2004 P.Missin - Details

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

Tempered diatonic harmonicas, Picardy Thirds, Diminished Tunings and more ratios and limits

First of all, I should perhaps note that not absolutely all diatonic harmonicas used to be tuned in Just Intonation, although almost all were. I believe that a few Hohner models were available in Equal Temperament even before they brought out the Golden Melody - if my memory serves me correctly, the Hohner Orchester I was tuned in ET, although it wasn't available in the US until the late 1960s. Also, the various Solo tuned diatonics that have been made over the years were usually tuned in 12TET, just like the typical solo tuned chromatics. However, those chromatics that used the same tuning as the Richter models (such as the Koch Chrom and some of the older Hohner 260s) were usually tuned to JI.

I was also asked to explain a little more about the Picardy Third, or Tierce de Picardie. This has nothing to do with taking Captain Picard to warp speed, it is the common term for the device of ending a minor key piece of music with a parallel major chord (ie a major chord rooted on the same tonic as the minor key). This was a popular device during the baroque period (and crops up in quite a few pieces in Johann Sebsastian Bach's "Well Tempered Clavier", as well as more recent stuff, such as "And I Love Her" by the Beatles). One reason for the Picardy Third has to do with difference tones. For those who don't know what a difference tone is, you might want to read this page.

The Picardy Third became common during the period when the most common tuning used on keyboard instruments was some form of Meantone Temperament. Unlike our well known 12 Tone Equal Temperament where each semitone is the same size and every key is equally in or out of tune, meantone temperaments are unequal temperaments with more than one size of semitone and some "good" keys and some "bad" keys, the harmonies in the "good" keys being close to Just Intonation.

So, when you play two or more notes on an instrument capable of sustained tones, difference tones are produced. The difference tones produced by a perfectly just Dm triad in root position (D F A) are Bb (produced by the D and the F) and F (produced by the F and the A). These difference tones themselves produce another difference tone between them, an even lower Bb. So, when you play a Dm triad in JI, you hear a low Bb - not the ideal bass note for a Dm chord. However, when you play a perfectly tuned D major chord in root position (D F# A), the difference tones converge to produce a low D, two octaves below the root of the chord, which gives a nice sense of "rootedness" and finality.

However, as tastes changed and the preferred tuning became closer and closer to modern day 12TET, the difference tones produced by minor triads also shifted around as the tunings were tweaked. In 12TET, the minor triad in root position is very close to a 19-limit minor triad tuned in the ratios 16:19:24. A Dm triad tuned like this produces difference tones of A and F#, which in turn produce a difference tone of D, two octaves below the root of the chord. Thus the 12TET minor chord is comparatively stable, although it is a good deal less consonant than the pure (5-limit) minor chord. This meant that the Picardy Third gradually fell out of common use and composers became content to end minor key tunes with a minor chord.

Moving back to the harmonica for a while, this is also the reason why the draw chord on the solo tuned harmonica in 12TET sounds so much more satisfying compared to the blow chord and why third position blues sounds so cool on the chromatic.

Here is another way of looking at the whole issue of Just Intonation. Intervals and chords in Just Intonation are expressed by ratios. These are a little like harmonic "recipes". If a recipe says to use three cups of flour to two cups of water, those ingredients are said to be in the ratio of 3 to 2. If you want to make a bigger serving of whatever it is you are cooking, you could use 6 cups of flour to 4 cups of water, or if you are cooking for a real lot of people, you could use 24 buckets of flour to 16 buckets of water, etc. However, in all these cases, the ratio of flour to water is still 3 to 2.

Likewise, a perfect fifth interval in JI always has 3 units of one frequency played against 2 units of another frequency. So the difference between 300Hz and 200Hz, or 600Hz and 400Hz, 24Hz and 16Hz, etc., is always a perfect fifth.

The term "limit" refers to the highest prime number used as a factor in such a ratio. The perfect fifth of 3:2 is a 3-limit ratio, as 3 is the highest prime number in this ratio. The perfect fourth ratio of 4:3 is also a 3-limit interval. The difference between hole 5 draw and hole 2 draw of a trad JI diatonic harmonica is 7:4, so it is a 7-limit ratio. The interval between 9 draw and 6 draw on that same harp would be 14:9, but as the largest prime factor in this ratio is 7, then it is still a 7-limit ratio.

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