Tuning and temperament

Meantone and otherwise non- equal temperament tunings

A description: http://www.albany.edu/piporg-l/meantone.html

And a demonstration:

The just intonation for the white piano keys starting from C are in ratios of
9/8, 10/9, 16/15, 9/8, 10/9, 9/8, 16/15

Note that a pure major third has ratio 5/4 and a pure minor third has ratio 6/5. On the other hand, in equal temperament the major third has ratio 2^(4/12) = 1.2599 > 5/4 and a minor third has ratio 2^(3/12) = 1.1892 < 6/5.

Intuitive experience is that the sharper the interval, the brighter the tone, while the flatter the interval, the sadder the tone. So while a minor third being flatter than a major third obviously generates the characteristic associations of emotional qualities to major vs. minor keys, it is interesting to find that pure intonation intervals themselves don't actually make these qualities more pronounced... Indeed, a pure major third is supposed to be flatter than the equal temperament equivalent and a pure minor third sharper than its equal temperament equivalent. Perhaps counter-intuitively, bending towards Pythagorean tonal purity within the scale lessens major/minor third affinity to their respective qualities. It’s the wrong knob to turn!

However, what pure intonation suggests is that certain keys in pure intonation (and derived mean-tone) tunings have pronounced major/minor qualities due to exaggerated thirds. D minor begins with the rather flat D-F minor third interval with ratio 1.1852 < 2^(3/12) < 6/5, so D minor is a reasonable candidate for the saddest of all keys in my book. Probably not a coincidence that, in some traditions, the minor scale developed from the (degree 2) Dorian mode to begin with.

Here are some supposed key qualities that have become nonsense in the equal temperament world.

From this, I’m just guessing that in common tunings, the D-F# major third interval was sharp, which contributed to the D major key’s qualities. Maybe.

One application of this is in the string ensemble setting. This is so true about good quartets vs. just average ones. One example of a group that has tuning figured out is Orpheus, which I have had the privilege of hearing in person. They played a Haydn Quartet (No. 52) with a tonal quality that is just surreal and not replicated by any number of inferior recordings I’ve heard since then.


  1. March 5th, 2009 | 4:06

    How can you save this video to the computer hard drive??

    Any ideas ??

  2. me
    March 5th, 2009 | 14:17

    Try FlashCapture.

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