[HN Gopher] Music and Geometry: Intervals and Scales
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Music and Geometry: Intervals and Scales
Author : coloneltcb
Score : 181 points
Date : 2024-12-19 18:52 UTC (1 days ago)
(HTM) web link (roelsworld.eu)
(TXT) w3m dump (roelsworld.eu)
| rekado wrote:
| If you like this you may also be interested in Emmett Chapman's
| Offset Modal System:
|
| https://www.stick.com/method/articles/offsetmodal/
| https://www.stick.com/method/articles/parallel/
| mlochbaum wrote:
| My own take on relating scales geometrically:
| https://mlochbaum.github.io/BQN-Musician/theory/modulation.h...
|
| It does seem that I include all Chapman's scales (while saying
| nothing about chords), although oddly enough he's chosen to use
| the modes of harmonic major but not those of its inversion,
| harmonic minor?
|
| Edit: In fact I found the second link (first one's pretty vague
| and wasn't enough for me to follow the diagram) relevant enough
| that I added a paragraph to point out the connection!
| pohl wrote:
| I love this. Here's another interesting thing I encountered. It's
| a way of organizing chromatic subsets by brightness
|
| https://www.reddit.com/r/musictheory/comments/1etydas/i_made...
| hammock wrote:
| That is cool and different. Glancing at the bottom several rows
| I tend to agree with the classification, as a trained musician.
| I wonder though, what is (the mathematical principle) behind it
| that is causing the brightness/ darkness in the sound of these
| chords?
|
| Don't say "dissonance" (or explain what dissonance is)- that
| much is obvious, looking for something a bit more detailed,
| e.g. why 1-2-5 sounds brighter than 1-4-5
| dbcurtis wrote:
| It is going to be the relative amplitudes of the overtones.
| Pure tone (like flute) is bright, many overtones present with
| the appropriate mix can sound dark (french horn). Next, you
| are going to ask me for the coefficients, but I don't know
| that. Break into the nearest church with a proper pipe organ
| and start pulling and pushing stops to see what happens. (Or
| ask your friend the organist to take you so that you don't
| get arrested)
| recursive wrote:
| It's a reference to this particular chart, not the general
| idea of brightness of a sound. And it's regarding chords or
| scales, not timbres.
| recursive wrote:
| I don't really understand this chart at all, but I think it's
| based on the idea that an upward movement of a fifth is
| bright, and a downward movement is dark. 1-2-5 can be built
| on two upward movements of a fifth, whereas 1-4-5 can be
| built be one upwards movement, and one downward (from the
| tonic)
| tugu77 wrote:
| The 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.
| openrisk wrote:
| Besides geometry there is not a lot of music either, in the
| sense that even this simple symmetry is kinda fake, effectively
| a forced resolution of an essentially unsolvable "problem":
| hammering the intervals into place to inject some "logic" into
| the task of dividing the octave in heptatonic scales. Despite
| the unrepentant Pythogereans across all ages, our musical brain
| is not mathematical except in a very loose way.
|
| Besides aesthetics, though, there might be educational value
| given that the equal temperament tuning is a cornerstone of
| western music education.
| PaulDavisThe1st wrote:
| > the task of dividing the octave in heptatonic scales.
|
| that's not task. The octave has already been divided into
| twelve tones here.
|
| The task is to pick sets of those twelve tones to serve some
| aesthetic purpose. The sets may be of various sizes, though 7
| is common in a lot of european music.
|
| If the task was to generate heptatonic scales from within the
| octave, there are a huge number of possibilities not
| described here, and most of them are rarely used (though many
| more than the ones based on a 12TET system are).
| openrisk wrote:
| > The task is to pick sets of those twelve tones to serve
| some aesthetic purpose
|
| Aesthetic in the musical or visual sense? The visual aspect
| is based on the Z12 symmetry and it is pleasant - like all
| symmetries.
|
| The question is what does the visual experience have to do
| with the music experience?
|
| The first disconnect with the musical experience is that
| the 12TET itself is not what people would, e.g., choose to
| sing in [1].
|
| The second disconnect is that the Greek modes of the major
| scale are not remotely covering all the scales people
| enjoy, even adopting a Eurocentric point [2].
|
| [1] https://music.stackexchange.com/questions/41383/do-
| capable-h...
