Permutations, Mathematics of permutations |
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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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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  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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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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