes up for the avalanche of other problems it causes.
81/64 major thirds are ideal for melody and harmony. Look on my channel, I have many many examples.They are not too wide as so many love to chant on repeat. It is important to have understanding when it comes to music, and so long as you are messing around with making the 5/4 the major third, your understanding is going to be limited at best, when it otherwise would not be.
When musicians talk of keys like C major or E minor, it is very important for musicians to know what a "key" even is, and most dictionaries get it wrong. A key is 7 notes derived from an unbroken chain of six 3/2 fifths (or a close approximation of the 3/2, but ideally 3/2). That is it. So a major third is defined as the pitch class obtained from going up 4 perfect fifths. All musical pitches can be expressed as whole number ratios by some power of 3 in relation to some power of 2. A major third is 3 to the fourth power over 2 to the sixth power.
One you understand what a key is. They it is important to know what chromatic notes are available per each tonic triad. Most people think there are 12 chromatic pitches, this is wrong. There are 17 chromatic pitches per tonic chord. There are 7 naturals, 5 flats, and 5 sharps in C major. And these 17 notes modulate up or down the chain of fifths when the tonic triad changes from major to paralell minor (or any of the paralell modes), or when the tonic note itself changes. So for example, and you will see this in real sheet music today, in C major, you can have Gb through to A# in a chain of fifths. So the scale from lowest pitch to highest would be:
C, Db, C#, D, Eb, D#, E, F, Gb, F#, G, Ab, G#, A, Bb, A#, B, (C)
Like I said this scale modulates with the key. The above 17 note vicinity scale is not just for C major, but also for A minor, E Phrygian, D Dorian, etc., all the relative modes. But for major, it is from the b5 though to the #6 in the chain of fifths.
When you look at any of the renowned composers, especially the older ones, but not exclusively, they were careful to spell notes properly, because while they did not play in tune, they understood that the notes were informing of the underworkings of music and how tuning it supposed to be originally. You won't find an Cx in C major, you won't find an Fb in C major, but D and E respectively. But you will find both C# and Db in C major without pushing the ear to hear that the tonic chord has changed.
Thanks Inna. Very helpful to understand this a bit better.
You're very welcome - Glad it was helpful! =)
I seriously love your videos
Thank you so much!
I'd love to see you in concert. ❤
Great treatment to this fascinating topic!!!
Many thanks for watching & commenting!
Thanks Inna! That was a great explanation!
Glad it was helpful!
Brava. This is great information on this subject.
Thank you! =)
Best video on youtube about intonation. Thank you, also for the book recommendation.
So glad you enjoyed the video! Thank you! The book is truly excellent!
It's actually one of the worst and is totally wrong.
Inna, very good discussion.
Thank you!
Everything you explain is so amazing! I wish I understood anything you say! :D
😉
Great video!! Thank you for comparing these two systems. Really interesting comparison!
Thanks Darlene! ☺️
I m so glad i found you
I just watched this again. Well done, subscribed.
cheers from foggy Vienna, Scott
Thank you for watching & subscribing! =)
Streaming your channel now!!!
So beautiful and wonderful talent
Another AmaZing video!
Thanks Carol!
Great video, informative and very well done!!!
Thanks so much, Jason!
What system should I use when tunning open strings?
By default, most violinists tune to pure 5ths (i.e. Pythagorean). However in a string quartet it's common to very slightly brighten up G. And in baroque ensembles it might be completely different tuning (e.g. tuning each string individually to the harpsichord).
Thank for sharing,,,, 😍
My pleasure!
@@Violinna ruclips.net/video/yirMaINPTy4/видео.html
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).
41edo is also a viable alternative Pythagorean tuning system. Its fifth is at the interval 24\41, about 702.44 cents, only 0.5 cents sharper than pure 3:2 (although, 53edo does better than 41 in its fifth). Its fourth is at the interval 17\41, about 497.56 cents, 0.5 cents flatter than the pure 4:3 fourth. Its 5:4 approximation is slightly worse than 53edo's at 380.49 cents instead of the ideal 386.31 cents. Although 53edo does better than 41 in the 5-limit, 41 does better than 53 in the 7-limit and higher, so this could be an alternative tuning system to both Pythagorean tuning and 53edo tuning (it's also fewer notes).
