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.
Accumulation of Bone-Seeking Radionuclides in Liver Metastasis from Colon Carcinoma
Fifteen colon cancer patients with metastasis to the liver were examined by both liver and bone scans. Seven of these showed accumulation of bone seeking radionuclides in the liver at the site of a defect in the scan. Such accumulation was found in two diphosphonate, in one pyrophosphate and in six 18F scans. To our knowledge, this is the first series to report localization of 18F in intrahepatic metastasis from colon cancer. The precise mechanism for this non-skeletal accumulation is unknown.