Mental experiments, How do/can people solve problems?

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polymer
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Mental experiments, How do/can people solve problems?

Postby polymer » Fri Dec 19, 2008 6:02 am UTC

One of the things I've been curious about is how flexible the mind can be when it solves problems. I got somewhat interested in mathematics last year(currently 18), and very interested in music the year before. What got me interested in mathematics was the realization, when solving a problem, of how similar it was to programming in the since that it bubbled down to logical problem solving. After attempting to solve this and other problems I discovered to my delight that it wasn't that much harder to project relatively simple math onto more real problems. Later, after playing around on a piano, I re-realized that every major scale and probably every other mode could be reevaluated as long as you knew the number of half-steps for each step. I was told that years before, but mathematical thinking wasn't something I cared much for when I was 9.

This got me thinking, could it be possible to somehow abstract the piano in my mind as a list of numbers from 0 to 87, and then use the logical/mathematical thinking mechanisms to deal with the mathematical relationships? Assuming I could pull this off, transposing keys would be much easier, and much less would have to be memorized.

I've been really interested in "programming" my mind with things like that lately.

After observing how one could solve problems, it made me think about how one usually does solve problems. For example, could video games in general be making people smarter if they play versions that are unique enough, or are the skills necessary to play games localized and therefore generally irrelevant when it comes to intelligence. Could I play guitar hero, develop a visual layout of the keys in my mind, as well as a possible projection of all the possible button combinations and somehow use that visualization mechanism for piano?

I'm curious about how the people here solve problems of a similar nature. What process do you go through when looking at a new mathematical problem, or any other solvable problem?

Also, if you know of some tricks for exercising the mind in the context I've mentioned, I'd appreciate them! :D

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Re: Mental experiments, How do/can people solve problems?

Postby berk » Fri Dec 19, 2008 6:35 am UTC

polymer wrote:After observing how one could solve problems, it made me think about how one usually does solve problems. For example, could video games in general be making people smarter if they play versions that are unique enough, or are the skills necessary to play games localized and therefore generally irrelevant when it comes to intelligence.


I find this a very interesting point, because we generally assume (sarcastically and seriously) that people are getting stupid, due to how easy survival is now. However, I heard about a study (and resulting book) that contradicted that point, saying that multimedia, games, and modern culture are actually making people smarter. MY guess is that it is mostly due to games and the inherent critical-thinking and problem solving involved.

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Re: Mental experiments, How do/can people solve problems?

Postby crickets » Fri Dec 19, 2008 9:24 am UTC

I don't think in terms of math. I think in terms of information.

I'm very visual, so each bit of info gets charted away in my brain, and then i organize it in ways that make it easy to flow from one thing to the next. I actually had to re-design the whole menu board at my work to make it easier to understand. It's like seeing the world in an ever narrowing series of categories. It makes me a really spazztastic person to talk to, because i can follow just about any leaps in logic to different conclusions, which results in me making connections no one else can see.

I also play a lot of tetris.

Probably not what you were looking for.
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Re: Mental experiments, How do/can people solve problems?

Postby Luthen » Fri Dec 19, 2008 9:53 am UTC

Just as a blanket answer to your original post, it depends. Not everyone's mind works in the same way, so you may or may not be able to train yourself to think like that and there's the possibility of people who can't imagine thinking any other way.

polymer wrote:This got me thinking, could it be possible to somehow abstract the piano in my mind as a list of numbers from 0 to 87, and then use the logical/mathematical thinking mechanisms to deal with the mathematical relationships? Assuming I could pull this off, transposing keys would be much easier, and much less would have to be memorized.

As someone interested in both maths and music as well (a fairly common combination, possibly for the reasons you're pondering), I often examine music intervals and other bits an pieces with maths (and physics!). However, I haven't found this useful for transposition. I can just transpose on the fly, just how my brain works I guess.

polymer wrote:I'm curious about how the people here solve problems of a similar nature. What process do you go through when looking at a new mathematical problem, or any other solvable problem?

Generally brute force, give up, flash of inspiration, solved! Can't really say much more.

Though generally be I think very spatially, and have to sometimes resort to pushing and weaving figments in the air.

crickets wrote:It makes me a really spazztastic person to talk to, because i can follow just about any leaps in logic to different conclusions, which results in me making connections no one else can see.

I think very one thinks they talk like that, I know I do.
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Re: Mental experiments, How do/can people solve problems?

Postby polymer » Fri Dec 19, 2008 8:27 pm UTC

Luthen wrote:
As someone interested in both maths and music as well (a fairly common combination, possibly for the reasons you're pondering), I often examine music intervals and other bits an pieces with maths (and physics!). However, I haven't found this useful for transposition. I can just transpose on the fly, just how my brain works I guess.

