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Understanding Base-40 Arithmetic in Music Apps: History, Theory, and Applications

Explore the innovative Base-40 arithmetic system developed by Walter Hewlett, aimed at enhancing musical literacy and intervallic complementarity. This comprehensive method preserves enharmonic spelling, supports accurate analysis, and ensures correct transposition. Initially conceived in 1986 and published in 1992, the system was further elaborated in U.S. Patent 5,675,100. The Base-40 system provides consistent arithmetical results across musical applications, making it a vital tool for musicians and developers alike. Discover its significance and applications in music technology and education.

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Understanding Base-40 Arithmetic in Music Apps: History, Theory, and Applications

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  1. Base-40 arithmetic for music apps Music 253/CS 275A Stanford University

  2. Where did Base-40 come from? • Conceived by Walter Hewlett (1986); first pub 1992 • Goals: enharmonic spelling preservation, correct analysis, correct transposition • Reproduced at http://www.ccarh.org/publications/reprints/ • Further elaborated in U.S. Patent 5,675,100 (7 October 1997) http://www.google.com/patents/US5675100 2013 Eleanor Selfridge-Field

  3. Why Base-40? • Musical literacy • Tonal legibility (common-practice era) • Musical computation in integer arithmetic • Intervallic complementarity Base-10 complementarity: If interval = 3, complement = 7 If interval = 6, complement = 4 2013 Eleanor Selfridge-Field

  4. Review: Interval sizes and qualities 2013 Eleanor Selfridge-Field

  5. Review: Intervallic complementarity 2013 Eleanor Selfridge-Field

  6. Review: Intervallic complementarity in chords 2013 Eleanor Selfridge-Field

  7. Need for integer arithmetic • Binomial solutions: Brinkman, Böker-Heil • Required 3 params (pitch name, octave number, inflection) • Arbitrary mappings: C=10, D=20, E=30…. • Same intervals do not always produced same numbers (depends on endpoints: F-E = 10, Eb-D = 9) • Base-40 produces consistent arithmetical results (irrespective of endpoints and without binomials) 2013 Eleanor Selfridge-Field

  8. Physical instrument Cultural apparatus Letter names Base-7 (0 #s/bs) Octave numbers Base-12 (1#/b) Inflection names Base-21 (1#/b) Inflection names Base-40 (2#/b) Enharmonic-notation tiers 2013 Eleanor Selfridge-Field

  9. Third tier ## # - b bb Fourth tier ### ## # - b bb bbb Wider system: Enharmonic-notation tiers (7 x 5) + 5 C## / D / Ebb A# / Bb / Cbb D## / E / Fb 2013 Eleanor Selfridge-Field

  10. Base-40 Rule Simple rule: Where a whole step exists between two key names, a null token is used. 2013 Eleanor Selfridge-Field

  11. From Base-40 for enharmonic preservation MIDI to base-7 MIDI to base-12 MIDI to base-21 MIDI to base-40 Solution: Translate from symbolic code to MIDIPlus 2013 Eleanor Selfridge-Field

  12. What is MIDIPlus? • In MIDI file format, a binary implementation of base-40 • Replaces last 3 bits of velocity byte • Used to interpret key number 2013 Eleanor Selfridge-Field

  13. MIDIPlus in Printing Raw MIDI to Notation (Bach Prelude in E Minor, BWV 855 Translation from symbolic code (MuseData) to MIDIPlus to notation 2013 Eleanor Selfridge-Field

  14. Chords (intervallic complementarity) • Intervallic complementarity • Chord definitions 2013 Eleanor Selfridge-Field

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