INTRO TO ECONOMICS
INTRO TO ECONOMICS ECON 109
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This 39 page Class Notes was uploaded by Arno Leuschke on Thursday October 22, 2015. The Class Notes belongs to ECON 109 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 45 views. For similar materials see /class/227162/econ-109-university-of-california-santa-barbara in Economcs at University of California Santa Barbara.
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Date Created: 10/22/15
Additive Synthesis Direct Digital Synthesis Processing and Composition Music 1091A2091A MAT 2761A Xavier Amatriain Winter 3906 o i i i quot M S de 1 Additive Synthesis Part I Intro to Additive Synthesis x l o SEJe 2 I M 1 Additive Synthesis Introduction Additive synthesis is based on the addition of elementary waveforms to create a more complex one It is one of the first methods used to create rich spectrums that resembled the natural behaviour of sound The acoustic analysis of the audio signals shows that natural sounds are composed and interpreted by the ear of many simple components 39 In the case of pitched periodic sounds these components are in fact multiples of the fundamental frequency l ii it i l J Additive Synthesis Introduction The idea of additive synthesis goes back to the Middle Age Organs had a number of tubes each for a given pitch in conjunction with other tubes also used for 5ths and octaves Much later on it was used in the beginnings of electric and electronic music for instance in the Telharmonium and the Hammond SHJe 4 I i l it ll Additive SyntheSIs Additive Synthesis Any periodic waveform can be expressed as the sum of one or more sine waves 1 i44 If we have two sine waves where one 3 W 3 repeats with 3 times the frequency of the other 1 and we add them together the MN sum will be a new periodic wave 1 3 1 3 r i W i i J Additive Synthesis Additive Synthesis i45 Another example with 5 harmonic sine waves W CHEW 1 2 3 4 5 ii i i i W Air E 5 de6 twpm quot Additive Synthesis Additive Synthesis add a weighted sum of harmonic sine waves some harmonics are more important louder Additive Synthesis Additive Synthesis Nhar sample imE E aha time sin21tharf1time Phar har1 har harmonic number o f1 fundamental frequency har phase of the harmonic often 0 usually doesn39t affect the sound l llll l W Atr E 5M T1 1 W will Additive Synthesis i46 Synthesizing the Following Spectrum 25E EDEN 3 E quotIECPU E 1300 5m I cl l I I I l l u 1 2 3 a 5 6 T a 9 1a 11 Harmnnlr NumbEar lt M EV 5fide9 Additive Synthesis Additive Synthesis Example 10 note statements il il il il il il il il il il 5fide st 1 dur JNNNWWWlblbm 25 85 55 17 amp 2400 900 600 1000 180 400 250 90 90 55 I KDCDQCMU IIleWNH harm attk dec 25 28 03 031 032 033 034 035 036 037 05 048 047 044 043 039 035 031 028 025 Additive Synthesis tH t mg HMHW Additive gynthesis Example 0R 1 note statement and 10 orc statements the peak amps of the partials are proportional to the amplitude of lowest partial iampl iamp2 iamp3 iamp4 iampS iamp6 iamp7 iamp8 iamp9 iamplO 2400 iampl iampl iampl iampl iampl iampl iampl gtlgtlgtlgtlgtlgtlgtlgtl iampl iampl 375 25 4167 075 1667 1042 0375 0375 022 9 llHl l o 5fide Additive Synthesis Additive Synthesis Instruments Tenor instrument design the voice has harmonic partials additive synthesis 15 harmonics Amplitude SYGD 5350 4020 134C 2530 a 1 3 E Q 12 Harmanic SEJe 12 Additive Synthesis 15 lt i it irfi i J Tquot 1 viii JAE Additive Synthesis Instruments tenorsco one wavetable sine wave for fundamental and partials fl 0 16385 10 l tenororc additive synthesis instr ll tenor voice idur p3 duration iamp p4 amplitude ifreq p5 frequency inorm 1731852 normalization Additive Synthesis 5fide 5fide tenor orc Amplitudes and En veloped Signals iampl iamp2 iamp3 iamp4 iamp5 iamp6 iamp7 iamp8 iamp9 iamplO iampll iamp12 iampl3 iampl4 iampl5 asigl asigZ asigB asig4 asig5 asig6 asig7 asig8 asig9 asiglO asigll asig12 asigl3 asigl4 asing oscili oscili oscili oscili oscili oscil39 oscil oscil oscil oscil oscil39 oscili oscili oscili oscili v 391 gt9gtgtgtgtegtegtegte gtgtegtegtegtegte iwtl iwtl iwtl iwtl iwtl iwtl iwtl iwtl lO iwtl ll iwtl 12 iwtl l3 iwtl l4 iwtl l5 iwtl tenor OI C add the signals 2 ampenv lgtnseg O iattack l isus l idecay O l O asigs asigl asig2 asig3 asig4 asig5 asig6 asig7 asig8 asig9 asiglO a 39 ll asig12 asig13 asigl4 asig15iqffiigt endin SEJe 15 Additive Synthesis Additive Synthesis Advantages Very flexible Can control each partial individually Can represent any harmonic or nearlyharmonic sound 39 But not good for noisy tones eg drums Can be used in combination with spectrum analysis to reconstruct musical instrument tones i A l l l W Atr E 5 1e16 I I I TI v 39 Additive SyntheSIs Additive Synthesis Disadvantages Slow Many instruments require summing 40100 harmonics Can t play very many notes in realtime on current hardware For example the hardware may only be able to produce 4 note polyphony to keep up in realtime l l ll l W A t E TI i wa rw l Additive Synthesis Additive Synthesis Disadvantages Difficult to control group as a whole Many parameters which are difficult to control 40100 amplitude envelopes plus 40100 frequency envelopes where each envelope consists of about 1000 timepoints e ole 1ft I 2 file Ina5 i 9 III l l ll o lt All 5 1e18 in i l W Additive Synthesis Part II More on Additive Synthesis x l J Sade 19 e Tquot W WE Additive Synthesis Generating functions with Additive Synthesis For additive synthesis we usually make use of function generators GENOQ GEN1O and GEN19 Typically the resulting waveforms are then controlled by an oscillator These three functions allow to create composite waveforms in which the relative sinusoidal intensities are specified GEN1O adds all the harmonic partials that are in phase 39 The relative amplitudes of the partials order 123 are defined in the p5p6 p7 parameters SHJe I i l it ll Additive SyntheSIs Generating functions with Additive Synthesis GEN9 adds the possibility of defining inharmonic partials The order of the partial the amplitude and the phase are defined in groups of three parameters using gen9 for additive synthesis 10512911033331805207l431809lllO This table will be filled with a triangular waveform which has the property of being formed by odd harmonics the amplitude that is the inverse of the order of the harmonic the phase is alternatively O and 180 for each odd harmonic ll i l i ui o lt SEJe 21 Additive Synthesis Generating functions with Additive Synthesis 9 GEN19 is just like GEN9 but adds a parameter of DC offset Each partial is defined with 4 parameters using genl9 for additive synthesis f 2 O 1024 19 l 5 270 5 This table will be filled by an almost gaussian curve that starts at O and has a maximum at 1 It is obtained by generating a sinusoid of amplitude 05 with a 05 DC offset With the offset the min and max values become 0 and 1 SEJe 22 ll l l l Additive Synthesis Generating functions with Additive Synthesis It is important to note the sign before the 19 this avoids the automatic normalization between 1 and 1 0 This is an interesting wave for granular synthesis using genl9 for additive synthesis f 2 O 1024 19 l 5 270 5 side 23 w it Additive SyntheSIs The phase Although the phase the deviation angle in tO of each harmonic is not important for the timbre that will be perceived it does have a great importance in the visual look of the waveform Look at the following square waveform generated by additive synthesis Additive Synthesis it39ll ldi The phase 0 This is what happens if we set the fith harmonic with a 90 degree initial phase instead of O The waveform is very different althogh the spectral content is the same I 1quot ii 7 I 7 If 1 I Ir 139 i l 39I r I r I I I I I I 39I I I Temps i ilii i o 5 1825 I I I i li Additive SyntheSIs Variable Waveform Additive Synthesis In order to obtain rich and convincing sounds we want to be able to control the amplitude of the harmonics in time t i xx m quot atx i aquot 439 M V tawny w quotup 1V4 h i ilii i o Wage WM Additive SyntheSIs O SEJe 27 Variable Waveform Additive Synthesis Actually we know that the evolution over time of the harmonic is a very important feature of acoustical instruments Amplitude These are the trajectories of a trumpet notice how the fundamental is not the highest partial only the one that lasts longeH Additive Synthesis Variable Waveform Additive Synthesis Peak deviation Peak deviation Frequency Frequency envelope envelope Although it is CPU consuming additive synthesis with variable waveform is a very generic 139 method we can reproduce almost any sound Fourier dixit Amplitude envelope For every partial we use a sinusoidal oscillator in which we can control freq and mag independently and throu envelopes Additive output signal SEJe 28 lt i ii irfi i J Tquot 1 iii JAE Additive Synthesis Variable Waveform Additive Synthesis Easily the amount of parameters to control in this schema can become overwhelming In the order of 500 for each event for a 30 harmonic instrument We need to find ways of generating data algorithmically using random number generators using previously analyzed data SEJe 29 I i all Additive SyntheSIs Variable Waveform Additive Synthesis Example reducing control parameters algorithmically Remember instr kenvl kenv2 kenv3 kenv4 kenv5 kenv6 asgl asg2 asgB asg4 asgS asg6 endin l osci osci osci osci osci osci osoil osoil osoil osoil osoil osoil kr oscill idel ll 0 lp3 ll 0 lp3 ll 0 lp3 ll 0 lp3 ll 0 lp3 ll 0 lp3 lOOOOkenVl 6000kenv2 4500kenv3 2OOOkenV4 500kenv5 250kenv6 out asglasg2asgBasg4asgSasg6 p45 l p46 l kamp idur ifn 2 p3 duration of fn 52p32 25 2 p3 4 125 2 0625 2 duration of second harmonic duration of 3m39harmonic 0375 2 p4 l p42 l p43 l p44 l Fundamental H1 H2 H3 H l H5 H6 Variable Waveform Additive Synthesis Example reducing control parameters algorithmically O fl 0 8192 10 1 f2 0 512 7 O 10 l 502 0 il O 10 440 has a single sinusoid has a triangular envelope e 5fide Additive Synthesis lt i l ig irj Tquot 1 M1 Wm Part III Direct Digital Synthesis Table lookup l 4 i i e Additive Synthesis Direct Synthesis or Table lookup This kind of synthesis is a direct application of the sampling theory A wavetable is stored in memory and an increment in the reading pointer allows to access the required sample at each moment in time I Table 5a mple values l DAG 4F E q e Table index values 24 23 1939 20 l 18 14l15 21122 Time 7 w 39 o ll l it Hi Additive Synthesis 16l1T 5 6i 7 9110l11 12 13 0 2 3 B 1 3 4 5Ed ii3 Table lockup algorithms If we want to change the pitch of the stored signal we have to change the table lookup speed s For instance if in a table with 8192 positions we read at 8192 samplessec we will obtain a 1 Hz sinusoid If we read at 16348 the frequency will be 2 Hz If the readingsampling frequency is fixed we need to skip or repeat samples in order to change the frequency 1 11 1 W Afr E SEJe 34 I I I T 1 Wm Additive SyntheSIs Table Iookup algorithms The simplest table lookup algorithm for an Lsamples table is the following indexphase modLpreviousphase increment output AMP tableindexphase The first operation is an addition and a mod operation the index is always less or equal to the table length The second operation reads the table sample located at indexphase and multiplies the result by the overall gain AMP l llll l W Air E SEJe 35 I I I T I Wm Additive SyntheSIs Table lockup algorithms The obtained frequency depends only and directly on the value of our increment lf fs is the sampling frequency and fdthe frequency we want we have the following relation increment fd L fs or fd increment fs L For instance if we want a 20 Hz frequency and we have a table with 2048 values and a sampling frequency of 8192 the increment will be 20 20488192 5 l l ll l o WA rEw 5711236 i il li Additive Synthesis Table Iookup algorithms That means that we will read one every 5 samples positions 0 5 10 15 After position 2045 we will start from the beginning in position 2 because 20455 mod 2048 equals 2 xiii i W Atr E 5 de37 I I I TI I Additive SyntheSIs Problems with simple table lockup If the frequency we want corresponds to a noninteger increment we will introduce noise Note that this is a very common case and it means that the reading pointer is between to index On a very large table the noise introduced can be acceptable but if the rounding is significant we will introduce artifacts in the output signal 5mg 38 q if Additive SyntheSIs Lookup with interpolation The solution to this problem is to use reading with interpolation This method has more computational cost but it can have very good results on almost any table length The interpolation is performed by using the fractional increment part to compute the approximated value of a sample that is not in the table itself The process is done by reading two or more samples in the table and interpolating between them l l l l W Air E w iINyTW l l Additive Synthesis
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