|
| [2] https://en.wikipedia.org/wiki/Harmonic_minor_scale
| HPsquared wrote:
| On singers and violinists adjusting their harmony to just
| intonation by fine-tuning to zero beat frequency, I
| wonder if anyone has made a keyboard that can do that.
| PaulDavisThe1st wrote:
| 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 that can be retuned in this way,
| because there is no physical mechanism to adjust. The
| results are ... odd. It is still challenging to play them
| because in addition to performing the notes, you need to
| signal the key change/retuning points.
|
| Also, when singers and violinists do this, they are not
| "fine tuning to zero beat frequency". Either you sing in
| just intonation, in which case you cannot modulate
| between keys (because the Nth note of the scale has a
| different frequency depending on the root note), or you
| sing in some tempered scale (in which the frequencies of
| the notes have been adjusted to make modulation
| possible).
| bkazez wrote:
| 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 shows some beautiful
| shapes linked to 12s, has nothing to do with actually
| (justly) in tune music.
|
| Source: master's degree in the topic; am a professional
| singers specializing in music written before equal
| temperament was invented
| tugu77 wrote:
| For a professional musician you are oddly singer/violin
| focused. Any instrument which can physically detune while
| playing has their musicians do this. On wind instruments
| it's via the mouthpiece, any string instrument beyond
| violin has some flexibility etc. It's only the piano that
| doesn't, essentially everybody else does.
|
| But in practise, for many music styles, it doesn't really
| matter. Music is so much more than whether some chord is
| pitch perfect in tune.
|
| Source: Jazz musician on 6 instrument types part time
| professional for 25 years (other part is software
| engineer).
| Tor3 wrote:
| I go to a lot of choir concerts. What I've found is that
| I much prefer (good) choirs singing without accompanying
| instruments, because when there are instruments involved
| the harmonies always fall into equal temperament. There
| is a quality when they sing a cappella which simply isn't
| there when they don't.
| PaulDavisThe1st wrote:
| 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.
| black_knight wrote:
| Eivind Groven developed a mechanical piano for playing in
| just intonation: http://www.joranrudi.no/mediefiler/The%2
| 0Just%20Intonation%2...
| PaulDavisThe1st wrote:
| 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 :)
| Rochus wrote:
| > _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.
| jvvw wrote:
| 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'.
| Kye wrote:
| 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.
| Rochus wrote:
| 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.
| PittleyDunkin wrote:
| Whenever someone says "there's not much geometry going on"
| you've identified a person with little capacity for imagination
| tugu77 wrote:
| I'd love to see some imagination on that page beyond some
| star shapes.
| jvvw wrote:
| 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.
| fuhsnn wrote:
| 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
| TheOtherHobbes wrote:
| Tymoczko's take is far more interesting and educational than
| the article.
|
| (I'm being polite.)
| ChocMontePy wrote:
| I never realized until now that in the the two different circles
| pictured (the Chromatic Circle and the Circle of Fifths) the
| pairs of notes opposite each other are the same in each circle.
| For example in both circles B is opposite from F.
|
| 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.
| smitelli wrote:
| That interval (B-F) would be the tritone, arguably the most
| dissonant one in the toolbox.
| kian wrote:
| If you take the chromatic scale and then swap every other pair
| of notes on opposite sides of the circle, it yields the circle
| of fifths. You'll notice that on the circle of fifths notes
| that skip a step are a whole tone apart in the chromatic scale.
|
| Although there have been some claims in these comments to the
| contrary, harmony is particularly mathematical. Symmetry and
| the breaking of within the integers mod 12 form the
| foundational principles of harmony.
| Duanemclemore wrote:
| Amazing. A deeper dive of the methods John Coltrane used.
|
| https://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!
| ziofill wrote:
| I've been playing piano for 30+ years, and I only learned to use
| the circle of fifths this year. It's been short of a revelation
| and I can't recommend enough to practice scales and drills based
| on it.
| niobe wrote:
| I am a technical musician for 40 years and I couldn't understand
| the points he was trying to make... poorly explained
| brcmthrowaway wrote:
| How can I make music without knowing an instrument?
