KVR :: Music Theory » harmonic geometry [View Original Topic]
There are 19 posts in this topic.

Emerald Tablet - Thu Mar 22, 2012 3:02 am
Hi guys.

While searching for an old scale calculating method (which i havent found but know ive once seen)i stumbled upon this:

Now thats a great way to display an harmonic structure
I wondered where i can find more of such geometrical representations of harmonics.

Any ideas?
shankfiddle - Thu Mar 22, 2012 3:54 am
cool plugin,

start thinking about shapes intervals/chords make on the circle of fifths itself. you want geometric representations? well the chromatic scale on the circle is a 12-pointed star (here we just reorganize the circle to be chromatic and redraw the node lines). after a quick glance before work, I can't really tell if there is logic in the way each successive layer inwards is labeled. The first one seems to be a tritone, but successive layers seem to simply be a selected interval?

half-step/chromatic scale is a 12-point star (1 system, hits every note)
whole steps/wholetone scale is a hexagon (2 unique systems)
minor thirds/fully diminished chords are squares (3 unique systems)
major thirds/augmented chords are triangles (4 unique systems)

and 4ths/fifths make a circle one system, naturally

so if your mind happens to be geometrically-organized and can do these in your head, start figuring out what chords would look like, seventh chords as trapezoids, notice where the fifth relationship is, and how it supports the rest of the chord shape...

how does a major chord deviate from the perfect triangle of an augmented?
diminished to a half-dim? and etc

or you could think about it the way everyone else does, it would cause less confusion, you'd also not learn as much. there will be those who say "there's no practical application for representing pitches like that, so its a waste of time" and i guess for some people learning new ways to think is
Emerald Tablet - Thu Mar 22, 2012 5:31 am
Thanks for the info!
Especialy for the 12star hexagon square and triangle reference!
I only wish my mind would be more geometrically-organized
(Or just simply organised)
coquillo - Thu Mar 22, 2012 8:44 am
Quote:
A Geometry of Music provides an accessible introduction to Tymoczko's revolutionary geometrical approach to music theory. The book shows how to construct simple diagrams representing the relationships among familiar chords and scales. This gives readers the tools to translate between the musical and visual realms, revealing surprising degrees of structure in otherwise hard-to-understand pieces.

http://www.oup.com/us/companion.websites/9780195336672/?view=usa

http://dmitri.tymoczko.com/sciencearticle.html
KBSoundSmith - Thu Mar 22, 2012 2:01 pm
@ Emerald Tablet:

Not quite exactly what you seem to be looking for, but if you are interested in generally perceiving pitch relationships in terms of mathematics, you may be interested in reading this:

http://www.amazon.com/Introduction-Post-Tonal-Theory-3rd-Edition/dp/0131898906/ref=sr_1_1?ie=UTF8&qid=1332453542&sr=8-1

It's essentially a more usable version of Allen Forte's book:

http://www.amazon.com/Structure-Atonal-Music-Allen-Forte/dp/0300021208/ref=sr_1_1?s=books&ie=UTF8&qid=1332453601&sr=1-1

In particular, this approach uses set class theory to analyze the musical material of 20th century composition, and is quite useful if you are interested in music that goes beyond the tonal system.

@coquillo

interesting looking book, I may have to give that a read.
wrench45us - Fri Mar 23, 2012 4:19 am
I came across some analysis of 'Giant Steps' that used soem geometry with the cycle of 5ths to help explain what was going on with those chord changes vs mnore traditional chord changes.
It's out there, but I can't find it again.
shankfiddle - Fri Mar 23, 2012 7:42 am
I think Jan linked that analysis in an argument we were having in another thread.

so correct me if I'm wrong, but I remember a ii-V-I transposed around in a triangle (major thirds)?
jancivil - Sat Mar 24, 2012 12:31 pm
I'm familiar with that ind of analysis. one of these guitarists (more than one I'm sure), can't put my finger on is a proponent of that. Pat Martino?

I don't have the mindset that can make much of geometry in music... I go by ear really.
tapper mike - Sat Mar 24, 2012 1:24 pm

I've always found his playing to be aloof. and if that's what your going for enjoy it.
shankfiddle - Sat Mar 24, 2012 4:34 pm
jancivil wrote:

I don't have the mindset that can make much of geometry in music... I go by ear really.

