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> Permutations, Mathematics of permutations
Larry F
post Apr 5 2013, 10:18 PM
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I wonder if people here are interested in permutations as a practice tool. If you have a set of 3 elements (say, notes), there are 6 permutations of those. A set of 4 elements has 24 permutations. If you want to combine, say, rhythm or picking with notes, then you are looking at two sets of permutations. To understand how those would work for any number and size of sets, the field of mathematic group theory might be interesting to investigate. Groups are amazing resources in music.
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Gabriel Leopardi
post Apr 6 2013, 12:39 AM
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I don't really know what you are talking about but it sounds interesting! It would be great if you can post a deeper explanation of this method of practice.


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Cosmin Lupu
post Apr 6 2013, 12:44 PM
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I think I get it - it's like when you have 4 fingers available on your left hand, on the fretboard and you create various combinations for let's say a chormatic exercise such as:

on the E string:1 2 3 4, on the A string: 2 3 4 1, on the D string: 3 4 1 2, on the G string: 4 1 2 3

and so on from B onwards - I'd better shoot a vid and show you smile.gif



This post has been edited by Cosmin Lupu: Apr 6 2013, 04:50 PM


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Larry F
post Apr 6 2013, 10:35 PM
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QUOTE (Cosmin Lupu @ Apr 6 2013, 11:44 AM) *
I think I get it - it's like when you have 4 fingers available on your left hand, on the fretboard and you create various combinations for let's say a chormatic exercise such as:

on the E string:1 2 3 4, on the A string: 2 3 4 1, on the D string: 3 4 1 2, on the G string: 4 1 2 3

and so on from B onwards - I'd better shoot a vid and show you smile.gif



Yes, exactly. I participated in a National Stage Band Camp in 1971 or 72. For two weeks, teachers from the Berklee School of Music taught lessons on different instruments, taught theory, arranging, and improve, and conducted large stage band size ensembles. John LaPorta was there. He was one of the architects of the modal system of teaching improve. Under this method, a ii chord in the accompaniment would be interpreted as a dorian mode on the same root/tonic. A V chord would be a mixolydian mode, and so forth. He was responsible for that entire school of using modes in that way. It was related to, but not identical with, the modal music of Miles, McCoy Tyner, and others. Similarly, it was related to, but not identical with, the use of the church modes in early western European music.

The guitar class had 4 students and was taught by Jack Peterson. He may have also been responsible for the Berklee modal method of teaching improv. In the guitar class, he showed us the 24 permutations of 4 chromatic notes on one string. I have seen these permutations a lot on the internet. These permutations form a mathematical group. I won't define what a group is, because I would need to give some more background. Instead, I will list some of the groups that can be found in music. These are:

Z2 = Cyclic group of order 2. 2 permutations of 2 elements.
12
21

Z3 = Cyclic group of order 3. 3 permutations of 3 elements.
123
231
312

D3 = Dihedral group of order 6. 6 permutations of 3 elements.
123 132
231 213
312 321

Z4 = Cyclic group of order 4. 4 permutations of 4 elements.
1234
2341
3412
4123

D4 = Dihedral group of order 8. 8 permutations of 4 elements.
1234 1432
2341 2143
3412 3124
4123 4321

K4 = Klein 4-group. 24 permutations of 4 elements.
1234 2341 3412 4123
1243 2314 3421 4132
1342 2413 3142 4213
1324 2431 3124 4231
1423 2134 3241 4312
1432 2143 3214 4321

You can play these as the 4 chromatic notes on string in one position. You can also play them in a minor pentatonic scale where:
1 = the lower note on a given string and position
2 = the higher note on the same string and position
3 = the lower note on the next higher string in the same position
4 = the higher note on the same string and position

Instead of having 3 and 4 played on the next higher string, you can also play them on the next lower string.
Instead of having 1 and 3 as the lower notes in that position, you can play them as the higher notes, with the original higher notes now played as lower notes.
Instead of either of these changes, you can have one string with 1 as a lower note and 2 as a higher note, with the next higher or lower string with 3 as a higher note and 4 as a lower note.

Instead of using the minor pentatonic scale, these 4 numbers can correspond to any 4 notes on the guitar.

If you look at the group names and permutations above, you can do some mental substitutions and see than Z2 sits inside Z4, D4, and K4. It does not sit inside Z3, but it does sit inside D3.

Z2, Z3, Z4, and D4 sit inside K4.

You will need to substitute 1 number for another when comparing groups. For example, the first 6 elements in K4 can be seen as D3 if you substitute 2 for 1, 3 for 2, and 4 for 3. Notice that the first element of the first 6 permutations in K4 is always 1. In this can, we can say that 1 is frozen and that we will not consider it. Thus, we are only looking at the 6 permutations of 2, 3, 4, which are the same as the 6 permutations of D3, having the elements 1, 2, 3. Being able to substitute 1 element from one group for another element of another group is a key requirement in being able to understand how groups are inter-related.

The very best book on group theory for beginners is F.J. Budden's The Fascination of Groups. It is $115 on Amazon: http://www.amazon.com/Fascination-Groups-B...n/dp/0521080169

Hopefully, you can find it in a university math library.

The tricky part about learning group theory, is that the examples generally use different kinds of math that not everybody understands. It is common to use the numbers 1, 2, 3, ... or 0, 1, 2, 3, ... in group theory books. Also common are a, b, c, ... or e, a, b, c, ... Here, e is similar to 0.

If you really want to take the guitar apart and train your hands to be flexible, groups can be invaluable tools.

