Friday, April 23, 2010


This book review by Jan Swafford is a nice overview of the temperament question. Thanks to A. C. Douglas for pointing it out.

Here's the main problem:

As Pythagoras also realized in mathematical terms, if you start with a C at the bottom of a piano keyboard and tune a series of 12 perfect 3:2 fifths up to the top, you discover that where you expect to have returned to a perfect high C, that C is overshot, intolerably out of tune. In other words, nature's math doesn't add up. A series of perfect intervals doesn't end at a perfect interval from where you started. If you tune three perfect 5:4 major thirds, it should logically add up to an octave, but it doesn't; the result is egregiously flat.


  1. That's right. If you tune a piano using equal temperament, the frequency ratio of C-G is not 1:1.5. it is 1:1.4983.

    And here is the source of your issues with regard to the flat G on the Bb side. The frequency ratio of F-A ought to be 4:5 or 1:1.25. But in equal temperament it is 1:1.2599. So the open A on the Bb side is really quite distinctly flat as compared to equal temperament. Therefore, so is the G you play when you put down 1st valve. Assuming your first valve is tuned to take the pitch down an equal temperament tone, the actual pitch ratio you get for Open F to 1st-valve-G is 1.1136, whereas an equal temperament tone has a ratio of 1:1.1225. That is quite an audible difference.

    Of course, the horn not being a perfect resonating tube, the frequency ratios won't be exactly as I have described, but they will be pretty close.

  2. Martin I. Gaines30/4/10 19:44

    Chord tuning in ensemble playing is its own issue... root of any chord should plant at pitch (A-440 or whatever your tuning base), the fifth 2 cents sharp, the third (in major tonality) flattened a remarkable 13.7 cents will give you a waveless, "just" tempered chord. There is a machine I have been using with my young bands that produces this variance... It has changed the way we vertically tune. There is a demo of it here:

    Further example:

    Now, as cool as this is, the burden on the musicians is great. The result, however, is "sonically" unbelievable...

    Just sharing...

  3. Maestro - Thanks for that comment and those links. The thing I remember about you and tuning was a comment along the lines of needing to be "in tone" to be "in tune". Seems to me that without good tone, the effect of that "waveless, 'just' tempered chord" is lost. It's sort of a theme I've been on here recently, but since you left, the word "tone" has rarely been mentioned in band. And we've done nothing like those wonderful chorale like warm-ups you used to do, which were my favorite part of the rehearsal because it was an opportunity to work on tone and intonation.