The Syntonic Comma

[robcat2075]

This video explains that the syntonic comma is the difference between a Pythagorean major third and a just major third.

The just third is lower than an equal-tempered third while the Pythagorean third is higher.

A just third can be created by playing the fifth harmonic of a string and subtracting two octaves, resulting in a 5:4 frequency ratio or 1.25:1. On a G string the 5th harmonic sounds a B

A Pythagorean third is created by stacking four perfect fifths and subtracting two octaves... a ratio of 81/64 or 1.265625:1.

On a violin one could manufacture a Pythagorean third above the G string by going up three 5ths to the E string, then playing the third harmonic, again a B, and subtracting three octaves. (this presumes the violin strings have been tuned in exact fifths and not tempered-fifths)


I add that as preface to this demonstration on a violin which did a bit of a hand-wave about why she was playing the harmonics that she did. It's impressive that she can call up either version of a third at will.

I presume that's what you get from being locked in a tower with a violin since age three.




I will also note that the syntonic comma is not to be confused with the gin & tonic coma.

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[LeTromboniste]

Yup! The tuning system I play in ~90% of the time is one where pure major thirds are achieved without adjusting anything on the fly like we do in orchestras. If I play a major third on the organ in my teaching room, it's tuned pure already. You do that by making every fifth narrower than pure by a quarter of a syntonic comma. It's very neat because you get pure thirds and very sweet sounding chords everwyhere (although the fifths are not as good as in equal-temperament, they're not nearly as bad as equal-tempered thirds are, and major and minor chords are significantly more in tune as a reault), while keeping all the whole tones the same size. You get two very different sizes of semitomes, a very small chromatic semitone and very large diatonic semitone, which is very expressive melodically in chromatic passages. The trade-off is that enharmonic equivalency doesn't exist, and therefore a modern 12-note-per-octave keyboard doesn't give you all the notes you need.

[tbdana]

Wouldn't be a problem if everyone just played trombone. Just sayin'.

[robcat2075]

Yup! The tuning system I play in ~90% of the time is one where pure major thirds are achieved without adjusting anything on the fly like we do in orchestras. If I play a major third on the organ in my teaching room, it's tuned pure already. You do that by making every fifth narrower than pure by a quarter of a syntonic comma...

that can't be true for all thirds, is it? If I stack three pure major thirds... 1.25 x 1.25 x 1.25 ... I don't get 2.0 for the octave.

[AndrewMeronek]

For sure, the Syntonic Comma is very useful in performance practice and in music theory. I think that more music education nowadays is gravitating toward teaching this, but there is LOTS of work to be done.

[AndrewMeronek]

Yup! The tuning system I play in ~90% of the time is one where pure major thirds are achieved without adjusting anything on the fly like we do in orchestras. If I play a major third on the organ in my teaching room, it's tuned pure already. You do that by making every fifth narrower than pure by a quarter of a syntonic comma...

that can't be true for all thirds, is it? If I stack three pure major thirds... 1.25 x 1.25 x 1.25 ... I don't get 2.0 for the octave.

This is talking about meantone temperament. And yes, stacking major thirds does not produce an octave.

Unless your tuning system aligns mathematically.

[mbarbier]

Yup! The tuning system I play in ~90% of the time is one where pure major thirds are achieved without adjusting anything on the fly like we do in orchestras. If I play a major third on the organ in my teaching room, it's tuned pure already. You do that by making every fifth narrower than pure by a quarter of a syntonic comma...

that can't be true for all thirds, is it? If I stack three pure major thirds... 1.25 x 1.25 x 1.25 ... I don't get 2.0 for the octave.

Stacking justly tuned major or minor third does not give an octave. The difference is referred to as the Major (or minor) Schismas. I made the below tracks for my just intonation class if you're interested in hearing the difference. They build up the chord then play the resulting "wrong" octave an unison. "major" track is major thirds, minor track is 6:5 minor thirds, and septimal track is 7:6 minor thirds.

https://on.soundcloud.com/pWetbVViKLTUVqhi9

[LeTromboniste]

that can't be true for all thirds, is it? If I stack three pure major thirds... 1.25 x 1.25 x 1.25 ... I don't get 2.0 for the octave.

Yes, it can be true of all thirds: there is no inherent requirement to have only 12 notes per octave. Without enharmonic equivalency, you just get more different notes; you don't have a circle of fifths, you have a spiral of fiths that doesn't close and keeps going around. In practice that's no problem for a trombone, a violin, a voice. It does however mean that on a regular keyboard instrument, you're missing some notes i.e. you have access to Bb but not A#, to C# but not Db, etc. And no double sharps or double flats. So you have for example a a very nice Bb major with a pure third, but you can't play F# major because you have no A#. In an ensemble situation if you're missing the third, you'd play an open fifth and let the missing note be played or sung by someone who actually can. But it was quite common in the Renaissance and early Baroque for big organs to have 14 notes per octave instead of 12, having "split" keys for Eb and D#, and for G# and Ab. That covers pretty much all the notes that were regularly used in the modal system of the time, and allows for some transpositions that would not otherwise be practical. Some instruments did have 19 keys per octave, with all of the black keys split, and even an added E# and B#. Rarely, instruments with 24 or even 31 notes per octave were used.

