# can math be used to create scales with intervals/harmonies not found in a standard

scale. i know the chromatic scale like on a piano/guitar has mathematical relationships, and its no coincidence that the harmonies we find to be most apparently in harmony are the intervals that they are (fifths, thirds...), even if i dont actually know anything about the math.

but i figure when playing atonally you find harmonies, even if just between two notes, that have a "valid" sounding quality to it. even tho it isnt from an interval that could be found in a standard 12 or 24 (quarter tone) scale. and i figure it must have complimentary harmonies rooted in math like a standard scale. and if you found them youd have totally new scale.

is there anything out there related to this? am i just totally off base from how music is actually structured?
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2 frets divided by 1 fret,

but seriously dude, math? There are 12 notes in a scale, you're really not doing anything else but counting the number of half steps one note is away from another, name it, and use that to build chords and what not.
Sounds like you're a bit confused.

Ill give you the music-math relationship in a nutshell:

Intervals work because they correspond to ratios of frequencies found in the natural vibration pattern of a vibrating string (look up "harmonic series" and "overtones"). The chromatic scale is a brilliant invention that moves a bunch of these ratios around slightly so that they still sound like the original ratios but now they're all equally spaced apart (look up "equal temperament"), so you can play any scale or mode starting on any note and move the tonality around freely.

And you can definitely find harmonies not found in our chromatic scale and make some neat sounds, but the chromatic scale pretty much covers all the main ones that your auditory cortex is able to distinguish between. (Our hearing is far from perfect, some guitarists dont mind playing with their guitars out of tune. As you develop your ear you can start to notice more subtle differences, but theres a limit). For example the 7th (E to D) interval could be played as a frequency ratio of 7/4 or 9/5. You could construct a scale with both of these and tell me what you come up with.

Hope that clears things up a bit.
Quote by bouttimeijoined
Sounds like you're a bit confused.

Ill give you the music-math relationship in a nutshell:

Intervals work because they correspond to ratios of frequencies found in the natural vibration pattern of a vibrating string (look up "harmonic series" and "overtones"). The chromatic scale is a brilliant invention that moves a bunch of these ratios around slightly so that they still sound like the original ratios but now they're all equally spaced apart (look up "equal temperament"), so you can play any scale or mode starting on any note and move the tonality around freely.

And you can definitely find harmonies not found in our chromatic scale and make some neat sounds, but the chromatic scale pretty much covers all the main ones that your auditory cortex is able to distinguish between. (Our hearing is far from perfect, some guitarists dont mind playing with their guitars out of tune. As you develop your ear you can start to notice more subtle differences, but theres a limit). For example the 7th (E to D) interval could be played as a frequency ratio of 7/4 or 9/5. You could construct a scale with both of these and tell me what you come up with.

Hope that clears things up a bit.

i guess equal temperament is a sure sign that theres something special about the chromatic scale. but why cant there be "weirder" scales with completely different sets of ratios, still validated thru math? pretend we have the auditory cortex to appreciate it. is there anything to be learned from the relationships in the chromatic scale that can be applied to creating scales with different ratios. even if the less apparently musical they are the more abstract the scale is?

what do you mean by frequency ratio. i know harmonies are different frequencies over each other but not much more than that.
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Take a root note of 440hz. Take its octave, 880hz. That's a 1/2 ratio. Take 440hz and its fifth, 660hz~. That's a 2/3~ ratio. The simpler the ratio can be expressed, the more consonant the sound... Which is why 440hz and its minor second, 466hz~ is so dissonant, because it's a ratio of 220/233~.

That's just in our equal-tempered scale, 12-TET, twelve notes per octave. You can get different scales. Say, 17-, 19-, 24-, or 36-tone scales which all have different ratios. Do a search on Youtube for these different scales.
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Last edited by Dayn at Jan 18, 2012,
There are plenty of cultures that use quarter tones and less, I believe India, Japan, and Greece all have had music based upon quarter tone division or less.

Here's a song from Greece that uses such intervals. Appreciate it, it took me forever to find this exact recording.

http://www.4shared.com/mp3/BCP-025E/03_euripides-_orestes_tragedy_.html

And for the mathmatics involved in music, and octave has a ratio of 1:2, a Whole step has a ratio of 9:8, and IIRC, fourths are 4:3, and fifths are 3:2.

