The Universal Accidental

THE UNIVERSAL ACCIDENTAL RICARDO JAVIER RADEMACHER MENA 0. INTRODUCTION N THIS PAPER, WE TAKE A NOVEL APPROACH to music theory by studying modes and scales as referenced by sets of accidentals, such as C Ionian having all natural scale degrees but C Mixolydian having a attened seventh scale degree. In doing so, we uncover a universal mapping property connecting all heptatonic modes derived from the dodecaphonic scale and reveal a new musical structure called Accidental Space. Part one casts Accidental Space as an algebra acting on the scale degrees of a heptatonic mode; here we get accustomed to thinking of modes as states and accidentals as operators. Part two casts Accidental Space as a network with modes as nodes and accidentals as links; here we get a comprehensive view of this new space through an eight-scale, 56-mode network. Part three casts Accidental Space as a category with modes as objects and accidentals as morphisms; here we nally see how the Accidental Algebra and Accidental Network are both part of the Accidental Category. Before we dive into the world of music and accidentals, let us rst disambiguate the terminology and context of this paper. Denote heptatonic spellings as alphabetic, CDEFGAB; numeric, 1234567; or intervallic, WWHWWWH (alternatively, m2 for half-step, M2 for I 198 Perspectives of New Music whole-step, P5 for perfect fth, etc.); call the former notes, the middle scale degrees, and the latter intervals. A note will refer to a unique ordered set of frequencies (a.k.a., a pitch-class), a mode will refer to a unique ordered set of notes, and a scale will refer to a unique ordered set of modes. A key will refer to the root note of the rst mode of a scale; this note may also be referred to as the tonic for the scale.1 The scales chosen for this paper, shown in Example 1, are a representative sample re ecting the author’s opinion of the most common heptatonic scales used in western music. Octatonic, hexatonic, and pentatonic scales are not formally within the scope of this paper, but all heptatonic scales built from minor second, major second, and minor third intervals are.2 Proposed statements in this paper re ect conjectures that have not been rigorously proven but will be considered valid within the context of Accidental Space. These propositions as well as new terminology are consistently italicized throughout the paper and are listed in Appendices 1 and 2 for quick reference. It is not the purpose of this paper to explain or comment on voice leading, functional harmony, triads, seventh chords, or any other rule or advanced concept of Western music theory. Nor is there an attempt at comparison with any other music theory, such as traditional, set, topos, or neo-Riemannian. Only basic music theory concepts like octave equivalence and intervals will be used to explore Accidental Space. We also don’t rely on any previous musical visualization or organization and thus there is no need for staff notation. This paper’s purpose is to de ne a proposed new mathemusical structure and give evidence of its use. It is the author’s hope that once this structure’s validity and coherence is con rmed, it can be used as a model where the canonical rules and elements of Western music theory will either naturally emerge or be cleanly integrated. Preliminary investigations along these lines can be found at www.orijikan.com and www.KeyChordMode.com. The Universal Accidental 199 Major Harmonic Major|Maj1〉 |000000〉 |0#000#〉d |Hmj1〉 |0000b0〉 |0#00b#〉d|Maj2〉 |0b000b〉 |000000〉d |Hmj2〉 |0bb00b〉 |00b000〉d|Maj3〉 |bb00bb〉 |b000b0〉d |Hmj3〉 |bbb0b0〉 |b0b0b#〉d|Maj4〉 |00#000〉 |0##00#〉d |Hmj4〉 |0b#000〉 |00#00#〉d|Maj5〉 |00000b〉 |0#0000〉d |Hmj5〉 |b0000b〉 |b#0000〉d|Maj6〉 |0b00bb〉 |0000b0〉d |Hmj6〉 |#0###0〉 |######〉d|Maj7〉 |bb0bbb〉 |b00bb0〉d |Hmj7〉 |bbb0bb〉 |b0b0b0〉d Melodic Minor (Jazz) Neapolitan Minor|Jaz1〉 |0b0000〉 |00000#〉d |Nmn1〉 |bb00b0〉 |b000b#〉d|Jaz2〉 |bb000b〉 |b00000〉d |Nmn2〉 |00#0#0〉 |0##0##〉d|Jaz3〉 |00##00〉 |0###0#〉d |Nmn3〉 |000#0b〉 |0#0#00〉d|Jaz4〉 |00#00b〉 |0##000〉d |Nmn4〉 |0b#0bb〉 |00...