Well I don't CONSCIOUSLY use it in writing or performing either, but merely being aware of this extra layer of beauty, that it exists makes the whole experience of music a little more magical for me.

and THAT has powerful effects on my writing and performing
Emerald Tablet - Sun Mar 25, 2012 9:13 am
KBSoundSmith wrote:
@ Emerald Tablet:

Not quite exactly what you seem to be looking for, but if you are interested in generally perceiving pitch relationships in terms of mathematics, you may be interested in reading this:

http://www.amazon.com/Introduction-Post-Tonal-Theory-3rd-Edition/dp/0131898906/ref=sr_1_1?ie=UTF8&qid=1332453542&sr=8-1

It's essentially a more usable version of Allen Forte's book:

http://www.amazon.com/Structure-Atonal-Music-Allen-Forte/dp/0300021208/ref=sr_1_1?s=books&ie=UTF8&qid=1332453601&sr=1-1

In particular, this approach uses set class theory to analyze the musical material of 20th century composition, and is quite useful if you are interested in music that goes beyond the tonal system.

@coquillo

interesting looking book, I may have to give that a read.

Another esoteric thing i stumbled upon is the marcotone system

Studies in cymatics and vedic math and pythagorian math lead me to believe
there is more shape to music than our senses reveal to us.
Emerald Tablet - Sun Mar 25, 2012 9:23 am
[img]www.youreallfree.com/music/mt.jpg[/img]
jancivil - Mon Mar 26, 2012 2:03 pm
shankfiddle wrote:
jancivil wrote:

I don't have the mindset that can make much of geometry in music... I go by ear really.

Well I don't CONSCIOUSLY use it in writing or performing either, but merely being aware of this extra layer of beauty, that it exists makes the whole experience of music a little more magical for me.

and THAT has powerful effects on my writing and performing
I started to say I buy 'triangles' out of the Giant Steps scheme. I get it, but I don't have the type of intelligence that would allow me to quite understand your post above. I don't get a circle out of a perfect fourth or fifth at all (or minor thirds as squares), my 'feelings' of shape aren't anything I could formulate like that. I would be silly if I tried to make anything of it.

visualizing music doesn't seem like a musician's job directly.
shankfiddle - Mon Mar 26, 2012 2:41 pm
I'm sure you've heard that improvising/writing music (not performing precomposed pieces) activates more parts of the brain than any other activity. You are using your visual cortex whether you like it or not. even if you're not making colors and fractals in your head, I'm sure you visualize in different ways. If I say "c half-diminished", I bet you have images of the piano and guitar finger pattern pop in your head almost instantaneously. You also likely have visual ways to interpret intervals. as a guitar example, I use shapes to find a fourth(up a string) tritone (one more than that) major 3 (up a string, down one fret). these are spatial measurements in the mind. and naturally the spatial measurements are very different on violin tuned in fifths. if i didn't visualize in this way going back and forth would be almost impossible, doncha think?

So exactly analagous to this triangle in giant steps but applied slightly differently:
When I say 3 "systems" of diminished chords I mean that if you ignore inversions and only look at pitch classes, there are only three dim chords: the pitches G#, B, D, F could be labeled as either G#o, Bo, Do, OR Fo, right? so there are only three unique pitch sets each of which we can define in one of four ways.

So thinking in these shapes can be useful for modulating. If we're in the key of A (i usually go back and forth borrowing chords from both minor and major) we can play a viio (G#) and the same chord could function as the vii of a new key... you can even use a dim chord to modulate a tritone--which bisects the circle, a quite jarring feel, great in the right places So once we've played a G#o you can switch inversions and naturally modulate to either keys C, Eb, F#(or gb) because this same dim chord CAN function as the new viio. so if I'm using a dim chord as the pivot chord to modulate, the possible "target keys" make a square. (in addition to the pitches themselves in the dim chord making a square)

similarly augmented chords are great for modulating a major third up or down. and your possible target keys form a triangle.

Now, modulating up a whole step has a distinctly different feel to modulating a minor third, or a major third, modulating a fifth or fourth has a smooth round feel to me... and I've learned to link THESE feelings to these shapes described, does that clarify a little bit? It may be completely arbitrary, but, hell, a lot of people have trouble comprehending modulations, and this idea might help.
jancivil - Tue Mar 27, 2012 1:33 pm
well, maybe it will help somebody. if it shores things up for you, good job.

