Group theory 101: How to play a Rubik’s Cube like a piano - Michael Staff
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Mathematics explains the workings of the universe, from particle physics to engineering and economics. Math is even closely related to music, and their common ground has something to do with a Rubik's Cube puzzle. Michael Staff explains how group theory can teach us to play a Rubik’s Cube like a piano.
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How we mapped the notes onto the cube:
1. Find 6 chords that make up a progression. For example, write down the names of chords used in a chord progression within a song. (Note: you may have to look to Classical music for an example, as many modern songs do not contain enough chords to do this)
2. Assign each chord to a different face of the Rubiks cube. (There are even blank cubes available to make it look even better)
3. Have each row and column on a face represent a different chord position. The top row for C Major, for example, would be the chord in root position: C, E, G. The middle row on the same face would be the first inversion: E, G, C. The last row would be the second inversion: G, C, E.
One example: Using the outer harmonic progression from Liszt's eighth transcendental etude:
Face 1: C minor:
row 1: (c, e-flat, g)
row 2: (e-flat, g, c)
row 3: (g, c, e-flat)
Face 2: B-flat Major:
row 1: (b-flat, d, f)
row 2: (d, f, b-flat)
row 3: (f, b-flat, d)
Face 3: A-flat Major:
row 1: (a-flat, c, e-flat)
row 2: (c, e-flat, a-flat)
row 3: ( e-flat, a-flat, c)
Face 4: G Major:
row 1: (g, b, d)
row 2: (b, d, g)
row 3: (d, g, b)
Face 5: F minor:
row 1: (f, a-flat, c)
row 2: (a-flat, c, f)
row 3: (c, f, a-flat)
Face 6: D-flat Major:
row 1: (d-flat, f, a-flat)
row 2: (f, a-flat, d-flat)
row 3: (a-flat, d-flat, f)
Another fun part of solving a cube using notes is that some notes are found in more than just one chord (for example, a-flat is in both the A-flat Major and D-flat Major chords), and so it adds to the challenge by having to figure out which permutation uses the note in a given position for that chord.
Group theory can explain much more in music than just basics. Using the properties of groups, we can treat music just like a mathematical field.
If we count the number of notes from C to the next C, we get twelve notes. If we then substitute each note with an integer, starting with 0, we get the set of integers 0 through 11. This also forms a group under addition by satisfying the four axioms. Musically, there are 12 intervals, or distances, from the starting note as well. For example, the first interval, represented by the integer 0, is just the distance from C to itself. The next interval is the distance from C to the next note up, C#, and is represented by the integer 1. The last interval before starting the whole sequence again is 11, and corresponds to the distance between C and the B above it.
We can combine intervals (which is how chords are created) just like adding numbers. The interval from C to E is a Major third, and counting up from 0, this would correspond to 4. The interval from C to A is a Major sixth, and counting up from 0, this is the same as the integer 9. Now we can add the two intervals, a Major third plus a Major sixth. To add the Major sixth to the Major third, we must count 9 up from E instead of C. A Major sixth from E is C-sharp (and is the same distance as from C to A). So a Major third plus a Major sixth, starting from C, brings us back to C-sharp. The interval from C to C-sharp is 1 (since it comes right after 0). In other words 4 + 9 = 1. At first this may not make a whole lot of sense. But in a group, the sequence starts all over again once we reach the highest element in the group, which in this case, is 11. So counting 9 from 4, while it would normally give us 13, gives us 1 (just like adding 2 and 2 in a group of 0, 1, and 2 gives us 1 not 4). This is the closure axiom, and is the result of something called modular arithmetic. If you've ever heard of binary, the language computers use, then you probably know that it uses just 0 and 1. This is a simple example, but in binary, 1 plus 1 is 0, since there is no 2 in the system and the next number in the sequence is 0.
There is also a fifth group axiom as well, which determines, not if something is a group, but what type of group it is. There is a special kind of group called an Abelian group, and interestingly, the musical intervals is an Abelian group. There are other types of groups as well, and many of them can be seen in music.
By following these basic ideas, we can verify that the notes form a group. By taking certain group elements and checking to see if the axioms still hold, we can make subgroups, which are what symmetrical chords, like the diminished seventh, are. Subgroups are groups, but they are groups taken from a larger group while still satisfying the same axioms that the group satisfies. The diminished seventh is not the only chord that is a symmetrical subgroup. Can you figure out what other chord(s) can be formed while satisfying the group axioms?
