Modal playing on an instrument designed for just intonation in the major scale (specifically dulcimer)
Sorry if this gets long-winded; I've only recently ventured down the JI rabbit-hole and want to spell out my reasoning in a way that will expose gaps in my understanding.
For those not familiar with the mountain dulcimer, it's an American folk instrument, a fretted zither, traditionally set up with one string tuned to the tonic of a major scale and two strings tuned to the dominant - CGG or equivalent. It has a diatonic, 'gapped' fretboard, and traditionally only one of the strings, the 'G' closest to the player, would ever be fretted, in order to play major-scale/Ionian mode melodies over a 1-5 drone.
The fret layout gives the following series of whole and half steps, so that the notes of the major scale start on the 3rd fret: WWHWWHWWWHWWH...
Traditionally (that word again!) the frets would have been set by ear in order to maximise consonance between the tonic drone and each note of the scale - i.e. an attempt at 'just intonation'. Which seems appropriate, since concerns about awkward key changes or harmonic progressions don't crop up.
BUT. It has also become common practice to re-tune the dulcimer in order to play in different modes on that diatonic fretboard - e.g. a '1-5-4' tuning like CGA would be used to play in Dorian mode with the scale starting at the fourth fret, or a 1-5-8 tuning like CGC to play in Mixolydian mode with the scale starting on the open string.
Which would be all very well with ET fret spacing, but doesn't seem to really work with JI fret spacing.
My understanding is that a standard JI layout of the fretboard described above in terms of simple whole and half steps would look like this, where T is a 204-cent greater whole tone, t is a 182-cent lesser whole tone, and S is a 112-cent semitone:
tTSTtSTtTSTt…
So treating the note at the fourth fret as the first note of the Dorian modal scale means you get the following intervals for that scale:
tSTtTSTt
which is surely not what you'd come up with if you set out to define a JI Dorian scale. The major second should be a 204-cent greater whole tone, shouldn't it, meaning a 316-cent minor third and a 702-cent perfect fifth... but instead there's a 182-cent major second, a 294-cent minor third and a 680-cent fifth - a 'wolf fifth', I believe.
Similarly, the Aeolian scale starting on the first fret has (I think!) an 'acute fourth' (?) of 520 cents. The Mixolydian scale starting on the open string just has - again, I think! - the wrong kind of major second, 182 cents instead of 204.
So - a question at last! - how 'wrong' are tunes played on those modal scales going to sound? I'm guessing the slightly 'off' second in Mixolydian isn't going to be a real problem, but how about the fourth in Aeolian? And what about Dorian - the third looks no worse than ET, really, but just how hideous is that fifth going to sound, especially against a drone on the same note but justly tuned?
One 'tweak' of the JI diatonic scale I've come across, here:
www.music.bracker.uk/_wiki/index.php?n=Music/Just-Intonation
swaps the intervals between the fifth and sixth, and sixth and seventh, degrees of the major/Ionian scale, which translated on to the dulcimer fretboard would give you this layout:
TtSTtSTTtSTt…
meaning (I think!) you get a Pythagorean 906-cent major sixth rather than a 5-limit JI 884-cent major sixth, but thereby get a just 702-cent fifth in Dorian, a true 498-cent fourth in Aeolian, and a spot-on JI scale in Mixolydian.
Is that right? And if so it a sensible trade-off; i.e. is that 'off' major sixth not so bad compared to the 'off' fifth and fourth it gets rid of? Am I missing anything about this 'tweaked' scale that makes it unsuitable for playing over drones on the tonic and dominant - some unpleasant interval lurking somewhere?
I'm probably way overthinking this, but I'm trying to figure whether my next dulcimer should be an off-the-peg JI-fretted instrument, an off the peg ET-fretted instrument, or an instrument with a custom JI scale designed as a 'best fit' for several modal tunings. Any thoughts, ideally expressed at 'newbie' level, very welcome.
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I'm no expert on mountain dulcimers, but your quest for an instrument with a fixed fretboard that play several scales in just intonation may indeed by a journey down a rabbit hole.
Don't forget the basic mathematical issue with JI: to create a complete diatonic scale and maintain "small" frequency ratios, you need two different sizes of whole tone.
That unfortunately means you also get two different sizes of perfect fifth, one of which sounds badly out of tune.
The numbers are easy to calculate: if C has frequency 1, G is 3/2, D is a fourth lower, 3/2 x 3/4 = 9/8; E is 5/4, and A is a fourth higher 5/4 x 4/3 = 5/3.
So the fifth between D and A is 5/3 x 8/9 = 40/27. Oops, that doesn't equal 3/2!
You can fix D-A by making C to D a "small whole tone" of 10/9. But then D to G isn't a perfect fourth.
Personally, I would be inclined to copy what was done in the 16th century, and tune a dulcimer to meantone temperament. That will give you pure major thirds, almost pure 4ths and 5ths, and the gross intonation problems described above disappear so long as you don't venture into extreme keys or transposed modes.
The mathematical basis of quarter-comma meantone is simple: you take a JI major third (5/4) and divide it into two equal whole tones (hence the name). That means that C:D has the frequency ratio 1 : (square root of 5)/2 = 1.118 approximately, compared with the narrow and wide whole tones of 1.111 and 1.125. Few people can hear the difference, because whole tones are not very consonant however you tune them.
You complete the scale by dividing the 9th, C to D plus an octave, into two equal fifths. That gives C:G as 1.495, which is near enough a JI perfect fifth - in fact the slow beats are audible, but they sound like a bit of vibrato, not like "this is out of tune."
You then have enough ratios to fix all the other notes using the same sized intervals.
Alternatively, do what was already being done in the 16th century with guitars, and just go for equal temperament. The "practical" method of setting the fret positions used a ratio of 18/17 for a semitone, which isn't exactly right (1.0588 compared with 1.0592) but the height of the guitar action compensates for that by making the higher frets sound sharper.
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