2004 P.Missin - Details

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

Intonational considerations for the Solo Tuned, Harmonic Minor, Natural Minor and Melody MakerTM harps

As I have mentioned before, with the exception of octaves (and to some extent perfect fourths and fifths) you can't have two identically sized Just intervals next to each other without introducing dissonance. This is the main cause of problems when tuning the typical major diatonic layout, where you have a minor third between 3 draw and 4 draw followed by another minor third between 4 draw and 5 draw. In order to get the smoothest possible dominant chord, the 7-limit tuning reduces the interval to a subminor third, making 5 draw sound rather "flat" when used melodically. As the Solo Tuning (as used on most standard chromatics) has the same notes as those found in holes 4 to 7 of standard major diatonic, it shares that same problem. The smoothest overall tuning of the draw notes is achieved by treating it as a rootless dominant ninth chord and tuning it in the 7-limit. Of course, this gives similar problems to the standard major diatonic tuning, specifically the F (assuming we are on an instrument in C) is substantially "flat" in a melodic context. Also the difference tones tend to reinforce the G dominant tonality, rather than a D minor tonality - not so useful for playing the typical third position blues style.

One alternative is to tune the minor triad in the 5-limit. As mentioned in my ramblings on the Picardy Third, this is the most consonant tuning of this triad, but because of the difference tones it generates, it is not necessarily the most stable when used as a tonic chord in a minor key. Not too surprisingly, my preferred JI tuning for the Solo layout if I am mostly going to be using it for 3rd position blues playing, is to use something similar to what I use in holes 4 to 7 of my "general purpose" major diatonic tuning. I tune the blow chord to a Just major triad (on a Solo tuned harp in C, that would place the C notes at +/-0 cents, the E at about -14 and the G at about +2 cents), then tune the minor triad in the ratio 16:19:24 (assuming your D is at +/-0 cents, then the F would be about -2.5 cents and the A at +2 cents). There are a couple of options for the 6th of the draw chord (B on a C harp). I alternate between tuning it a pure major third beneath the D (if the D is at 0, then this would make the B -14 cents) and tuning it to make the chord 16:19:24:27 (which would put the B at about +6 cents). This latter tuning is particularly close to 12TET, but sounds just a little more cohesive when you hit that big minor sixth chord.

The Harmonic Minor tuning has this same problem several times over, with minor thirds between 3 draw and 4 draw, 4 draw and 5 draw, 5 draw and 6 draw, 7 draw and 8 draw, 9 draw and 10 draw. In many ways, the simplest solution is to decide that holes 3 to 10 draw are one big diminished sevenths chord and simply tune it all in equal quarter octaves, or equally tempered "minor thirds". This is simple in concept and in execution (assuming you are using a tuner rather than tuning by ear), but the results can't really be described as consonant.

The traditional solution is to tune holes 1 to 5 draw the same as on the traditional 7-limit major diatonic tuning (for example, if you take 2 draw as zero, then 3 draw would be around -14 cents, 4 draw around +2 cents and 5 draw about -31 cents with holes 7 draw, 8 draw and 9 draw tuned similarly), then tune 6 draw and 10 draw about 5 cents sharper than equal. this puts the draw chord rooted on hole 2 (a dominant seventh chord with a flat ninth) in the ratio 8:10:12:14:17 - a 17-limit tuning. The difference tones produced by this strongly reinforce the root of the chord - the difference tones generated by 2 draw and 3 draw, 3 draw and 4 draw, 4 draw and 5 draw are equal to the note one octave below that produced by 2 draw; those produced by 5 draw and 6 draw, 6 draw and 7 draw are equal in pitch to the note produced by 1 draw; and so on all the way up the harp. This is the smoothest overall tuning for the draw chord and is great for tunes that are built around the minor chord and its dominant, although that 5 draw can sound just as "flat" melodically as the 5 draw on the traditionally tuned major key harps. Also several of the individual intervals can sound quite rough when they are not supported by the full chord.

