2004 P.Missin - Details

This is an edited version of a post written to harp-l in February 2004.

Intonation and the Diminished Tuning

Considering the fact that the Diminished Layout (say,
for a chromatic) has nothing but minor 3rds next to
each other, wouldn't this also be a big problem with
that kind of tuning scheme?

Exactly.

Does one have to compromise a lot between "in tune" single notes and
intervals that sound good in Dimi? (i'd test it myself, but i don't have "the ear")

Really, the diminished seventh chord is a product of 12TET, at least in its most common usages.

As a (hopefully) simple explanation, consider the following. A diminished seventh chord rooted on C has the notes C Eb Gb A (actually it's Bbb, but let's ignore that for now) and let's add another C on top of it.

A typical pure minor third (5-limit) is a nice consonant interval, so let's try to build a C dim7th with it. It has the ratio 6/5, which works out as being about 316 cents. So working our way up the C dim7th chord, C to Eb is 316 cents, C to Gb is 632 cents, C to A is 946 cents and finally, C to C is 1264 cents.

That's not good. An octave should be 1200 cents, but if you build one by stacking up minor thirds , you wind up with one that is more than a quartertone sharp. None the other various alternative JI tunings for a minor third interval add up to an octave either: the 7-limit subminor third gives you an octave that is actually flatter than a major 7th; the 19-limit minor third is quite close in value to the 12TET version, but still leaves you about 10 cents short of an octave when you add them up.

There are some JI interpretations of the diminished seventh, one common example being the chord tuned in the ratio of 10:15:14:17. This is in fact the traditional JI tuning for holes 3, 4, 5 and 6 draw of the harmonic minor tuning. This adds up nicely to an octave, but it leaves you with each "minor third" interval having a different size to the others. This is fine when the chord is used to suggest a rootless 7b9 chord, or as a passing chord in an otherwise simple diatonic progression. However, most commonly the dim7th is used in a way that relies upon it being equally spaced in all its inversions, diving the octave perfectly into four equal parts and allowing the same dim7th chord to be rooted on any of its four tones.

For use as the basis of a harmonica that is to be played chromatically, the only practical choice for the diminished is 12TET. The same goes for the augmented triad (which divides the octave equally into thirds) and the wholetone layout (which divides the octave into sixths).

The good news is that minor thirds in 12TET sound somewhat less harsh (to most ears) than major thirds in 12TET.


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