Here's the keyboard of a very nice 16th century organ in northern Italy :

[LeTromboniste]

Wouldn't be a problem if everyone just played trombone. Just sayin'.

Yes, much less of a problem!


I know it was tongue-in-cheek, but check this out.

Even then, playing perfectly in tune on any instrument still requires an underlying fixed scale for reference, for either the bass or the roots of the chords to follow, and upon which the other voices tune, which requires other voices to alter the pitch of their held or repeated notes as their harmonic role changes. If instead you keep held or repeated notes at the same ptich and tune the moving voices to them, you end up with a shift or drift in pitch. It's one of the reasons even the best choirs drift in pitch when singing a capella.

Here are two 16th century musical puzzles exemplifying exactly this.



Looking just at the first one, it is a very simple and short and trivial progression of four chords on repeat. Yet if you play/sing it perfectly in tune, you'll end up a semitone higher than you started after only 5 repetitions. If you sing only pure intervals, the middle voice's starting D will have to be purely in tune to the other voice's G. Then the D and A in the outer voices will be pure to the held D. Then the lower two voices tune their C and E or the top voice's held A. The top voice then tunes their G to the held C and E. And finally the lower voices' G and E will have to match that held G. We're back to the starting position, but everyone is now singing a comma higher than they started (about 20 cents), and after 5 times, we're a semitone higher.

It's very easy to try out with three trombones. The requirements are play everything perfectly in tune (i.e. pure, no beats), and when holding a note, do not move the slide. You'll find it very quickly gets extremely uncomfortable.

[AndrewMeronek]

Yet if you play/sing it perfectly in tune, you'll end up a semitone higher than you started after only 5 repetitions.

I once wrote up some junk on a common modern sequence that does the opposite: syntonic comma-based pitch drift that lowers. There are a few simple playbacks among the words for examples.

https://andrewmeronek.com/music-tools/u ... -drifting/

[AndrewMeronek]

Concerning how to avoid pitch drifting, I have a visualization that I like, using the Tonnetz Lattice:

https://en.wikipedia.org/wiki/Tonnetz

This lattice is adaptable to lots of tuning systems, including just intonation. Thus: in just intonation if you find yourself wanting to jump across the lattice, you have pitch drift. If you write counterpoint in a way that doesn't push these jumps, you'll avoid pitch drift, at least from causes of syntonic commas and related things.

Max's first example puzzle basically starts on the lattice on the left with G-D-A (in his first measure) and crawls to the right and you eventually get to the 2nd G-D-A on the right, and this repeats. This implies that if the sequence were a bit different and after arriving at C major it instead moves to include maybe F and B-flat (etc.) then it cycles back to the original location of that G-D-A; and viola, pitch drift is corrected. The so-inclined can check the math against the Tonnetz to see if that's exactly right. I did so a while ago; I know that's correct.

[robcat2075]

What I found most interesting about the video was their distinction between the third in harmony and the third in melody. He says he likes the "leading-tone" feel to the higher third when ascending to the fourth and that he was doing that as a preference before he knew anything about the theory behind it.

Prior to this video I didn't know a Pythagorean third was ever a desirable thing if you had any other choice. I had always read of it as merely an expedient way to derive a third for a keyboard instrument.

Not discussed... descending from four to three. Should that still be small? When i'm playing cello I find myself wanting to make half-steps as large as possible... ascending or descending.

To me, it sounds squishy when I do the small half step leading to the fourth or octave, especially in a slur. But the player doesn't sound squishy when she plays.

Maybe I am squishy. Real string players have callouses like rocks on the ends of their fingers and I don't have that.

[Kbiggs]

What happens when you start working in more remote keys in Pythagorean tuning? I can imagine that C, G, D, A (perhaps even E) would still sound pleasant with agreeable intervals. F, B-flat, and E-flat would probably sound okay. But surely, as you spiral (up? down?) away from the theoretical center of C:, the more remote keys would sound less and less pleasant.

[AndrewMeronek]

What I found most interesting about the video was their distinction between the third in harmony and the third in melody. He says he likes the "leading-tone" feel to the higher third when ascending to the fourth and that he was doing that as a preference before he knew anything about the theory behind it.

I think this is mainly a performance practice thing in classical tradition. I don't know for sure, but I suspect the idea to raise the leading tone is related to the more general tendency for a melodic soloist (especially with vibrato) to play slightly sharp for more "brightness".

What happens when you start working in more remote keys in Pythagorean tuning? I can imagine that C, G, D, A (perhaps even E) would still sound pleasant with agreeable intervals. F, B-flat, and E-flat would probably sound okay. But surely, as you spiral (up? down?) away from the theoretical center of C:, the more remote keys would sound less and less pleasant.

It's actually all the same intervals in Pythagorean tuning until you actually roll over far enough that you start seeing the Pythagorean comma (i.e, C vs. B-sharp and etc.) popping up on a keyboard.