The reason why it should be easy, is because every scale should look the same when looking at it in the way I described above, If sheet music was presented as numerical intervals for example, you could start wherever you wanted on the piano since the intervals represent the same thing no matter where you start.

Luthen wrote:Just as a blanket answer to your original post, it depends. Not everyone's mind works in the same way, so you may or may not be able to train yourself to think like that and there's the possibility of people who can't imagine thinking any other way.
Is this fundamental difference the result of people developing their own way to look at problems? Or is it because they exercised different parts of the brain when solving problems? Perhaps it's genetics, although I doubt it. I really don't know what to expect, which is part of why I asked the question.

crickets wrote:I don't think in terms of math. I think in terms of information.

I'm very visual, so each bit of info gets charted away in my brain, and then i organize it in ways that make it easy to flow from one thing to the next. I actually had to re-design the whole menu board at my work to make it easier to understand. It's like seeing the world in an ever narrowing series of categories. It makes me a really spazztastic person to talk to, because i can follow just about any leaps in logic to different conclusions, which results in me making connections no one else can see.

I also play a lot of tetris.

Probably not what you were looking for.


As I mentioned before, I don't know what to expect. That makes it kind of hard to define a wrong answer. :)

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Re: Mental experiments, How do/can people solve problems?

Postby Luthen » Sat Dec 20, 2008 1:49 am UTC

polymer wrote:
Luthen wrote:
As someone interested in both maths and music as well (a fairly common combination, possibly for the reasons you're pondering), I often examine music intervals and other bits an pieces with maths (and physics!). However, I haven't found this useful for transposition. I can just transpose on the fly, just how my brain works I guess.

The reason why it should be easy, is because every scale should look the same when looking at it in the way I described above, If sheet music was presented as numerical intervals for example, you could start wherever you wanted on the piano since the intervals represent the same thing no matter where you start.

I don't have any trouble reading sheet music as presenting the intervals. Sheet music just is a slightly compressed version of what you seem to be after. As most western music uses octave scales the staff is set up to cope with that. The key signatures are sort of a "compression" description, telling you how the twelve tones in an octave have been mapped to the staff.

To get the staff to represent every scale evenly, it would have to have spaces for each semitone. Thus you could transpose easily between things as every note just moves up or down a certain number of semitone steps. However, the drawback is more difficulty reading it as there will be many unnecessary gaps in most music's scores.
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Re: Mental experiments, How do/can people solve problems?

Postby Matsi » Sat Dec 20, 2008 2:00 am UTC

polymer wrote:(...)If sheet music was presented as numerical intervals for example, you could start wherever you wanted on the piano since the intervals represent the same thing no matter where you start.


This just doesn't work. There is a difference between something played in one octave or another, higher octave. Or any other interval. It is not just a difference in height of the notes, but also an emotional difference. Imagine, if you will, the opening to Mozart's fifth, but shifted up 2 or 3 octaves. Would it still work as well? I say it doesn't, the exact notes played matter a lot here to evoke the emotional response intended.

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Re: Mental experiments, How do/can people solve problems?

Postby polymer » Sat Dec 20, 2008 2:27 am UTC

It would sound like something else entirely of course, but if you didn't hear the original song you wouldn't find anything wrong with it.
Edit: That's besides the point though, transposing is generally really hard to do quickely, part of what made this system of looking at the problem interesting was the fact that transposing would become a non-issue.

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Re: Mental experiments, How do/can people solve problems?

Postby mrbaggins » Sat Dec 20, 2008 4:49 am UTC

Octaves, sure, you could listen to it fresh and not pick a difference.

But because you have gaps in the flats, moving to most of the intervals in between would stuff it up.

IE: C is CEG, or on a piano: [C] C# [D] D# [E] F F# G G# A A# B C

Moving that say an interval of 2 gives C C# [D] D# [E] F [F#] G G# A A# B C
Which is a D, with a 9th, and no 5th. Which just sounds bad.
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Re: Mental experiments, How do/can people solve problems?

Postby Luthen » Sat Dec 20, 2008 5:19 am UTC

mrbaggins wrote:But because you have gaps in the flats, moving to most of the intervals in between would stuff it up.

IE: C is CEG, or on a piano: [C] C# [D] D# [E] F F# G G# A A# B C

Moving that say an interval of 2 gives C C# [D] D# [E] F [F#] G G# A A# B C
Which is a D, with a 9th, and no 5th. Which just sounds bad.

That's just a D major tonic, you hate D major? Wait, your example doesn't match your illustration.

CEG=[C] C# D D# [E] F F# [G] G# A A# B C
up two semitones:
DFFA=C C# [D] D# E F [F#] G G# [A] A# B C
C major and D major tonics, lovely sounding.

Your piano diagrams where for CDE and DEF#, both of which don't sound nice.
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Re: Mental experiments, How do/can people solve problems?