Wow, I had no idea I would need this kind of information after a single day of my Music history class.
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Quote by rabbittroopsux
i guess equal temperament is a sure sign that theres something special about the chromatic scale. but why cant there be "weirder" scales with completely different sets of ratios, still validated thru math?

There can but they'd either be approximating the same ratios we get from the chromatic scale or sound like poop. There are other scales with different sets of ratios, most are just a subset of the ones found in the chromatic scale though.

Lets take the following ratios: 5/5 6/5 7/5 8/5 9/5 10/5. In order these correspond approximately to the unison, minor third, tritone, minor sixth, seventh, and octave found in the chromatic scale.

I suppose you could play with 7/7 8/7 9/7 10/7 11/7 12/7 13/7 and 14/7. See where that takes you.

pretend we have the auditory cortex to appreciate it. is there anything to be learned from the relationships in the chromatic scale that can be applied to creating scales with different ratios. even if the less apparently musical they are the more abstract the scale is?

No. The fact that the chromatic scale works so well suggests that there really are limitations on how "abstract" a scale you can make before things start matching up with the chromatic scale.

what do you mean by frequency ratio. i know harmonies are different frequencies over each other but not much more than that.

frequency is the number of vibrations per second (measured in hertz). A is generally 440 Hz. To get an octave, you take a frequency ratio of 2/1. 2 x 440 = 880 Hz, is also an A. 440 / 2 = 220 Hz is also an A. A fifth is a ratio of 3/2. 440 x 3 / 2 = 660 Hz is an E. 660 / 2 / 2 / 2 = 82.5 Hz is also an E. The correct frequency of the E string on your guitar is 85.41 Hz. This is because equal temperament gives us slightly imperfect frequency ratios.
Quote by Life Is Brutal
There are plenty of cultures that use quarter tones and less, I believe India, Japan, and Greece all have had music based upon quarter tone division or less.

Here's a song from Greece that uses such intervals. Appreciate it, it took me forever to find this exact recording.

http://www.4shared.com/mp3/BCP-025E/03_euripides-_orestes_tragedy_.html

Thats pretty neat, but you can hear how it sounds like a lot of the same intervals we have in our 12 tone chromatic scale, with a few sort of "detuned" sounding notes added in.
Quote by bouttimeijoined
Thats pretty neat, but you can hear how it sounds like a lot of the same intervals we have in our 12 tone chromatic scale, with a few sort of "detuned" sounding notes added in.

That's because the system of music we've established over the past several thousand years works, so you're always going to hear similarities. The Indian Sa-Re-Ga-Ma-Pa-Dah-Ni-Sa solfege system used in Ragas is almost exactly the same as our major scale, yet the two have developed almost completely independent from each other.

The same applies for traditional African music, where the intervals and notes used can all be comprehended and utilized through our current means of temperament, although for both prior cases some slight adjustment/rounding of tones would be needed.

For this instance, the distance from Sa to Pa is essentially a fifth, but by some aural traditions it may be slightly lower or higher than what we interpret as a fifth. But despite some very slight frequency differences, its essentially a fifth.
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It's been a long time but I recall learning something about Pythagoras and his students played music based on math, mostly because the Greeks had a hard-on for math in general.
^^The above is a Cryptic Metaphor^^

"To know the truth of history is to realize its ultimate myth and its inevitable ambiguity." Everything is made up and the facts don't matter.

Quote by rockingamer2
It's been a long time but I recall learning something about Pythagoras and his students played music based on math, mostly because the Greeks had a hard-on for math in general.

Pythagoras actually discovered the Major 2nd, Perfect Fourth/Fifth and Octave ratios. He did this by attaching a chord/string of some kind along a room, and then he hung weights off of it and struck it with a hammer. When he doubled the weight it made an octave to the original, and he then found consonant weight ratios for the other pitches.

mostly because the Greeks had a hard-on for math in general.

Music was taught right alongside arithmetic and astronomy due to it being seen as a completely mathematical construct that had similar relationships to cosmological bodies.

There's actually quite a bit of good philosophy I've read recently, there's a certain point I really liked but can't remember ATM.

Also, is your name Rocking-Amer, or Rockin'-Gamer? Its probably the latter.
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Quote by Life Is Brutal

Also, is your name Rocking-Amer, or Rockin'-Gamer? Its probably the latter.

Yep, the latter. Such is the pain of being thirteen when registering.
^^The above is a Cryptic Metaphor^^

"To know the truth of history is to realize its ultimate myth and its inevitable ambiguity." Everything is made up and the facts don't matter.

