Somewhat unrelated: I’m looking for a comprehensive overview of why the CAGED system works on guitar. I see lots of mechanical explanations of how to use it to play various chords down the neck, but nothing explaining the theory behind it.
Music and Geometry: Intervals and Scales
41–50 of 65 posts
Re: Music and Geometry: Intervals and Scales
#42Somewhat unrelated: I’m looking for a comprehensive overview of why the CAGED system works on guitar. I see lots of mechanical explanations of how to use it to play various chords down the neck, but nothing explaining the theory behind it.
CAGED is descriptive, ie it's the result of someone noticing "we all know these chord shapes, and they have some useful properties which can contribute to your fretboard knowledge". Those properties are:
- those 5 chords are the first you learn, and mechanically easy to play in open position, so you know them by heart
See this diagram (https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...):
- the CAGED 'order' (C then A then G etc) matches the order in which octaves of the same root appear as you go up the neck, therefore CAGED is a good way of visualising octaves (see how in the 'C-shapes' column the chords 'share' root notes when played in that order up the neck)
- each chord matches a root note position (C: top part of the neck box on the 5th string, A: bottom part of the neck box on the 5th string, etc), therefore if you're playing a scale, no matter what position you're playing it in, you can choose a CAGED chord to overlay on it and easily find the root, third, and fifths (see how in the 'C major scale' column, you can overlay each chord onto one way of playing the C major scale)
- learning these mnemonics should eventually help you 'unlock' the guitar neck, ie have an intuitive knowledge of what intervals you're playing and how to build melodic lines with them
Generally, music theory 'works' because it describes why things sound good. It's not theory that informs what sounds good, rather theory attempts to describe what sounds good and build patterns which will help theory learners, in turn, make music which sounds good.
Re: Music and Geometry: Intervals and Scales
#43The diagrams look nice, but in the end of the day, they are merely nice visualizations of what's fundamentally algebra. There is not much geometry going on besides a quite simple group structure of order 12.
Whenever someone says "there's not much geometry going on" you've identified a person with little capacity for imagination
Mathematicians regard 'symmetries' as algebra (as in group theory etc rather than high-school algebra) rather than geometry and I suspect this is about the use of the word geometry in part.
Re: Music and Geometry: Intervals and Scales
#44The diagrams look nice, but in the end of the day, they are merely nice visualizations of what's fundamentally algebra. There is not much geometry going on besides a quite simple group structure of order 12.
Geometry of Music by Dmitri Tymoczko is a fun book of visualizing chromatic music theory with geometry models, some of it also covered in author's papers https://dmitri.mycpanel.princeton.edu/publications.html
Tymoczko's take is far more interesting and educational than the article.
(I'm being polite.)
Re: Music and Geometry: Intervals and Scales
#45The diagrams look nice, but in the end of the day, they are merely nice visualizations of what's fundamentally algebra. There is not much geometry going on besides a quite simple group structure of order 12.
> There is not much geometry going on You can use the geometric representation ("necklace") to explain modulation of e.g. diatonic scales as axial mirroring. This can be regarded as a geometric operation. When you look at harmony in this way, some interesting insights open up. I've even built a tool to explore it: https://github.com/rochus-keller/MusicTools/tree/master .
This sounds interesting. As a mathematician (in the sense that I have a PhD in group theory), is there a good guide to music theory for mathematicians?
There seems to be lots of stuff along the lines of 'if you understand music, here is some mathematics to help you think about it' but not much 'if you understand mathematics, but not so much about music, here is how to think about music'.
Re: Music and Geometry: Intervals and Scales
#46Earlier quoted context omitted.
If you mean a keyboard which includes a mechanism for causing strings to vibrate, you can tune ANY such keyboard to use just intonation. What you cannot do is modulate between keys with a keyboard tuned to just intonation: it would have to be retuned for every key change. The scope of the mechanism that would be required to do this has not been implemented since the harpsichord was invented. There are synthesizers th…
Eivind Groven developed a mechanical piano for playing in just intonation: http://www.joranrudi.no/mediefiler/The%20Just%20Intonation%2...
Ah yes, of course. 36T ... increasing the number of pitches per octave is a different approach to the problem, and works (at some cost to the performer :)
Re: Music and Geometry: Intervals and Scales
#47Earlier quoted context omitted.