Postby mrbaggins » Sat Dec 20, 2008 5:28 am UTC

Yeah, I screwed that up. Translating between Piano keyboard > Note > Chord got screwed up. Ignore my last post.
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Re: Mental experiments, How do/can people solve problems?

Postby diotimajsh » Sat Dec 20, 2008 9:00 am UTC

polymer wrote:The reason why it should be easy, is because every scale should look the same when looking at it in the way I described above, If sheet music was presented as numerical intervals for example, you could start wherever you wanted on the piano since the intervals represent the same thing no matter where you start.
Hmm. I'm having a lot of trouble seeing how that would work in practice. Are you suggesting a notation where each interval is relative to the last? So maybe in halfsteps, we start with 0, and a melody might look like this: [0, 5, 0, -1, 1, 4, 0, -2];[5, -5, 0, -2, -1, -2, 2]. So the first four numbers are read as "up a perfect fourth, repeat that note, then down a minor second" and so on. (This is supposed to be the first phrase of "Hark the Herald Angels Sing"). I just tried that out from a random key, and it was actually not too bad to work through. In fact, it might be even good for ear and vocal training, to really emphasize intervals.

Another scheme might be to start from the tonic (first scale degree) and then jump always relative to that note. So we use 0 for the tonic of our key (that is, B=0 if we're in B major or minor; Eb=0 for Eb major or minor). Taking the above melody, that would be [-5, 0, 0, -1, 0, 4, 4, 2];[7, 2, 2, 0, -1, -3, -1]. Read as, "starting a fourth below tonic, go to the tonic, repeat it, then down a half-step, return to tonic, up a major third..."

Again, not too bad, for me at least. And interesting.

However, as far as notation for actual music goes (rather than scales and simplified melodies) , I think we immediately run into problems. Neither method supports large jumps very well (-35? 57? Wtf are those from my current note?). And, with the interval-from-the-previous-note scheme, if you're playing along and you make a mistake--well hey, you may have just jarringly modulated to an entirely new key, since you'll be playing the rest of the piece relative to that wrong note from now on. And imagine you made that mistake with only one of your hands, while you played the other correctly :shock:. Instant polytonality, and not the good kind. Worse, you may not be 100% certain where you got off track; how far would you need to backtrack to fix it? All the way to the start of the piece?

For my other suggested scheme, the "intervals from the the tonic," I think like this would still have problems with large leaps, and I'm not sure how it would handle atonal or highly chromatic music. Heck, even traditional modulation from one key to another might be kinda confusing. Our current notation system is already heavily biased toward writing in a fairly consistent diatonic key (as opposed to chromaticism and constant modulation); I feel like this way might be worse.

I also suspect that we'd run into complications when trying to transfer these ideas to physical playing, especially for non-piano instruments. But I'm interested to hear more thoughts on this subject.

ETA: Heh, my musical memory sucks, apparently =P. The melody I gave above is actually the second phrase from "Hark the Herald Angels Sing"--and I think the last few notes finish slightly differently. Whatever.
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Re: Mental experiments, How do/can people solve problems?

Postby polymer » Mon Dec 22, 2008 5:45 am UTC

It's that type of thinking when observing "problems" that I'm really curious about. Sort of like building diverse periodic table of elements to figure out how much more we can learn about elements.

-35 btw, is not that bad of an interval. If you know that an octave has 12 halfsteps, then you know that the 35th interval is one halfstep above the third octave. 57 is harder although not by much, 4 octaves and a sixth(48 + 9). A little practice with arithmetic and you could figure those systems out. The primary benefit of this wouldn't be notation(I don't think so at least). It's the fact that you wouldn't have to memorize every single scale for a mode, you would only have to know the intervals for the mode. When thinking of a chord, you wouldn't think of the letters associated with the keys, you would remember your root note and work off the intervals from there. That's something I'm going to experiment with in the future.

Something else I was going to experiment with, was if I label all the keys from 0 to 87(or in your case -39 to 48), if I started writing down all the intervals of chords, would I begin to find simple mathematical relationships that I could then memorize? Something obvious from the top of my head for example would be augmentations of [0,4,7] a C triad. I could recognize the inversion by not only observing where the root(0) is [-5,0,4], but I also get a feel for what is going on, especially since it's obvious that the -5 is in fact below the 0. This isn't as obvious with [G,C,E], although I'm willing to admit the possibility that I'm simply a poor musician :). With this I'll also admit It's a little harder to see that the -5 is the 7 an octave below unfortunately, you can kind of work around this by mentally subtracting -12 from -5, since the -12 is an octave below 0.

I could keep throwing ideas about how this might work, whether it'd be worth it to mentally label intervals with two numbers [octaves, half-steps](base 12 anyone?), if it's even useful to know that subtracting the root note from the chord sequence would give the intervals[5,9,12]-5=[0,4,7] etc.

