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On this third – and last – day of focusing mainly on intonation, my plan is to get a bit “heady.” And it might seem at first like it’s over-thinking things. But at the end of the process, I know I’ll have built up a specific spot for each note of the scale: perfect “Pythagorean” intonation!
“Pythagorean” intonation is the term I use to describe a kind of intonation where the place for each note of the scale is determined by perfect intervals. As the name implies, “perfect” intervals need to be tuned perfectly. There’s one specific distance between the two that sounds in tune, and anything else sounds out of tune. Contrast that with the major and minor intervals such as thirds and sixths. There’s a range of places that sound in tune for those intervals. So by using perfect intervals to find each note of the scale, we can create a specific, correct spot for each one. Perfect intonation!
But actually, not really perfect. There are many ways of establishing where a note is the most in-tune. The different ways lead to slightly different places for the notes – but they’re still all correct. That’s because there’s no such thing as perfect intonation. Or perhaps it’s better to say: that’s why there’s no such thing as perfect intonation. For any given note, there are at least a few possible places for it that are rational and which might sound in tune. In the end, we just have to train our ears to hear very precisely, and then having done that we go with the version that sounds best in context. It could depend on the key, the notes of the melody before and after it, the particular harmony, finding ring with the open strings, needing to adjust to another player’s line, etc.
So with that said, if it’s all just a judgement call ... what’s the point of getting so analytical about it? There are some very good reasons, it turns out.
Why do this one?
Intonation practice is mainly about training our ears (and a little about training our fingers). In order to do that, it’s extremely helpful to have a very specific spot to aim at for each note. It urges our ears to become more and more discerning or precise. So for that reason, even if you were never going to play any music with this kind of intonation it would still be helpful. But you will use it, because ...
This kind of intonation works really well in melodies. It leads to relatively high 3rds, which makes for a nice, tight half-step with the 4th note of the scale (the sub-dominant); and a relatively high leading tone (the 7th note of the scale), which creates a tight half-step into tonic (the main note). And those tight half-steps help the melody feel like it’s moving forward.
The Details
The notation below provides three ways of understanding Pythagorean Intonation. The first way is probably an easier way to understand the concept, but might be more difficult to apply in practice. The second way is a more practical way of building up the scale: gradually adding one note at a time, and using each new note to add another. The third way is also more practical, but also highlights the parallel placement of the fingers across the strings.
All three approaches can be helpful, and I recommend looking at all of them. Then decide which one seems more practical for you, and spend time practicing that way until your ear picks up on the placement for each note. For myself, I spent a lot of time building up just one octave, and only at the end did venture into a second octave.
In D Major, you’ll notice that for several notes the effect of this tuning approach is the same as matching all the open strings. But it goes beyond that, because it also gives us a specific spot for F#, B, and C#. And in the key of A-flat Major open strings alone are little help! So we need more than just open-string-matching to go on.
If you’ve tried today’s scales, I’d love to know what your experience was like! Just post a comment below.
D Major is a great key to start with, because you have backup help from so many open strings, and your violin will sound its very best when you really nail the intonation. After some experience with D Major, venture into other keys.
tomorrow:
I’m a violinist and private teacher in the Chicago area, and in a previous musical life I was in a professional string quartet. Teaching violin and chamber music are dear to my heart. Send me a note or leave a comment on a post — I’d love to hear from you.
If you’ve tried today’s scales, I’d love to know what your experience was like! Just post a comment below.