I was thinking about synthesizing drum loops and it occurred to me that Hadamard bases might be an interesting way to do it.
A simple approach to synthesizing a drum loop with a single instrument and up to 8 beats is to just have 16 variables for the volume and timing offset of each of the 8 beats. Some of the volumes might be 0, silencing those beats; some of the offsets might be nonzero, swinging those beats a bit.
A problem with this representation is that usually you don’t want to adjust just one beat at a time; you might want to do something to all the odd beats, for example, or all the even beats. A possible solution to this is to transform 8 volume variables and 8 timing-offset variables each through a 8x8 Walsh matrix, here given in “sequency order” rather than as a Hadamard matrix:
1 1 1 1 1 1 1 1
1 1 1 1 -1 -1 -1 -1
1 1 -1 -1 -1 -1 1 1
1 1 -1 -1 1 1 -1 -1
1 -1 -1 1 1 -1 -1 1
1 -1 -1 1 -1 1 1 -1
1 -1 1 -1 -1 1 -1 1
1 -1 1 -1 1 -1 1 -1
Given an appropriate constant scaling factor, this is an orthonormal basis for 8-vectors.
A more convenient form might be the ReLU of the same matrix, which you will note is still symmetric though no longer normalized:
1 1 1 1 1 1 1 1
1 1 1 1 0 0 0 0
1 1 0 0 0 0 1 1
1 1 0 0 1 1 0 0
1 0 0 1 1 0 0 1
1 0 0 1 0 1 1 0
1 0 1 0 0 1 0 1
1 0 1 0 1 0 1 0
So, for example with volumes, the first variable is a master volume control; the last variable is a master volume control for the major beats (assuming we’re in 4/4 time); the second variable controls the volume on the first half of the measure; the fourth variable controls the first and third beats (the “on” beats, giving you a backbeat if you turn them down); the fifth variable controls the first halves of the first and third beats, and the second halves of the second and fourth beats; and so on. It might make the most sense to have scale values that extend from zero amplitude to somewhere above 1.0, but do the matrix-multiply in decibels, so any single slider is capable of silencing either half or all the beats, and can either attenuate or amplify those beats from the overall reference volume level.
In this way you can adjust the volumes of the 8 beats to any desired combination of volumes, but every slider affects half the beats in what I speculate will be a more musically appealing way.
The same scheme can be used for adjusting timing, though perhaps there is no need for the first variable in that case. If the last timing variable (10101010) is pushed to an extreme, it could perhaps push the eighth notes to coincide with the quarter notes.
Here’s the corresponding 4x4 binary matrix:
1 1 1 1
1 1 0 0
1 0 0 1
1 0 1 0
And the corresponding 16x16 binary matrix:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1
1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0
1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1
1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0
1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0
1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1
1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0
1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0
1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1
1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0
1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
This was derived as follows:
def hadamard(n):
return ([[1]] if n < 1 else
[xs + xs for xs in hadamard(n-1)] +
[xs + [-x for x in xs] for xs in hadamard(n-1)])
def walsh(n):
return sorted(hadamard(n),
key=lambda xs: sum(1 for i in range(1, len(xs))
if xs[i] != xs[i-1]))
def walsh_order(n): # bit-reversed reflected binary Gray code
return ([0] if n < 1 else
[x * 2 for x in walsh_order(n-1) ] +
[x * 2 + 1 for x in reversed(walsh_order(n-1))])
def binrow(row):
return ('1' if x > 0 else '0' for x in row)
if __name__ == '__main__':
for row in walsh(4):
print(' '.join(binrow(row)))
It is possible to use the fast Hadamard transform to efficiently transform between the two representations — the set of user parameters and the set of music parameters — but that isn’t really important in this case.