In theory, notes with any frequency ratio could be played together to create harmony. Our ears, by nature and nurture, are conditioned to accept rational number frequency ratios (or close approximations) so anything seems like different degrees of weird. But if that's what you want you could experiment with simultaneous pitches and derive new intervals, then chords, etc. The guitar is not suitable for this purpose because of the frets.
Quote by rockingamer2
It's been a long time but I recall learning something about Pythagoras and his students played music based on math, mostly because the Greeks had a hard-on for math in general.

yes, but could they djent?
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Last edited by mdc at Jan 20, 2012,
Yeah, Steve Vai did it - I've unashamedly copy pasted this but I do actually own the magazine it was originally printed it which is the only reason I know about it - it's from Guitar World sometime in 1993.

GW: Leaving metaphysics aside, Sex And Religion is a harmonically adventurous album. You seem to be using modes that one doesn’t usually hear in rock and roll.

Vai: That’s another thing that I can’t help. You’re going to hear modes in there that you never heard on any other record or any other type of music, simply because I made them up out of synthetic scales. Like the end of “Deep Down Into The Pain” – that really weird birth sequence. What’s happening is that a child is coming out of the womb, you know? He’s hearing the voice of divinity and asking questions and all this weird stuff. But what you hear in the background is this wild music based on a scale I devised.

GW: A new scale?

Vai: Yeah, I call it the “Xavian” scale. What I did was take the 12-tone row and make sampled notes of it on the keyboard. Then what I like to do is experiment with different temperaments. [Ed. Note – The 12-note European tempered scale is only one way of dividing up the frequency range between octaves. Different systems exist in other cultures and in the work of composers like LaMonte Young and Wendy Carlos. Some modern synthesizers offer alternate temperaments.]

I have this book where I keep all these different scales, where I divided the octave up into different steps – like maybe 9 or 10 equal steps. I call these scales “fractals.” At the end of “Deep Down Into The Pain” I used a scale that’s based on dividing the octave into 16 equal steps, instead of the 12 steps of the conventional tempered scale. So each half-step within that is not quite a conventional half-step – it’s 60 microsteps as opposed to 100 microsteps. Instead of calling it a half-step, I call it a “quasar.” Then the “whole step” is 120 microsteps, instead of 200 microsteps. Instead of calling it a whole step, I call that a “nova.” All these different intervals create the Xavian scale, a 10-note scale that I extracted from this 16-note row. You take this scale and play chords with it and it’s like divine dissonance, because all the intervals are twisted.

Every six notes or so, you run across a tempered interval. But for the most part, there are not tempered intervals, so you get a whole structure of harmonics that is just eerie and unique. You know how every chord conjures up a different mood? Even to the most casual listener, a major ninth chord will create a different feeling than a minor ninth, or a major ninth with a sharp 11th. Imagine the twisted world of emotions you can open up from the Xavian scale! We human beings are so shaped by music in our evolution. I think that as more people get into experimenting with these fractals, a whole different emotional state of mind will result – one that is probably on a par with the way our evolution is going anyway. But I don’t think you’ll ever hear Metallica jamming on the Xavian scale.

GW: If they read this, maybe they’ll get into it.

Vai: I’ll lend Kirk my 16-fret guitar. You can’t do this stuff on a conventional fretted instrument. I have a guitar that has 16 frets to the octave. Steve Ripley built it for me years ago. He also built me one with 24 divisions to the octave.

GW: So you’ve been experimenting with this for some time?

Vai: Oh, yeah. He built me those guitars about eight years ago.

GW: Are there any other recordings of your with Xavian scales or the like on them?

Vai: No. But there are a couple of weird things. There’s a song called “Chronic Insomnia” on Flexable Leftovers, where I recorded eight different passes of the same melody. Each time I just tweaked the tape speed a little bit, so I ended up with a melody where each note spans and entire half-step. It’s a very dense, eerie-sounding thing. Incidentally, I’m probably going to be remixing Flexable Leftovers and all my other stuff from that period and putting it all onto one disc.

tl;dr
He splits the octave into a 16 equal fractions - usually we have 12. That gives him a completely different set of intervals that not only sound comletely alien to us, it's also impossible to play on a conventional guitar and you have to make one specially.

It's not used in the main melody but those weird ethereal sounds over the Deep Down into The Pain outro are the Xavian scale
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