If anybody has a similar way of looking at other problems, or their own cool way to look at problems, feel free to post! The fact that a problem can be looked at in so many ways is really impressive...
Last edited by polymer on Tue Dec 23, 2008 2:27 am UTC, edited 1 time in total.

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Re: Mental experiments, How do/can people solve problems?

Postby curious and questioning » Tue Dec 23, 2008 1:33 am UTC

[quote=polymer]It's the fact that you wouldn't have to memorize every single scale for a mode, you would only have to know the intervals for the mode. When thinking of a chords, you wouldn't think of the letters associated with the keys, you would remember your root note and work off the intervals from there.[/quote] reminds me of barre chords and scales on a guitar.

This way of thinking seems like it would make piano easier, and would also work on brass instruments (trombone's positions 1-7 as half steps, and valved brass have 0, 2, 1, 12, 23, 13, 123 going down chromatically) and I think other string instruments. I don't know about woodwind instruments, though- are there even patterns in their change of fingerings? I don't know much about woodwinds...
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Re: Mental experiments, How do/can people solve problems?

Postby polymer » Tue Dec 23, 2008 2:53 am UTC

I still want to observe how this or other methods would be done on a piano whose sharps and flats were separated. I imagine that could be enlightening as well. If I was on a string instrument however I would try looking at it in a different way anyway since they're not constricted to any particular temperament. Out of curiosity, can you adjust brass instruments' tuning on the fly?

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Re: Mental experiments, How do/can people solve problems?

Postby diotimajsh » Tue Dec 23, 2008 6:22 am UTC

Er, well isn't this really what music theory teaches already? I mean, if you ask me, as a music major, to name the intervals from the tonic for Dorian, I can rattle off "Major second; Minor third; Perfect fourth; Perfect fifth; Major sixth; Minor seventh" right off the top of my head (as I did just now). Similarly, I think of major chords as "root, major third, perfect fifth". My dad (also a musician) apparently does think of scales and modes as series of alternating half and whole steps, which is less intuitive for me, but eh. Musicians normally use terms like "major third" rather than the explicit number of half steps, since it's easier to parse that way. And yes, it certainly does make learning scales easier, although there's still quite a bit of physical memory involved that (for me) requires quite a bit of mechanical repetition to sink in. If thinking in terms of numbered half-steps makes it easier, then great :). I feel like they're roughly similar methods though, and traditional music theory already teaches this--maybe just less explicitly.

With that in mind, mathematical relationships between chords: yes, for all major triad inversions, you'll get the same intervals between members. (For every chord of the same type you'll get the same intervals; that's what gives it its identity. Well, and context, but anyway). Root position is "Bass note, major third, perfect fifth" [0, 4, 7]; First inversion is "Bass, minor third, minor sixth" [0, 3, 8]; Second inversion is "Bass, perfect fourth, major sixth" [0, 5, 9]. Add those triplets to any starting note--say, 24--et voila, we get [24+0, 24+4, 24+7]=[24, 28, 31] for root, [24, 27, 29] for first inv, [24, 29, 33] for second inv. Conversely, if we're given any three notes, we can subtract the first note from the second two, in this case giving [0, 5, 9]. Looking up above we see that the intervals matches a second inversion major chord, so we've identified it.

This also maps onto your example of [-5, 0, 4], since adding 5 to it yields my second inversion major triplet [0, 5, 9]: they're the same chord. I think it's easier, however, to analyze with the 0 as the first member, since in practice, when you're staring at a given collection of intervals, it won't be immediately clear which is the root of the chord. With my suggestion, you can just subtract values to find how the notes are offset from one another; then compare it to a memorized triplet. If it matches your root triplet, you'll know that the first member [0,,] is the root note. If it matches the first inversion intervals, then the third member (which will be [,,8]) is the root note; and for second inversion, the middle member (which will be [,5,]).

Example: we find ourselves looking at some notes like [58, 62, 67]. We know that the differences from the first value are [58-58, 62-58, 67-58] = [0, 4, 9], which we compare to our memorized collection of intervals (pretend we've memorized them all already); we find that [0, 4, 9] matches the first inversion minor chord; now we know it's a first inversion chord, which means the third member is the root; thus 67 is the root of the chord.

Maybe an easier way to think of inversions is to just remember that all you're doing is shifting the bass note up an octave. So a chord like [0, 3, 7] (minor root) becomes [(0+12)=12, 3, 7]. Put that in order, and you get get [3, 7, 12]; so you can see that the root of the chord has moved to the top. Repeat the process once more, and you get [(3+12)=15, 7, 12], which we put in order as [7, 12, 15]. Now our root, 12, has been moved down to second place, while the old bass, 3, moved to the top as 15. Finally, we add another octave to the 7, which takes us back to root position, only now the whole chord is an octave higher, [12, 15, 19].

