MathJax

Pages

Thursday, July 18, 2013

The Hauptmann Shuffle (1)






This is one of the most well known diagrams in the history of music in the West, the circle of fifths.[1] This diagram has also been augmented (at least since the mid 17th century) by a second circle of fifths inscribed either within the original circle or, as represented in the next illustration, by adding notes (the red dots with lower case labels) between the originals on the same circle:





This has the advantage of showing all 24 major and minor triads and all 24 major and minor diatonic scales. Any three consecutive nodes represent a triad whose root is the first of the three nodes (when moving clockwise), and any seven consecutive nodes can be rearranged to form a diatonic scale. For example, starting with F and proceeding clockwise to D generates the major triads F-a-C (IV, or subdominant triad in C major), C-e-G (I, or tonic triad in C major), and G-b-D (V, or dominant triad in C major) as well as the minor triads in the C-major scale, a-C-e (vi, submediant) and e-G-b (iii, mediant)




To get to the third remaining minor triad in C major, d-F-a (ii, supertonic) as well as the devilish diminished triad b-d-F (using the same logic), we can employ a more modern trick that amounts to the same thing as the often intricate explanatory strategies of older theorists. Looking at the diagram of the double circle of fifths above, think of snipping the circle just below the d on the left and below the D on the right. Then make a new circle  by joining the d and the D (the numbers indicate half-steps between each node):





We could follow several apparently divergent math paths at this point, not the least of which would begin by noting that this representation of the diatonic set is the maximally even 7-in-24, but right now I'd like to draw these well known concepts into the permutation theme. Remember, this is all about discovering deep connections. Just what does the circle of fifths have to do with spirals, sestinas, coils, perfect shuffles, parallel processing? Take a look at the next diagram from the mid 19th century found in Moritz Hauptman's treatise Die Natur der Harmonie und der Metrik (1853).[2]




What Hauptmann is doing here is tracing a path through his "triad of triads" (a linear representation which is essentially the same as the circular representation of the diatonic set above).[3]  Following this nicely symmetric path reveals the step-wise scale version of the set, for example,
F–G–a–b–C–D–e(–F). 
Bending a few of Hauptmann's curving lines upward, we can see it this way:






... the most familiar version of the coil permutation – the perfect shuffle. Here it's shuffling a deck of just seven cards/notes. In cyclic notation, this permutation is (F)(aGC)(ebd)[4], so we know that after three shuffles we will come back to the original arrangement:






The first note, F in this example, is the only fixed point and remains in the first position with each shuffle, while the other six notes permute as a perfect in shuffle.

But the most interesting thing to me is that you can get from any one of these three well-known orderings of the 7-note diatonic set to the others by applying just one permutation, the perfect out shuffle or its inverse, the outL pretzel – e.g., (F)(aGC)(ebd) or its inverse (F)(CGa)(dbe). A bonus here is that all three are maximally even sets (4343433 (7-in-24); (2221221 (7-in-12); (7777776 (7-in-48)), but after examining what happens when other maximally even sets are shuffled, it becomes obvious that a perfect shuffle of a maximally even set will not necessarily result in another max even set (try shuffling the octatonic as one counter example).


____________________
[1] I am fully aware of the centuries-old controversies concerning tuning problems, but here I will be assuming equal temperament and ignoring just intonation. I am doing this by fiat, not because of any personal preference or because I think ET is "right" and JI is "wrong," but simply to stay on topic to make a point that really has nothing to do with tuning. Tuning theory to one side, even the notes on a badly out-of-tune piano would still have the same underlying abstract relationships being described here, however awful they may sound to the sensitive ear.
[2] This is actually from the 1888 English edition, The Nature of Harmony and Metre, tr. & ed. W.E. Heathcote.
[3] Hauptmann's Roman numerals are not the same as the more familiar chord functions still in use, but explaining them here would take us too far off topic.
[4] If we rotate the circle to start with the scale (deck) arranged CeGbdFa, the result woud be the familiar major scale form CdeFGab and the permutation would be (C)(edG)(bFa), and so on for the other modes (arrangements of the deck).