Structural properties of multi-octave scales

Abstract

First and foremost, my new friend and brother in arms, Paco Gómez, for guiding me on this amazing journey and for opening my eyes to a fantastic world where there are so many things worth discovering. I am forever humbled by your knowledge and humanity. I look forward to many more adventures! Julián Ávila, for the gift of a marvellous software that turns scales expressed by a string of integer numbers indicating inter-onset distances into musical notation. You probably saved me about three months work on Appendix 2! Ander Garcia, for helping me actually put some of these scales into practice. This is merely the beginning! José Antonio García Sevilla, for the technical support with spreadsheets which enabled me to filter through tens of thousands of scale options.Luis Díez Antolinos, for the hours of conversation about the subject and for patiently undertaking countless preliminary experiments.Ramón Paús, for being a mentor and friend throughout my life and teaching me so much about art, life and music, including the existence of multi-octave scales.My dear Maud, Liam and Aaron, for all your love and support throughout the years. My wonderful parents, Jerry and Angela, for making me who I am and opening my heart to life, art, music and knowledge.My precious daughter, Alana, for being a continuous source of love and inspiration and for teaching me so much about myself and the world.A very special thanks to Pahola, my wife, friend and companion and so much more. Without your invaluable love, support, encouragement, patience and courageous go-for-it attitude towards everything you do, there is no way I would have completed this work (or started it). It's a privilege to share my adventures with you and I look forward to turning these scales into music together.And, last but not least, to you, dear reader. Whilst I am pretty sure that a tree falling in the forest with no one around to hear it probably does make a sound, I am positive that v vi PrefaceThe present research deals with multi-octave or non-octave-repeating scales. That is, scales in which pitch content differs in successive octaves. Throughout the text both terms are used indistinctly. While not exactly common, these structures have been employed in a variety of musical settings to organise pitch content in music from the middle ages to present day. Yet the literature dealing specifically with the topic is scarce and it would appear that a large number of musicians are unaware of their existence, or, if vaguely acquainted with the concept, have not considered the possibilities inherent in such scales. After discussing the musical cultures in which non-octave-repeating scales have existed for centuries and the music of composers and improvisors who have employed them in their work, we define a collection of two-octave scales based on musically desirable intervallic characteristics. We subsequently analyse their structural properties and provide a basic taxonomy. Finally, we look into these scales from the point of view of perception and cognition, considerin...