Er, I hope I'm not becoming overwhelmingly tedious here, with posts that are long and not too exciting o_O. Or that I'm just mindlessly going over things that are already obvious to you. Hopefully I'm at least giving you food for thought, though. And actually, I'm finding it interesting to work with notes in a more numerical manner. Haven't done this since studying "set theory" =P. (Which is really completely different from mathematical set theory. Silly music theoreticians and their names...)

(P.S. I'd be careful when using the word "augmentation" with respect to chords, since that calls to mind the "augmented chord," which I don't think is what you meant there (seems like you meant inversion?))
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Re: Mental experiments, How do/can people solve problems?

Postby polymer » Tue Dec 23, 2008 9:58 am UTC

yeah, augmentations simple meant some change. I was trying to emphasize that if I knew what the main tonic was, then I could get an idea as to what was changed. That was poor word choice though considering the subject...

Although this numerical way to explain modes is taught by standard music theory, actively practicing all the scales without keeping whats going on in mind can make it about as useful as a fact, that's what happened with me at least. What I was excited about was that after getting pretty good at math, I was able to relearn in a new light exactly what was going on. Major thirds, perfect 4ths, etc, abstract what happens independent of the scale you're talking about, which is similar, but it hides the logic of the temperament you're working with. If you told me a chord was composed of a root, a major third, and a perfect 5th, I may be able to figure out C since the keyboard is laid out nice :), but on any other key I'd be stuck guessing. Telling me that a major third includes 4 half-steps, and a perfect 5th includes 7 half steps gives me more to work with. Note, that doesn't mean I won't use the terminology, rather when I say major third I'll think 4 half steps in ET.


I agree, your trick does a better job at abstracting the particular inversions since there really isn't that many to remember, and more useful since my method does immediately assume you know where the root is.

However after thinking about this issue more, I realized that if you change the system to a base 12 number system with 0 being the lowest C(or the C below the piano's range), that less arithmetic would be necessary to read the intervals. The first digit would specify the octave, and the second digit would specify the interval from that octaves root.
So my original chord [-5,0,4] would instead be read as [2 7, 3 0, 3 4]. Instead of emphasizing the jumping octaves(which I like a lot actually), this emphasizes that the notes are all apart of the same fundamental tonic. It's like notation except with the mind :]. Add X to transpose it to whatever key is necessary. Also since you will know what the key will be, (some number 0 through 11 provided by the music) you can subtract that from the numbers to get the generic intervals, the octaves wouldn't get in the way. I'll add additional stuff later if you don't respond by the time I get back. It's currently 2 AM and I need sleep...
Last edited by polymer on Tue Dec 23, 2008 8:04 pm UTC, edited 1 time in total.

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Re: Mental experiments, How do/can people solve problems?

Postby diotimajsh » Tue Dec 23, 2008 4:17 pm UTC

Thinking in terms of half-steps is probably a good idea to cement the intervalic relationships. Yes, I agree that thinking about abstract scales while playing them is probably the best way to practice, since otherwise it is just rote and tedious memorization.

I really like using base-twelve for intervals; in fact, it seems like such an obviously good idea that I'm surprised it hasn't been tried before (I may just not have heard of it though :)). Seeing the octave as the first digit feels very convenient for me. I guess it's conceptually the same thing as specifying notes by "C-4," "Bb-2," or "G#-5," only with the "digit" positions swapped, and using letters and accidentals to show the interval above the octave rather a base-twelve digit. I like seeing it this way too though; and maybe it allows us to see the relationships between notes more quickly without needing to memorize how all the letters are related. ("1A minus 13 is 7 in base-twelve, so it's a perfect fifth" versus "Eb to the Bb above it is what again...? *Straining memory... straining memory...* Oh, yeah, a perfect fifth.").

Although I guess you still need to memorize what all the intervals are in terms of half-steps, but that's not necessarily a bad idea anyway.

Let me see if I'm understanding you about the latter part. You would say, then, that the first inversion major chord is [24, 27, 30], to emphasize that the root note is on top, right? Thus if we saw the notes [47, 4A, 53] and we knew our key is 3 semitones above C (or whatever our point of reference is), we'd simply subtract 3 to bring it back to C, giving [44, 47, 50], which we'd see (mainly by looking at the second digit) has the same relative intervals as [24, 27, 30]. Something like that?

(I hope I'm not making an idiot of myself by blundering base-twelve.)
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Re: Mental experiments, How do/can people solve problems?

Postby polymer » Tue Dec 23, 2008 8:33 pm UTC

Base 16 assigns letters the values of numbers A=10, B=11, etc. So yeah, you did it right as far as CS is concerned. I separated the numbers with a space [2 (11)] to avoid using letters, but meh you got away with it just fine so I'll keep it to current standards.