| Rochus wrote:
| E.g. with a tool like this: https://nodalmusic.com/
| brcmthrowaway wrote:
| How does this differ to Max/MSP or PureData
| Rochus wrote:
| It has nothing in common with Max nor Pd. The latter
| specifiy signal flow and operators on the signal. Nodal
| instead represents events and time distances between them;
| this way you can design musical patterns; time is two
| dimensional, so you can draw loops; then you can associate
| musical information with the events and add logical
| operations so that your loops vary in time. It's a very
| nice experimenting and composition tool, especially if you
| don't play an instrument. See e.g. here for a tutorial:
| https://www.youtube.com/watch?v=tQpJi0AkFBQ.
|
| And here are some nice composition created with Nodal:
| https://www.youtube.com/watch?v=y1BzGaz62PE,
| https://youtu.be/gi61bHLyDsU,
| https://www.youtube.com/watch?v=sTpyIT8dWIA.
| TheOtherHobbes wrote:
| With some persuasion, Max can do everything Nodal does.
| It's not _convenient_ and you have to use numbers instead
| of line lengths. But it 's untrue they have "nothing in
| common."
| Rochus wrote:
| Well, that requires a demonstration ;-)
|
| > _you have to use numbers instead of line lengths_
|
| That's the whole point of Nodal; you represent musical
| patterns (and control flow) with graphical means.
| Entering numbers is close to Midi event lists and has
| little to do with Nodal's approach. Max/Pd are great
| tools, but focus on the processing and signal flow aspect
| (i.e. the machine which generates the music/noise), not
| the representation of the music.
| shadowerm wrote:
| IMO the question doesn't make sense.
|
| You can't make music without learning an instrument.
|
| Maybe the instrument is a computer with PD or DAW software but
| it is still a musical instrument that you have to learn to
| play.
|
| There is no way around spending hours learning and practicing
| whatever instrument you pick.
|
| What makes a computer hard to learn to play is that it has too
| many options. You don't learn anything playing trumpet for a
| month, then changing to piano for 2 weeks, then trying a new
| guitar that just came out for 2 months. That though is
| basically what many computer musicians do.
|
| You need to pick the software setup and then spend a long time
| learning it and don't switch instruments.
| shermantanktop wrote:
| As a longtime guitarist, this is exactly the type of visual
| pattern crutch that the fretboard encourages and which is (for
| me) both a crutch and a trap. Geometry can help explain music but
| if it takes the lead, that's the definition of formulaic.
| hilbert42 wrote:
| _" Pythagorean Temperament"_
|
| '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
| f1shy wrote:
| There is a classical book by about the topic
| https://link.springer.com/book/10.1007/978-3-319-64364-9
| tambarskjelve wrote:
| I understand music and mathematics were much more closely related
| historically, to some extent practically regarded as the same
| subject, but new discoveries about this relationship are still
| happening in our time. One interesting finding is that the the
| Pythagoeran comma, i.e. tiny interval between to enharmonically
| equivalent notes can be constructed geometrically:
| https://link.springer.com/article/10.1007/s00283-022-10260-4
| BaculumMeumEst wrote:
| 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.
| lachlan_gray wrote:
| I was super obsessed with this for a while! When you have a
| string instrument tuned in 4ths, there are 2D patterns that
| emerge which you can use to "derive" or "extrapolate" what a
| scale shape/pattern will look like across the whole neck
|
| Using a 6-string bass as an example:
| https://bradleyfish.com/the-notes-on-the-6-string-bass-guita...
|
| You can find a 2D pattern in the white notes (green notes in
| the pic) that you can use to understand how the pattern will
| extend from a given point. For example notice EF+BC always
| appear in the same 2x2 box shape. Also how those boxes repeat
| in a diagonal line, and how boxes are connected vertically by a
| "strip" of 3 notes ADG
|
| The only difference for guitar is that you have to correct for
| the G/B strings which are separated by a 3rd instead of a 4th,
| by scooting the pattern on the B+E strings up by one fret
| derekerdmann wrote:
| Take a look at _Fretboard Theory_ by Desi Serna - it spends a
| lot of time on how different scales are constructed and
| relating different patterns and chord forms back to the
| underlying concepts.
| jrdres wrote:
| I 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-...
| giancaIta wrote:
| Oh that's my cup of tea, love this stuff!
|
| 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/
| aanet wrote:
| Thanks - this looks great! I might download / get one of the
| books!
|
| <3
| aanet wrote:
| Fantastic resource! Thanks for sharing this. I love both the
| math, the music, and math&music combo. Tickles my inner geek. <3
|
| 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).
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(page generated 2024-12-20 23:01 UTC)