I 'get' shapes in a way that's peculiar to me in a vague, impressionistic teenage acidhead quasi-synesthesia manner, but it isn't anything more than a side-effect to me.

I make a decision by ear, really. I remember when I used to think and plan and write. I got better.

now, teaching someone part-writing, I'd rely on formulation of whys and wherefores. Or if I was trying to write something that had to be good stylistic writing I can fall back on tools out of this or that discipline (where I have it). But I'm not worried about a pivot point, or like that, in my own creation.

Left to right, through-improvisation, action. Then there is moving stuff around and 'the shape of the line', a lot of editing and decisions. Lying around I come up with lines and ramifications and just go with it until the thread breaks. while I'm doing it, I *know* what the notes are; if my voice is up to it I could sing them, at once. That's the goal; I want to be rid of the middleman [eg., naming] really. It's some intervals. One can know them at once.

Spacial conceptualization and mechanical aptitude is a glaring weakness of mine. I think that missing part of my brain is part of my design, it removes - for me, this is just me - an ability that will be in the way. Instead I was supplied with language aptitude, so narrative flow, construction of arguments, came naturally.
jancivil - Tue Mar 27, 2012 1:43 pm
shankfiddle wrote:
Now, modulating up a whole step has a distinctly different feel to modulating a minor third, or a major third, modulating a fifth or fourth has a smooth round feel to me... and I've learned to link THESE feelings to these shapes described, does that clarify a little bit? It may be completely arbitrary, but, hell, a lot of people have trouble comprehending modulations, and this idea might help.
sure, I get it but 'up a fourth' might be a square or rectangle, if anything, to me. I couldn't reasonably say why. why do we agree on a triangle for a M3? Is a minor sixth a triangle? seems reasonable but not as instantly clickable for me. At this point, language or reason seems insufficient. My brain shifts gears...

there was a guy I lived with, a composer, during very formative years for both of us. He could suss the Rubik's Cube in no time at all, every time. I never once got it to happen. I did actually show him things I was working on on occastion, and that capacity would out over my... guesses. I got better and developed a kind of logic, 'arguments' to fill that void... had to.

it's good to collaborate with people that bring stuff to the table you don't have, though. We always endeavored to create a composition in real-time and after some years we could nail a lot of things, while developing a language.

We will have the facility to compose according to how well we can improvise. Improvising, coming up with viable material on the spot, does not allow time for wondering or second-guessing, or a lot of thought.
shankfiddle - Tue Mar 27, 2012 1:56 pm
jancivil wrote:

I make a decision by ear, really. I remember when I used to think and plan and write. I got better.

now, teaching someone part-writing, I'd rely on formulation of whys and wherefores. Or if I was trying to write something that had to be good stylistic writing I can fall back on tools out of this or that discipline (where I have it). But I'm not worried about a pivot point, or like that, in my own creation.

well as i said a couple posts above, I don't consciously think about any of this, I found these patterns and modulation tricks for a paper for my post-tonal composition class (ironic, eh?). But once you have pivot chords in your standard repertoire (as one of your fall back tools) THEN you can go on to just feeling it...

just like with all theory as you described. you don't think of individual notes in a chord, but you had to learn them at one point.

similarly thanks to these geometric patterns I no longer have to think about how to get to any key, if my ear wants to go here, we can, rotate the circle and it locks into place like a gear, and i have tools to fall back on so I have multiple options to prepare the modulation without trial/error or conscious thought. But I can only do that now because I consciously thought about it at one point... writing my paper. Then I got better
jancivil - Tue Mar 27, 2012 3:07 pm
the useful point out of this is, you want to learn the mechanics to the point you do not need to think about it.
jancivil - Tue Mar 27, 2012 3:09 pm
'comprehension' of a modulation or similar advanced device is going to be more facile if one has worked with music that modulated or what-have-you. Another reason playing music makes a musician.

If one never played music, at least the voice as an instrument, one is not a musician. "Music made by musicians will tend to be superior to music made by people that never were" is just not a real baffling concept.

There are 19 posts in this topic.