All equal temperaments are bad, especially when you could tune purely. 53 is among the best of the smaller number edos, but ONLY if you use it correctly, which basically no one does. People usually take 53 edo and make Fb the major third and make the E the diminished fourth above C, the exact opposite of what is just. The most important thing anyone can know about tuning is that Pythagorean just intonation (that is, with correct note spellings, acknowledging the 17 note chromatic vicinity encompassing each key in the chain of fifths) is True Intonation, is idea just intonation, in melodies AND chords. Check my channel for examples.
And the thing is, if you are already having 53 notes per octave, you are better off with pure Pythagorean imo, having 53 notes per octave, it will sound better, albeit quite subtly. And at the two ends of that chain of 53 fifths, is what is called a Mercator Comma. If D is in the exact center of your chain of 53 fifths, then on the flat end you will have Ebbbb and on the sharp end you will have Cxx. Ebbbb is less than 4 cents flat of a perfect 3/2 above Cxx. Ebbbb is enharmonic to Gxx, they are almost the same note. Like wise Cxx is enharmonic by the mercator comma nearly a perfect fifth below Ebbbb. Cxx is enharmonic, almost the same pitch as Abbbb. So if for some reason you wished to modulate really far, you could "jump" or modulate over that mercator comma and no one would hear that you are now on the other end of the chain of fifths.
So interesting again!
Thanks Bico!
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.
Yes, actually your solution is exactly what I discuss in part 2 of this, in rep examples. When I mentioned the word "paradox" in my title it was more along the lines of "there is no one single way to tune a note" and "there is no such thing as perfect intonation"
The solution has been under your nose all along. Play EVERYTHING in Pythagorean. 5 limit tuning is incoherent and is constantly creating puzzles that need solutions that are filled with compromise. It is an incoherent conception of intonation when it comes to music when you really put it to the test. Pythagorean Just Intonation, or what I more aptly call True Intonaitoin, sounds perfectly in tune in chords, melodies, and double stops. It is what is displayed in sheet music and what is ideal in tuning. This 5 limit thing is a more recent trend of the last few hundred years and is the reason why all music is out of tune.
There are examples of this tuning in tertian harmony on my channel.
Make sure when dealing with intonation to listen holistically. That is both melodically, each voice moving from one note to another processing that over time, AND at the same time listen to the sounding harmony every instance. If you can allow your brain to do this, and it must be trained to NOT do this, you will hear Pythagorean sounds perfectly in tune every time. Sadly we who become interetsed in intonation usually unknowingly condition ourselves to only listen vertically to harmony in an instance and in no way track the melodies that make up chords and process that information over time. We are stuck in the moment, but music is primarliy something which unfolds over time. That is the melodic intonation awareness that must be acknowledgeed even within chords. Pythagorean, or True Intonaiton, handles melody and harmony perfectly, 5 limit creates confusion and puzzles of compromise and commas.
Wonderful video!
Thank you so much!
THIS SHOULD HAVE A MILLION VIEWS
super great vid
Thank you very much!
BRAVISSSSIMOOOOO, I LOVE IT!!!
Thank you very much, Sam!
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?
By default, I'll keep the main melody (which is most often in the top voice) pythagorean, and have the other voice adjust.
pythagorean is a type of just intonation btw
It is the only just intonation, btw. The others are rational intonations, but not just in the ultimate sense. Just in a physics sense, but not a musical sense. Pythagorean just intonation should be used in all melodies and chords and double stops and tuned according to the score (unless there is a misspelling) as the note sits in the chain of 3/2 ratio fifths.
That depends on definition. Really it's a form of a well temperament I think, and not actually Just.
How is it that the octaves aren't in tune? Is that in relation to equal fitting tuning?
This needs more views!!
Thanks for watching and comment =)
This is such a great topic! Very interesting and useful!
Thank you, April!
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
You are correct! I actually talk about this exact problem in Bach Andante in part 2 of this topic, with repertoire examples (should be linked in description box somewhere). It's not possible to play this Bach opening "100% in tune" and different players interpret what sounds best, where there is more than one compromise that's possible - this is where intonation actually becomes subjective!