Here is a more in-depth look at abstract math, such as group theory, in music: http://www.michaelstaff.com/instruction.html
To learn about the history of group theory as a field of study, click here.
1. Find 6 chords that make up a progression. For example, write down the names of chords used in a chord progression within a song. (Note: you may have to look to Classical music for an example, as many modern songs do not contain enough chords to do this)
2. Assign each chord to a different face of the Rubiks cube. (There are even blank cubes available to make it look even better)
3. Have each row and column on a face represent a different chord position. The top row for C Major, for example, would be the chord in root position: C, E, G. The middle row on the same face would be the first inversion: E, G, C. The last row would be the second inversion: G, C, E.
One example: Using the outer harmonic progression from Liszt's eighth transcendental etude:
Face 1: C minor:
row 1: (c, e-flat, g)
row 2: (e-flat, g, c)
row 3: (g, c, e-flat)
Face 2: B-flat Major:
row 1: (b-flat, d, f)
row 2: (d, f, b-flat)
row 3: (f, b-flat, d)
Face 3: A-flat Major:
row 1: (a-flat, c, e-flat)
row 2: (c, e-flat, a-flat)
row 3: ( e-flat, a-flat, c)
Face 4: G Major:
row 1: (g, b, d)
row 2: (b, d, g)
row 3: (d, g, b)
Face 5: F minor:
row 1: (f, a-flat, c)
row 2: (a-flat, c, f)
row 3: (c, f, a-flat)
Face 6: D-flat Major:
row 1: (d-flat, f, a-flat)
row 2: (f, a-flat, d-flat)
row 3: (a-flat, d-flat, f)
Another fun part of solving a cube using notes is that some notes are found in more than just one chord (for example, a-flat is in both the A-flat Major and D-flat Major chords), and so it adds to the challenge by having to figure out which permutation uses the note in a given position for that chord.
Group theory can explain much more in music than just basics. Using the properties of groups, we can treat music just like a mathematical field.
If we count the number of notes from C to the next C, we get twelve notes. If we then substitute each note with an integer, starting with 0, we get the set of integers 0 through 11. This also forms a group under addition by satisfying the four axioms. Musically, there are 12 intervals, or distances, from the starting note as well. For example, the first interval, represented by the integer 0, is just the distance from C to itself. The next interval is the distance from C to the next note up, C#, and is represented by the integer 1. The last interval before starting the whole sequence again is 11, and corresponds to the distance between C and the B above it.
We can combine intervals (which is how chords are created) just like adding numbers. The interval from C to E is a Major third, and counting up from 0, this would correspond to 4. The interval from C to A is a Major sixth, and counting up from 0, this is the same as the integer 9. Now we can add the two intervals, a Major third plus a Major sixth. To add the Major sixth to the Major third, we must count 9 up from E instead of C. A Major sixth from E is C-sharp (and is the same distance as from C to A). So a Major third plus a Major sixth, starting from C, brings us back to C-sharp. The interval from C to C-sharp is 1 (since it comes right after 0). In other words 4 + 9 = 1. At first this may not make a whole lot of sense. But in a group, the sequence starts all over again once we reach the highest element in the group, which in this case, is 11. So counting 9 from 4, while it would normally give us 13, gives us 1 (just like adding 2 and 2 in a group of 0, 1, and 2 gives us 1 not 4). This is the closure axiom, and is the result of something called modular arithmetic. If you've ever heard of binary, the language computers use, then you probably know that it uses just 0 and 1. This is a simple example, but in binary, 1 plus 1 is 0, since there is no 2 in the system and the next number in the sequence is 0.
There is also a fifth group axiom as well, which determines, not if something is a group, but what type of group it is. There is a special kind of group called an Abelian group, and interestingly, the musical intervals is an Abelian group. There are other types of groups as well, and many of them can be seen in music.
By following these basic ideas, we can verify that the notes form a group. By taking certain group elements and checking to see if the axioms still hold, we can make subgroups, which are what symmetrical chords, like the diminished seventh, are. Subgroups are groups, but they are groups taken from a larger group while still satisfying the same axioms that the group satisfies. The diminished seventh is not the only chord that is a symmetrical subgroup. Can you figure out what other chord(s) can be formed while satisfying the group axioms?
Here is a more in-depth look at abstract math, such as group theory, in music: http://www.michaelstaff.com/instruction.html
To learn about the history of group theory as a field of study, click here.
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