2:3:4 Harmony within the tritave
In the Pythagorean tuning system, the fifth is used to generate a scale of 12 notes per octave. In this paper, we use the octave to generate a scale of 19 notes per tritave; one can play this scale on a traditional piano. In this system, the octave becomes a proper interval and the 2:3:4 chord a proper chord. We study harmonic properties obtained from the 2:3:4 chord, in particular composition elements using dominants, inversions, major and minor chords, and diminished chords. The Tonnetz (array notation) turns out to be an effective tool to visualize the harmonic development in a composition based on these elements. 2:3:4 harmony may sound pure, yet sparse, as we illustrate in a short piece.
Musimathics : the mathematical foundations of music
Volume 2 of Musimathics continues the story of music engineering begun in Volume 1, focusing on the digital and computational domain. Loy goes deeper into the mathematics of music and sound, beginning with digital audio, sampling, and binary numbers, as well as complex numbers and how they simplify representation of musical signals. Chapters cover the Fourier transform, convolution, filtering, resonance, the wave equation, acoustical systems, sound synthesis, the short-time Fourier transform, and the wavelet transform. These subjects provide the theoretical underpinnings of today's music technology. The examples given are all practical problems in music and audio. Additional material can be found at http://www.musimathics.com.
DESIGN CONSIDERATION TO PPM KLYSTRONS FOR INDUSTRIAL LINAC
Toshiba has been developing the C-band and X-band Periodic Permanent Magnet (PPM) focused klystron in collaboration with KEK and manufactured prototypes for the JLC project. [1] The PPM klystron has an advantage to enhance the total efficiency as a result of eliminating the electric power for the focusing magnet. From experience on the development the PPM focused klystrons for the JLC project, we plan to apply the PPM focused klystron to the medical and industrial linac. The periodic magnet focusing scheme has some technical difficulties such as stop band voltage. To design the PPM klystron for the medical and industrial linac, it is necessary to give heed to electron beam instability; so that the tube parameters of the PPM focused klystrons have some constraints. The size of the accelerator system is substantially reduced if a PPM focused klystron is used. The majority of medical and industrial linacs use frequencies of S-band range. We discuss the availability of the S-band PPM focused klystron in this paper.