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.
Tortuosity or aneurysm? The palaeopathology of some abnormalities of the vertebral artery
The anatomy of the vertebral artery (VA) and two pathological conditions which affect it—tortuosity and aneurysm—are described. The changes which these lesions produce in the cervical vertebrae are also described and we present two cases in which these changes have been found. One case is of a female aged at least 45 years at the time of death from a 15th century medieval cemetery site in Britain with what we consider to be tortuosity of the VA. The second case is of a male aged between 25 and 35 years at the time of death from a Coffin period site in Japan, dating to ca 1600 years BP. The changes in this skeleton are considered to be more in keeping with a diagnosis of aneurysm of the VA. Copyright © 2002 John Wiley & Sons, Ltd.