[JAllen]

Hey everyone.
Once again, thank you to all for your input and interest.

Balfour. I like your order! I agree, it would be awesome if pedals were in a nice easy CDEFGAB. That is even one of the orders I stumbled upon while overthinking my above post of mathematical permutations… It makes complete sense for all the reasons you give, plus it would be waaayyy easier to teach to students just starting on the pedal harp. The one issue I have with this order is that both feet must move left to right while moving around the circle of fifths, and right to left while moving around the circle of fourths. I prefer the more symmetrical approach as modulating is more similar to: middle pedals…Now, outside pedals… Okay, inside pedals….(or the inverse for order of flats) and the feet move in mirror as we modulate, instead of moving with the idea of, okay, now we are in the key of D and in order to get to E I need to move both feet to the right one pedal… I also think that might cause one to fell unbalanced while playing. I know I dislike resting on my B and A or D and E. Granted, that is a big assumption considering I’ve never tried that pedaling method and I’m sure with enough practice one could get used to almost anything.
In regards to the Dilling harp, the levers are designed in a manner that is perpendicular to the performer. Plus, all the levers can be easily manipulated by one hand (I am presuming that key changes are generally made with the right hand?). In this case I would believe that having the order of levers CDEFGAB makes sense as the levers are moving away from you. Plus they are closer together and you don’t have to worry about more than one limb (typically) attempting to change keys…

I agree completely with Deette Bunn’s and Paul Knoke’s explanations. (I hope the run of Peter Pan wasn’t too horrible transposed… I feel your pain). Making it so that modulations, especially via whole step, move across the feet so we have one pedal in each foot. Our current pattern fulfills this requirement. But so does DCE BFGA or CDE FBAG…

I spoke to my teacher about this and we looked at some prints from the Encyclopédie or Dictionnaire Raisonné des Sciences, des Arts et des Métiers that she has. You can check out an image some of the prints Here.
You can see that the pedals (shown from the bottom of the harp) are in our current order, DCB EFGA (notated in solfège). This means that our order has been standard since 1751.
I also stumbled across US patent 786275A from 1905 by Per Erik Ekman for the improvements on the harp. You can check it out Here.
From my brief skimming of diagrams I see no reason one could not make a harp with a different order… Halfway tempted to call up a harp manufacturer to see if it is possible…

Best,
Jenna

[JAllen]

Hmmm. Some tidbits here of harp history I don’t know… Thank you.
I think it is interesting to point out the tuning… I will look into this some more.

You are correct with the 5 pedal harp. My logic, for all practical purposes, collapses.
Something interesting I discovered while researching the Hochbrucker harp is that the order for the 7 pedal version was BCDEFGA as to keep the pedal rods from crossing.
I link the article http://www.beatwolf.ch/Portals/14/pdf/Report_Hochbrucker_EN.pdf
For any interested persons.

In regards to using the circle of 5ths as reasoning behind the pedals. This has always bothered me since simply changing modes would shift your pattern. This is why I continually look to the order of sharps/flats. However, it is definitely worth noting that the order does drop out running the alternating pattern on the Ionian mode starting at Eb moving around in fifths…

[JAllen]

Thank you for the responses!!
These have led me to the following explanation I have been pondering on all day. I post this for any other interested parties and because I am a bit of a math music nerd at heart.

I believe my following thought will corroborate with all the above explanations…

—

We have 7 pedals (and the occasional 8th for swell) that we must place around the base of the harp. Assuming that most harps will not have a swell pedal we still have 7 pedals that we must distribute around the base of the harp. This means we have 7 Permute 7 ways of distributing the pedals. This is equal to 7 factorial which is another way of writing 5040. That is a lot of possibilities.

How might we narrow this down?

It makes sense to separate the pedals into two different sets since human beings have two separate feet. Therefore, let us split 7 as evenly as possible into 2 sets: 3 and 4. But which pedals should we assign to which set?
Let us assume that the set of three shall be handled primarily by the left foot and the set of four will be handled primarily by the right foot. This is presumably because most harps will be constructed to be pulled back to the right shoulder resulting in a (slight!) angling of the harp meaning that there is less room for pedals on the left side of the harp. We will then assume to leave a gap between the two sets of pedals for a swell pedal which is out of the way and easy for both feet to reach.
Now, it would make sense to separate out the pedals as symmetrically by some sort of order. That way our harpists will not be required to play primarily with one foot. This is where the circle of fifths/fourths comes in. The circle is built, no matter what mode, with the order of flats being BEADGCF and the order of sharps being the opposite (FCGDAEB). Let us use the order of flats simply because the harp is tuned in all flats (C flat major). Separate out this order so that we have four pedals on the left and three on the right without favoring one foot over the other an you will get EDC on the left and BAGF (notice that I alternated the pedal distribution: B on right, E on left, A on right, etc. It you were to start on the left (B on left…) you would be forced to place two pedals in a row and get BAG on the left and EDCF on the right, note the CF together).

This cleans up our 5040 possibilities a lot! We know that EDC should be on the left, only 6 ways to arrange those pedals. And BAGF are the pedals we should place on the right, only 24 possibilities!
The total number of ways to combine the left and the right possibilities is 144! That is almost small enough to warrant listing them all out!

However, we can still yet define our sample to an even smaller set:
Assume that we still want the left foot and right foot to move in similar symmetric patterns as the instrument modulates through the circle. That means if the right foot pedals B and B is close to the center, then the next pedal, E should be (approximately) equidistant from the center.
There are only 12 ways to do this:
DCE BFGA
DCE BGAF
CED ABGF
CED FABG
DEC GBAF
DEC FGBA
EDC FGAB
EDC GABF
CDE BAGF
CDE FBAG
ECD AGBF
ECD FAGB

Now it would make sense to start the pattern close to the center, where the pedals are easiest to reach and rest your feet on…
That leaves us with DCE BFGA, DCE BGAF, CDE FBAG, and CDE FBAG

One pattern in particular stands out:
DCE BFGA Has the most consistent pattern (start inside then progress inward without having to double back) and the pedals are almost in order moving outward. If we were to swap the B and the E….

Voila! We have DCB EFGA, our symmetry is maintained, and while we may have sacrificed by placing two right foot pedals in a row, the pedals are in order as you move out from the center: BCD on the left and EFGA on the right.

—

This is quite possibly a long, and even painful, method to come to the same conclusion but I feel like it has somewhat sound reasoning of why we use the current symmetric pattern over another…
Please let me know if you think this is in anyway interesting, helpful, amusing or completely overthought.

Best,
Jenna

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