My calculations on Eight Tone Scales was way too manually intensive for me to trust that I got it right. I should check my work.
- generate the set of all strings of intervals of length 8.
- Generate an array of length 8 and populate it with all 1’s except the first value, which gets a 0.
- increment each digit. If the digit goes over five, overflow to the next digit. If the last digit overflows, we are done.
- Sum the digits. If they don’t add up to 8, throw it out
- Convert them to a canonical form. We can make an array of 8 digits into an eight digit integer. Take the highest valued integer for each mode of the scale.
- Convert the digits into a single eight digit integer.
- Rotate the array by one and convert it to an 8 digit number.
- Compare the results of the previous two steps and keep the higher value.
- Repeat until we’ve rotated through all values in the array.
- Put them into a Set which will only contain unique values to remove duplicates.
- Sort the set
Since I am moving on to a Python Project, I figured I’d code it up in python.
#!/usr/bin/python
def increment(intervals, index):
if index >= len(intervals):
return False
intervals[index] += 1
if intervals[index] > 5:
intervals[index] = 1
return increment(intervals, index + 1)
return True
def sums_to(intervals, sum):
total = 0
for val in intervals:
total += val
return sum == total
def int_form(intervals):
value = 0
for digit in intervals:
value *= 10
value += digit
return value
def rotate(intervals):
l = intervals[:]
n = len(l) - 1
head = l[:n]
l[:n] = []
l.extend(head)
return l
def max_int_form(intervals):
copy_of = intervals[:]
max = 0
for digit in intervals:
next_int_form = int_form(copy_of)
if next_int_form > max:
max = next_int_form
copy_of = rotate(copy_of)
return max
intervals = [ 0, 1, 1, 1, 1, 1, 1, 1]
scale_set = set()
count = 0
while increment (intervals,0):
if sums_to(intervals, 12):
scale_set.add(max_int_form(intervals))
scales = list(scale_set)
scales.sort()
for scale in scales:
count += 1
print scale
print count ," Scales generated"
And the Output:
[ayoung@ayoung python]$ ./eightone.py 21212121 22112121 22112211 22121121 22121211 22122111 22211121 22211211 22212111 22221111 31111122 31111212 31111221 31112112 31112121 31112211 31113111 31121112 31121121 31121211 31122111 31131111 31211112 31211121 31211211 31212111 31221111 31311111 32111112 32111121 32111211 32112111 32121111 32211111 33111111 41111112 41111121 41111211 41112111 41121111 41211111 42111111 51111111 43 Scales generated