My own preferred solution is to tune the first 5 draw reeds the same as my preferred 19-limit tuning for the major diatonic, then tune 6 draw so that it is a pure minor third above 5 draw (and tune holes 9 and 10 similarly). This not only sounds much stronger melodically, but it means that the minor subdominant can be strongly suggested by playing various combinations of holes 5, 6, 9 and 10 draw. It still leaves you with some rough intervals, but I can't change the laws of physics...

Thankfully, when it comes to the Natural Minor tuning, things are much more straightforward. This layout has no consecutive intervals of the same size, so the whole thing can be tuned perfectly in the 5-limit, giving perfectly consonant chords without any huge deviations from 12TET. The 5-limit tuning for the minor triad, assuming that the root is set at zero, places the minor third at +16 cents and the perfect fifth at +2 cents. The draw chord of the Natural Minor tuning actually has two minor triads, the tonic minor rooted on 2 draw and another minor triad rooted on 4 draw. (If your 2 draw is at +/ 0, then 4 draw should be around +2 cents, which means that 5 draw needs to be around +18 cents and 6 draw about +6.). One nice side effect of this tuning is that the major chord in holes 3, 4 and 5 draw also comes out perfectly pure, as does the major seventh chord in holes 3, 4, 5 and 6 draw. You also get a lovely minor ninth chord by playing all the draw notes from holes 2 to 6. These extended chords sound beautifully rich in Just Intonation, in a way that cannot be achieved by any tempered tuning.

A general rule of thumb regarding minor tunings is that those minor chords that are used as relative minor to major key, are usually best tuned in 5-limit JI. Although I most often use my natural minor harps to play in straightahead minor keys, because of the presence of both minor and relative major triads in the draw notes of the natural minor tuning, I always use the 5-limit tuning for this harp. Actually, although I typically play the harmonic minor tuning using the blow chord as the tonic in a straight minor piece, I also prefer the 5-limit tuning for this one too, rather than using the 19-limit triad that would reinforce the feeling of tonicity and rootedness. Go figure... As the 5-limit minor tunings involve a lot of notes that are sharp compared to their 12TET equivalent, I also tend to set the root notes of these tunings a little lower than I would if they were major tunings. I usually set the roots of my major harps at +/0 cents relative to A=443, whereas for minor tunings, I tend to root them relative to A=440. This means that both my major and my minor tuned harps have no notes that fall substantially below their 12TET values based on the so-called standard concert pitch of A=440Hz.

At last we come to the Melody MakerTM tuning. The more astute of you will have noticed that the Melody Maker is very closely related to the Natural Minor tuning. For example, holes 2 to 5 on a G Melody Maker has exactly the same notes as holes 3 to 6 on an Em Natural Minor tuning. Not surprisingly, this means that the Melody Maker can also be tuned completely purely using five limit intervals. Assuming your root is set as zero, then the major thirds are at about -14 cents and the perfect fifths at about +2 cents. Just as the Natural Minor has two minor triads a fifth apart (rooted on 2 draw and 4 draw respectively), the Melody Maker tuning has two major triads rooted a perfect fifth apart, rooted on 2 draw and 4 draw respectively. (If you have set 2 draw to zero on your tuner, then 4 draw will be set to +2; this means that 5 draw needs to be about -12 cents and 6 draw about +6.) Also, the blow chord contains a major sixth note as hole 3 blow. This needs to be tuned a perfect fourth above the major third, or -16 cents lower than its 12TET value.

So there you have it - the commonest alternate layouts in their Just Intonation forms. Obviously, you can tune any of them to your own compromise temperaments, simply by tweaking the major or minor thirds according to personal taste, somewhere between their 12TET values and their JI values. Keep in mind that the "serving suggestions" I have given in these pages mostly assume that you have your root notes set to zero on your tuner. If you set your roots a few cents higher or lower, then you will also need to tune the other notes the same amount sharp or flat. However, I must stress (yet again!) that simply tuning each individual note to a given number of cents on your tuner is not going to give anything like the results you can obtain by learning to use your ears, tuning chords as a whole by tuning out beats, utilising the various difference tones, etc, etc. This is nowhere near as difficult as it sounds, it merely takes a little practice. The method I use for doing all of this is explained step by step in Altered States (which can be downloaded for free from this page.

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