Scales with zero or negative intervals and other bs

Discussion in 'Music Theory, Lessons & Techniques' started by ncfiala, Apr 3, 2012.

  1. ncfiala

    ncfiala Silence you bastard

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    First, some definitions. An interval is a positive integer multiple of half steps. A scale is a sequence of intervals adding up to 12 half steps. Therefore, we can identify scales with (additive) compositions (ordered partitions) of the integer 12. We could take into account the playing of the same string on the same fret multiple times in a row by allowing an interval to be a non-negative integer multiple of half steps. Then scales could be identitified with weak compositions of 12. We could also take into account "backtracking" by allowing an interval to be any integer multiple of half steps. Finally, we could allow for scales that don't repeat every octave but instead repeat after some number of octaves by allowing a scale to be a sequence of intervals adding up to any integer multiple of 12. The concatenation of two scales is then again a scale. In this way the set of scales is given the structure of a non-commutative monoid (if we allow the empty scale). The subset of scales with only positive intervals, but that may not repeat every octave, would be a subsemigroup. What is this good for? Nothing but my mind runs rampant.
     
  2. PortalNathrakh

    PortalNathrakh DO NOT SPEAK

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    This thread in a nutshell.
     
  3. Thep

    Thep Blast & Sweep

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    Paragraphs, son.
     
  4. Gothic Headhunter

    Gothic Headhunter Born on a Monday

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    I'm sorry, what did you say? I was with you up till "First, some definitions."
     
  5. mr_rainmaker

    mr_rainmaker Resident Cherokee

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  6. pentecost

    pentecost SS.org Regular

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    which wouldn't accomplish anything... the whole idea of being 'additive' shouldn't be tossed to make the round hole more square. the original definitions you've put in play don't specify a number of notes in said scale, adding a null is redundant. it's already there.
     
  7. Augminished

    Augminished Augminished Studio

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    Does anyone else feel like they just got yelled at and lectured?

    Close to two years of theory in school and never have I been this confused.

    I feel like a just walked in to a door that had notes on it but they were negative. :lalala::blahblah:
     
  8. groovemasta

    groovemasta SS.org Regular

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  9. niffnoff

    niffnoff Just another SunBro

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    Hey guys what's goi- WHOAH WHAT THE FU-

    Seriously though, my head hurts now, what's the point in any of that OP :(
     
  10. JStraitiff

    JStraitiff Melodic Mamma Jamma

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    Perhaps if you formatted the original post a little we would be compelled to read. Separate the definitions using line breaks and then the rest below that.

    Paragraph per topic :p
     
  11. ncfiala

    ncfiala Silence you bastard

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    Sorry, I was pretty much just thinking out loud. Or rather thinking in print. As a mathematician, I try to abstract and generalize everything. I can't turn it off. The question is does the abstraction and generalization have any real world use or value. In this case, and in many cases, it probably doesn't. But that usually doesn't stop us from doing it anyway.
     
  12. ixlramp

    ixlramp SS.org Regular

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    I feel you might find microtonal and 'just-intonation' music theory interesting.
     
  13. Explorer

    Explorer He seldomly knows...

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    A friend of mine obsessed about what he called "spiral" scales, which didn't have all the notes in each octave, instead repeating the "scales" over two and three octaves.

    I'm ready to hear something compelling arising from this thinking. Otherwise, I'd say you just sound like you've been hitting the weed and daydreaming. "Hey... I was just thinkin'... isn't that cosmic?" *laugh*

    Short version: I would love a cupcake... I'm just thinkin' out loud... what do you think of that idea?
     
    Alberto7 and Augminished like this.
  14. fantom

    fantom Misses his 6 strings

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    So after reading that a few times... maybe I missed something. But for the sake of a fun mathematics discussion (in 1 paragraph), can you clarify something for me? How did you go from elements of the monoid being notes to scales? If you treat your operator as combining scales, you're forcing an ordered sequence to be elements. This pretty much generates all permutations of notes rather quickly with no purpose. If you just meant that notes are elements, then you have to drop the notion of a scale when dealing with a monoid, because all you can do is an single operator to combine two relative intervals to produce a third interval. So say what?
     
  15. mm66554

    mm66554 SS.org Regular

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    Sorry but you sound like you just started a qualification in either maths or computer programming and are hyped up to somehow prove your newly aquired knowledge (or words) on an unrelated internet forum.
    Instead of thinking scales, think in terms of notes, that in itself makes "negative intervals" obsolete as they are already encompassed within keys, inverted chords etc. If you want to create your own scales based off intervals the only way is to experiment and see what sounds best. Every scale will repeat at the octave if a semi-tones is your lowest incriment, that's just how it is, even if you force yourself not to play the octave another instrument such as bass almost certainly will.
     
  16. ixlramp

    ixlramp SS.org Regular

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    Interesting.
     
  17. Augminished

    Augminished Augminished Studio

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    I'm sorry but i'm still confused are we talking about scales or math?

    If its math I have to get into math mode and then I can conquer you ALL.

    PS I know shit about math so ya.
     
  18. ncfiala

    ncfiala Silence you bastard

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    Actually I have a Ph.D. in mathematics. And I certainly wouldn't call mathematics and music unrelated.

    And it's not true that any sequence of intervals will repeat every octave. A sequece of intervals will repeat every octave if and only if it adds up to a divisor of 12.
     
  19. ncfiala

    ncfiala Silence you bastard

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    The elements of the monoid are sequences (of intervals) and the operation is concatenation of sequences. It's just like when you form the free monoid on the strings over some alphabet.
     
  20. niffnoff

    niffnoff Just another SunBro

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    ... what in the hell am I reading and why does my head hurt and why do I feel like I'm about to explode...

    can someone break down the math into a simpler term for us who may not be at a PHD level.
     

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