The reason why I want to avoid using intervals like [0,5,9] is that they're more like sheet music and less like relationships. I'm trying to make what I memorize as fundamental as possible. Instead of knowing what particular inversion those notes are of, I am trying to keep in mind what the tonic is and where its notes are jumping around. Consider This, 0 is the root(so C) and you play two chords Left hand [24, 34] Right hand [47, 5O].
That is not a clean inversion, but you know it'll sound good since all the right digits are in the tonic of the chord. This example has a Major third :] octave in the left hand, with the root up an octave in the right(Edit: I'd say first inversion, except you don't know since the Major 3rd could be up an octave or in the same octave).

So with this strategy, the thing that has to be fundamentally memorized would be, the fundamental chords. Any tricks for determining those would be nifty too :).
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Re: Mental experiments, How do/can people solve problems?

Postby Alder » Tue Dec 23, 2008 9:06 pm UTC

Speaking as a pianist and piano teacher, I'm pretty sure this is how they'd make me play music in hell... :shock:



Certainly don't fancy explaining it to my 6 year old students, just note lengths tends to be enough maths for them. :D
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Re: Mental experiments, How do/can people solve problems?

Postby diotimajsh » Tue Dec 23, 2008 10:08 pm UTC

Well, okay. I guess I'm not seeing how this will apply to chords based on other roots though? I mean, what about when you have a tonic other than C? What about when you're wanting to create or identify an inversion where you can't just look for the 0 to see where the root is?

Or did I miss something?

Alder26 wrote:Speaking as a pianist and piano teacher, I'm pretty sure this is how they'd make me play music in hell... :shock:

Certainly don't fancy explaining it to my 6 year old students, just note lengths tends to be enough maths for them. :D

:lol: Yeah, that's true. But just think, if we can make music as complicated* and mathematical as possible, we can sneer at all the other disciplines for being too easy! Because we would not only require thousands of hours of practice and technique training, but absurdly convoluted mental gymnastics as well!

It'll be great.


* Arguably this is trying to simplify music not complicate it; it's just, math isn't always intuitive.
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Re: Mental experiments, How do/can people solve problems?

Postby polymer » Tue Dec 23, 2008 10:33 pm UTC

diotimajsh wrote:Well, okay. I guess I'm not seeing how this will apply to chords based on other roots though? I mean, what about when you have a tonic other than C? What about when you're wanting to create or identify an inversion where you can't just look for the 0 to see where the root is?

Or did I miss something?

Alder26 wrote:Speaking as a pianist and piano teacher, I'm pretty sure this is how they'd make me play music in hell... :shock:

Certainly don't fancy explaining it to my 6 year old students, just note lengths tends to be enough maths for them. :D

:¡This cheese is burning me!: Yeah, that's true. But just think, if we can make music as complicated* and mathematical as possible, we can sneer at all the other disciplines for being too easy! Because we would not only require thousands of hours of practice and technique training, but absurdly convoluted mental gymnastics as well!

It'll be great.


* Arguably this is trying to simplify music not complicate it; it's just, math isn't always intuitive.


See except you know what the root is, you can tell by the number of sharps/flats the music gives you. So with my example above in D, or 2.
Left hand [26, 36] Right hand [49, 52]. It's easy to spot the root, basic arithmetic is required to see the 9 is a perfect fifth and that the 6 is a major third.

The octaves don't get in the way like they would if it were a base-10 system, so you can largely ignore them for determining intervals(from the root). I should right this out for Bb though to see how well it works. :/

This system lends itself to tonics at least, IV and V I'm not sure about. *goes off to play with those*

Alder26: The key to this strategy is that I'm trying to memorize on a fundamental level. Math helps at this and other things, although I didn't know if there were even more ways to look at problems, that inspired this thread. At this point I'm not sure it would make sense to change topics anyways, heh.

How bout this, if you were simply taught that the notes were 0-11, and that you denote their octaves with a number in front 3(11), and played music knowing only those facts. How long do you suppose it'd take before you started noticing relationships and patterns? Would that not happen? I'm not positive, but I don't think so since it makes the relationships as plain as they can be. CDEFGABC is easy for memorizing C major, but as soon as you add sharps and flats it's really hard to see what's happening. 0(0),0(2),0(4),0(5),0(7),0(9),0(11),1(0) would be really hard at first, but after the initial struggle would it really be that much harder to see (C# major) 0(1),0(3),0(5),0(6),0(8),0(10),1(0), 1(1),? Maybe it would be...I really don't like memorizing unless it's essentially required though. Is there another way to look at it and still see what's going on?
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Re: Mental experiments, How do/can people solve problems?