Thanks for the answer! I think I’m going to stick with equal temp :)
@@esteban100 Always go with Pythagorean Just Intonation, not 5 limit tuning (which this video calls just intonation), and also not equal temperament. If you tune Pythagorean it will sound best, every time, and knowing which note to play and how it is tune will be much much simpler An E is an E is an E. The note C is always the same frequency in its given octave. Fb is Fb. B# is always B#. Each note name always has the same frequency in its given octave. Everything is then clear, not just in your mind, but it will reflect perfectly what is on the page, and will sound perfectly in tune to the audience. Check out some of the examples on my channel. True Intonation is Pythagorean Just Intonation.
best explainer video! thank you
This video is totally wrong. Pythagorean Just Intonation is what is just intonation and is always the right tuning, even for chords. 5 limit tuning is founded on a relatively new and backwards way of conceiving of harmony that has resulted in basically all music being out of tune in the world. 5 limit harmony, when put to the test is totally incoherent as a musical system of pitches, and is melodically badly out of tune. And using both Pythagorean and 5 limit tuning alternating is confusion because the note C is C. There are no pluses or minuses on the page, no up and down commas, all that is a recent invention. Even meantone tuning, invented and popularized by Jesuitical Roman Catholics, is based on a perversion. It just declares by fiat that the major third should be a 5/4 ratio, and then perverts and detunes the 3/2 perfect fifth to make that which is by nature a just major third, the 81/64 ratio, into the 5/4. The 5/4 is NOT a major third. All of these people are lying to you,, knowingly or unknowingly. The just major third is Pythagorean, it is 81/64.
This is such a great video with amazing insights!
Thank Kaya!
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 ?
Correct - depending on the context of the notes in relation to harmony, the tuning systems can all live within the same piece. For example, an F# as a passing tone in a melodic G major scale would likely be Pythagorean. But the same F# would be in Just Intonation for a held D major chord. Any part that doubles one of the hands in the piano would have to opt for the equal temperament (or whatever the piano is tuned to).
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?
Don't overthink it!! Just follow and trust your ear! :)
The exact opposite, you have to play in tune, but being in tune depends on the context. The notes of the 12-T system change based on what context you play in. Basically just use your ears to tune,
I so wish you had played musical examples. You provide a good explanation, but it's abstract.
Hi! There is a Part 2, with musical examples. Link is in the description.
@@Violinna Thanks!
I bought that book. Thanks
Enjoy
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.
It should always be the 81/64 Pythagorean for the major third. The 5/4 is not a major third, but simply the fifths harmonic. Check out my channel for examples of Pythagorean chords in real musical context. True Intonation is what the world calls Pythagorean Just Intonation.
@@RememberGodHolyBible I stand by what I said. If you're listening to a sustained tonic triad, such as at the end of a cadence, a 5/4 just third scale tone will fit the tonic and dominant tones, with fewer heterodyne beats than an 81/64 Pythagorean third scale tone. Singers and non-fretted string players make fine adjustments in their tuning depending on whether, as noted, they're tuning to a melodic line or to a sustained chord.
If you happen to prefer 81/64 at all times, happy listening. Some people prefer equal temperament; to each his own.
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.
Casals was a genius! I just read the book about him called "Art of Interpretation".. it's incredible.
Bravissima!
Grazie!
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.
Since the syntonic comma of 81/80 is tempered out in 12edo, most musicians (even those who have to check their intonation) think in terms of one whole tone and one half-step and not as two separate whole tones and half-steps separated by 81/80.
@@YoVariable It's true that 12-TET splits the difference. However, it doesn't do so evenly, like meantone. The 12-TET whole tone is negligibly lower than the Pythagorean one, while the half-step is indeed somewhere in the middle, but still a tiny bit closer to 256/243. This talk about JI being melodically out of tune just goes to show how the acoustic identity of the individual notes in the scale is tied to the Pythagorean system.
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.
Yes, it definitely depends on the situation! And different cultures are accustomed to hearing intonation differently. This video is a basic overview/introduction of the two types of tuning system and the general definition, if that makes sense.
Quick sub
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!
The concept is sound.
