In this album I have compiled the 5 pieces that I have composed following the harmonic pattern of the Tonnetz. Using the chromatic scale, there are 4 different tonnetz, what differentiates each of them is the determined number of notes that have their chords that can be 2, 3, 4, 6 notes respectively. Each of the first four pieces (all for piano) follows the scheme of one of these Tonnetz. In the album the pieces are ordered according to this number of notes that have their chords. In the fifth piece (which is orchestral) the four Tonnetz intervenes successively in its different parts.

In the following PDFs, one for each language, it is explained in some detail what Tonnetz is and the characteristics of the different types, for those who are interested or curious about the subject. Now that I have the information recently studied, it seemed convenient to make this summary.

Explanation in English language:

https://drive.google.com/file/d/1LxHW0Ly5Rl4RZjuPjzJj92SBHl8lzMgc/view?usp=sharing

Explicación en castellano:

https://drive.google.com/file/d/1QLABP3AOhZOl1-G_-BtHvhA5otWEg9Kv/view?usp=sharing

The 4 Tonnetz of the chromatic scale.

Walking by four-note chords.

This work can be considered a continuation (or a second track of the same album) of one of my previous works "Parsimonious Trichords".
This is the case since in the elaboration of its harmonies I have followed the same line of work but taking one more step in the current work. In the harmony of the two works I have used the Tonnetz. This comes from the German term "Tonnetzwerk" which means network of tones. The prototypical case is the one that is formed by the major and minor chords (of three notes) organized in such a way that those that are contiguous only differ by a single note (with only a semitone or tone of difference) and on this network we can go chaining the chords following different itineraries or paths thus defining the harmonic progression. In the work "Parsimonious Trichords" I used this Tonnetz with three-note chords. The present work tries to go a step further and use four-note chords to define the Tonnetz. To achieve this I have applied the ideas developed by Dmitri Tymoczko in his excellent article "The Generalized Tonnetz" (dmitri.mycpanel.princeton.edu/tonnetzes.pdf). The objective will be the same: to get a network that organizes the chords (in this case of four notes) in such a way that those that are contiguous continue to be differentiated by a single note. In this case, to achieve this, it is necessary to use 12 chords of dominant seventh, 12 chords of half-diminished seventh, 12 chords of minor seventh and 6 chords of French sixth. To achieve To achieve the harmonic progression, on this network of chords I have defined 7 itineraries (one for each part of the work) of the so-called Hamiltonians, which means that they go through all 42 Tonnetz chords once and only once. As can be assumed, the chord succession that is achieved is also very parsimonious in the sense that there is very little variation between a chord and the one that follows it.

The work of creating the Tonnetz of four-note chords and as well as the elaboration of the itineraries I have done it by programming a series of algorithms in Python language.

Regarding the rhythmic aspect of the work, I have used an archetype: the rhythmic "motif of the walker" (half, quarter, quarter) for two reasons: one, which is quite evident, to reinforce the concept of itinerary or path and the other as a small tribute to the recently published book "Paisajes del Romanticismo musical" by the composer and theorist Benet Casablancas since it is one of the many characteristics that he analyzes about the music of the Romantic period.

For all these reasons, the exhaustive title of this work could have been "Walking through the Hamiltonian itineraries of the graph (of minimal variation) of four-note chords." which I have not dared to use for obvious reasons.

Regarding the interpretation, it should be said that it is done through NotePerformer playback engine.
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Traducción de esta introducción al castellano:

https://drive.google.com/file/d/1daCJQkXXW8_1gj9H7pzbkHWQPANP6T7L/view?usp=sharing

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Traducció d'aquesta introducció al català:

https://drive.google.com/file/d/1DByRKuTZAfrodnbwF_1muzsVB3pV62FY/view?usp=sharing
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Walking by four-note chords

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.
  
In the following links you will find the PDF files with the theoretical presentation of the current piece:

Explanation in English language:
https://drive.google.com/file/d/1-cz65Td04fTxB_LMbT46GTB8E7dLI7dh/view?usp=sharing

Explicació en català:
https://drive.google.com/file/d/16DUwMUx9ScDbrZiATfhkXAMf7P46EjHd/view?usp=sharing

Explicación en castellano:
https://drive.google.com/file/d/1d0bnSIus1h5v7za1y_oqH8sTv5RG2Hhf/view?usp=sharing

Six-note chords

Here ends my adventure with the Tonnetz! This is the piece that closes the series of four works that I have composed based on it.


In the following links you will find the PDF files, where I explain it in detail with the theoretical presentation of the current piece.