Postby Alder » Tue Dec 23, 2008 10:52 pm UTC

diotimajsh wrote:But just think, if we can make music as complicated* and mathematical as possible, we can sneer at all the other disciplines for being too easy! Because we would not only require thousands of hours of practice and technique training, but absurdly convoluted mental gymnastics as well!
It'll be great.

Heh...and then I can make a fortune from my patented "E-Zee Music" Course...! Bwahaha...


Adding a bit 'cause of the previous post:

polymer wrote:How bout this, if you were simply taught that the notes were 0-11, and that you denote their octaves with a number in front 3(11), and played music knowing only those facts. How long do you suppose it'd take before you started noticing relationships and patterns? Would that not happen? I'm not positive, but I would think so since it makes the relationships as plain as they can be. CDEFGABC is easy for memorizing C major, but as soon as you add sharps and flats it's really hard to see what's happening. 0(0),0(2),0(4),0(5),0(7),0(9),0(11),1(0) would be really hard at first, but after the initial struggle would it really be that much harder to see (C# major) 0(1),0(3),0(5),0(6),0(8),1(0), 1(1),? Maybe it would be...I really don't like memorizing unless it's essentially required though. Is there another way to look at it and still see what's going on?

O............kay. Goes off and comes back with a Beethoven Sonata and a notebook and pencil...

Right. Well. First off, you need to pick which note is 0. You're starting with C, because we're taught that C is the easiest scale/key, it tends to be where we begin to learn music, but - in this format, with the first number um... [note to all maths people, I'm bad at this]... 'X' being the octave and the second '(Y)' denoting the note of the scale, there's no need for 0 to equal C. In fact, if you're dealing with the piano primarily, the lowest note on the keyboard is A. But I digress. I suppose the A, Bb, B below the lowest C could be... -1 when it came to describing their octave.

I'm going to go with that for the time being.
Edit - wait, no I'm not. Turns out - which I'd forgotten 'cause I use it so rarely - the octaves are already numbered... The A, Bb, B are 0. The first C starts at 1. I'm gonna go with that instead.

So the lowest C on the piano becomes 1(0).

Let's take bar 5 of "Larghetto maestoso - Allegro assai", the first movement from Sonata in F minor (should that be "5 minor"?)

Ok, well, already I have problems, since I have two hands playing separate strands of music at the same time. The right hand is playing 2 notes at the same time. The left hand is moving in semiquavers (16th notes).I have no idea how to represent this, so I'll go with the right hand first -

4(5), 5(5) [2 beats] then....oh. A rest. Um....I'll gloss over that...then 4(5), 5(5),4(5), 5(5),4(5), 5(5) [quavers/8th notes]
While the left hand is going -
1(5), 2(5), 2(4), 2(5), 2(8), 2(5), 2(4), 2(5), 1(5), 2(5), 2(4), 2(5), 2(8), 2(5), 2(4), 2(5), [in semiquaver/16th notes]

Laying aside that you still need a way of representing the individual note lengths...that's just a bar. There are 83 of them in this movement alone. Maybe I'm just more visual than mathematical, but at speed, I can pick out scale patterns almost without thinking, I know what they're supposed to be doing, and I can follow it. Recognising patterns in a string of numbers, while better for analysis, at speed in the middle of a sonata...I don't know...I'm just really glad we're not using it...:D Looks impressively code-like, though.
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Re: Mental experiments, How do/can people solve problems?

Postby polymer » Wed Dec 24, 2008 1:06 am UTC

Was messing around on the piano and found that my method thus far doesn't work well for telling the difference between tonics,their inversions, with the fourth and fifths inversions in play. Determining whether something is an inversion by using diotiMajsh's trick, and then using the root of that inversion to find if it's a tonic, 4th, or 5th might work...but that's becoming more unwieldy mentally.

Alder26: the reason I haven't brought up rhythm, is because it seems like that is much more of a human variable(except for perhaps keeping tempo.) It's very hard to call a particular rhythm wrong. It's (relatively)easy to observe an "ideal" system with harmony, and understand what rules are "broken." So, at this point at least, I'm thinking in terms of the instances I make chords.

Although after trying to play the song you presented a bit I'm glad you did bring up rhythm...Seeing all the notes as independent blocks makes it harder to see where they're going since literally anything can happen...The way I'm looking at things, it seems the only advantage would be that I'd know what I can add to chords, which is okay for composing, not great for actively playing though :/. Still...

If we were talking about sheet music you could present it something like this [15 --|--> 25--->24--->25-|->24]
Where the vertical dashes demonstrate that the sub dashes are even. You could tell if it was 1-4 beats(I know there are more, I'm just throwing something together) visually. The previous example presented has 12 dashes, with...perhaps the quarter note as 4 dashes and the down beat, so...this would be 3/4 time. I really doubt you could think with something like this on the fly though, and it isn't that much more revealing then our current system...Any other proposals perhaps? Again more interested in things that contain few steps that are easy to mentally swallow.

