Music and Geometry: Intervals and Scales
The blog post examines the link between music theory and geometry, detailing 13 musical intervals, their geometric representations, and the relationship between equal temperament and perfect geometric shapes.
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The blog post "Music & Geometry - Intervals & Scales" by Roel Hollander explores the relationship between music theory and geometry, particularly focusing on musical intervals and scales. It identifies 13 intervals in Western music, ranging from unison to octave, and discusses their geometric representations through tone circles, specifically the Chromatic Circle and the Circle of Fifths. The Chromatic Circle visualizes the 12-tone equal temperament, while the Circle of Fifths illustrates the stacking of perfect fifths. The article emphasizes that perfect geometric shapes can only be achieved using equal temperament, as just intonated intervals lead to imperfect shapes. Various geometric shapes, such as lines, triangles, squares, hexagons, and dodecagons, are used to visualize the connections between tones. The Tetractys, a significant Pythagorean symbol, is also mentioned as a geometric representation of musical interval ratios. The post concludes that while just intonated intervals may sound more consonant, they do not yield perfect geometric shapes, suggesting a complex relationship between music, mathematics, and geometry.
- The blog discusses 13 musical intervals in Western music theory.
- It highlights the geometric representations of these intervals using tone circles.
- Perfect geometric shapes are only achievable with equal temperament tuning.
- Various geometric shapes illustrate the connections between musical tones.
- The Tetractys is presented as a significant geometric representation of musical ratios.
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16 commentshttps://www.openculture.com/2024/12/john-coltrane-draws-a-pi...
And
https://www.openculture.com/2017/10/john-coltrane-draws-a-my...
With plenty of great links to dive deeper in both!
https://www.stick.com/method/articles/offsetmodal/ https://www.stick.com/method/articles/parallel/
A few months ago, mathematician John Baez had a series on the mathematics of various temperament and keys. Of course he knows his math, but also music thanks to being a member of rather famous musical family. (More math in the second link.)
https://johncarlosbaez.wordpress.com/2024/01/11/well-tempera...
https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-...
https://www.reddit.com/r/musictheory/comments/1etydas/i_made...
I've created a 3d guitar fretboard here, where the height of the blocks corresponds to the height of the pitches: https://www.fachords.com/guitar-fretboard-3d/
And here are the shapes of the different chord qualities in the Circle Of Fifths: https://www.fachords.com/circle-of-fifths-chord-shape/
I'd love to see if anyone has done this geometric / visualization for Indian classical (specifically, Hindustani) music??
Perhaps there's specific shapes / visualizations in certain Ragas that naturally emanate?
Note that in the Indian classical (Hindustani) music system, the Ragas are a "framework" for melody, not really a mode (as in Western music theory).
And if you move around the Chromatic Circle, swapping every second pair of notes with its opposite on the other side of the circle, you have the Circle of Fifths.
'Pythagorean Temperament' involves the Pythagorean Comma (aka Comma of Pythagoras). Whilst mentioned, it's not spelled out as such here: https://en.m.wikipedia.org/wiki/Pythagorean_comma
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The paper "Generalized Carlos Scales" by Andrew V. Sills generalizes Wendy Carlos's musical scales, emphasizing major and minor thirds, and corrects typographical errors from its first version.
Werckmeister Temperment
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A music theory teacher is developing an Electron app with interactive games for skills enhancement, focusing on note recognition and interval identification, and seeking tools for future score-interactive games.