Today's signal of the day deals with one of the most revolutionary chords in Western music history and it is called the Tristan Chord. So Tristan Chord is the first chord in Wagner's opera Tristan und Isolde. First performed in 1865 this opera tells a story of impossible love between a Cornish knight, Tristan and an Irish princess, Isolde married to Tristan King. Their passion only finds a tragic resolution in the death of two characters at the end of the opera. Unfortunately, Wagner is not only famous for his music, but also for his controversial political and social beliefs. The fact that we refer to him and his opera in this example is for purely signal processing reason and it does not imply our acceptance of those beliefs which we strongly condemn. The Tristan Chord first appears in the introductory bars or the prelude of the opera. [MUSIC] It may not sound revolutional, but it must have surprised the first listeners. The Tristan Chord is a dissonant chord. It does not sound pleasing to the ear. Instead of evolving to a consonant chord, so these ones that is pleasing to the ear like musical convention of the time would have expected. It evolves towards another dissonant chord. It just creates a never ending tension of longing and desire, which reflects in possible love between Tristan and Isolde. Before further analyzing this chord, let us make a quick detour through Western music conventions. Since a time of Johann Sebastian Bach, the reference note in the Western music system is A4, which is associated with frequency 440 hertz. There exists various intervals between two notes, and the most basic one is the octave that corresponds to a doubling or housing of the original frequency. For example, A3 has a frequency 220Hz, and A5 one of 880Hz, or a quadrupling of the frequency that is two octaves. Each octave is evenly distributed in 12 half-tones. Thus, the frequency of each note can be calculated by multiplying the reference frequency to the 12th root of 2 raised by the power of the number of half-tones separating the two frequencies. Let us now apply the fast Fourier transform to this chord to analyze its frequency components. From this analysis we see that there are many components. So we will need to do educated guess which ones are really part of the orchestration. We can see that there is an F3, a B3, a D4 sharp, and a G4 sharp. In addition, there appears to be an F4, which is a regular octave. But also B4. Now that we know the frequency components of this chord we can try to synthesize it. For example using the Karplus-Strong algorithm we have studied during the lecture. This algorithm allows to synthesize each note separately by setting the frequency at which the algorithm works. So we can synthesize in DLC F3, [SOUND] in the B3, [SOUND] D4# [SOUND] G4 sharp [SOUND] To synthesize a chord we use four Karplus-Strong algorithms in parallel. Each for producing a specific note and simply adds the a result to attain the chord. Let us listen to the result. [SOUND] The chord does not sound as nice as in the original version. The reason for this is that we're only reproducing one element of the chord, the frequency components, but we have ignored an important element the timbre of an instrument. The timbre characterises the way the frequencies components decay over several multiples of the base frequencies. They depend on the way the chord is excited. Despite having the same frequency a chord will produce a completely different sound depending on whether it is plucked, pinched, or pressed. Using an electromechanical system it is even possible to excite a chord in such a way that its timbre resembles that of a wind instrument for example. The Tristan Chord is so important in music theory that it has its own entry in Wikipedia. This is certainly not the first time this chord was used by a composer. But it is a context in which it appears and its resolution that makes it so special. So Tristan Chord has had a huge influence on music and is seminal in the evolution of the use of harmony in the 20th century classical music. Many composers, such as Ravel, Debussy or Mahler have used a Tristan Chord as a tribute to Wagner's genius. The influence of the Tristan Chord goes far beyond the universe of classical music, and spans all sorts of arts. The famous Dutch painter, van Gogh, said that he had been influenced by it. More recently, the film director, Lars Von Trier, was so touched, that it became a recurring pattern in the soundtrack of his movie, Melancholia. Coming back to Wagner, the Tristan chord reappears all over the opera as a light motif but it's resolution only appears at the end. We now leave the stage to Walter Meyer interpreting this grand finale at the Bashaud Festival. [MUSIC]
