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Old 04-03-2012, 07:56 PM   #1
ncfiala
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Scales with zero or negative intervals and other bs

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.
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Old 04-03-2012, 08:45 PM   #2
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Originally Posted by ncfiala View Post
What is this good for? Nothing but my mind runs rampant.
This thread in a nutshell.
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Old 04-03-2012, 08:55 PM   #3
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Quote:
Originally Posted by ncfiala View Post
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.
Paragraphs, son.
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Old 04-03-2012, 08:58 PM   #4
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I'm sorry, what did you say? I was with you up till "First, some definitions."
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Old 04-03-2012, 09:20 PM   #5
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none of this matters unless you resolve your musical statment at the end... ie: cadence.
Cadence (music) - Wikipedia, the free encyclopedia

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Old 04-03-2012, 09:38 PM   #6
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Originally Posted by ncfiala View Post
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.
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.
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Old 04-03-2012, 10:11 PM   #7
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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.

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Old 04-03-2012, 10:20 PM   #8
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Old 04-03-2012, 11:26 PM   #9
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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

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Old 04-04-2012, 01:17 AM   #10
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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

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Old 04-04-2012, 09:26 AM   #11
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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.
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Old 04-04-2012, 12:18 PM   #12
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I feel you might find microtonal and 'just-intonation' music theory interesting.
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Old 04-04-2012, 08:18 PM   #13
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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?

If you don't care enough to research your own question, why should anyone else care more?

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Old 04-05-2012, 04:24 AM   #14
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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?
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Old 04-05-2012, 07:15 AM   #15
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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.

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Old 04-05-2012, 10:07 AM   #16
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Quote:
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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.
Interesting.
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Old 04-05-2012, 10:23 AM   #17
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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 .... about math so ya.

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Old 04-05-2012, 11:01 AM   #18
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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.
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.
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Old 04-05-2012, 11:08 AM   #19
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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?
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.
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Old 04-05-2012, 11:43 AM   #20
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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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Old 04-05-2012, 12:14 PM   #21
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^ I agree it sounds interesting but I have no ....ing clue what any of it means.

ncfiala break it down in to simpler terms and it might make for a great topic.

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Old 04-05-2012, 12:23 PM   #22
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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.
It really doesn't matter dude. It just helps me to put things into an abstract context that I'm familiar with. It helps me to think about things in different ways. You can never have too many different perspectives on any subject. But it's probably not going to be useful to anyone else, or even to me for that matter.

Also, the math is actually really simple but I don't think it's worth your time to figure it. Just go practice, which is what I need to go do just as soon as I finish painting this beat up Donkey Kong Jr. cab.
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Old 04-05-2012, 02:19 PM   #23
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But it's probably not going to be useful to anyone else, or even to me for that matter.

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Old 04-05-2012, 06:45 PM   #24
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Sounds like you're thinking along the lines of synthetic scales and non-octave tunings to me.

Extreme Syncopation =/= Poly-rhythmic

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Old 04-05-2012, 09:14 PM   #25
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I just realized you're the same guy who was posting about about theory and obsessing about learning everything about theory.


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