The Polish Duda

:: An Intervallic Analysis Of The Polish Duda

My acquaintance Rafał shared some pages from a book showing that the notes on the chanter of the Polish duda, a kind of bagpipe, are quite microtonal.

The tones are conventionally notated [D, F#, G, A, B, C, D, E] and they are separated by roughly [350, 85, 190, 190, 120, 140, 180] cents. We can see that these are only precise to 5 or 10 cents.

I tried to find frequency ratios to make sense of this. I looked for ratios that

1) Match the interval sizes, 

2) Are simple (have small numerators and denominators), 

3) Multiply together with adjacent ratios to be even simpler. 

Here's the scale that I came up with. The ratios only have factors of 2, 3, 5, and 7.

D to F# :: ~(350 cents) -> 60/49  (351 cents)

F# to G :: ~(85 cents) -> 21/20 (84 cents)

G to A :: ~(190 cents) -> 10/9 (182 cents)

A to B :: ~(190 cents) -> 28/25 (196 cents)

B to C :: ~(120 cents) -> 15/14 (119 cents)

C to D :: ~(140 cents) -> 49/45 (147 cents)

D to E :: ~(180 cents) -> 10/9 (182 cents)

Here are some nice compound intervals produced by this tuning:

Low D to G: 9/7

F# to A: 7/6

A to C: 6/5

B to high D: 7/6

G to C: 4/3

Low D to A: 3/2

A to High D: 7/5

If there were an intermediate tone "E" between the low D and F#, it would probably be 15/14 (119 cents) over the low D, leaving 8/7 (231 cents) to reach F#. The interval from E to G would then be 6/5, a just minor third.

The reference from Rafał also showed how each scale tone deviates from 12-TET, with the {A natural} being the only note tuned the same. This makes me think that I should write all the intervals relative to {A natural}, perhaps a drone {A natural} an octave below the one in the scale. This starts out well but goes a little crazy at the end:

[Sbd5, P5, SpM6, m7, P8, Sbd10, m10, SbSbdd12, SbSbd13]

[7/5, 3/2, 12/7, 9/5, 2/1, 56/25, 12/5, 196/75, 392/135]

Note that I've included the "E" tone above low D because I like it. I don't have much faith that Polish bagpipes use  intervals like "SbSbdd12" and "SbSbd13".

If the second to last interval (between C and High D) were 120 cents instead of 140 cents, we could interpret it as 15/14 and we'd have this very nice scale:

[Sbd5, P5, SpM6, m7, P8, Sbd10, m10, SpM10, SpA11]

[7/5, 3/2, 12/7, 9/5, 2/1, 56/25, 12/5, 18/7, 20/7]

I'm still looking for other ways to make sense of the scale.

If you add up all the cents between A (in the scale, not the theoretical drone) and the high D, you get 

190 + 120 + 140 = 450 cents

which looks like 

Sb4 # 35/27 _ 449c

This would mean that the D is

(35/27) / (6/5) =  175/162 at  134c

over C, which is the just tuning for SbM2, an odd duck but fairly close to the 140 that the reference specified. If the ratio above that really is 10/9, then the E is a SbGr5 over A, which is justly tuned to 350/243 at 632c. If we could change that 10/9 to a 9/8, we'd have a fairly nice Sb5 at 35/24. Alas that interval is supposed to be about 180 cents, and 10/9 is 182c while 9/8 is a 204c.

For the tuning of the high E, the reference says that it's about 630 cents over A. The sub grave fifth justly tuned to 350/243 is 632c, but also a diminished 5th, justly tuned to 36/25, is 631 cents.

Anyway, I don't feel that I've fully solved this, but I've done pretty well with the data that I have.