No, no its not. Its not sound or right at all. Pythagorean just intonation is what is desirable in every instance. 5 limit tuning, what she calls just intonation here is based on a wrong conception of harmony and creates all sorts of confusion, and is the reason why all our music is out of tune these days. Without people seeking after 5 limit and rejecting Pythagorean thirds, we would never have been sold on 12 tet. You can have perfect just intonation (Pythagorean) on acoustic keyboard instruments, AND modulate indefinitely. The issue is that people forsook that which was right because they became enchanted, seeking after this carrot, the 5/4 as the major third, which they can never reach, because by its very nature it causes problem after problem. It is outside the key yet people treat it like it is a diatonic note, and it is so close to the real diatonic note, that the ear hears it as that, even though it is not, and the confusion it has wrought is too much to really fathom. Because of this incoherence which cannot be solved in any truly satisfying way, it then opens the door to "anything goes" tuning or ever increasing amounts of temperaments to try or x-limit tunings to try. All which keep you on the merry-go-round, and for what , a tuning system that sounds bad and is also uneccessarily complicated.
A key is 7 notes derived from an unbroken chain of six 3/2 fifths. Not the confusion described in 5 limit. There are infinte notes in the diatonic scale in 5 limit. It is confusion.
The only way off and to land in something good, intonatioin wise, is to realize that Pythagorean tuning, 17 notes per octave per tonic triad derived from an unbroken chain of fifths, that is just intonation, and sounds by far the best in chords, melodies, and double stops, AND it stands up to scrutiny and testing in various pieces of varying complexity, styles, and timbres.
So they make clickbait books too nowadays? Cool.
Last I checked, clickbait book titles existed before the internet ;) Keep an open mind
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.
While Pythagorean tuning has its strengths, especially in melodic contexts and with certain interval purity, the claim that it universally provides “true intonation” in all situations overlooks some key aspects:
5-Limit Just Intonation and Harmonic Consonance: In harmonic contexts, especially with triads and chords, 5-limit just intonation’s 5/4 major third is often preferred because it more closely aligns with the harmonic series and provides a purer consonance compared to the 81/64 major third in Pythagorean tuning. The 5/4 ratio gives a smoother harmonic blend, which is why it has been a cornerstone of Western harmony for centuries. Dismissing it as "out of tune" ignores its actual harmonic effectiveness.
Beating and Dissonance in Pythagorean Thirds: While the 81/64 third produces a specific type of beating, it’s inaccurate to suggest this is inherently different or “better” than the beating in other tuning systems. Pythagorean thirds can create noticeable dissonance in harmonic contexts because they diverge from the harmonic series, particularly in music emphasizing smooth consonance.
Practicality of Extended Pythagorean Tuning: The suggestion of using a 53-tone octave based on a chain of fifths may work in theory but is not practical for most musicians. Additionally, using a large microtonal scale doesn't inherently align with Pythagorean tuning principles used in traditional Western music, which typically adheres to a 12-tone system. The complexity added by microtonal options does not resolve the challenges of consonance in harmonic contexts.
Historical Context of Tuning Systems: The assertion that 5-limit tuning led to the development of equal temperament is inaccurate. Equal temperament was developed to balance key modulation needs, enabling musicians to move between keys with minimal tuning inconsistencies. This allowed harmonic and melodic flexibility beyond what strictly Pythagorean or 5-limit systems could offer.
In summary, each tuning system has its ideal uses, and claiming that Pythagorean tuning is “true intonation” in all situations oversimplifies the nuances of harmony and the different purposes of various tuning systems.
@@jenshoffmann2210 Man, the music speaks for itself. All your theory is one thing, but the music on my channel and in other places as well testifies to the contrary. Pythagorean thirds and intervals in general are not just usable in chords, they are ideal. This is not the case with 5 limit intervals, they sound okay in isolation but woefully flat when tested in various pieces of music and the listener is listening both for melodic and harmonic purity simultaneously.
Western music is not based off of 12 notes per octave, regardless of what everyone else says repeatedly, believe your own eyes, not what parrots tell you. If you ar in C major, or any key, really there will be 17 possible notes available. This is what you will see when you study scores. In C major these are C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B. These are all the notes in the vicinity of C major in all western music. You won't find an Ebb in C major. You won't find B# or E# or Cb or Fb in C major, Those notes exist in other keys, but not C major. Every key has a chromatic vicinity of 17 notes per octave. The 53 note scale I mentioned is a larger scale that the 17 notes can modulate within. Not so much for all the notes to be used as one big chromatic scale at once.
