Summer of Math Exposition

The video presents the math needed to expand “Tonal Space”, where a well-defined, extended set of musical intervals resides. Numerous “Rational Intervals” and “Spectral Chords” are illustrated with Manim animations and audio examples. Intervals and chords are the building blocks of musical harmony. They constitute tonality within a piece of music. If harmony is to be evolved, supplementary chords must be identified and appropriate strategies for playing have to be developed. Thus, the musical chord becomes our central entity for further research. Assuming that some “extended harmony” exists outside of our familiar cadences, the video illustrates how to identify attractive complementary chords and how to integrate them into a musical instrument for live performance. While countless timbres have been created since the early days of electronic sound design, keyboard instruments with *dynamic intonation control* have not gained much popularity on the stages of the world (with the exception of MIDI pitchbend, which affects all ringing notes by the same interval). Perhaps one reason is that more than twelve pitches per octave appear difficult to handle in a live situation. Maybe the benefits of enriched harmony have not been successfully communicated to composers and to performing musicians. Therefore, the second part of the video deals with the concept of a real instrument with enhanced intonation capabilities, designed to … • minimize dissonance and beating in polyphonic music • render an extended set of consonant musical chords • freely modulate through 12 chromatic keys • implement innovative strategies for changing the pitch of multiple notes (individually and simultaneously) over a wide range - continuously or in discrete steps • maintain the best possible compatibility with twelve tone equal temperament (12tet) • be suitable for live performance A piano keyboard (that can simultaneously address up to twelve pitches per octave) is a good starting point for explaining a process we call “dynamic pitch mapping”, which describes the assignment of a given “sequence of intervals” to successive keys at performance time. For best tonal compatibility with instruments tuned in 12tet, the twelve equally tempered pitches per octave (forming a stack of “irrational intervals”) are an integral part of the instrument's design. It's up to the player to assign one (at a time) of the 12tet-pitches to the appropriate key during live performance. While such an assignment slightly retunes all pitches, the actual intervals between the keys remain unaffected. The subject of this video is at the intersection of art, math and engineering. It would be awesome to take the ideas that have been prototyped so far and turn them into an instrument available to anyone interested.   *Video Chapters* 00:00 Introduction 00:50 Should I watch? 02:25 Frequency Ratio and Musical Interval 03:39 Expanding a Plane of Tonal Space 05:16 Definition of Terms 07:02 Triads in Tonal Space 09:10 3D-Tonal Space 09:45 Balance - A Property of Spectral Chords 10:22 Added Notes and Extended Chords 11:38 Hyperbolic Stretching of Intervals 12:09 Hyperbolic Stretching of Chords 13:08 A Real Musical Instrument 14:58 Dynamic Mapping of Intervals 17:57 Musical Score 18:28 Proof of Concept 19:28 Polar Art 20:30 Intermediary Chords in Standard Chord Progressions 21:43 Coda (until 22:36) *Acknowledgements* Two decades ago, I was lucky enough to find Tim Thompson's “Keykit” on the Internet, a programming environment for experimental real-time processing and creation of MIDI messages. The Keykit language features an ingenious generic “phrase” data type, that stores a tempo-related sequence of MIDI events. The software of the instrument described here was written entirely in Keykit. Keykit is still available on GitHub. Stephan Schmitt, co-founder of Native Instruments and director of Nonlinear Labs in Berlin encouraged me to put my ideas on paper and produce the first sound examples. Stephan is a brilliant motivator. Grant Sanderson's Manim library was a major inspiration for the dynamic visualization of tonal relationships. In a last-minute move, Joachim Schmidt, Hamburg, mastered the audio track of the video. Thanks to all of you, to the 3b1b team and of course to the Manim CE community! *Resources* Sethares, William A. Tuning Timbre Spectrum Scale. London: Springer Verlag, 1999. Helmholtz, Hermann. On the Sensations of Tone. New York: Dover Publications, Inc., 1954. Eskelin, Gerald. Lies my music teacher told me. Woodland Hills CA, USA: Stage 3 Publishing, 1997 Numerous contributors of royalty-free photography and audio at pixabay.com – thank you so much for sharing your art. This video was created by Holger Stoltenberg (Hamburg, DE) for the community-based Summer of Math Exposition (SoMEpi), 2024. You can contact the author at [email protected]