Some of this stuff can also be found in Joseph Schillinger's The Mathematical Basis of the Arts (on Amazon for $79 at http://www.amazon.com/Mathematical-Basis-P...basis+for+music )

It is a classic and should be found at a university library. BB King has used this book for years.
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Cosmin Lupu
post Apr 7 2013, 12:27 PM
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Wow! Man, thanks for all this valuable info smile.gif

While in highschool and in the first 2 years of college, I have studied algebra and implicitly, groups. I also always known and believed that music is maths with a big soul, but I never took that extra step to compare the stuff that I have learned in college with the stuff I knew about music.

Even if those books have threatening prices, I would be tempted to buy them.

Tell me, can you use these concepts at will? would you record some examples for us?

All the best,

Cosmin


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Larry F
post Apr 8 2013, 12:12 AM
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Here's a recording of my band in 1974. I was starting to explore permutations and groups then. I didn't know about groups, instead thought of them as subsets of permutations and classes or permutations, which I later learned where isomorphic (same structure) as groups. I think groups fit right in with adventuresome guitar playing.

https://www.youtube.com/watch?v=aJTTXrR9New
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Larry F
post Apr 8 2013, 06:05 AM
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QUOTE (Cosmin Lupu @ Apr 7 2013, 11:27 AM) *
Tell me, can you use these concepts at will? would you record some examples for us?

All the best,

Cosmin


I could try to write out some things and maybe play them, but it would have to be in a month or so. I have internalized these to the where I can compose with them without having to write anything down in a table or anything like that.

I think it would be a tremendous practice tool, due to the different types of symmetries these produce. To an extent, I believe it is useful to train your hands to be neutral, so that certain moves don't overshadow other moves. Symmetry in practice can be good for that.

larry
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HungryForHeaven
post Apr 8 2013, 06:09 AM
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Cool stuff, Larry. I was thinking of making a "Math for Guitarists" series; there are so many things in music in general, and guitar in particular, that can be formalized and explained with mathematics.
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Cosmin Lupu
post Apr 8 2013, 09:55 AM
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I see its excellent use in practicing and in creating some phrases which involve this sort of concepts but it would be a bit too much for me to use it as a main weapon when making music.

What happened to Air Strike? And I would be interested to see the way you approach practice using these concepts, of course smile.gif


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Larry F
post Apr 8 2013, 05:41 PM
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QUOTE (Cosmin Lupu @ Apr 8 2013, 08:55 AM) *
I see its excellent use in practicing and in creating some phrases which involve this sort of concepts but it would be a bit too much for me to use it as a main weapon when making music.

What happened to Air Strike? And I would be interested to see the way you approach practice using these concepts, of course smile.gif


I agree that its main value would be for developing technique. I remember doing the K4 permutations on one string, but having the first note played on an adjacent string. This would be combined with permutations of down and up strokes. If you do all of the permutations in a situation like this, it can take a very long time to do. The solution is to use subgroups of a group. A subgroup is a subset of a group of an order (number of elements) that evenly divides the order of the group. This is called the LaGrange Theorem.

12-tone composers, such as Arnold Schoenberg, Anton Webern, Alban Berg, and Milton Babbitt use groups in their music, although Babbitt was the only who was aware of the mathematical concept. He wrote tons of articles on this. I'll bet that a lot of guys on this forum would like his music, which can be YouTubed, thankfully. He has a couple of guitar pieces, one called Sheer Pluck. I'm on the editorial board of the Journal of Mathematics and Music, which is a Taylor and Francis publication. You'll see a lot of group theory there.

Not long after we recorded Air Strike, we went our separate ways, and I focused on jazz for a few years, until tendonitis popped up, and I then went the academic route. That was always my backup plan. 25 years later, I started playing again, blues, and have reformed my old high school band, which includes the bass player on the Air Strike recording. We are trying to track down the drummer, but no luck yet. Those were some of the best days of my life. And we could make a living playing gigs, believe it or not. What has happened to the music scene for bands since then seems practically criminal. Playing country western for 5 hours a night, 4-6 nights a week really gets the rhythm into your bones.
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Cosmin Lupu
post Apr 9 2013, 09:22 AM
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QUOTE (Larry F @ Apr 8 2013, 04:41 PM) *
I agree that its main value would be for developing technique. I remember doing the K4 permutations on one string, but having the first note played on an adjacent string. This would be combined with permutations of down and up strokes. If you do all of the permutations in a situation like this, it can take a very long time to do. The solution is to use subgroups of a group. A subgroup is a subset of a group of an order (number of elements) that evenly divides the order of the group. This is called the LaGrange Theorem.

12-tone composers, such as Arnold Schoenberg, Anton Webern, Alban Berg, and Milton Babbitt use groups in their music, although Babbitt was the only who was aware of the mathematical concept. He wrote tons of articles on this. I'll bet that a lot of guys on this forum would like his music, which can be YouTubed, thankfully. He has a couple of guitar pieces, one called Sheer Pluck. I'm on the editorial board of the Journal of Mathematics and Music, which is a Taylor and Francis publication. You'll see a lot of group theory there.

Not long after we recorded Air Strike, we went our separate ways, and I focused on jazz for a few years, until tendonitis popped up, and I then went the academic route. That was always my backup plan. 25 years later, I started playing again, blues, and have reformed my old high school band, which includes the bass player on the Air Strike recording. We are trying to track down the drummer, but no luck yet. Those were some of the best days of my life. And we could make a living playing gigs, believe it or not. What has happened to the music scene for bands since then seems practically criminal. Playing country western for 5 hours a night, 4-6 nights a week really gets the rhythm into your bones.


If I am not mistaken, I have studied the LaGrange Theorem, but it's all too blurry right now to remember smile.gif I went like a hurricane through college because I wanted to play guitar and I also wanted to finish and graduate, not abandon it.

On the other hand, nothing beats country music biggrin.gif

Looking forward to the examples when you have some free time smile.gif


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