If you mean a keyboard which includes a mechanism for causing strings to vibrate, you can tune ANY such keyboard to use just intonation. What you cannot do is modulate between keys with a keyboard tuned to just intonation: it would have to be retuned for every key change. The scope of the mechanism that would be required to do this has not been implemented since the harpsichord was invented. There are synthesizers th…
This is not true. Singers and violinists can and do adjust intonation so each chord sounds (justly) in tune. The exception is if they were trained with equal tempered instruments (which is common nowadays - see Duffin, “How Equal Temperament Ruined Harmony”) or if they are playing with pre-quantized (fretted/keyed) instruments, in which case they would match the existing temperaments. So the linked article, while it…
I did speak a bit loosely and too simplistically. What I was trying to get to was the point you're making which is the same point the GP was making: when performers have the ability (because of the instrument they are using, including the voice), they will adjust to reduce beating.
My words on this were wrong and misleading.
Re: Music and Geometry: Intervals and Scales
#48Earlier quoted context omitted.
> There is not much geometry going on You can use the geometric representation ("necklace") to explain modulation of e.g. diatonic scales as axial mirroring. This can be regarded as a geometric operation. When you look at harmony in this way, some interesting insights open up. I've even built a tool to explore it: https://github.com/rochus-keller/MusicTools/tree/master .
This sounds interesting. As a mathematician (in the sense that I have a PhD in group theory), is there a good guide to music theory for mathematicians? There seems to be lots of stuff along the lines of 'if you understand music, here is some mathematics to help you think about it' but not much 'if you understand mathematics, but not so much about music, here is how to think about music'.
It depends on your goal.
Music theory is a way to encode and share the practice of music. The practice is largely unconcerned with and unaware of math. Any mathematical treatment that gets too far from the practice won't help you understand music.
If you want to understand and practice music, it's safest to limit your exposure to the body of work we call theory to scales, chords, and the circle of fifths and carefully expand from there. Theory can be useful, but the practice of theory can become too about itself and lose sight of the music.
Being too about theory is how you get people saying, confidently, that songs which use that common four chord progression are boring/hackish even though all the examples are of famous and beloved songs.
Re: Music and Geometry: Intervals and Scales
#49I don't know enough music to tell if this is insightful, or just neat pattern-matching.
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-...
Re: Music and Geometry: Intervals and Scales
#50Earlier quoted context omitted.
> There is not much geometry going on You can use the geometric representation ("necklace") to explain modulation of e.g. diatonic scales as axial mirroring. This can be regarded as a geometric operation. When you look at harmony in this way, some interesting insights open up. I've even built a tool to explore it: https://github.com/rochus-keller/MusicTools/tree/master .
This sounds interesting. As a mathematician (in the sense that I have a PhD in group theory), is there a good guide to music theory for mathematicians? There seems to be lots of stuff along the lines of 'if you understand music, here is some mathematics to help you think about it' but not much 'if you understand mathematics, but not so much about music, here is how to think about music'.
There are many with various mathematical depth:
- Fauvel et al., Music and Mathematics - From Pythagoras to Fractals, 2003, Oxford UP
- Loy, Musimatics Volume 1, 2006 MIT Press
- Tymoczko, A Geometry of Music, 2011, Oxford UP
- Walker, Mathematics and Music, 2013, CRC Press
- Toussaint, The Geometry of Musical Rhythm, 2013, CRC Press
- Chew, Mathematical and Computational Modeling of Tonality, 2014, Springer
- Hook, Exploring Musical Spaces, 2023, Oxford UP
From my point of view, all titles can be appreciated by non-musicians with mathematical background (though I'm an engineer, not a mathematician, and very much involved with non-classical music). But for your specific requirement, maybe Loy is suited, but personally I consider the later books more interesting, especially Tymoczko and Hook. Book recommendations are always very subjective.
Also note that the music theory commonly taught at high schools and universities is barely able to describe music, or only a small fraction of it. And only a fraction of this theory has a mathematical fundament. Most of it is just a heuristic projection of existing music, only useful for recognizing and classifying elements, and not for deriving new music. In recent years, however, new theories have emerged that allow for both a more formal and a more practical approach.