I’ve said a lot about quarter-comma meantone and its great properties. It’s almost time to start exploring the vast realm of ‘well-tempered’ tuning systems that flourished starting around 1690.
But there’s one last thing I want to say about quarter-comma meantone. If you use this system—or any of the systems I’ve talked about except equal temperament—there are some advantages to having your scale start at D rather than C! For example, Wikipedia presents quarter-comma meantone starting from D here:
• Quarter-comma meantone: 12-tone scale.
It’s not an arbitrary decision! This confused me at first, but Matt McIrvin straightened me out and I think I get it now. It’s all about white keys versus black keys on the piano—or harpsichord, or clavichord, or organ.
The advantage of D-based tuning
If we straighten out the circle of fifths shown above, putting C at the middle, we get this picture:
There are 13 notes here, since we need an odd number of notes if we want one to be in the middle. Thus, I’m writing F♯ and G♭ as two separate notes, even though some tuning systems consider them as the same. In C-based Pythagorean tuning, just intonation and quarter-comma meantone they are different.
Notice how asymmetrical this picture is. To emphasize the asymmetry I’ve marked the flat notes in red and the sharp ones in blue. Though C is in the middle, it’s much closer to the flat notes than the sharp ones!
You may complain that any flat note can be rewritten as a sharp one. That’s true. So here’s a more precise way to make my point. Say a note is an accidental if it has either a flat sign or a sharp sign. Then: though C is in the middle, there are more accidentals to its left than to its right!
But here’s the weird part. If we straighten out the circle of fifths putting D at the middle, this asymmetry evaporates:
Puzzle. Why is this true? Why, even though the white notes on a piano form a major scale starting at C, does
• the number of notes you go up the circle of fifths before hitting a black note
equal
• the number of notes you go down the circle of fifths before hitting a black note
only when you start at D?
Of course a ‘why’ question can have many different answers, including that’s just how things are! But there are a few enlightening answers to this question. I’ll just mention that I’m not concerned here with our conventions concerning letter names for notes.
Yes, it’s odd that we say the white notes on a piano form a major scale starting with C rather than something more logical like A. That convention ultimately goes back to a decision made by Boethius shortly after 500 AD, long before pianos or harpsichords existed:
• Wikipedia, Musical note: history of note names.
But if we changed our letter names for notes, my puzzle would persist, with different names for things.
Now to the point. All along in my discussion of quarter-comma meantone I’ve started my scales at C. This puts all the problems connected to the tritone at the bottom of this circle, between F♯ and G♭:
But there are more accidentals on the left side of this circle than on the right! There are five on the left and just one on the right.
If instead we start our scale at D, this asymmetry disappears:
Now we’ve got 3 flat notes on the left of the circle and 3 sharps on the right.
This has implications for practical music. Basically, we’d like to hide the lesser diesis 128/125, or the wolf fifth that appears when we eliminate the lesser diesis, as deeply as possible among the black notes! That will make the scales that are mainly white notes sound better.
Even if you haven’t followed all the details, I hope you’ve seen that there’s a lot of richness to the tuning systems I’ve discussed so far. They kept musicians happy until around 1690. But then a large number of ‘well-tempered’ systems burst onto the scene, which exploited the basic principles we’ve seen so far in new ways. So that’s what I want to talk about next!
For more on Pythagorean tuning, read this series:
For more on just intonation, read this series:
For more on quarter-comma meantone tuning, read these:
• Part 1. Review of Pythagorean tuning. How to obtain quarter-comma meantone by expanding the Pythagorean major thirds to just major thirds.
• Part 2. How Pythagorean tuning, quarter-comma meantone and equal temperament all fit into a single 1-parameter family of tuning systems.
• Part 3. What ‘quarter-comma’ means in the phrase ‘quarter-comma meantone’: most of the fifths are lowered by a quarter of the syntonic comma.
• Part 4. Omitting the diminished fifth or augmented fourth from quarter-comma meantone. The relation between the Pythagorean comma, lesser diesis and syntonic comma.
• Part 5. The sizes of the two kinds of semitone in quarter-comma meantone: the chromatic semitone and diatonic semitone. The size of the tone, and what the ‘meantone’ means in the phrase ‘quarter-comma meantone’.
• Part 6. What happens to quarter-comma meantone when you change it from a 13-tone scale to a more useful 12-tone scale by removing the diminished fifth.
• Part 7. Why it’s better to start the quarter-comma meantone scale on D rather than C.
For more on equal temperament, read this series: