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Proceedings of the 7th WSEAS International Conference on SIGNAL PROCESSING, ROBOTICS and AUTOMATION (ISPRA '08) University of Cambridge, UK, February 20-22, 2008

Simulation and Spectral Analysis of Inharmonic Tones of Musical Instruments

VARSHA SHAH 1, Dr.REKHA S. PATIL 2 Asst. Prof., Rizvi College of Engineering 1, Principal, J.J.Magdum College of Engineering 2 University of Mumbai 1, Shivaji University 2 INDIA

Abstract: - Simulation and spectral analysis of and inharmonic pure tones of musical instruments is undertaken. It is shown how the basic wave such as sine can be used to simulate various harmonic and inharmonic signals of different instruments generating different . It is shown that even the modest value of coefficient is sufficient to raise the of inharmonic by a partial step more than that of the harmonic spectrum thus deviating the spectrum from strictly harmonic case. Listening tests of the simulated tones are also conducted which proves that the perception of inharmonicity depends on the number of partials present in the sound.

Keywords: - Inharmonic Signal, Inharmonicity, Partial, Synthesis, Spectrum Analysis.

1. Introduction iv) Wavetable Synthesis: It is used in Basic wave shape such as sine, square and digital musical instruments (synthesizers) to triangle play an important role in digital produce natural . The sound of a signal processing applications. Various existing instrument sampled and stored synthesizing techniques use these waves to inside a wavetable. By repeatedly playing simulate the sound wave. samples from this table in a loop the Various synthesizing techniques are original sound is imitated classified in the following manner [2] Additive synthesis method is widely i) Additive Synthesis: In this method used in synthesizing musical simple each sound signal is represented by linear musical tones. We have used the said combination of sinusoids i.e. with variable method to generate simple as well as frequency and amplitude over time. complex harmonic and inharmonic musical ii) Frequency modulation synthesis (or tones by incorporating the inharmonicity FM synthesis): It is a form of audio coefficient. synthesis where the of a simple waveform is changed by frequency modulating it with a modulating frequency2 2 Difference Harmonic and that is also in the audio range, resulting in a Inharmonic Tones more complex waveform and a different- sounding tone.[3] A simple tone has only one frequency, iii) Subtractive synthesis: It is technique while the complex tone consists of one or which creates musical timbres by filtering more simple tones. A complex tone can be complex waveforms generated by harmonic or inharmonic. oscillators. Subtractive synthesis is usually A complex signal is called the harmonic (but not exclusively) associated with analog signal if all the of partials are voltage controlled synthesizers. It can integer multiples of fundamental produce very natural changes in a sound, frequencies that is the lowest frequency of owing to the intuitive way in which it the signal. Various of this signal works. can be calculated using following equation:

ISSN: 1790-5117 261 ISBN: 978-960-6766-44-2 Proceedings of the 7th WSEAS International Conference on SIGNAL PROCESSING, ROBOTICS and AUTOMATION (ISPRA '08) University of Cambridge, UK, February 20-22, 2008

fn = nf0 equation (1) coefficient was varied in the range of 50. Where n = number of harmonic 10-6 to 600. 10-6. The numbers of partials present in the sound were also varied in the In case of inharmonic signal the range of 5 to 25. Different fundamental frequencies of harmonics are not exact frequencies in the range of 30 Hz to integral multiples of the fundamental 1500Hz were used to simulate harmonic frequencies. Hence they are called as and inharmonic signals. To simulate and partials [7] .For e.g. the sound waves analyze signals and their generated from the struck or plucked type MATLAB 6.5 software and its toolboxes of musical instruments such as piano. This are used. is due to the stiffness of the strings of the instrument. The various partials can be calculated using the equation of the 4 Spectrum Analysis of stretching partials of piano tones which Simulated Tones relates the partial frequency, fn, with the Various simulated sounds are analyzed by f0 in the presence of plotting their frequency spectrums using an inharmonicity coefficient B [1]. DSP technique. Graphical representation of

2 0.5 various spectrums is shown. They are as fn=nf0(1+Bn ) equation (2) follows:

3 4 2 1. Figure 1 to 4 show frequency spectrums B = π Qd / ( 64l T) equation (3) of 2 different sets of harmonic and Where n = partial number inharmonic signals. In this the tones are Q = Young’s modulus, generated by keeping inharmonicity d = the diameter coefficient constant = 300*10^-6, number l = the length of partials=25 for two different fundamental T = the tension of the string frequencies of 100Hz and 200Hz. f0 = fundamental frequency

Typical values of B for piano strings lie roughly between 0.00005 for low bass tones and 0.015 for the high treble tones.[7] Timber is that parameter of the sound wave which allows us to differentiate the sounds of two different instruments of same pith and the amplitude.

3 Implementation of Simulation of Musical tones Simulation of complex inharmonic tones: b=300 * 10^-6; No. of partials: 25 Harmonic and inharmonic signals were Fig.1: Harmonic with fun.freq. = 100; generated by varying number of partials, Fig2: Inharmonic with fun.freq. = 100; fundamental frequency & inharmonicity Fig.3: Harmonic with fun.freq. = 200; coefficient. Fig.4: Inharmonic with fun.freq. = 200; Initially simple tones of single

frequency are simulated. Equation 1 is used 2. Figures 5 to 8 show the spectrum of one to simulate complex harmonic sound of a harmonic and 3 inharmonic spectrums in musical instrument viz. harmonium which which fundamental frequency is taken as consists of more than one frequency. 450Hz, inharmonicity coefficient is taken as Equation 2 is used to simulate complex 100 * 10^-6 and the number of partials are inharmonic pure tones of musical taken as 5, 15 and 30. instrument viz. piano. The inharmonicity

ISSN: 1790-5117 262 ISBN: 978-960-6766-44-2 Proceedings of the 7th WSEAS International Conference on SIGNAL PROCESSING, ROBOTICS and AUTOMATION (ISPRA '08) University of Cambridge, UK, February 20-22, 2008

3. Figures 9-12 show the spectrums of 1 harmonic signal and 3 inharmonic signals. Inharmonicity coefficients of 3 different values are selected to generate inharmonic signals. In this number of partials are taken as 25 for all the four signals.

5 Perceptual Analysis Five subjects having musical training participated in listening tests of the simulated sounds. Different harmonic and inharmonic sounds were simulated and the wave files were played to find the effects of No.of partials=25; fun.freq.=80Hz. different inharmonicity coefficients, Fig.9 Harmonic signal fundamental frequencies and number of Fig10:Inharmonic with b=50 * 10^-6; partials present in the sound. Repeated tests Fig.11:Inharmonic with b=200 * 10^-6; were conducted to confirm the subjective Fig.12:Inharmonic with b=600 * 10^-6; effect of various sounds. iii. After listening to the inharmonic sounds the opinion of the subjects were as follows: a. when the inharmonicity coefficient was 6 Observations and results varied by keeping the fundamental frequency constant and keeping the number 6.1 Results of Audibility test of partials same the subjects heard the i.It is the opinion of all the subject listeners metallic touch to the sounds. that the simple notes timber sound like a b. They also felt that up to approximately broken note of any wind instrument. 600*10^-6 of inharmonicity coefficient ii. Subjects heard various harmonics sound with the fundamental frequency of 50 to and found them as the pure tones 450 Hz the inharmonic sound was pleasant generated from the wind instrument like to hear than harmonic one. harmonium. 6.2 Results of Spectral Analysis of the Simulated Signals i. It is observed that even the modest b of 300*10^-6 increases the frequency of the inharmonic spectrum by a partial step. It is observed from figure 1 and figure 2 that due to inharmonicity the frequency of the inharmonic spectrum deviates in such a way that frequency of 20th partial of harmonic signal is shifted to 21st partial in case of inharmonic signal. ii. It is observed from figures 5 to 8 that as the fundamental frequency increases [b=100 * 10^-6; fun.freq.=450Hz. keeping the inharmonicity coefficient Fig.5: Harmonic with no. of partials = 5; constant the deviation of the spectrum from Fig 6: Inharmonic with no. of partials= 5; the strictly harmonic case also increases. Fig.7: Harmonic with no. of partials=15; iii. It is observed in figures 9 to 12 that as Fig.8: Inharmonic with no. of partials=30; the inharmonicity coefficient increases the deviation of the frequencies of the spectrum of inharmonic signal also increase & the

ISSN: 1790-5117 263 ISBN: 978-960-6766-44-2 Proceedings of the 7th WSEAS International Conference on SIGNAL PROCESSING, ROBOTICS and AUTOMATION (ISPRA '08) University of Cambridge, UK, February 20-22, 2008

spectrum moves away from the strictly [9] V. Valimaki, M. Karjalainen, T. Tolono, harmonic case. and C. Erkut, “Nonlinear modeling and synthesis of the kantele – a traditional Finnish string instrument,” in Proc. Int. 7 Conclusion Computer MusicConf., (Beijing, China), It was seen from the analysis of simulated pp. 220–223, 1999. musical tones that the inharmonic sound is [10] Serra, X. and Smith, J. O., 1990. found to be natural than the harmonic sound "Spectral Modeling Synthesis: A Sound if adequate amount of inharmonicity is Analysis/Synthesis System Based on a present. It is also observed that if Deterministic plus Stochastic inharmonicity is introduced the partial decomposition", Computer Music J., Vol. frequencies are deviated from their 14, No. 4, pp. 12-24. harmonic values. It is concluded that if the [11] Horner, A. 2000. "Low Peak inharmonic signal is found to be unpleasant Amplitudes for Wavetable Synthesis," due to higher value of inharmonicity some IEEE Transactions on Speech and Audio method has to be adopted to reduce this Processing, 8(4), 467-470. deviation of spectrum from harmonic to [12] H. Fletcher, E. D. Blackham, and R. inharmonic nature. Stratton,“Quality of piano tones,” J. Acoust. Soc. Amer., vol. 34, no. 6, 1961. Reference: [1] Robert W.Young, “Inharmonicity of Plain Wire Piano Strings” The Journal of the Acoustical Society of America, Volume 24, No. 3, MAY 1952 [2] http://en.wikipedia.org/wiki/synthesis [3]J. Chowning, "The Synthesis of Complex Audio Spectra by Means of Frequency Modulation," Journal of the Audio Engineering Society 21(7), 1973 [4] Askill, Jhon, “Physics of Musical Sounds”. [5]H.A.Conklin, “Generation of partials due to nonlinear mixing in stringed instruments”,J.Acoust.Soc.Am.,vol.105,no. 1,pp.536-545,1999. [6]Alexander Galembo and Aders Askenfelt “Signal representation and Estimation of spectral parameters by inharmonic comb filter with application to the piano” IEEE Tran. on Speech and Audio Processing, vol.7, No.2, March 1999, 00 197-2020 [7] J. O. Smith, Physical Audio Signal Processing: Digital Waveguide Modeling of Musical Instruments and Audio Effects, http://ccrma.stanford.edu/~jos/pasp04/, August 2004. [8]Dai, H. (2000), ‘On the relative influence of individual harmonics on pitch judgment’, J. Acoust. Soc. Am 107(2), 953- 959

ISSN: 1790-5117 264 ISBN: 978-960-6766-44-2