Generating Musical Symbols to Perform Expressively
by approximate functions

Kenzi NOIKE, Nobuo INUI, Takashi NOSE, Yoshiyuki KOTANI

Department of Computer Science
Tokyo University of Agriculture and Technology
e-mail: {noike, nobu, nose, kotani}@cc.tuat.ac.jp

Abstract
We designed the transcription system generate musical score have musical symbols to perform expressively as well as notes from human expressive performance. In this paper, we propose a method which generate musical symbols to perform expressively by approximate functions. We use three parameters, the local metronomic speed value, the ratio of a performance time and musical score time, and the dynamics value, to express performance. We analyzed the correspondences between expressions in human performance and musical symbols to perform expressively by our performance model, and made approximate functions to infer and to generate musical symbols. Our experimental results show that all of the five fermata's on the original musical score were generated correctly.


 1. Introduction
 We thought that humans can expressive Performers must understand some kinds of musical symbols to vary tempos and to add dynamics of sounds by order of composers. Besides this, they can perform expressively because they are recognizing whole musical structures. A transcription system takes away these expressions from human expressive performance to infer duration of each note on scores.
There are the researches of [AAFR, 1994], [DH, 1994], [David, 1992] and so on, in the related research of this field. These methods do not generate information about removed or recognized human expressions in musical score as a result.
We thought that a transcription system should generate these expressions as musical symbols in a musical score. Based on this thought, we designed the transcription system generate musical score have musical symbols to perform expressively as well as notes from human expressive performance. To generate musical symbols, we use approximate functions made by analysis of the correspondence between human performance and musical symbols.




2. Analysis of the correspondence between human performance and musical symbols
 2.1 Performance model
 To analyze human performance, we use our performance model defined by three following parameters.

Parameter M(i)
 M(i) is the local metronomic speed value of the i-th note in a musical score.
Parameter A(i)
 A(i) is the ratio of between a performance time (a time of key-on to key-off) of the i-th note in a musical score and it's musical score time(a time of i-th note's key-on to it's next note's key-on).
Parameter V(i)
 V(i) is the dynamics value of the i-th note in a musical score. V(i) is equal to the key-on velocity value of MIDI.

2.2 Making of approximate functions
 An approximate functions which generates musical symbols is made by following procedures.

(1) Extract the sequence of values of the performance model parameters M(i), A(i) and V(i) from human performance.
(2) Get the coefficients and the sum of square error of approximate functions of it's sequence.
We prepared seven types approximate functions as follows.

ax + b
ax2 + bx + x
ax3 + bx2 + cx + d
a exp(bx)
a + b log2(x)
axb
a + b / x

"x" expresses the time which is based on the 32nd note, not equal to i of "i-th note in a musical score".

3. Generation procedure
 Generation procedure of musical symbols is described as follows.

Procedure 1.
 Calculate the sequence of values of M(i), A(i) and V(i) of human performance. The plausible sequence of M(i) is calculated under the assumption that tempos of human performance do not vary suddenly. This calculation of the plausible sequence correspond to the tempo tracking.
Procedure 2.
 Get the sum of square error of between the sequence calculated at the procedure 1 and a sequence calculated by approximate function.
Procedure 3.
 If the sum of square error at the procedure 2 is equal to or less than the sum of square error at the making of a approximate function, generate the musical symbols which made the approximate function.
Our approximate functions deal with four beats range. But We will adjust this range according to analysis of human performance and result of generation experiment.




4. Results of experiment
 The aim of our experiment is to find as follows :
1) which approximate functions are best.
2) the musical symbol "fermata" is recognize.
Results are summarized below:
Result 1.
 All of the five fermata's on the original musical score were generated correctly. Result 2.
 Non-existing nine fermata's on the original musical score were generated. Result 3. The most useful form of approximate function to generate the musical symbol "fermata" is a fraction form a + b / x.

5. Conclusion
 We proposed a method which generate musical symbols to perform expressively by approximate functions.
To generate, we analyzed the correspondence between human performance and musical symbols, and represented human performance expression by approximate functions.
In the brief experiment, we got a result that all of fermata on the original musical score were generated correctly.
Our next theme is to develop a method to self-organize approximate functions from examples of human expressive performance. These functions will express characteristics of performers explicitly.

Reference
 [AAFR, 1994] Carlos Agon, Gerard Assayag, Joshua Fineberg, Camilo Rueda, "Kant: a Critique Pure Quantification. ICMC 94 Aarhus.", Proceedings of the ICMC, Aarhus, pp. 52 - 59, 1994.
[David, 1994] David Rosenthal, "Intelligent rhythm tracking", Proceedings of the ICMC, San Jose, pp. 227 - 230, 1992.
[DH, 1994] Peter Desain & Henkjan Honing, "Advanced issue in beat induction modelling: syncopation, tempo and timing", Proceedings of the ICMC, Aarhus, pp. 92 - 94, 1994.