Now because keyboards and guitars in 12 tet have taken over, people just wrongly assumed that western music is based on 12 notes, even though when they read sheet music there are more than 12 notes per key per octave. Because of the keyboard and edication based on the keyboard deemed as definitive, people's understanding has become very dark.
The reason why 5 limit is responsible to a large degree for 12 tet is because: people seeking after the 5/4, and being told lies repeatedly about the 81/64, and not being able to acheivethe 5/4 in many keys for modulation, they sought temperaments, they temperaments usually detuned the more fundamental 3/2 interval to make that which is by nature an 81/64 into something resembling or exactly, a 5/4, by a chain of detuned fifths. But because these meantone temperaments led wolf intervals and still limited modulation, and the fifths were dull from being detuned, people sought well temperaments, all with the wrong presupposition that the 81/64 is not ideal. Then finally because of the Jesuit Order, a crucial part of Babylon the Great, the Mother of Harlots and Abominations of the Earth, 12 tet was brought over from China where it was rejected, and then it was heavily promoted in the west. And because the major thirds are closer to the 5/4 and the fifths are better than meantone AND you can modulate in a circle with only 12 notes, people were hooked, bewitched by it.
But if people never fell for the lie of the 5/4 major third, keyboards would have been developed for more than 12 notes peroctave that can play perfectly in tune. There is not just one layout that can play Pythagorean perfectly in tune on acoustic instruments, there are many, and most of them are better ergonomically than the keyboard we have today, for scales, chords, transposition, and they can modulate indefinitely. One does not even need to have all 53 notes available on the keyboard at once. You can have 24 or 17 or 12, and then with foot pedals modulate the keys (via a fretting mechanism is the string based keyboards). This can be done on piano instruments, organs, harpsichords, and certainly with electric instruments.
But because people fell for the lie (as I did for 9 years), none of that was ever developed and even now people are still drinking the kool-aid of the 5/4 major third and still thinkning the 81/64 does not sound good or ideal in chords.
The problem is that people who are in to intonation, understandably examine intervals outside of musical context. And because they like the sound outside the context, in a vacuum, they think that extends over into real musical context. But music is not made in an instance of time. It is made over time. It is not one moment, it is information of sound processed over time. And in the "over time" part 5 limit is found to be incoherent and impractical both, audibly and conceptually.
There are 7 notes in True Intontion in the diatonic scale, 17 with chromatic notes. How many notes are there in the diatonic scale in 5 limit? Think about it. There is no defined answer. Its not 7, not 9, not 12, there is no limit to the amount of notes in the diatonic scale, and there is a comma in the diatonic scale. And then think about the diatonic modes, it is chaos. Dorian has two different tonics and the scale at minimum straddles two different syntonic comma scales. There is no consistency of pitch or harmony. And even if you find a way to make it work conceptually, it always finds situations that require compromise to keep the pitch ffrom drifting, or to keep notes somewhat more melodically related to the tonic. Putting it to the test, 5 limit doesn't work. Sure with very simple music you could pass it off as acceptable. But when you really start working with it, it both does not work and is not practical and sounds bad, when one is listening holistically (vertically and horizontally simultaneuosly.
I too was VERY resistent to this idea I am telling you now. I invested 9 years of my life into 5 limit. But when I stopped long enough to put these things to the test, I was astounded and also glad at the findings. Just intonation, True Intonaiton, is possible and practical for acoustic instruments and keyboards. But it must start with proper education, both in terms of the 17 notes per octave per tonic chord, and how that all works, and with understanding the right things around intonation, and getting people who are already into intonation to retrain their ears to hear as they once did before all the 5 limit conditioning was imparted.
Please stop talking about it and demonstrate what it sounds like. Sounds not speech.
Most certainly! In fact, here is Part 2, with a few examples: ruclips.net/video/R07c1Dfx0RA/видео.html
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.
Thank you for watching. You're welcome to change playback speed in youtube settings to slow it down =)
LOL what a stupid comment. Maybe you cant process what she says like normal intelligent people and maybe you are projecting your insecurities on her
this isn't fast