Explanation in English language:

https://drive.google.com/file/d/1YbM9PspOSuMw0mqS46Eyzq1bCTQJEZUr/view?usp=sharing

Explicación en castellano:

https://drive.google.com/file/d/1ezUhGhdC13jp7eCEq1SnBujtXhLKryhK/view?usp=sharing

Explicació en català:

https://drive.google.com/file/d/1H1dI8ZNRwx9iVITabbns0v4bAok1bbgN/view?usp=sharing

I have also created an album where I have gathered the 5 pieces that I have composed following the pattern of the Tonnetz. Each of the first four pieces (all of them for piano) follow the scheme of a different Tonnetz, what differentiates each one of them is the certain number of notes that their chords have. The pieces are presented ordered according to this number of notes that their chords have: 2, 3, 4, 6 notes respectively. In the fifth piece (which is orchestral) the four Tonnetz intervene successively in their different parts.

https://soundcloud.com/ramon-capsada-blanch/sets/the-4-tonnetz-of-the-chromatic

Fifth chords chained with common notes

This is a piece for piano based on Neo-Riemannian theory. It is a MIDI version performed by a VST piano from the Vienna Symphonic Library. You can also hear a real version played by Edward Cohen, a pianist from Scotland: 
https://www.youtube.com/watch?v=pn0t5N1mXdA

In the following links there are the PDF files with the explanation of what the fundamentals of this piece are:

Explanation in English language:
https://drive.google.com/file/d/1Le66JNBocRH9bBr3d-mRrqqa-uS4nYz4/view?usp=sharing

Explanation in Spanish language:
https://drive.google.com/file/d/1t1-xbA49NjBk-WL9LEtPrqxhYZs67-Ko/view?usp=sharing

Explanation in Catalan language:
https://drive.google.com/file/d/1GR9_w7Lw_8CjItXDzzLjpKw_Stx7xaoj/view?usp=sharing

Parsimonious Trichords

With respect to form, this piece follows the line of work started in my previous work "Parsimonious Chords", which was based on the "Tonnetz" graph formed by the notes of the major and minor triads. The current work is based on the "generalized Tonnetz" that includes the previous Tonnetz but also other similar graphs defined in spaces formed by chords of any number of notes. This concept is precisely defined by Dmitri Tymoczko in his excellent article "The Generalized Tonnetz" (https://dmitri.mycpanel.princeton.edu/tonnetzes.pdf). In my specific case I have worked with four Tonnetz: those formed by chords of two, three, four and six notes. In each of them I have used Hamiltonian paths (those that visit all the chords of the graph only once) to define the harmony and the conduction of voices.

With respect to the expressive character, this work is influenced by the harshness of the health crisis that the world is experiencing and also by the atmosphere of the Christmas dates in which it was written.

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Perdidos por caminos de espacios de acordes

En cuanto a la forma, esta pieza sigue la línea de trabajo iniciada en mi anterior obra "Parsomonious Chords", que estaba basada en el grafo "Tonnetz" formado por las notas de las tríadas mayores y menores. La obra actual está basada en el "Tonnetz generalizado" que incluye al anterior Tonnetz pero también otros grafos similares  definidos en espacios formados por acordes de un número cualquiera de notas. Este concepto lo define con precisión Dmitri Tymoczko en su excelente artículo "The Generalized Tonnetz" (https://dmitri.mycpanel.princeton.edu/tonnetzes.pdf). En mi caso concreto he trabajado con cuatro Tonnetz: los formados por acordes de dos, tres, cuatro y seis notas. En cada uno de ellos he utilizado caminos hamiltonianos (aquellos que visita todos los acordes del grafo una sola vez) para definir la harmonía y la conducción de voces. 

En cuanto al carácter expresivo, esta obra está influenciada por la dureza de la crisis sanitaria que el mundo está viviendo i también por el ambiente de las fechas navideñas en la que ha sido escrita.

Lost on paths of chord spaces

A piece of music is a mentally constructed entity, of which scores and performances are partial representations by which the piece is transmitted.
(Fred Lerdahl & Ray Jackendoff)

Any tone can succeed any other tone, any tone can sound simultaneously with any other tone or tones, and any group of tones can be followed by any other group of tones, just as any degree of tension  or nuance can occur in any medium under any kind of stress or duration. Successful projection will depend upon the contextual and formal conditions that prevail, and upon the skill and soul of the composer.
(Vincent Persichetti)

Music is a system of signs composed of complex forms which may be represented by physical sounds, and which in this way mediate between mental and psychic contents. To deal with such a system, semiotics are naturally evoked. To describe these forms, mathematics is the adequate language. To represent these forms on the physical level, physics are indispensable. And to understand the psychic contents, psychology is the predestined science.
(Guerino Mazzola)

The 4 Tonnetz of the chromatic scale.

Ramon Capsada Blanch