by Kraig Grady and Juno Kim [29-5-2013]
The purpose of this paper is to provide an excel sheet as a quick reference guide to a set of primary and secondary sequences. This can be used to find recurrent sequences that coincide with given generators, multiple sequences that result in the same generator, and possible fundamental spacings of proportional triads . More commonly it can be used to find recurrent series that are extremely close to others providing possibilites of other spacings and proportional harmonies worthy of investigation for similar types of resonance or near resonances.
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The excel chart below serves as a companion to the primary series found in meruthree.pdf pages 2-8.
It computes all the secondary patterns where the resultant generator is less than 5.5 octaves and the two seeding generators are never more than 5 octaves apart. These extreme limits hopefully will cover most musical application.
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Here is the basic formula used to find the primary sets
(1 + G^C)^(1/D)
which represents an Hn recurrent sequence this way
Hn=Hn -D + (Hn -D +C).
G is the answer that is fed back into the formula. To start it is seeded with 1.
I wish to point out to those not familiar with like formulas that 'C' might cause confusion as it is the number of steps above the beginning term and not a distance from the ending generator. for example
(1 + G^3)^(1/4) is the sequence
Hn= Hn-4 + Hn-1. (note that Hn-1 is 3 above -4)
[and not Hn-4 +Hn-3 which might seem to be so]
Each Secondary Recurrent Series can involve the raising of either or both of the seeds one or more octaves. Hence we modify our above formula to this form by adding the variables A and B (A designates the octave of D and B the octave of C). A - D now coincides with the columns of our excel sheet. The formula thus becomes
(A + (B x G^C)) ^ (1/D) as listed in row 1
which translates to Hn = (AHn-D) + (Bhn - (D - C))
Here is an example which coincides with row 5131
A=1 B =4 C=9 D=14
this becomes
(1 + (4G^9))^(1/14)
which is the formula for this sequence
Hn = Hn-14 + 4Hn-5 (once again -5 being 9 steps above 14)
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Meruthree-secondary-patterns.csv
Here is the complete Excel Formula created by Juno Kim
----------------
#
# Kraig's Secondary Recurrent Generator
#
gcd <- function (a, b)
{
while (a != b)
{
if (a > b)
{
a = a - b
} else
{
b = b - a
}
}
return (a)
}
gcd2 <- function (a, b)
{
if (a == b)
{
return (1)
}
while (a != b)
{
if (a > b)
{
a = a - b
} else
{
b = b - a
}
}
return (a)
}
comp <- function (aa, bb, cc, dd)
{
G = 1
i = 0
j = 0
while (abs(G - i) > 0.0000000000001)
{
i = G
G = (aa+(bb*G^cc))^(1/dd)
j = j + 1
}
return (G)
}
A = c(1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192) #A variables 1,2,4,8,16,32,64,128,256
B = c(1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192) #B variables 1,2,4,8,16,32,64,128,256
D = 2:25
odat = NULL
upper = NULL
for (a in A)
{
b = 1
if (a >= 2^5)
{
b = a/(2^5)
}
upper = a*2^5
if (upper > 8192)
{
upper = 8192
}
while (b <= upper)
{
for (d in D)
{
for (cc in 1:(d-1))
{
if (gcd2(cc, d) == 1)
{
G = comp(a,b,cc,d)
rdat = c(a, b, cc, d, G)
odat = rbind(odat, rdat)
}
}
}
b = b * 2
}
}
#C variables 1-24 C < D
#D variables 1-25
out = cbind(odat, log(odat[,5], base=2))
write.csv(out, file="C:\\Users\\u\\Documents\\Kraig\\coprime4.csv")