[harrisonreed]

What I found most interesting about the video was their distinction between the third in harmony and the third in melody. He says he likes the "leading-tone" feel to the higher third when ascending to the fourth and that he was doing that as a preference before he knew anything about the theory behind it.

Prior to this video I didn't know a Pythagorean third was ever a desirable thing if you had any other choice. I had always read of it as merely an expedient way to derive a third for a keyboard instrument.

Not discussed... descending from four to three. Should that still be small? When i'm playing cello I find myself wanting to make half-steps as large as possible... ascending or descending.

To me, it sounds squishy when I do the small half step leading to the fourth or octave, especially in a slur. But the player doesn't sound squishy when she plays.

Maybe I am squishy. Real string players have callouses like rocks on the ends of their fingers and I don't have that.

I've written about this multiple times on the forum and I've been dismissed like this wasn't a thing. Those half steps should be small in melodic playing, absolutely! You hear all the time college students playing JI thirds in their etudes and the melodies sound tone-deaf. Dull and flat. In Pythagorean tuning most of the scale will actually be raised, and it sounds more exciting and alive.

Most modern melodic singing is actually based on Pythagorean tuning! The higher major third in Pythagorean tuning is generally preferred to equal temperament and just intonation in melodic lines when you ask people to pick their favorite. Violinists have been trained to play melodies like this for a long time, and it is not strange or new to them.

For modern music you have horizontal intonation, which is Pythagorean. And you have vertical intonation, which is generally 5-limit just intonation. The two are not compatible and you need to make compromises for it to work. Over sustained chords with a moving melodic line, you can use both systems at the same time but if anything is sustained in the melody someone has to give in.



In any case, for melodies you should use 12TET (like a piano) at the very least, if you can't get on board with Pythagorean tuning. JI is flat out (pun intended) bizarre sounding for melodies.

[VJOFan]

The professor gives away the game at about 3:00 when he says its something everyone does and he even taught the sounds before he knew the theory. The theory of any music describes practice, it doesn't direct it.

If we follow our ears, things will work out pretty nicely.

[LeTromboniste]

High thirds are absolutely a thing in solo playing. It makes sense. Andrew, you are correct, this has to do with the brightness of it, and the fact that blend is not actually desirable when you're a violin or cello soloist in front of an orchestra. Anything that helps you stick out from the texture is good. To what extent a trombone needs that help to stick out when playing solo is another question, but nonetheless, Pythagorean-leaning thirds have their place in solo playing, for sure – unless the keyboard instrument has a tuning that leans towards pure thirds (then it'll sound awful).

My big beef with it is that many, many, many string players don't know anything about the theory of it. They've just been told that leading tones need to be high, that sharps are high and flats are low, by their conservatory teacher who's been told by their conservatory teacher, who's been told by, who's been told by.....And then they apply that everywhere, including any situation in ensemble and orchestra playing. I've seen them teach it, and I've seen them raise their eyebrows in puzzlement or even disagreement when enlightened conductors request pure thirds, with generally low sharps and high flats.

What happens when you start working in more remote keys in Pythagorean tuning? I can imagine that C, G, D, A (perhaps even E) would still sound pleasant with agreeable intervals. F, B-flat, and E-flat would probably sound okay. But surely, as you spiral (up? down?) away from the theoretical center of C:, the more remote keys would sound less and less pleasant.

All keys sound pretty bad in Pythagorean tuning for music that has third-based harmonies (which is most music composed in the last 500+ years). The thirds are really quite awful, that's why they were considered a dissonance in medieval times. But all chords of the same nature sound equally bad, because all the fifths are the same size. Except on a keyboard instrument, once you involve the "wolf" fifth, that's even extra worse, but it's just for harmonies that cross over that particular fifth. Similarly with ¼-comma meantone (what I described in my first post), all chords sound equally good, until you try to use enharmonics, which are basically unusable.

[Savio]

I must admit that I don't understand all of the discussion here so please be kind if I'm completely lost.

But all notes, intervals, scales have a function. And they are different in different contexts.

Thirds + sevenths are special because they often lead to the next note or chord. And in some contexts there can be many more notes leading. Especially jazz but also something as simple as a fourth leading to a third in church music or as in Bach's compositions.

I can't jump and dance about playing clean all the time. But if you play with a piano you have to follow the piano. You can't lower or raise notes no matter chord, scale or melodic progresjon.

When we play in a trombone quartet, we lower the third. But not always. If the third is in the bass voice we don't do that. I know that strings also do the same as us in a string quartet.

In an orchestra I have no idea but as a bass trombonist I love playing major thirds! Not so much minor thirds. In a solo I'm not sure what I do. In a major scale, both the third and the major seventh lead up to a root note. But we all know that it feels right to lower the third in a trombone quartet?

WE have to use our ears all the time, no matter the context. The theories are confusing because there are so many.

What's interesting here are those with absolute pitch. My daughter has it. She doesn't think anything about thirds or sevenths. Not about lowering or raising notes. She doesn't even know the names of all the chords but can play them immediately after hearing them on her keyboard. She's only 13 years old and dont want me to teach her anything.

OK, I said I might be a little off topic here.

Leif