1. The Implementation of chronotonic similarity within an applet
Ludger Hofmann-Engl
Development – Croydon Family Groups
Axmedis 2008, Floerence, Italy, 17 – 19 November 2008

2. Traditional Notation of Rhythms (I) Basic note values
The absolute tempo is fixed as: MM (Mälzel 's Metronome)
e.g.: G... = 60 means 60 quarter notes in a minute

3. Traditional Notation of Rhythms (II) Other note values and example 3
Note + dot = Note value + half note value e.g. G.... = G... + H (¼ + 1/8)
Notes can be split into other ratios other than e.g. G... = H H H (triplets 1/12 each)
Example: G... H H F H H G... G... F 3

4. Atomic Notation and c-chains
The rhythm: G... H H F H H G... G... F
can be re-written using fractions.
We get: ¼ 1/8 1/8 ½ 1/8 1/8 ¼ ¼ ½
The smallest fraction of this sequence is 1/8.
Instead of 1/8 we write: [1](1/8)
Instead of ¼ we write: [2, 2] (1/8)
Instead of ½ we write: [4, 4, 4, 4] (1/8)
Finally, we get the c-chain:
c-chain = [2, 2; 1; 1; 4, 4, 4, 4; 1; 1; 2, 2; 2, 2; 4, 4, 4, 4](1/8)

5. Graphical Representation of a c-chain
Example:
c-chain = [2, 2; 1; 1; 4, 4, 4, 4; 1; 1; 2, 2; 2, 2; 4, 4, 4, 4](1/8)

6. Representing 2 c-chains
and
c-chain2 = [4, 4, 4, 4; 2, 2; 2, 2; 1; 1; 4, 4, 4, 4; 1; 1; 2, 2](1/8)

7. Tempo differences
c-chain1 = [2, 2; 1; 1; 4, 4, 4, 4; 1; 1; 2, 2; 2, 2; 4, 4, 4, 4](1/8)
and
c-chain2 = [2, 2; 1; 1; 4, 4, 4, 4; 1; 1; 2, 2; 2, 2; 4, 4, 4, 4](1/2)
with ln2(1/8 : ½) = ln2(1/2 : 1/8) = 1.92
we obtain the similarity factor (with 0.15 as an empirical constant):
𝐹 3 = 𝑒 −015∗1.92 =0.75
and generally:
𝐹 3 = 𝑒 −015∗ ln 2 𝑎 𝑏

8. Computing Chronotonic Distance
c-chain1 = [2, 2; 1; 1; 4, 4, 4, 4; 1; 1; 2, 2; 2, 2; 4, 4, 4, 4](1/8)
and
c-chain2 = [4, 4, 4, 4; 2, 2; 2, 2; 1; 1; 4, 4, 4, 4; 1; 1; 2, 2](1/8)
computing the logarithmic distances at all points as:
Computing Chronotonic Distance
𝑠 𝑖 =ln 𝑐 1i 𝑐 2i
and inputed into:
∥ 𝐹 1 ∥= 𝑖=1 𝑛 𝑒 − 𝑘 1 𝑠 2 𝑖 𝑛
we get (k1 = 12.8):
∥ 𝐹 1 ∥=0.341

9. The functional applet is here
The GUI of the Applet
The functional applet is
here

10. Processing Tempo Similarity
for (int k = 0; k < i; k++) {
SumV1 = SumV1 + v1[k]; }
for (int k = 0; k < j; k++)
SumV2 = SumV2 + v2[k];
v = 0.15 * Math.pow(Math.log(SumV1/SumV2),2);
F3 = Math.pow(2.17, -v);
F3 = Math.round(F3*10000);
F3 = F3/10000;

11. Equalizing the Length of the c-chains
Before the chronotonic distance can be calculated the two chains need to be made possessing equal length. The relevant passage reads:
if (SumV2 > SumV1) {
L = SumV2/SumV1;
for (int m = 0; m < i; m++)
v1[m] = L*v1[m]; }
if (SumV1 > SumV2)
L = SumV1/SumV2;
for (int m = 0; m < j; m++)
v2[m] = L*v2[m];

12. Quasi Atomic Notation The relevant passage for chain c1 reads:
Instead of finding the lowest common factor in order to generate a c-chain based on an atomic interval, we set the quasi atomic interval k = 0.005
The relevant passage for chain c1 reads:
for (int m = 0; m < i; m++) {
for (double k= 0.005; k < v1[m] ; k+=0.005)
al1++;
a1[al1] = v1[m]; }

13. Computation of the Chronotonic Distance
The computation of the chronotonic distance is now trivial, and the passage reads:
for (int m = 1; m < al1; m++) {
T[m] = Math.log(a1[m]/a2[m])/Math.log(2); }
for (int m = 1; m <al1; m++)
o[m] = Math.pow(T[m],2);
p[m] = Math.pow(2.17,(-12.8*o[m]));
T1 = T1 + Math.pow(p[m],2);
T1 = Math.sqrt(T1/al1);
T1 = Math.round(T1*1000+1);
T1 = T1/1000;

14. Chronotonic Similarity
where Sc is the chronotonic similarity, F3 the tempo similarity and ||F1|| the chronotonic distance.
𝑆 𝑐 = 𝐹 3 ∗∥ 𝐹 1 ∥

15. Applications of the Chronotonic Similarity
Integration into other applications (e.g. aural training)
MIR – Search Engines
Automated composer (producing variations)
Musical Analysis
Classification of Ethnomusicological Material

16. The Implementation of chronotonic similarity within an applet
Ludger Hofmann-Engl
Development – Croydon Family Groups
Axmedis 2008, Floerence, Italy, 17 – 19 November 2008