Half rambling at this point. :/

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Re: Mental experiments, How do/can people solve problems?

Postby Alder » Wed Dec 24, 2008 2:12 am UTC

polymer wrote:Alder26: the reason I haven't brought up rhythm, is because it seems like that is much more of a human variable(except for perhaps keeping tempo.) It's very hard to call a particular rhythm wrong. It's (relatively)easy to observe an "ideal" system with harmony, and understand what rules are "broken." So, at this point at least, I'm thinking in terms of the instances I make chords.

Hmmm...I'd probably take issue with the thought that you can't call a rhythm wrong, you definitely can. If you play - say "Yesterday" by the Beatles, and play a bunch of different notes, it stops being "Yesterday" and becomes something else. Same with the rhythm.
Unless you're talking about that absolute point of composition, then, yeah, you might get good/bad/indifferent, but not wrong, as such.
But that's also true about harmony, you get things that sound good, things that sound terrible, but at least in theory, not wrong. (Provided you can justify it in some way!) Even a mad, random note cluster, as you get in some modern music, isn't wrong.

polymer wrote:If we were talking about sheet music you could present it something like this [15 --|--> 25--->24--->25-|->24]
Where the vertical dashes demonstrate that the sub dashes are even. You could tell if it was 1-4 beats(I know there are more, I'm just throwing something together) visually. The previous example presented has 12 dashes, with...perhaps the quarter note as 4 dashes and the down beat, so...this would be 3/4 time.

Heh...
I throw into the mix, breves [double whole notes?] dotted notes of various kinds [a dot = 50% extra time, it's a shorthand) and hemidemisemiquavers (64th notes). And that's without tied notes, where various lengths are joined together to get a particular length.
Oh! and ornaments - trills, turns, upper and lower mordents, appoggiaturas, and acciaccaturas. Enjoy....:P

I'm going to go off and think about chords, though.
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Re: Mental experiments, How do/can people solve problems?

Postby polymer » Wed Dec 24, 2008 2:43 am UTC

Alder26 wrote:
polymer wrote:Alder26: the reason I haven't brought up rhythm, is because it seems like that is much more of a human variable(except for perhaps keeping tempo.) It's very hard to call a particular rhythm wrong. It's (relatively)easy to observe an "ideal" system with harmony, and understand what rules are "broken." So, at this point at least, I'm thinking in terms of the instances I make chords.

Hmmm...I'd probably take issue with the thought that you can't call a rhythm wrong, you definitely can. If you play - say "Yesterday" by the Beatles, and play a bunch of different notes, it stops being "Yesterday" and becomes something else. Same with the rhythm.
Unless you're talking about that absolute point of composition, then, yeah, you might get good/bad/indifferent, but not wrong, as such.
But that's also true about harmony, you get things that sound good, things that sound terrible, but at least in theory, not wrong. (Provided you can justify it in some way!) Even a mad, random note cluster, as you get in some modern music, isn't wrong.
What you can say though is that the random jumble of notes step away from the piano's best approximation of the harmonic series, and build your logic around that. In the system we've been experimenting with currently though it's a little different. It emphasizes that the notes from different octaves are in fact an octave apart in accordance with Equal temperament, and that their approximation of equal temperament give them a similar character despite the key. Kind of...At any rate the point is you can see how it varies from an "ideal" case, rhythm is very different. You can demonstrate with physics why some notes are in fact in tune. The only thing necessary for rhythm to be correct as far as I can see is if its downbeat is consistent. Although I'll admit with a fixed amount for the BBM where you put the down beat can make the interpretations of the music in question interesting...Actually, I'll take back the comment regarding an incorrect rhythm, perhaps you could draw a line for an "ideal" rhythm and build a system around it.

polymer wrote:If we were talking about sheet music you could present it something like this [15 --|--> 25--->24--->25-|->24]
Where the vertical dashes demonstrate that the sub dashes are even. You could tell if it was 1-4 beats(I know there are more, I'm just throwing something together) visually. The previous example presented has 12 dashes, with...perhaps the quarter note as 4 dashes and the down beat, so...this would be 3/4 time.

Heh...
I throw into the mix, breves [double whole notes?] dotted notes of various kinds [a dot = 50% extra time, it's a shorthand) and hemidemisemiquavers (64th notes). And that's without tied notes, where various lengths are joined together to get a particular length.
Oh! and ornaments - trills, turns, upper and lower mordents, appoggiaturas, and acciaccaturas. Enjoy....:P
Is that a challenge :p? Give me a day to burn and I might come up with something creative :], I just don't see what you can pull out of it that'd be particularly useful...
I'm going to go off and think about chords, though.


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