# Six-note chords



## RamonC (Jun 7, 2018)

For a couple of years I have had a line of work open that is the composition of piano pieces whose harmony is based on the "Tonnetz". The one I'm publishing here is the third one that I compose (the other two are "Parsimonious Trichords" and "Walking by four-note chords") and there would still be a fourth to finish this chapter and that I think I will compose soon.


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https://soundcloud.com/ramon-capsada-blanch%2Fsix-note-chords



Next, I attach the introduction that explains what is the "Tonnetz" and the theoretical context of this piece:


*Six-note chords*

This work is the third in a series of piano pieces that I have composed following the harmony that defines the _Tonnetz_. The first was _Parsimonious_ _Trichords_ and the second was _Walking_ _by_ _four_-_note_ _chords_. As I explained in the introduction to these two works, the _Tonnetz_ comes from the German term _Tonnetzwerk_ which means network of tones. The prototypical case is the network or graph in whose nodes each of the 12 notes of the chromatic scale are placed in such a way that the arcs that join these nodes form the chords of three notes, major and minor, distributed in such a way that the chords that are contiguous are only differentiated by a single note with only a semitone or tone difference. This little variation in the transit of one chord to its neighbor has caused them to be called _parsimonious chords_, in the sense of scarce, thrifty. On this network we can chain the chords following different itineraries or paths, thus defining the harmonic progression. In the piece _Parsimonious Trichords_ I used this Tonnetz with three note chords. But it is also possible to define other _Tonnetz_ with chords of more notes that maintain this parsimonious structure in which the contiguous chords have this minimum difference of half or one tone. This process of definition has been well studied by Dmitri Tymoczko in his excellent article _The Generalized Tonnetz_ (_dmitri.mycpanel.princeton.edu/tonnetzes.pdf_). In the second piece _Walking by four-note chords_ I worked with a _Tonnetz_ of four-note chords and in the present piece I have used the _Tonnetz_ with six-note chords. It is only possible to build _Tonnetz_ that meet the condition that the number of notes that form the chords is a divisor of the total number of notes available in the musical scale used; therefore, in the case of the 12-note chromatic scale, only four different _Tonnetz_ can be built, those with 2-, 3-, 4-, or 6-note chords. (In my case, I would need to compose a piece with the 2-note chord _Tonnetz_, which I will probably do in the near future).

The evolution of the size and complexity of the _Tonnetz_ is remarkable. We are going from the 24 major and minor chords that make up the _Tonnetz_ of 3-note chord, to the 42 chords of the _Tonnetz_ of 4-note chord (dominant seventh, half-diminished seventh, minor seventh and French sixth chords), to reach the 124 chords that make up the _Tonnetz_ of 6-note chords. In this latter case, the chords do not have their own name, but they can be classified into 12 different types according to the distribution of the amplitudes of the intervals that appear between the contiguous notes that vary between one, two or three semitones. Another significant difference is that the harmony provided by 6-note chords is more dense and more dissonant than that obtained by seventh chords and much more than the diaphanous harmony obtained by triads. The third difference to highlight is the notable increase in parsimony, of the little variation that exists between the contiguous 6-note chords; this increase is due to the fact that we went from a proportion of 1/3 to another of 1/6 of different notes with respect to the total number of notes of the chord.

As I did in the first two works, I have defined the harmonic progression on the _Tonnetz_ by choosing the contiguous chords that follow a Hamiltonian path, that is, a path that passes through all the chords of the _Tonnetz_ but only once. Specifically, I have used 3 different Hamiltonian paths, one for each of the three parts of the work.

The construction work of the _Tonnetz_ of 6-note chords is laborious and I have done it by programming diverse algorithms in Python language. I have done the same for the search for the Hamiltonian itineraries, but in this case the difficulty has been much greater because this involves solving a problem that, in mathematics, is called a NP-complete type problem that requires practically infinite time to be solved, when the size of the set where it arises has a considerable size. What was very easy in a set of 12 elements (Tonnetz of 3-note chords) and also in the set of 42 chords of 4 notes, has become practically impossible in the current set of 124 chords of 6 notes. It has not been possible to write or find an algorithm that computes a Hamiltonian path in a 124-element graph in a time that is practicable with the computing tools at my disposal. This has forced me to make some approximations to get paths that were not purely Hamiltonian, but almost. The itineraries used have an 8% deviation from the Hamiltonian itineraries; but perhaps this small deviation has a favourable effect: to give a little more variety to the harmony and alleviate the possible excessive monotony due to the great parsimony that the 6-note chords would have if they were part of a perfect Hamiltonian itinerary.


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